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Curricular Standards
for
Mathematics
Kansas State Board of Education
Revised July 2003
Kansas Curricular Standards for Mathematics
Table of Contents
Kansas Curricular Standards for Mathematics Writing Committee .......................................................
Kansas Curricular Standards for Mathematics Writing Committee – Special Education .......................
Introduction to the Kansas Curricular Standards for Mathematics ........................................................
Mission Statement ....................................................................................................................
Vision Statement ......................................................................................................................
Purpose of this Document ........................................................................................................
Prior History ..........................................................................................................................
The Revision Process...............................................................................................................
Format of the Kansas Curricular Standards .........................................................................................
Standards, Benchmarks and Indicators ....................................................................................
Grade Levels Where Benchmarks and Indicators Have Been Developed .................................
Special Note Regarding Seventh and Tenth Grade Benchmarks and Indicators ......................
Definitions Which Will Be Helpful In Understanding the Document ...........................................
Uses of this Document .........................................................................................................................
Development of Local Curriculum and Assessments ................................................................
Development of the Kansas Mathematics Assessments ...........................................................
Grade Levels to be Assessed .......................................................................................
Kansas Assessment Scores to be Reported .................................................................
The Use of Calculators on Kansas Mathematics Assessments .....................................
Prioritization of Indicators to be Assessed .....................................................................
Supplemental Documents ....................................................................................................................
One Page Summary of the Standards and Benchmarks ......................................................................
Curricular Standards by Standard and Benchmark ..............................................................................
Number and Computation
Number Sense ..............................................................................................................
Number Systems and Their Properties .........................................................................
Estimation .....................................................................................................................
Computation .................................................................................................................
Algebra
Patterns ........................................................................................................................
Variables, Equations, and Inequalities ..........................................................................
Functions ......................................................................................................................
Models ..........................................................................................................................
continued on next page
Kansas Curricular Standards for Mathematics – July 2003
Curricular Standards by Standard and Benchmark (continued from previous page)
Geometry
Geometric Figures and Their Properties .......................................................................
Measurement and Estimation .......................................................................................
Transformational Geometry ..........................................................................................
Geometry from an Algebraic Perspective ......................................................................
Data
Probability .....................................................................................................................
Statistics .......................................................................................................................
Assessment Framework .......................................................................................................................
Appendix 1 - Information on Mathematical Communication,
Reasoning, and Problem Solving....................................................................................
Appendix 2 – Curricular Standards Grade by Grade ............................................................................
Appendix 3 – Glossary of Terms ..........................................................................................................
Appendix 4 – National Standards in Personal Finance .........................................................................
Appendix 5 – Selected Bibliography .....................................................................................................
Appendix 6 – Helpful Resources ..........................................................................................................
Appendix 7 – Template for Pattern Blocks ...........................................................................................
Appendix 8 – Scope and Sequence .....................................................................................................
Kansas Curricular Standards for Mathematics – July 2003
The Kansas Curricular Standards for Mathematics Writing Committee
The writing committee would like to thank the more than 1,600 individuals and/or groups who
submitted written responses to the first and sixth working drafts of this document as well as the more
than 100 persons who provided input at work sessions and at public meetings held throughout the
state. The committee thoughtfully read and considered each of the responses received and felt this
input was invaluable to the development of this document.
In addition, the committee would like to thank the teachers, school administrators, parents, and
community members who have and will continue to work toward improving mathematics education in
Kansas.
Co-Chair: Betsy Wiens, Auburn Washburn Public Schools (Topeka), USD 437
Co-Chair: George Abel, Emporia Public Schools, USD 253
Bill FaFlick, Wichita Public Schools, USD 259
Michelle Flaming, ESSDACK (Hutchinson)
Pat Foster, Oskaloosa Public Schools, USD 341
Ann Goodman, Garden City Public Schools, USD 457
Margie Hill, Blue Valley Public Schools (Overland Park), USD 229
Nancy McDonnell, Prairie View Public Schools (LaCygne), USD 362
Sarah Meadows, Topeka Public Schools, USD 501
Jo Musselwhite, Salina Public Schools, USD 306
Cheryl Murra, Leavenworth Public Schools, USD 453
Joan Purkey, Newman University (Wichita)
Judy Roitman, University of Kansas (Lawrence)
Teresa San Martin, Maize Public Schools, USD 266
Allen Sylvester, Geary County Public Schools (Junction City), USD 475
Ethel Edwards, KSDE – School Improvement and Accreditation
Canda Mueller Engheta, KSDE – Planning and Research
Lynnett Wright, KSDE – Student Support Services
Kansas Curricular Standards for Mathematics – July 2003
The Kansas Curricular Standards for Mathematics Writing Committee – Special Education
Chris Butler, Victoria Public Schools, USD 432
Melissa Donaldson, Columbus Public Schools, USD 493
Debra Esau, SCKSEC (Pratt)
Larry Finn, Kansas State School for the Deaf (Olathe)
Linda Hickey, Blue Valley Public Schools (Overland Park), USD 229
Marjorie Hill, Blue Valley Public Schools (Overland Park), USD 229
Sharon Laverentz, Perry Public Schools, USD 343
Charlene Lueck, Geary County Public Schools (Junction City), USD 475
Sheila R. Nigh, Wichita Public Schools, USD 259
Sally Richardson, Wichita Public Schools, USD 259
Cecilia Rijfkogel, Garden City Public Schools, USD 475
Deb Stephenson, Blue Valley Public Schools (Overland Park), USD 229
K. Nadine Voth, Hesston Public Schools, USD 460
Mary Kay Woods, Three Lakes Educational Cooperative (Quenemo)
Ethel Edwards, KSDE – School Improvement and Accreditation
Scott Smith, KSDE – School Improvement and Accreditation
Lynnett Wright, KSDE – Student Support Services
Kansas Curricular Standards for Mathematics – July 2003
Introduction to the Kansas Curricular Standards for Mathematics
Mission Statement
The mission of Kansas mathematics education is for all Kansas students to learn mathematical
content and skills that are used to solve a variety of problems.
Vision Statement
The vision of mathematics education in Kansas is to work toward the following:




Kansas mathematics education will be recognized as one of the premier
programs in the United States.
Kansas mathematics education will be equally effective for all students,
irrespective of gender, race, or socioeconomic background.
Kansas families will broadly recognize the importance of and be encouraged
to participate actively in their child(ren)’s mathematics learning.
Technology will be a fundamental part of mathematics teaching and learning.
The Purpose of this Document
The standards, benchmarks, and indicators in this document have been created to assist
Kansas educators in developing local curricula and assessments, as well as to serve as the
basis for the development of the state assessments in mathematics. The committee strove to
recommend high, yet reasonable expectations for all students.
High, yet reasonable
expectations for all students are components of fairness in education. All students includes:
those who choose to attend college, those who choose technical preparation, those who will
enter the workforce, those from various socio-economic backgrounds, those who have been
identified as gifted in the area of mathematics, those who have been identified with learning
disabilities, those who have previously been successful with mathematics, and those who have
struggled with mathematics sometime in the past.
Students may need additional support both within and outside the regular classroom to meet
those expectations. Teachers should be given the professional development and resources
necessary to enable them to help all students strive to meet or exceed these expectations. This
may seem a daunting task, but the alternative is not acceptable.
Prior History
The Kansas State Board of Education developed and adopted the Kansas Mathematics
Improvement Program in 1989. As part of that plan, Kansas Mathematics Curriculum Standards
were developed by Kansas educators in 1990. A new Kansas mathematics assessment
program began as a statewide pilot in 1991. In 1992, the Kansas legislature directed that
further work be done in the development of the mathematics curriculum standards, so that
benchmarks would be established at a minimum of three grade levels. In 1993, the State Board
adopted revised standards for mathematics. The 1993 Mathematics Curriculum Standards then
served as the basis for the Kansas Mathematics Assessment program through the end of the
1998-99 school year. In August 1997, the State Board directed that additional work be done in
Kansas Curricular Standards for Mathematics – July 2003
each of the curriculum areas. The mathematics writing committee convened in October 1997
and worked through February 1999. At the March 1999 meeting of the Kansas State Board of
Education, the Kansas Curricular Standards for Mathematics were approved. These standards
served as the basis for the Kansas Mathematics Assessments that were administered for the
first time during the spring of 2000. The Standards approved in March 1999 serve(d) as the
basis for the Kansas Mathematics Assessments through the spring of 2005.
The Revision Process
At the May 2002 meeting of the Kansas State Board of Education, the State Board directed that
academic standards committees composed of stakeholders from throughout Kansas were to be
convened for the curriculum areas defined by state law, and, at this time, only reading, writing,
and mathematics. The mathematics committee was charged to:
1)
2)
3)
4)
5)
6)
review the current standards document.
review modified and extended standards for inclusion in the document.
review the format to ensure usability.
determine the level of specificity of skills assessed.
recommend essential indicators to be assessed.
review previous assessment results and review past KSDE studies as they related to
student achievement.
The mathematics writing committee began its work in June 2002. Members worked more than
thirty days on revising the standards. Members of the committee reviewed various research
articles, examples of mathematics curriculum standards from other states, and reviews of the
current Kansas Mathematics Curriculum Standards by various individuals and
organizations. Because earlier editions of the Kansas Mathematics Curriculum
Standards had been used to guide curriculum, instruction, and assessment throughout the state
since 1993, the committee felt it was very important to build upon existing efforts to improve
mathematics education. The committee unanimously elected to use the 1999 Standards
document as the basis of their work.
At their initial meeting, the writing committee reached consensus on the following revisions for
the first draft of the revised Kansas Curricular Standards for Mathematics:
1) definitions of curriculum standards remain the same.
2) greater clarity and specificity to the benchmarks and indicators by grade level so
teachers know what they should teach and what students should learn (grades K-8
and high school).
3) format and language revisions that make the standards and the benchmarks easier
for non-mathematics educators to understand, as well as aligning the formatting of
the mathematics standards document with standards documents from the other
curriculum areas.
4) expectations for students were to be high, yet reasonable to achieve.
5) prioritize the state curriculum indicators to be assessed by the state assessments
and provide advice regarding the assessment of the state curricular standards.
During meetings throughout the summer of 2002, members of the mathematics revision
committee and subcommittees worked to bring specificity and clarity to the benchmarks in the
current document by revising the benchmarks and indicators outlined in the 1999 Kansas
Mathematics Curriculum Standards. In August 2002, the first working draft was mailed to
Kansas Curricular Standards for Mathematics – July 2003
schools throughout the state to obtain feedback on the draft. The committee continued to work
during the fall and winter months reviewing and discussing the feedback from schools, revising
grade level expectations, learning about and discussing various assessment issues, and
brainstorming possible assessment priorities. By February of 2003, a sixth working draft was
mailed to educators, community members, and professional organizations. Public meetings
were held in March throughout the state to receive public input on the sixth draft. Summaries of
the public meetings and written feedback were compiled for the committee. At the same time,
the sixth draft was submitted to the State Board’s designated outside reviewer, Mid-Continent
Regional Educational Laboratory, for review. In March and April, the writing committee met to
make revisions based on recommendations from McREL’s review and to further prioritize
indicators that would be used as the basis for assessment items on the revised Kansas
Mathematics Assessments. In May, the committee met to refine the document. Grade level
expectations were revised and further prioritization of benchmarks and indicators to be used in
developing the Kansas Mathematics Assessments were evaluated. The tenth working draft was
prepared for submission to the State Board and discussed at their June meeting.
After the June 2003 meeting of the State Board, subcommittees met to refine their work
including additional examples, completing the Financial Literacy connection, and designing a
glossary. In addition, during June and July, committee members were facilitating Summer
Mathematics Standards Academies across the state; nine such academies were held. The
committee's recommendations, as presented in the eleventh draft, were acted upon and
approved at the July 2003 meeting of the Kansas State Board of Education. Kansas
Mathematics Assessments, based on these standards, are to be administered for the first time
during the spring of 2006.
Kansas Curricular Standards for Mathematics – July 2003
To be completed for standards document:
Format of the Kansas Curricular Standards for Mathematics
Uses of this Document
Supplemental Documents
One Page Summary of the Standards and Benchmarks
Curricular Standards by Standard and Benchmark
Assessment Framework
Kansas Curricular Standards for Mathematics – July 2003
Standard 1: Number and Computation
KINDERGARTEN
Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 1: Number Sense – The student demonstrates number sense for whole numbers, fractions, and money
using concrete objects in a variety of situations.
Kindergarten Knowledge Base Indicators
Kindergarten Application Indicators
The student...
The student...
1. establishes a one-to-one correspondence with whole numbers from
0 through 20 using concrete objects and identifies, states, and
writes the appropriate cardinal number (2.4.K1a) ($).
2. compares and orders whole numbers from 0 through 20 using
concrete objects (2.4.K1a) ($).
3. recognizes a whole, a half, and parts of a whole using concrete
objects (2.4.K1a,c) ($), e.g., half a pizza, part of a cookie, or the
whole school.
4. identifies positions as first and last (2.4.K1a).
5. identifies pennies and dimes and states the value of the coins using
money models (2.4.K1d) ($).
1. solves real-world problems using equivalent representations and
concrete objects to compare and order whole numbers from 0
through 10 (2.4.A1a) ($).
K-9
January 31, 2004
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Number sense refers to one’s ability to reason with numbers and to work with numbers in a flexible way. The ability to compute
mentally, to estimate based on understanding of number relationships and magnitudes, and to judge reasonableness of answers are all involved in
number sense.
When we say that someone has good number sense, we mean that he or she possesses a variety of abilities and understandings that include an
awareness of the relationships between numbers, an ability to represent numbers in a variety of ways, a knowledge of the effects of operations,
and an ability to interpret and use numbers in real-world counting and measurement situations. Such a person predicts with some accuracy the
result of an operation and consistently chooses appropriate measurement units. This “friendliness with numbers” goes far beyond mere
memorization of computational algorithms and number facts; it implies an ability to use numbers flexibly, to choose the most appropriate
representation of a number for a given circumstance, and to recognize when operations have been correctly performed. (Number Sense and
Operations: Addenda Series, Grades K-6, NCTM, 1993)
Mathematical models such as concrete objects, pictures, number lines, or unifix cubes are necessary for conceptual understanding and should be
used to explain computational procedures. If a mathematical model can be used to represent the concept, the indicator in the Models benchmark
is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a
(process models). Then, the indicator in the Models benchmark lists some of the mathematical models that could be used to teach the concept. In
addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For
example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1
(Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
Standard 1: Number and Computation
KINDERGARTEN
K-10
January 31, 2004
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 2: Number Systems and Their Properties – The student demonstrates an understanding of whole
numbers with a special emphasis on place value in a variety of situations.
Kindergarten Knowledge Base Indicators
Kindergarten Application Indicators
The student...
The student...
1. reads and writes whole numbers from 0 through 20 in numerical
form ($).
2. represents whole numbers from 0 through 20 using place value
models (2.4.K1b) ($), e.g., ten frames, unifix cubes, straws bundled
in 10s, or base ten blocks.
3. counts (2.4.K1a) ($):
a. whole numbers from 0 through 20,
b. whole numbers from 10 to 0 backwards,
c. subsets of whole numbers from 0 through 20.
4. groups objects by 5s and by 10s (2.4.K1a).
5. uses the concept of the zero property of addition (additive identity)
with whole numbers from 0 through 20 and demonstrates its
meaning using concrete objects (2.4.K1a) ($), e.g., 4 apples and no
(zero) other apples are 4 apples.
1. solves real-world problems with whole numbers from 0 through 20
using place value models (2.4.A1b) ($), e.g., group the class into
tens, count by tens; then continue counting by ones to find the total.
2. counts forwards and backwards from a specific whole number using
a number line from 0 through 10 (2.4.A1a).
K-11
January 31, 2004
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: From the Mathematics Dictionary and Handbook (Nichols Schwartz Publishing, 1999), property as a mathematical term means a
characteristic (an attribute) of a number, geometric shape, mathematical operation, equation, or inequality. To give an example:
 Property of a number: 8 is divisible by 2.
 Property of a geometric shape: Each of the four sides of a square is of the same length.
 Property of an operation: Addition is commutative. For all numbers x and y, x + y = y + x.
 Property of an equation: For all numbers a, b, and c, if a = b, then a + c = b + c.
 Property of an inequality: For all numbers a, b, and c, if a > b, then a – c > b – c.
Mathematical models such as concrete objects, pictures, number lines, or unifix cubes are necessary for conceptual understanding and should be
used to explain computational procedures. If a mathematical model can be used to represent the concept, the indicator in the Models benchmark
is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a
(process models). Then, the indicator in the Models benchmark lists some of the mathematical models that could be used to teach the concept. In
addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For
example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1
(Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
K-12
January 31, 2004
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
KINDERGARTEN
Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 3: Estimation – The student uses computational estimation with whole numbers in a variety of situations.
Kindergarten Knowledge Base Indicators
The student...
1. determines if a group of 20 concrete objects or less has more, less,
or about the same number of concrete objects as a second set of
the same kind of objects (2.4.K1a).
Kindergarten Application Indicators
The student...
1. compares two randomly arranged groups of 10 concrete objects or
less and states the comparison using the terms: more, less, about
the same (2.4.A1a).
K-13
January 31, 2004
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Estimate, as a verb, means to make an educated guess based on information in a problem or to give an answer
close to the exact number. Estimation is used when an exact answer is not needed, as in many real-life situations for which “ballpark”
computations are acceptable. Good number sense enables one to estimate a quantity, estimate a measure, or estimate an answer.
Estimation serves as an important companion to computation. It provides a tool for judging the reasonableness of computational
methods including mental math, paper and pencil, and appropriate technology. However, being able to compute does not
automatically lead to an ability to estimate or judge reasonableness of answers. Frequent modeling by the teacher helps students
develop a range of estimation strategies. (Principles and Standards for School Mathematics, NCTM, 2000)
In order for students to become more familiar with estimation, estimation should be introduced with examples where rounded or
estimated numbers are used emphasizing real-world examples where only estimation is required, e.g., about how many hours do you
sleep a night? Using the language of estimation is important, so students begin to realize that a variety of estimates (answers) are
possible.
Mathematical models such as concrete objects, pictures, number lines, or unifix cubes are necessary for conceptual understanding
and should be used to explain computational procedures. If a mathematical model can be used to represent the concept, the
indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the
mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to
the other indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6,
1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked
at three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the
Appendix.
Standard 1: Number and Computation
KINDERGARTEN
Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 4: Computation – The student models, performs, and explains computation with whole numbers using
concrete objects in a variety of situations.
K-14
January 31, 2004
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Kindergarten Knowledge Base Indicators
The student...
1. adds and subtracts using whole numbers from 0 through 10 and
various mathematical models (2.4.K1a) ($), e.g., concrete objects,
number lines, or unifix cubes.
2. uses repeated addition (multiplication) with whole numbers to find
the sum when given the number of groups (three or less) and given
the same number of concrete objects in each group (five or less)
(2.4.K1a), e.g., two nests with three eggs in each nest means 3 + 3
= 6 or 2 groups of 3 makes 6.
3. uses repeated subtraction (division) with whole numbers when given
the total number of concrete objects in each group to find the
number of groups (2.4.K1a), e.g., there are 9 pencils. If each
student gets 2 pencils, how many students get pencils? 9 – 2 – 2 –
2 – 2 or 9 minus 2 four times means four students get 2 pencils
each and there is 1 pencil left over. or There are eight cookies to be
shared equally among four people, how many cookies will each
person receive?
Kindergarten Application Indicators
The student...
1. solves one-step real-world addition or subtraction problems with
whole numbers from 0 through 10 using concrete objects in various
groupings and explains reasoning (2.4.A1a) ($), e.g., seven apples
are in a basket and five students each take an apple; how many
apples are left in the basket?
Teacher Notes: The definition of computation is finding the standard representation for a number. For example, 6 + 6, 4 x 3, 17 –
5, 24 ÷ 2 are all representations for the standard representation of 12.
Mathematical models such as concrete objects, pictures, number lines, or unifix cubes are necessary for conceptual understanding
and should be used to explain computational procedures. If a mathematical model can be used to represent the concept, the
indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the
mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to
the other indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6,
1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked
at three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the
Appendix.
K-15
January 31, 2004
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
K-16
January 31, 2004
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
KINDERGARTEN
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 1: Patterns – The student recognizes, describes, extends, develops, and explains relationships in
patterns using concrete objects in a variety of situations.
Kindergarten Knowledge Base Indicators
The student...
1. uses concrete objects, drawings, and other representations to work
with types of patterns (2.4.K1a):
a. repeating patterns, e.g., an AB pattern is like red-blue, red-blue,
…; an ABC pattern is like dog-horse-pig, dog-horse-pig, …; or
an AAB pattern is like Δ-Δ-Ο, Δ-Δ-Ο, …;
b. growing (extending) patterns, e.g., 5, 6, 7, … is an example of a
pattern that adds one to the previous number to continue the
pattern.
2. uses these attributes to generate patterns:
a. whole numbers (2.4.K1a), e.g., 2, 4, 6, …;
b. geometric shapes with one attribute change (2.4.K1e), e.g., Δ,
Ο, Δ, Ο, Δ, Ο, …;
c. things related to daily life (2.4.K1a), e.g., breakfast, lunch, and
dinner.
3. identifies and continues a pattern presented in various formats
including numeric (list or table), visual (picture, table, or graph),
verbal (oral description), and kinesthetic (action) (2.4.K1a) ($).
4. generates (2.4.K1a):
a. repeating patterns for the AB pattern, the ABC pattern, and the
AAB pattern;
b. growing (extending) patterns that add 1, 2, or 10 to continue the
pattern.
5. classifies and sorts concrete objects by similar attributes (2.4.K1a)
($).
Kindergarten Application Indicators
The student...
1. generalizes the following patterns using pictorial, and/or oral
descriptions including the use of concrete objects:
a. repeating patterns for the AB pattern, the ABC pattern, and the
AAB pattern (2.4.A1a) ($);
b. patterns using geometric shapes with one attribute change
(2.4.A1c).
2. recognizes multiple representations of the AB pattern (2.4.A1a),
e.g., big- little, big-little, big-little, ... and 1-2, 1-2, 1-2, ..., or AB, AB,
AB, ....
3. uses concrete objects to model a whole number pattern (2.4.A1a):
a. counting by ones:  ,, , …;
b. counting by twos: ,  
, 
, …;
c. counting by tens:
,  
, 
, …
K-17
January 31, 2004
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Working with patterns is an important process in the development of mathematical thinking. Patterns can be based
on geometric attributes (shapes, regions, angles); measurement attributes (color, texture, length, weight, volume, number); relational
attributes (proportion, sequence, functions); and affective attributes (values, likes, dislikes, familiarity, heritage, culture). (Learning to
Teach Mathematics, Randall J. Souviney, Macmillan Publishing Company, 1994)
This process (working with patterns) can be used to develop or deepen understandings of important concepts in number theory, whole numbers,
measurement, geometry, probability, and functions. Working with patterns provides opportunities for students to recognize, describe, extend,
develop, and explain.
Mathematical models such as concrete objects, pictures, number lines, or unifix cubes are necessary for conceptual understanding
and should be used to explain computational procedures. If a mathematical model can be used to represent the concept, the
indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the
mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to
the other indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6,
1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked
at three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the
Appendix.
K-18
January 31, 2004
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
KINDERGARTEN
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 2: Variables, Equations, and Inequalities – The student solves addition equations using concrete objects
in a variety of situations.
Kindergarten Knowledge Base Indicators
Kindergarten Application Indicators
The student...
The student...
1. finds the unknown sum using the basic facts with sums through 10
using concrete objects and pictures (2.4.K1a) ($), e.g., 5 marbles +
5 marbles = .
1. describes real-world problems using concrete objects and pictures
and the basic facts with sums through 10 (2.4.A1a) ($), e.g., given
some marbles, Sue says: There are 3 red marbles and 3 blue
marbles. Altogether, there are 6 marbles.
Teacher Notes: Understanding the concept of variable is fundamental to algebra. In the early grades, students use various
symbols, including letters and geometric shapes, to represent unknown quantities that both do and do not vary. Quantities that are
not given and do not vary are often referred to as unknowns or missing elements when they appear in equations, e.g., 2 + 4 = Δ.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks
are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can
be used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a)
refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the
Models benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each indicator in the
Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For example, process
models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number
Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked
at three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the
Appendix.
K-19
January 31, 2004
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
KINDERGARTEN
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 3: Functions – The student recognizes and describes whole number relationships using concrete
objects in a variety of situations.
Kindergarten Knowledge Base Indicators
The student...
1. locates whole numbers from 0 through 20 on a number line
(2.4.K1a).
Kindergarten Application Indicators
The student...
1. represents and describes mathematical relationships for whole
numbers from 0 through 10 using concrete objects, pictures, and
oral descriptions (2.4.A1a) ($).
Teacher Notes: Functions are relationships or rules in which each member of one set is paired with one, and only one, member of
another set. A number line (a mathematical model) is a diagram that represents numbers with equal distances marked off as points
on a line, an example of one-to-one correspondence (a relation). A number line can be used as a visual representation of numbers
and operations. In addition, a number line used horizontally and vertically is a precursor to the coordinate plane.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks
are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can
be used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a)
refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the
Models benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each indicator in the
Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For example, process
models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1
(Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked
at three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the
Appendix.
K-20
January 31, 2004
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
KINDERGARTEN
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 4: Models – The student uses mathematical models including concrete objects to represent, show, and
communicate mathematical relationships in a variety of situations.
Kindergarten Knowledge Base Indicators
The student...
1. knows, explains, and uses mathematical models to represent
mathematical concepts, procedures, and relationships. Mathematical
models include:
a. process models (concrete objects, pictures, number lines, unifix
cubes, measurement tools, or calendars) to model computational
procedures and mathematical relationships, to compare and
order numerical quantities, and to represent fractional parts
(1.1.K1-4, 1.2.K3-5, 1.3.K1, 1.4.K1-3, 2.1.K1, 2.1.K2a, 2.1.K2c,
2.1.K3-5 2.2.K1, 2.3.K1, 3.1.K2, 3.2.K1-3, 3.3.K1-2, 3.4.K1-2) ($);
b. place value models (ten frames, unifix cubes, bundles of straws,
or base ten blocks) to represent numerical quantities (1.2.K2) ($);
c. fraction models (fraction strips or pattern blocks) to represent
numerical quantities (1.1.K3) ($);
d. money models (base ten blocks or coins) to represent numerical
quantities (1.1.K5) ($);
e. two-dimensional geometric models (geoboards, dot paper, or
attribute blocks), three-dimensional geometric models (solids),
and real-world objects to compare size and to model attributes of
geometric shapes (2.1.K1a, 3.1.K3);
f. two-dimensional geometric models (spinners), three-dimensional
geometric models (number cubes), and concrete objects to model
probability (4.1.K1-2) ($);
g. graphs using concrete objects, pictographs, and frequency tables
to organize and display data (4.2.K1-3) ($).
2. uses concrete objects, pictures, drawings, diagrams, or
dramatizations to show the relationship between two or more things
($).
Kindergarten Application Indicators
The student...
1. recognizes that various mathematical models can be used to
represent the same problem situation. Mathematical models
include:
a. process models (concrete objects, pictures, number lines,
unifix cubes, measurement tools, or calendars) to model
computational procedures and mathematical relationships, to
compare and order numerical quantities, and to model
problem situations (1.1.A1, 1.2.A2, 1.3.A1, 1.4.A1, 2.1.A1a,
2.1.A2-3, 2.2.A1, 2.3.A1, 3.1.A3, 3.2.A1-2, 3.3.A1-2, 3.4.A1)
($);
b. place value models (ten frames, unifix cubes, bundles of
straws, or base ten blocks) to represent numerical quantities
(1.2.A1) ($);
c. two-dimensional geometric models (geoboards, dot paper, or
attribute blocks), three-dimensional geometric models (solids),
and real-world objects to compare size and to model attributes
of geometric shapes (3.1.A1-2);
d. two-dimensional geometric models (spinners), threedimensional geometric models (number cubes), and concrete
objects to model probability (4.1.A1);
e. graphs using concrete objects, pictographs, and frequency
tables to organize and display data (4.1.A1, 4.2.A1) ($).
K-21
January 31, 2004
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: For assessment purposes, the mathematical modeling process appropriate to the indicator may be included as part of the item
being assessed.
The mathematical modeling process involves:
a. selecting key features and relationships within the real-world situation and representing these concepts in mathematical terms through
some sort of mathematical model,
b. performing manipulations and mathematical procedures within the mathematical model,
c. interpreting the results of the manipulations within the mathematical model,
d. using these results to make inferences about the original real-world situation.
The use of mathematical models is necessary for conceptual understanding. The ways in which mathematical ideas are represented is
fundamental to how students understand and use those ideas. As students begin to use multiple representations of the same situation, they begin
to develop an understanding of the advantages and disadvantages of the various representations/models. For example, comparing the number of
boys and girls in the classroom can be represented by lining them up in two different lines. The same situation also can be represented by pictures
of the children (pictograph), a bar graph, or by using two different colors of the same manipulative (unifix cubes or color tiles).
Many mathematical models are listed in this benchmark. The indicator lists some of the mathematical models that could be used to teach a
concept. Each indicator in this benchmark is linked to other indicators in other benchmarks; those indicators are identified in the parentheses. For
example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1
(Number Sense), and Knowledge Indicator 3. In addition, the indicator in the other benchmarks identifies, in parentheses, the Models’ indicator.
For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models).
For assessment purposes, the mathematical modeling process appropriate to the indicator may be included as part of the item being assessed.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
K-22
January 31, 2004
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
KINDERGARTEN
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 1: Geometric Figures and Their Properties – The student recognizes geometric shapes and their
attributes using concrete objects in a variety of situations.
Kindergarten Knowledge Base Indicators
The student...
1. recognizes circles, squares, rectangles, triangles, and ellipses
(ovals) (plane figures/ two-dimensional figures) (2.4.K1e).
2. recognizes and investigates attributes of circles, squares,
rectangles, triangles, and ellipses using concrete objects, drawings,
and/or appropriate technology (2.4.K1a,e).
3. sorts cubes, rectangular prisms, cylinders, cones, and
spheres (solids/three-dimensional figures) by their attributes
using concrete objects (2.4.K1e).
Kindergarten Application Indicators
The student...
1. demonstrates how several plane figures (circles, squares,
rectangles, triangles, ellipses) can be combined to make a new
shape (2.4.A1c).
2. sorts by one attribute real-world geometric shapes that are
representations of the solids (cubes, rectangular prisms, cylinders,
cones, spheres) (2.4.A1c), e.g., boxes can be sorted as rectangular
prisms, cans can be sorted as cylinders, some ice cream cones can
be sorted as cones, and some balls can be sorted as spheres.
3. recognizes (2.4.A1a):
a. circles, squares, rectangles, triangles, and ellipses (plane
figures) within a picture;
b. cubes, rectangular prisms, cylinders, cones, and spheres
(solids) within a picture.
K-23
January 31, 2004
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Geometry is the study of shapes, their properties, and their relationships to other shapes. Symbols and numbers are used to
describe their properties and their relationships to other shapes. The fundamental concepts in geometry are point (no dimension), line (onedimensional), plane (two-dimensional), and space (three-dimensional). Plane figures are referred to as two-dimensional and solids are referred to
as three-dimensional.
From the Mathematics Dictionary and Handbook (Nichols Schwartz Publishing, 1999), property as a mathematical term means a characteristic
(an attribute) of a number, geometric shape, mathematical operation, equation, or inequality. To give an example:
 Property of a number: 8 is divisible by 2.
 Property of a geometric shape: Each of the four sides of a square is of the same length.
 Property of an operation: Addition is commutative. For all numbers x and y, x + y = y + x.
 Property of an equation: For all numbers a, b, and c, if a = b, then a + c = b + c.
 Property of an inequality: For all numbers a, b, and c, if a > b, then a – c > b – c.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra),
Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the
mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other
indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3
referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
K-24
January 31, 2004
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
KINDERGARTEN
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 2: Measurement and Estimation – The student estimates and measures using standard and nonstandard
units of measure with concrete objects in a variety of situations.
Kindergarten Knowledge Base Indicators
Kindergarten Application Indicators
The student...
1. uses whole number approximations (estimations) for length using
nonstandard units of measure (2.4.K1a) ($), e.g., the classroom
door is about two kindergartners high or this paper is about two
pencils long.
2. compares two measurements using these attributes (2.4.K1a) ($):
a. longer, shorter (length);
b. taller, shorter (height);
c. heavier, lighter (weight).
d. hotter, colder (temperature).
3. reads and tells time at the hour using analog and digital clocks
(2.4.K1a).
The student...
1. compares and orders concrete objects by length or weight
(2.4.A1a) ($).
2. locates and names concrete objects that are about the same length
or weight as a given concrete object (2.4.A1a) ($).
Teacher Notes: The term geometry comes from two Greek words meaning “earth measure.” The process of learning to measure at the early
grades focuses on identifying what property (length, weight) is to be measured and to make comparisons. Estimation in measurement is
defined as making guesses as to the exact measurement of an object without using any type of measurement tool. Estimation helps students
develop a relationship between the different sizes of units of measure. It helps students develop basic properties of measurement, and it gives
students a tool to determine whether a given measurement is reasonable.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra),
Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the
mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other
indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3
referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
K-25
January 31, 2004
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
KINDERGARTEN
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 3: Transformational Geometry – The student develops the foundation for spatial sense using concrete
objects in a variety of situations.
Kindergarten Knowledge Base Indicators
The student...
1. describes the spatial relationship between two concrete objects
using appropriate vocabulary (2.4.K1a), e.g., behind, above, below,
on, or under.
2. identifies two like objects or shapes from a set of four objects or
shapes (2.4.K1a).
Kindergarten Application Indicators
The student...
1. shows two concrete objects or shapes are congruent by physically
fitting one object or shape on top of the other (2.4.A1a).
2. follows directions to move concrete objects from one location to
another using appropriate vocabulary (2.4.A1a), e.g., up, down,
behind, or above.
Teacher Notes: Transformational geometry is another way to investigate geometric figures by moving every point in a plane figure to a new
location. To help students form images of shapes through different transformations, students can use concrete objects, figures drawn on graph
paper, mirrors or other reflective surfaces, or appropriate technology.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra),
Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the
mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other
indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3
referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
K-26
January 31, 2004
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
KINDERGARTEN
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 4: Geometry From An Algebraic Perspective – The student identifies one or more points on a number
line in a variety of situations.
Kindergarten Knowledge Base Indicators
The student...
1. locates and plots whole numbers from 0 through 20 on a horizontal
number line (2.4.K1a).
2. counts forwards and backwards from a given whole number from 0
through 10 on a number line (2.4.K1a).
Kindergarten Application Indicators
The student...
1. solves real-world problems involving counting whole numbers from
0 through 20 using a number line (2.4.A1a) ($), e.g., if Bill has 8
pieces of candy and his dad gives him 4 more pieces, how many
pieces of candy does he have now?
Teacher Notes: A number line (a mathematical model) is a diagram that represents numbers with equal distances marked off as points on a
line, and is an example of one-to-one correspondence (a relation). A number line can be used as a visual representation of numbers and
operations. In addition, a number line used horizontally and vertically is a precursor to the coordinate plane.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra),
Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the
mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other
indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3
referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
K-27
January 31, 2004
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 4: Data
KINDERGARTEN
Standard 4: Data – The student uses concepts and procedures of data analysis in a variety of situations.
Benchmark 1: Probability – The student applies the concepts of probability using concrete objects in a variety of
situations.
Kindergarten Knowledge Base Indicators
The student...
1. recognizes whether an event is impossible or possible (2.4.K1f) ($),
e.g., the possibility of a person having ten heads is impossible,
while the possibility of a person having red hair is possible.
2. recognizes and states whether a simple event in an experiment or
simulation including the use of concrete objects can have more
than one outcome (2.4.K1a,f).
Kindergarten Application Indicators
The student...
1. conducts an experiment or simulation with a simple event and
records the results in a graph using concrete objects or frequency
tables (tally marks) (2.4.A1a,d-e).
Teacher Notes: Ideas from probability reinforce concepts in the other Standards, especially Number and Computation and Geometry. In the
early grades, students need to develop an intuitive concept of chance – whether or not something is unlikely or likely to happen. Beginning
probability experiences should be addressed through the use of concrete objects, coins, and geometric models (spinners or number cubes).
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra),
Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the
mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other
indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3
referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels – grades 4, 8, and 12 – and are grouped into four major categories – Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
K-28
January 31, 2004
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 4: Data
KINDERGARTEN
Standard 4: Data – The student uses concepts and procedures of data analysis in a variety of situations.
Benchmark 2: Statistics – The student collects, records, and explains numerical (whole numbers) and non-numerical
data sets including the use of concrete objects in a variety of situations.
Kindergarten Knowledge Base Indicators
The student...
1. records numerical (quantitative) and non-numerical (qualitative)
data including concrete objects, graphs, and tables using these
data displays (2.4.K1a,g) ($):
a. graphs using concrete objects,
b. pictographs with a whole symbol or picture representing one
(no partial symbols or pictures),
c. frequency tables (tally marks).
2. collects data related to familiar everyday experiences by counting
and tallying (2.4.K1a,g) ($).
3. determines the mode (most) after sorting by one attribute
(2.4.K1a,g) ($), e.g., color, shape, or size.
Kindergarten Application Indicators
The student...
1. communicates the results of data collection from graphs using
concrete objects and frequency tables (2.4.A1e) ($), e.g., there are
sixteen kindergartners. Using themselves as concrete objects, the
six students wearing tennis shoes line up in a row. The ten students
wearing sandals line up in a row. The kindergartners become the
bar graph. Then someone says: There are less kids wearing tennis
shoes than kids wearing sandals.
K-29
January 31, 2004
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Graphs (data displays) are pictorial representations of mathematical relationships and are used to tell a story. Emphasizing the
importance of using equal-sized pictures or intervals is critical to ensuring that the data display is accurate. Graphs take many forms: pictographs
and graphs using concrete objects compare discrete data, frequency tables show how many times a certain piece of data occurs using tally marks
to record the data, circle graphs (pie charts) model parts of a whole, and line graphs show change over time.
The measures of central tendency (averages) of a data set are mean, median, and mode. Conceptual understanding of mean, median, and
mode is developed through the use of concrete objects that represent the data values.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra),
Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the
mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other
indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3
referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
K-30
January 31, 2004
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
FIRST GRADE
Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 1: Number Sense – The student demonstrates number sense for whole numbers, fractions, and money
using concrete objects in a variety of situations.
First Grade Knowledge Base Indicators
The student...
1. knows, explains, and represents whole numbers from 0 through 100
using concrete objects (2.4.K1a) ($).
2. compares and orders ($):
a. whole numbers from 0 through 100 using concrete objects
(2.4.K1a),
b. fractions with like denominators (halves and fourths) using
concrete objects (2.4.K1a,c).
3. recognizes a whole, a half, and a fourth and represents equal parts
of a whole (halves, fourths) using concrete objects, pictures,
diagrams, fraction strips, or pattern blocks (2.4.K1a,c) ($).
4. identifies and uses ordinal numbers first (1st) through tenth (10th)
(2.4.K1a).
5. identifies coins (pennies, nickels, dimes, quarters) and currency
($1, $5, $10) and states the value of each coin and each type of
currency using money models (2.4.K1d) ($).
6. recognizes and counts a like group of coins (pennies, nickels,
dimes) (2.4.K1d) ($).
First Grade Application Indicators
The student...
1. solves real-world problems using equivalent representations and
concrete objects to compare and order whole numbers from 0
through 50 (2.4.A1a) ($).
2. determines whether or not numerical values using whole numbers
from 0 through 50 are reasonable (2.4.A1a) ($), e.g., when asked if
40 dictionaries will fit inside the student’s desk, the student answers
no and explains why.
3. demonstrates that smaller whole numbers are within larger whole
numbers using whole numbers from 0 to 30 (2.4.A1a) ($), e.g., if
there are five pigs in a pen, there are also three pigs in the pen.
1-31
January 31, 2004
N – Noncalculator
($) – Financial Literacy
YEAR.
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL
Teacher Notes: Number sense refers to one’s ability to reason with numbers and to work with numbers in a flexible way. The ability
to compute mentally, to estimate based on understanding of number relationships and magnitudes, and to judge reasonableness of
answers are all involved in number sense.
When we say that someone has good number sense, we mean that he or she possesses a variety of abilities and understandings that
include an awareness of the relationships between numbers, an ability to represent numbers in a variety of ways, a knowledge of the
effects of operations, and an ability to interpret and use numbers in real-world counting and measurement situations. Such a person
predicts with some accuracy the result of an operation and consistently chooses appropriate measurement units. This “friendliness
with numbers” goes far beyond mere memorization of computational algorithms and number facts; it implies an ability to use numbers
flexibly, to choose the most appropriate representation of a number for a given circumstance, and to recognize when operations have
been correctly performed. (Number Sense and Operations: Addenda Series, Grades K-6, NCTM, 1993)
Mathematical models such as concrete objects, pictures, number lines, or unifix cubes are necessary for conceptual understanding
and should be used to explain computational procedures. If a mathematical model can be used to represent the concept, the
indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the
mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to
the other indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6,
1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked
at three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the
Appendix.
1-32
January 31, 2004
N – Noncalculator
($) – Financial Literacy
YEAR.
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL
Standard 1: Number and Computation
FIRST GRADE
Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 2: Number Systems and Their Properties – The student demonstrates an understanding of whole
numbers with a special emphasis on place value and recognizes, applies, and explains the concepts
of properties as they relate to whole numbers in a variety of situations.
First Grade Knowledge Base Indicators
The student...
1. reads and writes whole numbers from 0 through 100 in numerical
form ($).
2. represents whole numbers from 0 through 100 using various
groupings and place value models (place value mats, hundred
charts, or base ten blocks) emphasizing ones, tens, and hundreds
(2.4.K1b) ($), e.g., how many groups of tens are there in 32 or how
many groups of tens and ones in 62?
3. counts subsets of whole numbers from 0 through 100 both forwards
and backwards (2.4.K1a) ($).
4. writes in words whole numbers from 0 through 10.
5. identifies the place value of the digits in whole numbers from 0
through 100 (2.4.K1b) ($).
6. identifies any whole number from 0 through 30 as even or odd
(2.4.K1a).
7. uses the concepts of these properties with whole numbers from 0
through 100 and demonstrates their meaning using concrete objects
(2.4.K1a) ($):
a. commutative property of addition, e.g., 3 + 2 = 2 + 3,
b. zero property of addition (additive identity), e.g., 4 + 0 = 4.
First Grade Application Indicators
The student...
1. solves real-world problems with whole numbers from 0 through 50
using place value models (place value mats, hundred charts, or
base ten blocks) and the concepts of these properties to explain
reasoning (2.4.A1a-b) ($):
a. commutative property of addition, e.g., group 5 students into a
group of 3 and a group of 2, add to find the total; then reverse
the order of the students to show that 2 + 3 still equals 5;
b. zero property of addition, e.g., have students lay out 11
crayons, tell them to add zero (crayons). Then ask: How many
crayons are there?
1-33
January 31, 2004
N – Noncalculator
($) – Financial Literacy
YEAR.
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL
Teacher Notes: From the Mathematics Dictionary and Handbook (Nichols Schwartz Publishing, 1999), property as a mathematical term means a
characteristic (an attribute) of a number, geometric shape, mathematical operation, equation, or inequality. To give an example:
 Property of a number: 8 is divisible by 2.
 Property of a geometric shape: Each of the four sides of a square is of the same length.
 Property of an operation: Addition is commutative. For all numbers x and y, x + y = y + x.
 Property of an equation: For all numbers a, b, and c, if a = b, then a + c = b + c.
 Property of an inequality: For all numbers a, b, and c, if a > b, then a – c > b – c.
Mathematical models such as concrete objects, pictures, number lines, or unifix cubes are necessary for conceptual understanding and should be
used to explain computational procedures. If a mathematical model can be used to represent the concept, the indicator in the Models benchmark
is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a
(process models). Then, the indicator in the Models benchmark lists some of the mathematical models that could be used to teach the concept. In
addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For
example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1
(Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
1-34
January 31, 2004
N – Noncalculator
($) – Financial Literacy
YEAR.
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL
Standard 1: Number and Computation
FIRST GRADE
Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 3: Estimation – The student uses computational estimation with whole numbers in a variety of situations.
First Grade Knowledge Base Indicators
The student...
1. estimates whole number quantities from 0 through 100 using
various computational methods including mental math, paper and
pencil, concrete objects, and appropriate technology (2.4.K1a) ($).
2. estimates to check whether or not results of whole number
quantities from 0 through 100 are reasonable (2.4.K1a) ($).
First Grade Application Indicators
The student...
1. adjusts original whole number estimate of a real-world problem
using whole numbers from 0 through 50 based on additional
information (a frame of reference) (2.4.A1a) ($), e.g., an estimate is
made about the number of tennis balls in a shoebox; about half of
the tennis balls are removed from the box and counted. With this
additional information, an adjustment of the original estimate is
made.
1-35
January 31, 2004
N – Noncalculator
($) – Financial Literacy
YEAR.
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL
Teacher Notes: Estimate, as a verb, means to make an educated guess based on information in a problem or to give an answer
close to the exact number. Estimation is used when an exact answer is not needed, as in many real-life situations for which “ballpark”
computations are acceptable. Good number sense enables one to estimate a quantity, estimate a measure, or estimate an answer.
Estimation serves as an important companion to computation. It provides a tool for judging the reasonableness of computational
methods including mental math, paper and pencil, concrete objects, and appropriate technology. However, being able to compute
does not automatically lead to an ability to estimate or judge reasonableness of answers. Frequent modeling by the teacher helps
students develop a range of estimation strategies. Students should be encouraged to frequently explain their thinking as they
estimate. As with exact computation, sharing estimation strategies allows students access to others’ thinking and provides
opportunities for class discussion. Identifying the estimation strategy by name is not critical; however, explaining the thinking behind
the strategy to make a valid estimation is important. (Principles and Standards for School Mathematics, NCTM, 2000)
Mental math and estimation are distinct but related mathematical skills. Proficiency in mental math contributes to increased skill in
estimation. Students develop mental math skills easier when they are taught specific strategies and specific strategies based on
understanding also need to be developed for estimation. An estimation strategy for the early grades is front-end; the focus is on the
“front-end” or the leftmost digit because these digits are the most significant for forming an estimate. In order for students to become
more familiar with estimation, teachers should introduce estimation with examples where rounded or estimated numbers are used.
Emphasis should be placed on real-world examples where only estimation is required, e.g., about how many hours do you sleep a
night? Using the language of estimation is important, so students begin to realize that a variety of estimates (answers) are possible.
Mathematical models such as concrete objects, pictures, number lines, or unifix cubes are necessary for conceptual understanding
and should be used to explain computational procedures. If a mathematical model can be used to represent the concept, the
indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the
mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to
the other indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6,
1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked
at three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the
Appendix.
1-36
January 31, 2004
N – Noncalculator
($) – Financial Literacy
YEAR.
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL
Standard 1: Number and Computation
FIRST GRADE
Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 4: Computation – The student models, performs, and explains computation with whole numbers using
concrete objects in a variety of situations.
1-37
January 31, 2004
N – Noncalculator
($) – Financial Literacy
YEAR.
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL
First Grade Knowledge Base Indicators
The student...
1. computes with efficiency and accuracy using various computational
methods including mental math, paper and pencil, concrete objects,
and appropriate technology (2.4.K1a) ($).
2. N states and uses with efficiency and accuracy basic addition facts
with sums from 0 through 10 and corresponding subtraction facts ($).
3. skip counts by 2s, 5s, and 10s through 50 (2.4.K1a).
4. uses repeated addition (multiplication) with whole numbers to find the
sum when given the number of groups (ten or less) and given the
same number of concrete objects in each group (ten or less)
(2.4.K1a), e.g., three plates of cookies with 10 cookies on each plate
means 10 + 10 +10 = 30 cookies.
5. uses repeated subtraction (division) with whole numbers when given
the total number of concrete objects in each group to find the number
of groups (2.4.K1a), e.g., there are 9 pencils. If each student gets 2
pencils, how many students get pencils? 9 – 2 – 2 – 2 – 2 or 9 minus
2 four times means four students get 2 pencils each and there is 1
pencil left over. or There are 30 pieces of candy to put equally into five
bowls, how many pieces of candy will be in each bowl? 30 – 5 – 5 – 5
– 5 – 5 – 5 means there are six in each bowl.
6. performs and explains these computational procedures (2.4.K1a-b):
a. adds whole numbers with sums through 99 without regrouping
using concrete objects, e.g., 42 straws (bundled in 10s) + 21
straws (bundled in 10s) = 63 straws (bundled in 10s);
b. subtracts two-digit whole numbers without regrouping using
concrete objects, e.g., 63 cubes – 21 cubes = 42 cubes.
7. shows that addition and subtraction are inverse operations using
concrete objects (2.4.K1a) ($).
First Grade Application Indicators
The student...
1. solves one-step real-world addition or subtraction problems with
various groupings of two-digit whole numbers without regrouping
(2.4.A1a-b) ($), e.g., Jo has 48 crayons and 16 markers in her
desk. How many more crayons does she have than markers?
This problem could be solved using base 10 models or a number
line or by saying 48 – 10 = 38 and 38 – 6 = 32.
8. reads and writes horizontally and vertically the same addition
expression, e.g., 5 + 4 is the same as 4
+ 5.
1-38
January 31, 2004
N – Noncalculator
($) – Financial Literacy
YEAR.
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL
Teacher Notes: Efficiency and accuracy means that students are able to compute single-digit numbers with fluency. Students
increase their understanding and skill in single-digit addition and subtraction by developing relationships within addition and
subtraction combinations and by counting on for addition and counting up for subtraction and unknown-addend situations. Students
learn basic number combinations and develop strategies for computing that makes sense to them. Through class discussions,
students can compare the ease of use and ease of explanation of various strategies. In some cases, their strategies for computing
will be close to conventional algorithms; in other cases, they will be quite different. Many times, students’ invented approaches are
based on a sound understanding of numbers and operations, and these invented approaches often can be used with efficiency and
accuracy. (Principles and Standards for School Mathematics, NCTM, 2000)
The definition of computation is finding the standard representation for a number. For example, 6 + 6, 4 x 3, 17 – 5, and 24 ÷ 2 are all
representations for the standard representation of 12. Mental math is mentally finding the standard representation for a number – calculating in
your head instead of calculating using paper and pencil or technology. One of the main reasons for teaching mental math is to help students
determine if a computed/calculated answer is reasonable, in other words, using mental math to estimate to see if the answer makes sense.
Proficiency in mental math contributes to a better understanding of place value, mathematical operations, and basic number properties and an
increased skill in estimation. Students develop mental math skills easier when they are taught specific strategies. Mental math strategies for the
early grades include counting on, counting back, counting up, doubling numbers (doubles), and making ten. Other mental math strategies for the
early grades include skip counting and using number patterns.
One way to improve the understanding of numbers and operations is to encourage children to develop computational procedures that are
meaningful to them. Invented procedures promote the idea of mathematics as a meaningful activity. This is not to suggest that algorithm invention
is the only way to promote student understanding or that children should be discouraged from using a standard algorithm if that is their choice.
Different problems are best solved by different methods. For example, solving 7000 – 25 by counting down by tens and fives, yet solving 41 – 25 by
adding up, or decomposing, the numbers. Another example is adding 52 + 45; a student may count, “52, 62, 72, 82, 92, plus 5 is 97.” (“Invented
Strategies Can Develop Meaningful Mathematical Procedures” by William M. Carroll and Denise Porter, Teaching Children Mathematics, March
1997, pp. 370-374)
Regrouping refers to the reorganization of objects. In computation, regrouping is based on a “partitioning to multiples of ten” strategy.
For example, 46 + 7 could be solved by partitioning 46 into 40 and 6, then 40 + (6 + 7) = 40 + 13 (and then 13 is partitioned into 10
and 3) which then becomes (40 + 10) + 3 which becomes 50 + 3 = 53 or 7 could be partitioned as 4 and 3, then 46 + 4 (bridging
through 10) = 50 and 50 + 3 = 53. Before algorithmic procedures are taught, an understanding of “what happens” must occur. For
this to occur, instruction should involve the use of structured manipulatives. To emphasize the role of the base ten-numeration system
in algorithms, some form of expanded notation is recommended. During instruction, each child should have a set of manipulatives to
work with rather than sit and watch demonstrations by the teacher. Some additional computational strategies include doubles plus
one or two (i.e., 6 + 8, 6 and 6 are 12, so the answer must be 2 more or 14), compensation (i.e., 9 + 7, if one is taken away from 9, it
leaves 8; then that one is given to 7 to make 8, then 8 + 8 = 16), subtracting through ten (i.e., 13 – 5, 13 take away 3 is 10, then take
2 more away from 10 and that is 8), and nine is one less than ten (i.e., 9 + 6, 10 and 6 are 16, and 1 less than 16 makes 15).
(Teaching Mathematics in Grades K-8: Research Based Methods, ed. Thomas R. Post, Allyn and Bacon, 1988)
1-39
January 31, 2004
N – Noncalculator
($) – Financial Literacy
YEAR.
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL
Mathematical models such as concrete objects, pictures, number lines, or unifix cubes are necessary for conceptual understanding
and should be used to explain computational procedures. If a mathematical model can be used to represent the concept, the
indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra),
Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of
the mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back
to the other indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6,
1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator
3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked
at three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the
Appendix.
1-40
January 31, 2004
N – Noncalculator
($) – Financial Literacy
YEAR.
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL
Standard 2: Algebra
FIRST GRADE
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 1: Patterns – The student recognizes, describes, extends, develops, and explains relationships in
patterns using concrete objects in a variety of situations.
1-41
January 31, 2004
N – Noncalculator
($) – Financial Literacy
YEAR.
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL
First Grade Knowledge Base Indicators
The student...
1. uses concrete objects, drawings, and other representations to work
with types of patterns (2.4.K1a):
c. repeating patterns, e.g., an AB pattern is like 1-2, 1-2, …; an
ABC pattern is like dog-horse-pig, dog-horse-pig, …; an AAB
pattern is like Δ-Δ-Ο, Δ-Δ-Ο, …;
d. growing (extending) patterns, e.g., 1, 2, 3, …
2. uses the following attributes to generate patterns:
a. counting numbers related to number theory (2.4.K1.a), e.g.,
evens, odds, or skip counting by 2s, 5s, or 10s;
b. whole numbers that increase (2.4.K1a) ($), e.g., 11, 21, 31, ... or
like 2, 4, 6, …;
c. geometric shapes (2.4.K1f), e.g., ▲, ▀, ◊, ▲, ▀, ◊, …;
d. measurements (2.4.K1a), e.g., counting by inches or feet;
e. the calendar (2.4.K1a), e.g., January, February, March, …;
f. money and time (2.4.K1d) ($), e.g., 10¢, 20¢, 30¢, … or 1:00,
1:30, 2:00, ...;
g. things related to daily life (2.4.K1a), e.g., seasons, temperature,
or weather;
h. things related to size, shape, color, texture, or movement
(2.4.K1a); e.g., tall-short, tall-short, tall-short, …; or snapping
fingers, clapping hands, or stomping feet (kinesthetic patterns).
3. identifies and continues a pattern presented in various formats
including numeric (list or table), visual (picture, table, or graph),
verbal (oral description), kinesthetic (action), and written (2.4.K1a)
($).
First Grade Application Indicators
The student...
1. generalizes the following patterns using pictorial, oral, and/or written
descriptions including the use of concrete objects:
a. whole number patterns (2.4.A1a) ($);
b. patterns using geometric shapes (2.4.A1c);
c. calendar patterns (2.4.A1a);
d. patterns using size, shape, color, texture, or movement
(2.4.A1a).
2. recognizes multiple representations of the same pattern (2.4.A1a), e.g.,
the AB pattern could be represented by clap, snap, clap, snap, … or
red, green, red, green, … or square, circle, square, circle, ….
3. uses concrete objects to model a whole number pattern (2.4.A1a):
d. counting by ones:  ,, , …
e. counting by twos: ,  
, 
, …
f. counting by fives: xxxxx, xxxxx xxxxx
xxxxx, xxxxx
xxxxx, …
g. counting by tens:
,  
, 
, …
4. generates (2.4.K1a):
a. repeating patterns for the AB pattern, the ABC pattern, and the
AAB pattern;
b. growing patterns that add 1, 2, 5, or 10.
Teacher Notes: Working with patterns is an important process in the development of mathematical thinking. Patterns can be based on geometric
attributes (shapes, regions, angles); measurement attributes (color, texture, length, weight, volume, number); relational attributes (proportion,
sequence, functions); and affective attributes (values, likes, dislikes, familiarity, heritage, culture). (Learning to Teach Mathematics, Randall J.
Souviney, Macmillan Publishing Company, 1994)
This process (working with patterns) can be used to develop or deepen understandings of important concepts in number theory, whole numbers,
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measurement, geometry, probability, and functions. Working with patterns provides opportunities for students to recognize, describe, extend,
develop, and explain.
Number theory is the study of the properties of the counting numbers (positive integers), their relationships, ways to represent them, and patterns
among them. Number theory includes the concepts of odd and even numbers, factors and multiples, primes and composites, and greatest common
factor and least common multiple.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that
could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are
identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and
Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
1-43
January 31, 2004
N – Noncalculator
($) – Financial Literacy
YEAR.
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL
Standard 2: Algebra
FIRST GRADE
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 2: Variable, Equations, and Inequalities – The student solves addition and subtraction equations using
concrete objects in a variety of situations.
First Grade Knowledge Base Indicators
The student...
1. explains and uses symbols to represent unknown whole number
quantities from 0 through 20 (2.4.K1a).
2. finds the unknown sum or difference of the basic facts using
concrete objects (2.4.K1a) ($), e.g., 12 dominoes – 5 dominoes = Δ
dominoes or Δ cubes = 2 cubes + 4 cubes.
3. describes and compares two whole numbers from 0 through 100
using the terms: is equal to, is less than, is greater than (2.4.K1a-b)
($).
First Grade Application Indicators
The student...
1. represents real-world problems using concrete objects, pictures, oral
descriptions, and symbols and the basic addition and subtraction
facts with one operation and one unknown (2.4. A1a) ($), e.g., given
some marbles, Sue says: 3 red marbles and 3 blue marbles equal 6
marbles. Sue also shows and writes the problem and solution: 3 + 3 =
 or RRR + BBB = , 3 + 3 = 6.
2. generates and solves problem situations using the basic facts to find
the unknown sum or difference with concrete objects (2.4.A1a), e.g.,
a student generates this problem: I have 6 marbles. My sister has 4.
How many do we have altogether? The student shows 6 + 4 = ,
and 6 + 4 = 10.
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THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL
Teacher Notes: Understanding the concept of variable is fundamental to algebra. In the early grades, students use various symbols including
letters and geometric shapes to represent unknown quantities that both do and do not vary. Quantities that are not given and do not vary are often
referred to as unknowns or missing elements when they appear in equations, e.g., 2 + 4 = Δ.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that
could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are
identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and
Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
Standard 2: Algebra
FIRST GRADE
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 3: Functions – The student recognizes and describes whole number relationships using concrete
objects in a variety of situations.
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THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL
First Grade Knowledge Base Indicators
The student...
1. plots whole numbers from 0 through 100 on segments of a number
line (2.4.K1a).
2. states mathematical relationships between whole numbers from 0
through 50 using various methods including mental math, paper and
pencil, and concrete objects (2.4.K1a) ($), e.g., every time a hand is
added to the set, five more fingers are added to the total.
3. states numerical relationships for whole numbers from 0 through 50
in a horizontal or vertical function table (input/output machine,
T-table) (2.4.K1e) ($), e.g.,
Number of bicycles
Total number of wheels
1
2
2
4
3
6
4
8
First Grade Application Indicators
The student...
1. represents and describes mathematical relationships for whole
numbers from 0 through 50 using concrete objects, pictures, oral
descriptions, and symbols (2.4.A1a) ($).
2. recognizes numerical patterns (counting by 2s, 5s, and 10s) through
50 using a hundred chart (2.4.A1a).
…
…
5
10
The student states: For every bicycle added, you add two more
wheels.
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Teacher Notes: A number line (a mathematical model) is a diagram that represents numbers with equal distances marked off as points on a line,
an example of one-to-one correspondence (a relation). There are many kinds of relations – one-to-many, many-to-one, many-to-many, and one-toone. A function is a special kind of relation.
Functions are relationships or rules in which each member of one set is paired with one, and only one, member of another set (an
ordered pair). The concept of function can be introduced using function machines. Any number put in the machine will be changed
according to some rule. A record of inputs and corresponding outputs can be maintained in a two-column format. Function tables,
input/output machines, and T-tables may be used interchangeably and serve the same purpose.
Function concepts should be developed from growing patterns. Each term in a number sequence is related to its position in the sequence – the
functional relationship. The pattern – 4, 7, 10, 13, 16, 19, and so on – is an arithmetic sequence with a difference of 3. The pattern could be
described as add 3 meaning that 3 must be added to any term after the first to find the next. This is an example of a recursive pattern. In a
recursive pattern, after one or more consecutive terms are given, each successive term in the sequence is obtained from the previous term(s).
In the pattern 1, 4, 9, 16, 25, …, 225; there is no common difference. This sequence is not arithmetic or geometric (no common ratio between
geometric terms). Neither is it a combination of the two; however, there is a pattern and the missing terms between 25 and 225 can be found. To
find the term value, square the number of the term. The next missing terms would be 36, 49, 64, 81, 100, 121, and 144. This is an example of an
explicit pattern. In an explicit pattern, each term in the sequence is defined by the term’s number in the sequence.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
1-47
January 31, 2004
N – Noncalculator
($) – Financial Literacy
YEAR.
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL
Standard 2: Algebra
FIRST GRADE
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 4: Models – The student uses mathematical models including concrete objects to represent, show, and
communicate mathematical relationships in a variety of situations.
First Grade Knowledge Base Indicators
The student...
1. knows, explains, and uses mathematical models to represent
mathematical concepts, procedures, and relationships. Mathematical
models include:
a. process models (concrete objects, pictures, diagrams, number
lines, unifix cubes, hundred charts, measurement tools, or
calendars) to model computational procedures and
mathematical relationships, to compare and order numerical
quantities, and to represent fractional parts (1.1.K1-4, 1.2.K3,
1.2.K6-7, 1.3.K1-2, 1.4.K1, 1.4.K2-7, 2.1.K1, 2.1.K1d-h, 2.1.K2ab, 2.2.K3-4, 2.3.K1-2, 3.2.K1-6, 3.3.K1-3, 3.4.K1-3 4.2.K3-4) ($);
b. place value models (place value mats, hundred charts, or base
ten blocks) to compare, order, and represent numerical
quantities and to model computational procedures (1.2.K2,
1.2.K5, 1.4.K6, 2.2.K3) ($);
c. fraction models (fraction strips or pattern blocks) to compare,
order, and represent numerical quantities (1.1.K2-3) ($);
d. money models (base ten blocks or coins) to compare, order, and
represent numerical quantities (1.1.K5-6, 2.1.K2f) ($);
e. function tables (input/output machines, T-tables) to model
numerical relationships (2.3.K3) ($);
f. two-dimensional geometric models (geoboards, dot paper,
pattern blocks, tangrams, or attribute blocks), three-dimensional
geometric models (solids), and real-world objects to compare
size and to model attributes of geometric shapes (2.1.K1c,
3.1.K1-3);
g. two-dimensional geometric models (spinners), three-dimensional
geometric models (number cubes), and concrete objects to
model probability (4.1.K1-2) ($);
First Grade Application Indicators
The student…
1. recognizes that various mathematical models can be used to
represent the same problem situation. Mathematical models
include:
f. process models (concrete objects, pictures, diagrams, number
lines, unifix cubes, measurement tools, or calendars) to model
computational procedures and mathematical relationships, to
compare and order numerical quantities, and to model problem
situations (1.1.A1-3, 1.2.A1, 1.3.A1, 1.4.A1, 2.1.A1a, 2.1.A1c-d,
2.1.A2-3, 2.2.A1-2, 2.3.A1-2, 3.2.A1-3, 3.3.A1-2, 3.4.A1, 4.2.A2)
($);
g. place value models (place value mats, hundred charts, or base
ten blocks) to compare, order, and represent numerical
quantities and to model computational procedures (1.2.A1,
1.4.A1) ($);
h. two-dimensional geometric models (geoboards, dot paper,
pattern blocks, tangrams, or attribute blocks), three-dimensional
geometric models (solids), and real-world objects to compare
size and to model attributes of geometric shapes (2.1.A1b,
3.1.A1-2);
d. two-dimensional geometric models (spinners), three-dimensional
geometric models (number cubes), and concrete objects to
model probability (4.1.A1) ($);
e. graphs using concrete objects, pictographs, frequency tables,
and horizontal and vertical bar graphs to organize, display, and
explain data (4.1.A1, 4.2.A1) ($).
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h. graphs using concrete objects, pictographs, frequency tables,
horizontal and vertical bar graphs, and Venn diagrams or other
pictorial displays to organize, display, and explain data (4.1.A1,
4.2.A1-2) ($);
i. Venn diagrams to sort data (4.2.K4).
2. uses concrete objects, pictures, diagrams, drawings, or
dramatizations to show the relationship between two or more things
($).
Teacher Notes: For assessment purposes, the mathematical modeling process appropriate to the indicator may be included as part of the item
being assessed.
The mathematical modeling process involves:
a. selecting key features and relationships within the real-world situation and representing these concepts in mathematical terms through some
sort of mathematical model,
b. performing manipulations and mathematical procedures within the mathematical model,
c. interpreting the results of the manipulations within the mathematical model,
d. using these results to make inferences about the original real-world situation.
The use of mathematical models is necessary for conceptual understanding. The ways in which mathematical ideas are represented is
fundamental to how students understand and use those ideas. As students begin to use multiple representations of the same situation, they begin
to develop an understanding of the advantages and disadvantages of the various representations/models. For example, comparing the number of
boys and girls in the classroom can be represented by lining them up in two different lines. The same situation also can be represented by pictures
of the children (pictograph), a bar graph, or by using two different colors of the same manipulative (unifix cubes or color tiles).
Many mathematical models are listed in this benchmark. The indicator lists some of the mathematical models that could be used to teach a
concept. Each indicator in this benchmark is linked to other indicators in other benchmarks; those indicators are identified in the parentheses. For
example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1
(Number Sense), and Knowledge Indicator 3. In addition, the indicator in the other benchmarks identifies, in parentheses, the Models’ indicator. For
example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models).
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
1-49
January 31, 2004
N – Noncalculator
($) – Financial Literacy
YEAR.
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL
Standard 3: Geometry
FIRST GRADE
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 1: Geometric Figures and Their Properties – The student recognizes geometric shapes and
describes
their attributes using concrete objects in a variety of situations.
First Grade Knowledge Base Indicators
The student...
4. recognizes and draws circles, squares, rectangles, triangles, and
ellipses (ovals) (plane figures/two-dimensional figures) (2.4.K1f).
2. recognizes and investigates attributes of circles, squares,
rectangles, triangles, and ellipses (plane figures) using concrete
objects, drawings, and appropriate technology (2.4.K1f).
3. recognizes cubes, rectangular prisms, cylinders, cones, and
spheres (solids/three-dimensional figures) (2.4.K1f).
First Grade Application Indicators
The student...
1. demonstrates how (2.4.A1c):
a. a geometric shape made of several plane figures (circles,
squares, rectangles, triangles, ellipses) can be separated to
make two or more different plane figures;
b. several plane figures (circles, squares, rectangles, triangles,
ellipses) can be combined to make a new geometric shape;
c. several solids (cubes, rectangular prisms, cylinders, cones,
spheres) can be combined to make a new geometric shape.
2. sorts plane figures and solids (circles, squares, rectangles, triangles,
ellipses, cubes, rectangular prisms, cylinders, cones, spheres) by a
given attribute (2.4.A1c).
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Teacher Notes: Geometry is the study of shapes, their properties, and their relationships to other shapes. Symbols and numbers are used to
describe their properties and their relationships to other shapes. The fundamental concepts in geometry are point (no dimension), line (onedimensional), plane (two-dimensional), and space (three-dimensional). Plane figures are referred to as two-dimensional and solids are referred to
as three-dimensional.
From the Mathematics Dictionary and Handbook (Nichols Schwartz Publishing, 1999), property as a mathematical term means a characteristic (an
attribute) of a number, geometric shape, mathematical operation, equation, or inequality. To give an example:
 Property of a number: 8 is divisible by 2.
 Property of a geometric shape: Each of the four sides of a square is of the same length.
 Property of an operation: Addition is commutative. For all numbers x and y, x + y = y + x.
 Property of an equation: For all numbers a, b, and c, if a = b, then a + c = b + c.
 Property of an inequality: For all numbers a, b, and c, if a > b, then a – c > b – c.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
1-51
January 31, 2004
N – Noncalculator
($) – Financial Literacy
YEAR.
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL
Standard 3: Geometry
FIRST GRADE
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 2: Measurement and Estimation – The student estimates and measures using standard and nonstandard
units of measure with concrete objects in a variety of situations.
First Grade Knowledge Base Indicators
The student...
1. uses whole number approximations (estimations) for length and
weight using nonstandard units of measure (2.4.K1a) ($), e.g., the
width of the chalkboard is about 10 erasers long or the weight of
one encyclopedia is about five picture books.
2. compares two measurements using these attributes (2.4.K1a) ($):
a. longer, shorter (length);
b. taller, shorter (height);
c. heavier, lighter (weight);
d. hotter, colder (temperature).
3. reads and tells time at the hour and half-hour using analog and
digital clocks (2.4.K1a).
4. selects appropriate measuring tools for length, weight, volume, and
temperature for a given situation (2.4.K1a) ($).
5. measures length and weight to the nearest whole unit using
nonstandard units (2.4.K1a) ($).
6. states the number of days in a week and months in a year
(2.4.K1a).
First Grade Application Indicators
The student...
1. compares and orders concrete objects by length or weight (2.4.A1a)
($).
2. compares the weight of two concrete objects using a balance
(2.4.A1a).
3. locates and names concrete objects that are about the same length,
weight, or volume as a given concrete object (2.4.A1a) ($).
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THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL
Teacher Notes: The term geometry comes from two Greek words meaning “earth measure.” The process of learning to measure at the early
grades focuses on identifying what property (length, weight) is to be measured and to make comparisons. Estimation in measurement is defined
as making guesses as to the exact measurement of an object without using any type of measurement tool. Estimation helps students develop a
relationship between the different sizes of units of measure. It helps students develop basic properties of measurement and it gives students a tool
to determine whether a given measurement is reasonable.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
1-53
January 31, 2004
N – Noncalculator
($) – Financial Literacy
YEAR.
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL
Standard 3: Geometry
FIRST GRADE
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 3: Transformational Geometry – The student develops the foundation for spatial sense using concrete
objects in a variety of situations.
First Grade Knowledge Base Indicators
The student...
1. describes the spatial relationship between two concrete objects
using appropriate vocabulary (2.4.K1a), e.g., behind, above, below,
on, under, beside, or in front of.
2. recognizes that changing an object's position or orientation does
not change the name, size, or shape of the object (2.4.K1a).
3. describes movement of concrete objects using appropriate
vocabulary (2.4.K1a), e.g., right, left, up, or down.
First Grade Application Indicators
The student...
1. shows two concrete objects or shapes are congruent by physically
fitting one object or shape on top of the other (2.4.A1a).
2. gives and follows directions to move concrete objects from one
location to another using appropriate vocabulary (2.4.A1a), e.g.,
right, left, up, down, behind, or above.
Teacher Notes: Transformational geometry is another way to investigate geometric figures by moving every point in a plane figure to a new
location. To help students form images of shapes through different transformations, students can use concrete objects, figures drawn on graph
paper, mirrors or other reflective surfaces, or appropriate technology.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
1-54
January 31, 2004
N – Noncalculator
($) – Financial Literacy
YEAR.
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL
Standard 3: Geometry
FIRST GRADE
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 4: Geometry From An Algebraic Perspective – The student identifies one or more points on a number
line in a variety of situations.
First Grade Knowledge Base Indicators
The student...
1. locates and plots whole numbers from 0 through 100 on a segment
of a number line (horizontal/vertical) (2.4.K1a), e.g., using a
segment of a number line from 45 to 60 to locate the whole number
50.
2. describes a given whole number from 0 to 100 as coming before or
after another number on a number line (2.4.K1a).
3. uses a number line to model addition and counting using whole
numbers from 0 to 100 (2.4.K1a).
First Grade Application Indicators
The student...
1. solves real-world problems involving counting and adding whole
numbers from 0 to 50 using a number line (2.4.A1a) ($), e.g., Nancy
has 23¢. She finds 18¢ more in her pocket. How much money does
she now have?
Teacher Notes: A number line (a mathematical model) is a diagram that represents numbers with equal distances marked off as points on a line,
and is an example of one-to-one correspondence (a relation). A number line can be used as a visual representation of numbers and operations.
In addition, a number line used horizontally and vertically is a precursor to the coordinate plane.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra),
Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the
mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other
indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3
referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
1-55
January 31, 2004
N – Noncalculator
($) – Financial Literacy
YEAR.
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL
Standard 4: Data
FIRST GRADE
Standard 4: Data – The student uses concepts and procedures of data analysis in a variety of situations.
Benchmark 1: Probability – The student applies the concepts of probability using concrete objects in a variety of
situations.
First Grade Knowledge Base Indicators
The student...
1. recognizes whether an outcome of a simple event in an experiment
or simulation is impossible, possible, or certain (2.4.K1g) ($).
2. recognizes and states whether a simple event in an experiment or
simulation including the use of concrete objects can have more
than one outcome (2.4.K1g).
First Grade Application Indicators
The student...
1. makes a prediction about a simple event in an experiment or
simulation, conducts the experiment or simulation, and records the
results in a graph using concrete objects, a pictograph with a
symbol or picture representing only one, or a bar graph (2.4.A1d-e).
Teacher Notes: Ideas from probability reinforce concepts in the other Standards, especially Number and Computation and Geometry. In the
early grades, students need to develop an intuitive concept of chance – whether or not something is unlikely or likely to happen. Beginning
probability experiences should be addressed through the use of concrete objects, coins, and geometric models (spinners or number cubes).
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra),
Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the
mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other
indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3
referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
1-56
January 31, 2004
N – Noncalculator
($) – Financial Literacy
YEAR.
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL
Standard 4: Data
FIRST GRADE
Standard 4: Data – The student uses concepts and procedures of data analysis in a variety of situations.
Benchmark 2: Statistics – The student collects, displays, and explains numerical (whole numbers) and non-numerical
data sets including the use of concrete objects in a variety of situations.
First Grade Knowledge Base Indicators
The student...
1. displays and reads numerical (quantitative) and non-numerical
(qualitative) data in a clear, organized, and accurate manner
including a title, labels, and whole number intervals using these
data displays (2.4.K1h) ($):
a. graphs using concrete objects,
b. pictographs with a whole symbol or picture representing one
(no partial symbols or pictures),
c. frequency tables (tally marks),
d. horizontal and vertical bar graphs,
e. Venn diagrams or other pictorial displays, e.g., glyphs.
2. collects data using different techniques (observations or interviews)
and explains the results (2.4.K1h) ($).
3. identifies the minimum (lowest) and maximum (highest) values in a
data set (2.4.K1a) ($).
4. determines the mode (most) after sorting by one attribute
(2.4.K1a,i) ($).
5. sorts and records qualitative (non-numerical, categorical) data sets
using one attribute (2.4.K1a) ($), e.g., color, shape, or size.
First Grade Application Indicators
The student...
1. communicates the results of data collection and answers questions
(identifying more, less, fewer, greater than, or less than) based on
information from (2.4.A1e) ($):
a. graphs using concrete objects,
b. a pictograph with a whole symbol or picture representing only
one (no partial symbols or pictures),
c. a horizontal or vertical bar graph.
2. determines categories from which data could be gathered (2.4.A1a)
($), e.g., categories could include shoe size, height, favorite candy
bar, or number of pockets in clothing.
1-57
January 31, 2004
N – Noncalculator
($) – Financial Literacy
YEAR.
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL
Teacher Notes: Graphs (data displays) are pictorial representations of mathematical relationships and are used to tell a story. Emphasizing the
importance of using equal-sized pictures or intervals is critical to ensuring that the data display is accurate. Graphs take many forms: pictographs
and graphs using concrete objects compare discrete data, frequency tables show how many times a certain piece of data occurs using tally marks
to record the data, circle graphs (pie charts) model parts of a whole, line graphs show change over time, and Venn diagrams show relationships
between sets of objects.
The measures of central tendency (averages) of a data set are mean, median, and mode. Conceptual understanding of mean, median, and
mode is developed through the use of concrete objects that represent the data values.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
1-58
January 31, 2004
N – Noncalculator
($) – Financial Literacy
YEAR.
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL
Standard 1: Number and Computation
SECOND GRADE
Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 1: Number Sense – The student demonstrates number sense for whole numbers, fractions, and money
using concrete objects in a variety of situations.
Second Grade Knowledge Base Indicators
Second Grade Application Indicators
The student…
The student...
1. ■ knows, explains, and represents whole numbers from 0 through
2. solves real-world problems using equivalent representations and
1,000 using concrete objects (2.4.K1a) ($).
concrete objects to ($):
2. compares and orders:
a. compare and order whole numbers from 0 through 1,000
a. whole numbers from 0 through 1,000 using concrete objects
(2.4.A1b), e.g., using base ten blocks, represent the students
(2.4.K1a) ($);
in each class in the school; represent the numbers using digits
b. fractions greater than or equal to zero with like denominators
(24) and compare and order in different ways;
(halves, fourths, thirds, eighths) using concrete objects
b. add and subtract whole numbers from 0 through 100
(2.4.K1a,c).
(2.4.A1b), e.g., using base ten blocks, represent the number of
3. uses addition and subtraction to show equivalent representations for
students in each class in the school; find the total of all
whole numbers from 0 through 100 (2.4.K1a-b), e.g., 8 – 5 = 2 + 1 or
students in grades K, 1, and 2 and the total of all of the
20 + 40 = 70 – 10.
students in grades 3, 4, and 5 and then subtract to find the
4. identifies and uses ordinal positions from first (1st) through twentieth
difference between the primary and intermediate grades;
(20th) (2.4.K1a).
c. compare and order a mixed group of coins to $1.00 (2.4.A1c),
5. ▲identifies coins, states their values, and determines the total value to
e.g., use actual coins to show 2 different amounts; students
$1.00 of a mixed group of coins using pennies, nickels, dimes,
write: 47¢ is more than 31¢;
quarters, and half-dollars (2.4.K1d) ($).
d. find equivalent values of coins to $1.00 without mixing coins
6. counts a like combination of currency ($1, $5, $10, $20) to $100
(2.4.A1c), e.g., 50 pennies = 2 quarters, 5 dimes = 2 quarters,
(2.4.K1d) ($).
or 10 nickels = 2 quarters.
2. determines whether or not numerical values that involve whole
numbers from 0 through 1,000 are reasonable (2.4.A1a-b) ($), e.g.,
if there are 26 children, plus 10 more children, is it reasonable to
say there are 50 children?
2-59
January 31, 2004
▲ – Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Number sense refers to one’s ability to reason with numbers and to work with numbers in a flexible way. The ability
to compute mentally, to estimate based on understanding of number relationships and magnitudes, and to judge reasonableness of
answers are all involved in number sense.
When we say that someone has good number sense, we mean that he or she possesses a variety of abilities and understandings that
include an awareness of the relationships between numbers, an ability to represent numbers in a variety of ways, a knowledge of the
effects of operations, and an ability to interpret and use numbers in real-world counting and measurement situations. Such a person
predicts with some accuracy the result of an operation and consistently chooses appropriate measurement units. This “friendliness
with numbers” goes far beyond mere memorization of computational algorithms and number facts; it implies an ability to use numbers
flexibly, to choose the most appropriate representation of a number for a given circumstance, and to recognize when operations have
been correctly performed. (Number Sense and Operations: Addenda Series, Grades K-6, NCTM, 1993)
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks
are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can
be used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a)
refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the
Models benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each indicator in the
Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For example, process
models are linked to 1.1.K3, 1.2.K6, 1.3.K1, with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number
Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked
at three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the
Appendix.
2-60
January 31, 2004
▲ – Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
SECOND GRADE
Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 2: Number Systems and Their Properties – The student demonstrates an understanding of whole
numbers with a special emphasis on place value and recognizes, uses, and explains the concepts
of properties as they relate to whole numbers in a variety of situations.
Second Grade Knowledge Base Indicators
Second Grade Application Indicators
The student...
1. reads and writes ($):
a. whole numbers from 0 through 1,000 in numerical form, e.g., 942
is read as nine hundred forty-two and is written in numerical form
as 942;
b. whole numbers from 0 through 100 in words, e.g., 76 is read as
seventy-six and is written in words as seventy-six.
c. whole numbers from 0 through 1,000 in numerical form when
presented in word form, e.g., nine hundred forty-six is read as nine
hundred forty-six and is written as 946.
2. ▲ represents whole numbers from 0 through 1,000 using various
groupings and place value models emphasizing 1s, 10s, and 100s;
explains the groups; and states the value of the digit in ones place,
tens place, and hundreds place (2.4.K1b) ($), e.g., in 385, the 3
represents 3 hundreds, 30 tens, or 300 ones; the 8 represents 8 tens
or 80 ones; and the 5 represents 5 ones.
3. counts subsets of whole numbers from 0 through 1,000 forwards and
backwards (2.4.K1a) ($), e.g., 311, 312, …, 320; or 210, 209, …, 204.
4. ▲ identifies the place value of the digits in whole numbers from 0
through 1,000 (2.4.K1b) ($).
5. identifies any whole number from 0 through 100 as even or odd
(2.4.K1a).
uses the concepts of these properties with whole numbers from 0 through
100 and demonstrates their meaning including the use of concrete
objects (2.4.K1a) ($):
a. commutative property of addition, e.g., 5 + 6 = 6 + 5;
The student...
1. solves real-world problems with whole numbers from 0 through
100 using place value models and the concepts of these
properties to explain reasoning (2.4.A1a-b) ($):
a. commutative property of addition, e.g., group 17 students into
a 9 and an 8, add to find the total, then reverse the students
to show 8 + 9 still equals 17;
b. zero property of addition, e.g., have students lay out 22
crayons, tell them to add zero (crayons). How many crayons?
22 + 0 = 22.
2. performs various computational procedures with whole numbers
from 0 through 100 using these properties and explains how they
were used (2.4.A1b):
a. commutative property of addition (5 + 6 = 6 + 5), e.g., given
6 + 5, the student says: I know that the answer is 11 because
5 + 6 is 11 and the order you add them in does not matter;
b. zero property of addition (17 + 0 = 0 + 17), e.g., given 17 + 0,
the student says: I know that the answer is 17 because
adding 0 does not change the answer (sum).
2-61
January 31, 2004
▲ – Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
zero property of addition (additive identity), e.g., 4 + 0 = 4;
c. associative property of addition, e.g., (3 + 2) + 4 = 3 + (2 + 4);
d. symmetric property of equality applied to basic addition and
subtraction facts, e.g., 10 = 2 + 8 is the same as 2 + 8 = 10 or
7 = 10 – 3 is the same as 10 – 3 = 7.
b.
Teacher Notes: From the Mathematics Dictionary and Handbook (Nichols Schwartz Publishing, 1999), property as a mathematical term means a
characteristic (an attribute) of a number, geometric shape, mathematical operation, equation, or inequality. To give an example:
 Property of a number: 8 is divisible by 2.
 Property of a geometric shape: Each of the four sides of a square is of the same length.
 Property of an operation: Addition is commutative. For all numbers x and y, x + y = y + x.
 Property of an equation: For all numbers a, b, and c, if a = b, then a + c = b + c.
 Property of an inequality: For all numbers a, b, and c, if a > b, then a – c > b – c.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, with 1.1.K3 referring to Standard 1
(Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
2-62
January 31, 2004
▲ – Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
SECOND GRADE
Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 3: Estimation – The student uses computational estimation with whole numbers
and money in a variety
of situations.
Second Grade Knowledge Base Indicators
Second Grade Application Indicators
The student...
The student...
1. estimates whole number quantities from 0 through 1,000 and
monetary amounts through $50 using various computational
methods including mental math, paper and pencil, concrete
objects, and appropriate technology (2.4.Ka-b,d) ($).
2. uses various estimation strategies to estimate whole number
quantities from 0 through 1,000 (2.4.K1a) ($).
1. adjusts original whole number estimate of a real-world problem using
numbers from 0 through 1,000 based on additional information (a
frame of reference) (2.4.A1a) ($), e.g., given a pint container and told
the number of marbles it has in it, the student would estimate the
number of marbles in a quart container.
2. estimates to check whether or not the result of a real-world problem
using whole numbers from 0 through 1,000 and monetary amounts
through $50 is reasonable and makes predictions based on the
information (2.4.A1a-c) ($), e.g., in the lunchroom, good behavior that
day can earn the class an extra 5 minutes of recess. Is it reasonable
to think you can earn an hour of extra recess in one week? After
answering the first question, then ask: About how many days would it
take?
3. selects a reasonable magnitude from three given quantities, a onedigit numeral, a two-digit numeral, and a three-digit numeral (5, 50,
500) based on a familiar problem situation and explains the
reasonableness of the selection (2.4.A1a), e.g., could the basket of
fruit on the kitchen table hold 7 apples, 70 apples, or 700 apples? The
student chooses 7 apples because apples are about the size of
baseballs and 7 will fit in a basket on the kitchen table.
2-63
January 31, 2004
▲ – Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Estimate, as a verb, means to make an educated guess based on information in a problem or to give an answer
close to the exact number. Estimation is used when an exact answer is not needed, as in many real-life situations for which “ballpark”
computations are acceptable. Good number sense enables one to estimate a quantity, estimate a measure, or estimate an answer.
Estimation serves as an important companion to computation. It provides a tool for judging the reasonableness of computational
methods including mental math, paper and pencil, concrete objects, and appropriate technology. However, being able to compute
does not automatically lead to an ability to estimate or judge reasonableness of answers. Frequent modeling by the teacher helps
students develop a range of estimation strategies. Students should be encouraged to frequently explain their thinking as they
estimate. As with exact computation, sharing estimation strategies allows students access to others’ thinking and provides
opportunities for class discussion. Identifying the estimation strategy by name is not critical; however, explaining the thinking behind
the strategy to make a valid estimation is important. (Principles and Standards for School Mathematics, NCTM, 2000)
Mental math and estimation are distinct but related mathematical skills. Proficiency in mental math contributes to increased skill in
estimation. Students develop mental math and estimation easier when they are taught specific strategies. An estimation strategy for
the early grades is front-end. The focus is on the “front-end” or the leftmost digit because these digits are the most significant for
forming an estimate. In order for students to become more familiar with estimation, teachers should introduce estimation with
examples where rounded or estimated numbers are used. Emphasis should be placed on real-world examples where only estimation
is required, e.g., About how many hours do you sleep a night? Using the language of estimation is important, so students begin to
realize that a variety of estimates (answers) are possible. In addition to front-end estimation and rounding, another estimation
strategy is compatible “nice” numbers.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks
are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can
be used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a)
refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the
Models benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each indicator in the
Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For example, process
models are linked to 1.1.K3, 1.2.K6, 1.3.K1, with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number
Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked
at three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the
2-64
January 31, 2004
▲ – Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Appendix.
2-65
January 31, 2004
▲ – Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
SECOND GRADE
Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 4: Computation – The student models, performs, and explains computation with whole numbers and
money using concrete objects in a variety of situations.
Second Grade Knowledge Base Indicators
Second Grade Application Indicators
The student...
The student...
1. computes with efficiency and accuracy using various computational
methods including mental math, paper and pencil, concrete objects,
and appropriate technology (2.4.K1a) ($).
2. N states and uses with efficiency and accuracy basic addition facts
with sums from 0 through 20 and corresponding subtraction facts
(2.4.K1a) ($).
3. skip counts by 2s, 5s, and 10s through 100 and skip counts by 3s
through 36 (2.4.K1a).
4. uses repeated addition (multiplication) with whole numbers to find
the sum when given the number of groups (ten or less) and given
the same number of concrete objects in each group (twenty or less)
(2.4.K1a) ($), e.g., five classes of 15 students visit the zoo; 15 + 15 +
15 + 15 + 15 = 75.
5. uses repeated subtraction (division) with whole numbers when given
the total number of concrete objects in each group to find the
number of groups (2.4.K1a) ($), e.g., there are 25 cookies. If each
student gets 3 cookies, how many students get cookies? 25 – 3 – 3
– 3 – 3 – 3 – 3 – 3 – 3 or 25 minus 3 eight times means eight
students get 3 cookies each and there is 1 cookie left over.
6. fair shares/measures out (divides) a total amount through 100
concrete objects into equal groups (2.4.K1a-b), e.g., fair sharing 48
eggs into four groups resulting in four groups of 12 eggs or
measuring out 48 eggs with 12 eggs in each group resulting in four
groups of 12 eggs.
1. solves one-step real-world addition or subtraction problems with
various groupings of ($):
a. two-digit whole numbers with regrouping (24A1a-b), e.g., for the
food drive, the class collected 64 cans (cylinders) and 28 boxes
(rectangular prisms). How many did they collect in all? This
problem could be solved with base 10 models, or by saying 64
+ 20 = 84 and 84 + 8 = 92 or 60 + 20 = 80 and 4 + 8 = 12 and
80 + 12 = 92 or with the traditional algorithm;
b. monetary amounts to 99¢ with regrouping (2.4.A1a-c), e.g., an
extra carton of milk costs 25¢. If three students want an extra
carton, how much money should the teacher collect? The
student could solve by using coins (q + q + q or d + d + n + d +
d + n + d + d + n) or by counting by 25s or by drawing or using
base 10 models or with the traditional algorithm.
2. generates a family of basic addition and subtraction facts given one
fact/equation (2.4.A1a), e.g., given 9 + 8 = 17; the other facts are
8 + 9 = 17, 17 – 8 = 9, and 17 – 9 = 8.
2-66
January 31, 2004
▲ – Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
7. ▲ N performs and explains these computational procedures:
a. ■ adds and subtracts three-digit whole numbers with and without
regrouping including the use of concrete objects (2.4.K1a-b),
b. adds and subtracts monetary amounts through 99¢ using cent
notation (25¢ + 52¢) and money models (2.4.K1a-b,d) ($).
8. ▲N identifies basic addition and subtraction fact families (facts with
sums from 0 through 20 and corresponding subtraction facts)
(2.4.K1a).
9. reads and writes horizontally and vertically the same addition or
subtraction expression e.g., 6 – 3 is the same as 6.
-3
Teacher Notes: Efficiency and accuracy means that students are able to compute single-digit numbers with fluency. Students increase their
understanding and skill in single-digit addition and subtraction by developing relationships within addition and subtraction combinations and by
counting on for addition and counting up for subtraction and unknown-addend situations. Students learn basic number combinations and develop
strategies for computing that makes sense to them. Through class discussions, students can compare the ease of use and ease of explanation of
various strategies. In some cases, their strategies for computing will be close to conventional algorithms; in other cases, they will be quite different.
Many times, students’ invented approaches are based on a sound understanding of numbers and operations, and these invented approaches often
can be used with efficiency and accuracy. (Principles and Standards for School Mathematics, NCTM, 2000)
The definition of computation is finding the standard representation for a number. For example, 6 + 6, 4 x 3, 17 – 5, and 24 ÷ 2 are all
representations for the standard representation of 12. Mental math is mentally finding the standard representation for a number —
calculating in your head instead of calculating using paper and pencil or technology. One of the main reasons for teaching mental
math is to help students determine if a computed/calculated answer is reasonable, in other words, using mental math to estimate to
see if the answer makes sense. Students develop mental math skills easier when they are taught specific strategies. Mental math
strategies for the early grades include counting on, counting back, counting up, doubling numbers (doubles), and making ten. Other
mental math strategies for the early grades include skip counting and using number patterns.
One way to improve the understanding of numbers and operations is to encourage children to develop computational procedures that are
meaningful to them. Invented procedures promote the idea of mathematics as a meaningful activity. This is not to suggest that algorithm invention
is the only way to promote student understanding or that children should be discouraged from using a standard algorithm if that is their choice.
Different problems are best solved by different methods. For example, solving 7000 – 25 by counting down by tens and fives, yet solving 41 – 25 by
adding up, or decomposing, the numbers. Another example is adding 52 + 45; a student may count, “52, 62, 72, 82, 92, plus 5 is 97.” (“Invented
Strategies Can Develop Meaningful Mathematical Procedures”) by William M. Carroll and Denise Porter, Teaching Children Mathematics, March
1997, pp. 370-374)
2-67
January 31, 2004
▲ – Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Regrouping refers to the reorganization of objects. In computation, regrouping is based on a “partitioning to multiples of ten” strategy.
For example, 46 + 7 could be solved by partitioning 46 into 40 and 6, then 40 + (6 + 7) = 40 + 13 (and then 13 is partitioned into 10
and 3) which then becomes (40 + 10) + 3 becomes 50 + 3 = 53 or 7 could be partitioned as 4 and 3, then 46 + 4 (bridging through 10)
= 50 and 50 + 3 = 53. Before algorithmic procedures are taught, an understanding of “what happens” must occur. For this to occur,
instruction should involve the use of structured manipulatives. To emphasize the role of the base ten numeration system in
algorithms, some form of expanded notation is recommended. During instruction, each child should have a set of manipulatives to
work with rather than sit and watch demonstrations by the teacher. Some additional computational strategies include doubles plus
one or two (i.e., 6 + 8, 6 and 6 are 12, so the answer must be 2 more or 14), compensation (i.e., 9 + 7, if one is taken away from 9, it
leaves 8; then that one is given to 7 to make 8, then 8 + 8 = 16), subtracting through ten (i.e., 13 – 5, 13 take away 3 is 10, then take
2 more away from 10 and that is 8), and nine is one less than ten (i.e., 9 + 6, 10 and 6 are 16, and 1 less than 16 makes 15).
(Teaching Mathematics in Grades K-8: Research Based Methods, ed. Thomas R. Post, Allyn and Bacon, 1988)
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks
are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can
be used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a)
refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the
Models benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each indicator in the
Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For example, process
models are linked to 1.1.K3, 1.2.K6, 1.3.K1, with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number
Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
2-68
January 31, 2004
▲ – Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
SECOND GRADE
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 1: Patterns – The student recognizes, describes, extends, develops, and explains relationships in
patterns using concrete objects in a variety of situations.
Second Grade Knowledge Base Indicators
Second Grade Application Indicators
The student...
The student...
1. uses concrete objects, drawings, and other representations to work
with types of patterns (2.4.K1a):
a. repeating patterns, e.g., an AB pattern is like left-right, left-right,
…; an ABC pattern is like dog-horse-pig, dog-horse-pig, …; an
AAB pattern is like ↑↑→, ↑↑→, …;
b. growing (extending) patterns, e.g., 7, 9, 11, … where the rule
could be add 2 or the odd numbers beginning with 7.
2. uses the following attributes to generate patterns:
b. counting numbers related to number theory (2.4.K1a), e.g.,
evens, odds, or skip counting by 3s, or 4s;
b. whole numbers that increase or decrease (2.4.K1a) ($), e.g., 11,
22, 33, ... or 98, 88, 78, …;
c. geometric shapes (2.4.K1f), e.g., Δ-Ο-Ο, Δ-Ο-Ο, …;
d. measurements (2.4.K1a), e.g., 1”, 3”, 5”, … or 5 lbs, 10 lbs, 15
lbs, …;
e. the calendar (2.4.K1a), e.g., Sunday, Monday, Tuesday, …;
f. money and time (2.4.K1a,d) ($), e.g., $5, $10, $15, ... or 1:15,
1:30, 1:45, …;
g. things related to daily life (2.4.K1a), e.g., seasons, temperature,
or weather;
h. things related to size, shape, color, texture, or movement
(2.4.K1a), e.g., ,, , … ; or snapping fingers, clapping
hands, or stomping feet or over, under, or behind using a bean
bag toss (kinesthetic patterns).
3. ▲ ■ identifies and continues a pattern presented in various formats
including numeric (list or table), visual (picture, table, or graph), verbal
(oral description), kinesthetic (action), and written (2.4.K1a) ($).
1. generalizes these patterns using a written description:
a. whole number patterns(2.4.A1a) ($);
b. patterns using geometric shapes (2.4.A1d);
c. calendar patterns (2.4.A1a);
d. money and time patterns (2.4.A1a,c) ($);
e. patterns using size, shape, color, texture, or movement
(2.4.A1a);
recognizes multiple representations of the same pattern (2.4.A1a),
e.g., the ABB pattern could be represented by clap, snap, snap, …
or red, blue, blue, … or square, circle, circle, ….
uses concrete objects to model a whole number patterns (2.4.A1a),
e.g.,
counting by twos: ,  
, 
, …;
counting by fives: xxxxx, xxxxx xxxxx
xxxxx, xxxxx
xxxxx, …;
counting by tens:
,  
, 
, …;
counting by twenty-fives:
,
,
, …
2-69
January 31, 2004
▲ – Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
generates (2.4.K1a):
repeating patterns, e.g., 1-2, 1-2, 1-2, … where the elements
repeat; growing (extending) patterns, e.g., 1, 4, 7, …where the
rule is add 3.
Teacher Notes: Working with patterns is an important process in the development of mathematical thinking. Patterns can be based on geometric
attributes (shapes, regions, angles); measurement attributes (color, texture, length, weight, volume, number); relational attributes (proportion,
sequence, functions); and affective attributes (values, likes, dislikes, familiarity, heritage, culture). (Learning to Teach Mathematics, Randall J.
Souviney, Macmillan Publishing Company, 1994)
This process (working with patterns) can be used to develop or deepen understandings of important concepts in number theory, whole numbers,
measurement, geometry, probability, and functions. Working with patterns provides opportunities for students to recognize, describe, extend,
develop, and explain.
Number theory is the study of the properties of the counting numbers (positive integers), their relationships, ways to represent them, and patterns
among them. Number theory includes the concepts of odd and even numbers, factors and multiples, primes and composites, and greatest
common factor and least common multiple.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, with 1.1.K3 referring to Standard 1
(Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
2-70
January 31, 2004
▲ – Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
SECOND GRADE
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 2: Variables, Equations, and Inequalities – The student uses symbols and whole numbers to solve
addition and subtraction equations using concrete objects in a variety of situations.
Second Grade Knowledge Base Indicators
The student...
1. explains and uses symbols to represent unknown whole number
quantities from 0 through 100 (2.4.K1a).
2. finds the sum or difference in one-step equations with : ($)
a. whole numbers from 0 through 99 (2.4.K1a-b), e.g., 32 + 19 = Δ
or Δ = 79 – 46;
b. up to two different coins (2.4.K1d), e.g., nickel + penny = Δ¢.
3. finds unknown addend or subtrahend using basic addition and
subtraction facts (fact family) (2.4.K1a) ($), e.g., 12 = Δ + 7 or 12 – Δ
= 7.
4. describes and compares two whole numbers from 0 through 1,000
using the terms: is equal to, is less than, is greater than (2.4.K1a-b)
($).
Second Grade Application Indicators
The student...
1. represents real-world problems using symbols and whole numbers
from 0 through 30 with one operation (addition, subtraction) and
one unknown (2.4.A1a) ($), e.g., when asked to give the total
number of students in class today, the students write: 14 boys and
9 girls = 
students.
2. generates (2.4.A1a) ($):
a. addition or subtraction equations to match a given real-world
problem with one operation and one unknown using whole
numbers from 0 through 99, e.g., a boy has 45 stickers. How
many more stickers does he need to have 80 stickers? This is
represented by 45 + n = 80 or 80 – 45 = n.
b. a real-world problem to match a given addition or subtraction
equation with one operation using the basic facts, e.g., the
student is given the addition equation, 9 +  = 17 and writes
this problem situation: You have 9¢ and a piece of candy costs
17¢. How much more money do you need to buy the candy?
2-71
January 31, 2004
▲ – Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Understanding the concept of variable is fundamental to algebra. In the early grades, students use various
symbols including letters and geometric shapes to represent unknown quantities that both do and do not vary. Quantities that are not
given and do not vary are often referred to as unknowns or missing elements when they appear in equations, e.g., 2 + 4 = Δ.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks
are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can
be used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a)
refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the
Models benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each indicator in the
Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For example, process
models are linked to 1.1.K3, 1.2.K6, 1.3.K1, with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number
Sense), and Knowledge Indicator 3.]
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked
at three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the
Appendix.
2-72
January 31, 2004
▲ – Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
SECOND GRADE
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 3: Functions – The student recognizes and describes whole number relationships using concrete
objects in a variety of situations.
Second Grade Knowledge Base Indicators
Second Grade Application Indicators
The student...
The student...
1. states mathematical relationships between whole numbers from 0
through 100 using various methods including mental math, paper and
pencil, and concrete objects (2.4.K1a) ($), e.g., every time a dog is
added to the pack, 2 more ears are added to the total.
2. finds the values and determines the rule that involve addition or
subtraction of whole numbers from 0 through 100 using a horizontal
or vertical function table (input/output machine, T-table) (2.4.K1e),
e.g.,
In
Out
after looking at the function table,
9
11
different students might respond
2
4
that the rule is In + 2 equals Out, the
13
15
rule is N + 2, or the rule is plus 2.
42
44
57
6
72
N
1. represents and describes mathematical relationships between
whole numbers from 0 through 100 using concrete objects,
pictures, oral descriptions, and symbols (2.4.A1a) ($).
2. finds the rule, states the rule, and extends numerical patterns with
whole numbers from 0 through 100 (2.4.A1a), e.g., given 1, 3, 5, 7,
9 and continues with 11, 13, 15, 17, … recognizing that the pattern
could be the odd numbers.
59
?
?
?
..
3. generalizes numerical patterns using whole numbers from 0 through
100 with one operation (addition, subtraction) by stating the rule using
words, e.g., if a set of numbers is 2, 4, 6, 8,10, …; the rule is add two.
2-73
January 31, 2004
▲ – Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Functions are relationships or rules in which each member of one set is paired with one, and only one, member of another set
(an ordered pair). The concept of function can be introduced using function machines. Any number put in the machine will be changed according
to some rule. A record of inputs and corresponding outputs can be maintained in a two-column format. Function tables, input/output machines,
and T-tables may be used interchangeably and serve the same purpose.
Function concepts should be developed from growing patterns. Each term in a number sequence is related to its position in the
sequence — the functional relationship. The pattern – 4, 7, 10, 13, 16, 19, and so on – is an arithmetic sequence with a difference of
3. The pattern could be described as add 3 meaning that 3 must be added to the previous term to find the next. This pattern is
explained by using the recursive definition for a sequence. The recursive definition for a sequence is a statement or a set of
statements that explains how each successive term in the sequence is obtained from the previous term(s).
In the pattern 1, 4, 9, 16, 25, …, 225; there is no common difference. This sequence is not arithmetic or geometric (no common ratio
between geometric terms). Neither is it a combination of the two; however, there is a pattern and the missing terms between 25 and
225 can be found. To find the term value, square the number of the term. The next missing terms would be 36, 49, 64, 81, 100, 121,
and 144. This pattern is explained by using the explicit formula for a sequence. The explicit formula for a sequence defines a rule for
finding each term in the number sequence related to its position in the sequence. In other words, to find the term value, square the
number of the term — the 5th term is 52, the 8th term is 82, ….
Patterns themselves are not explicit or recursive. The RULE for the pattern can be expressed explicitly or recursively and MOST
patterns can be explained using either format especially IF that pattern reflects either an arithmetic sequence or geometric sequence.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, with 1.1.K3 referring to Standard 1
(Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
2-74
January 31, 2004
▲ – Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
SECOND GRADE
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 4: Models – The student uses mathematical models including concrete objects to represent, show, and
communicate mathematical relationships in a variety of situations.
Second Grade Knowledge Base Indicators
Second Grade Application Indicators
The student...
1. knows, explains, and uses mathematical models to represent
mathematical concepts, procedures, and relationships. Mathematical
models include:
a. process models (concrete objects, pictures, diagrams, number
lines, unifix cubes, hundred charts, or measurement tools) to
model computational procedures and mathematical relationships,
to compare and order numerical quantities, and to represent
fractional parts (1.1.K1-4, 1.2.K3, 1.2.K5-6, 1.3.K1-2, 1.4.K1-8,
2.1.K1, 2.2K1, 2.1K1a-b, 2.1K1d-h, 2.1.K3-4, 2.2.K2a, 2.2.K3-4,
2.3.K1, 3.2.K1-5, 3.3.K1, 3.4.K1-3, 4.2.K3-5) ($);
b. place value models (place value mats, hundred charts, or base ten
blocks) to compare, order, and represent numerical quantities and
to model computational procedures (1.1.K3, 1.2.K2, 1.2.K4,
1.3.K1, 1.4.K6-7, 1.4.K7a, 2.2.K2a, 2.2.K4) ($);
c. fraction models (fraction strips or pattern blocks) to compare,
order, and represent numerical quantities (1.1.K2b) ($);
d. money models (base ten blocks or coins) to compare, order, and
represent numerical quantities (1.1.K5-6, 1.3.K1, 1.4.K7b, 2.1.K1f,
2.2.K2b) ($);
e. function tables (input/output machines, T-tables) to model
numerical relationships (2.3.K2) ($);
f. two-dimensional geometric models (geoboards, dot paper, pattern
blocks, tangrams, or attribute blocks) to model perimeter and
properties of geometric shapes and three-dimensional geometric
models (solids) and real-world objects to compare size and to
model attributes of geometric shapes (2.1.K2c, 3.1.K1-6, 3.3.K23);
The student...
1. recognizes that various mathematical models can be used to
represent the same problem situation. Mathematical models
include:
a. process models (concrete objects, pictures, diagrams,
number lines, unifix cubes, hundred charts, or measurement
tools) to model computational procedures and mathematical
relationships, to compare and order numerical quantities, and
to model problem situations (1.1.A1a-b, 1.1.A2, 1.2.A1-2,
1.3.A1, 1.4.A1-2, 2.1.A1a, 2.1.A1c-e, 2.2.A1-2, 2.3.A1-2, ,
3.2.A1-4, 3.3.A1-2, 3.4.A1, 4.2.A2) ($);
b. place value models (place value mats, hundred charts, or
base ten blocks) to compare, order, and represent numerical
quantities and to model computational procedures (1.1.A1ab, 1.1.A2, 1.2.A1-2, 1.3.A2, 1.4.A1a) ($);
c. money models (base ten blocks or coins) to compare, order,
and represent numerical quantities (1.1.A1c-d, 1.3.A2,
1.4.A1b, 2.1.A1d) ($);
d. two-dimensional geometric models (geoboards, dot paper,
pattern blocks, tangrams, or attribute blocks) to model
perimeter and properties of geometric shapes and threedimensional geometric models (solids) and real-world objects
to compare size and to model attributes of geometric shapes
(2.1.A1b, 3.1.A1-3);
e. two-dimensional geometric models (spinners), threedimensional geometric models (number cubes), and process
models (concrete objects) to model probability (4.1.A1) ($);
2-75
January 31, 2004
▲ – Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
f.
g. two-dimensional geometric models (spinners), three-
dimensional geometric models (number cubes), and process
models (concrete objects) to model probability (4.1.K1-2) ($);
h. graphs using concrete objects, representational objects, or
abstract representations, pictographs, frequency tables,
horizontal and vertical bar graphs, Venn diagrams or other
pictorial displays, and line plots to organize and display data
(4.1.K2, 4.2.K1, 4.2.K2) ($);
graphs using concrete objects, representational objects, or
abstract representations, pictographs, horizontal and vertical
bar graphs (4.1.A1, 4.2.A1-4) ($).
2. selects a mathematical model that is more useful than other
mathematical models in a given situation.
i. Venn diagrams to sort data.
2. creates a mathematical model to show the relationship between
two or more things, e.g., using pattern blocks, a whole (1) can
be represented using
a
two
three
six
(1/1) or
(2/2) or
(3/3) or
(6/6).
2-76
January 31, 2004
▲ – Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: For assessment purposes, the mathematical modeling process appropriate to the indicator may be included as part of the item
being assessed.
The mathematical modeling process involves:
a. selecting key features and relationships within the real-world situation and representing these concepts in mathematical terms through
some sort of mathematical model,
b. performing manipulations and mathematical procedures within the mathematical model,
c. interpreting the results of the manipulations within the mathematical model,
d. using these results to make inferences about the original real-world situation.
The use of mathematical models is necessary for conceptual understanding. The ways in which mathematical ideas are represented is
fundamental to how students understand and use those ideas. As students begin to use multiple representations of the same situation, e.g.,
comparing the number of boys and girls in the classroom can be represented by lining them up in two different lines. The same situation also can
be represented by pictures of the children (pictograph), a bar graph, or by using two different colors of the same manipulative (unifix cubes or color
tiles). As students work with the different representations of the same situation, they begin to develop an understanding of the advantages and
disadvantages of the various representations/models.
Many mathematical models are listed in this benchmark. The indicator lists some of the mathematical models that could be used to teach a
concept. Each indicator in this benchmark is linked to other indicators in other benchmarks; those indicators are identified in the parentheses. For
example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1
(Number Sense), and Knowledge Indicator 3. In addition, the indicator in the other benchmarks identifies, in parentheses, the Models’ indicator.
For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models).
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked
at three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the
Appendix.
2-77
January 31, 2004
▲ – Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
SECOND GRADE
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 1: Geometric Figures and Their Properties – The student recognizes geometric shapes and describes
their properties using concrete objects in a variety of situations.
Second Grade Knowledge Base Indicators
The student...
1. recognizes and investigates properties of circles, squares,
rectangles, triangles, and ellipses (ovals) (plane figures/twodimensional shapes) using concrete objects, drawings, and
appropriate technology (2.4.K1f).
2. ■ recognizes, draws, and describes circles, squares, rectangles,
triangles, ellipses (ovals) (plane figures) (2.4.K1f).
3. recognizes cubes, rectangular prisms, cylinders, cones, and
spheres (solids/three-dimensional figures) (2.4.K1f).
4. recognizes the square, triangle, rhombus, hexagon, parallelogram,
and trapezoid from a pattern block set (2.4.K1f).
5. compares geometric shapes (circles, squares, rectangles, triangles,
ellipses) to one another (2.4.K1f).
6. recognizes whether a shape has a line of symmetry (2.4.K1f).
Second Grade Application Indicators
The student...
1. solves real-world problems by applying the properties of plane
figures (circles, squares, rectangles, triangles, ellipses) (2.4.A1d),
e.g., which shape could be used to completely cover the lid of a
pencil box with no overlapping?
2. demonstrates how (2.4.A1d):
a. ▲ plane figures (circles, squares, rectangles, triangles, ellipses)
can be combined or separated to make a new shape;
b. solids (cubes, rectangular solids, cylinders, cones, spheres) can
be combined or separated to make a new shape.
3. identifies the plane figures (circles, squares, rectangles, triangles,
ellipses) used to form a composite figure (2.4.A1d).
2-78
January 31, 2004
▲ – Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Geometry is the study of shapes, their properties, and their relationships to other shapes. Symbols and numbers are used to
describe their properties and their relationships to other shapes. The fundamental concepts in geometry are point (no dimension), line (onedimensional), plane (two-dimensional), and space (three-dimensional). Plane figures are referred to as two-dimensional and solids are referred to
as three-dimensional.
From the Mathematics Dictionary and Handbook (Nichols Schwartz Publishing, 1999), property as a mathematical term means a characteristic (an
attribute) of a number, geometric shape, mathematical operation, equation, or inequality. To give an example:
 Property of a number: 8 is divisible by 2.
 Property of a geometric shape: Each of the four sides of a square is of the same length.
 Property of an operation: Addition is commutative. For all numbers x and y, x + y = y + x.
 Property of an equation: For all numbers a, b, and c, if a = b, then a + c = b + c.
 Property of an inequality: For all numbers a, b, and c, if a > b, then a – c > b – c.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, with 1.1.K3 referring to Standard 1
(Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
Pattern blocks are a collection of six geometric shapes in six colors. Each set contains 250 pieces — 50 green triangles, 25 orange squares, 50
blue rhombi, 50 tan rhombi, 50 red trapezoids, and 25 yellow hexagons. The blue rhombus and the tan rhombus also are parallelograms, and the
orange square is a parallelogram. The blocks are designed so that their sides are all the same length with the exception that the trapezoid has one
side twice as long. This feature allows the blocks to be nested together and encourages the exploration of relationships among the shapes.
Activities with pattern blocks help students explore patterns, functions, fractions, congruence, similarity, symmetry, perimeter, area, and graphing.
A pattern block template can be found in the Appendix.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
2-79
January 31, 2004
▲ – Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
SECOND GRADE
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 2: Measurement and Estimation – The student estimates and measures using standard and nonstandard
units of measure with concrete objects in a variety of situations.
Second Grade Knowledge Base Indicators
The student...
1. uses whole number approximations (estimations) for length, weight,
and volume using standard and nonstandard units of measure
(2.4.K1a) ($), e.g., the height of the classroom door is 14 chalkboard
erasers laid end to end or 7 feet high or an apple weighs about 42
unifix cubes.
2. ▲ reads and tells time by five-minute intervals using analog and
digital clocks (2.4.K1a).
3. selects and uses appropriate measurement tools and units of
measure for length, weight, volume, and temperature for a given
situation (2.4.K1a) ($).
4. measures (2.4.K1a) ($):
a. ▲ length to the nearest inch or foot and to the nearest whole
unit of a nonstandard unit;
b. weight to the nearest nonstandard unit;
c. volume to the nearest cup, pint, quart, or gallon;
d. temperature to the nearest degree.
5. states (2.4.K1a):
a. the number of minutes in an hour,
b. the number of days in each month.
Second Grade Application Indicators
The student...
1. compares the weights of more than two concrete objects using a
balance (2.4.A1a) ($).
2. solves real-world problems by applying appropriate measurements
(2.4.A1a):
a. length to the nearest inch or foot, e.g., a cookie is almost how
many inches wide?
b. length to the nearest whole unit of a nonstandard unit, e.g., how
many paper clips long is a candy bar?
3. estimates to check whether or not measurements or calculations for
length in real-world problems are reasonable (2.4.A1a) ($), e.g., is it
reasonable to say that you measured your thumb and it is 2 feet
long?
4. adjusts original measurement or estimation for length and weight in
real-world problems based on additional information (a frame of
reference) (2.4.A1a), e.g., I estimated that the stapler is 20
paperclips long. Then I lay out 4 paper clips next to the stapler. I
realize that since I am half done, my estimate is too high; so I adjust
my estimate to 8 paper clips.
2-80
January 31, 2004
▲ – Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: The term geometry comes from two Greek words meaning “earth measure.” The process of learning to measure at the early
grades focuses on identifying what property (length, weight, volume) is to be measured and to make comparisons. Estimation in measurement is
defined as making guesses as to the exact measurement of an object without using any type of measurement tool. Estimation helps students
develop a relationship between the different sizes of units of measure. It helps students develop basic properties of measurement, and it gives
students a tool to determine whether a given measurement is reasonable.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, with 1.1.K3 referring to Standard 1
(Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
2-81
January 31, 2004
▲ – Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
SECOND GRADE
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 3: Transformational Geometry – The student recognizes and shows one transformation on simple
shapes and concrete objects in a variety of situations.
Second Grade Knowledge Base Indicators
The student...
1. knows and uses the cardinal points (north, south, east, west)
(2.4.K1a).
2. recognizes that changing an object's position or orientation
including whether the object is nearer or farther away does not
change the name, size, or shape of the object (2.4.K1f).
3. recognizes when a shape has undergone one transformation
(flip/reflection, turn/rotation, slide/translation) (2.4.K1f).
Second Grade Application Indicators
The student...
1. shows two concrete objects or shapes are congruent by physically
fitting one shape or object on top of the other (2.4.A1a).
2. follows directions to move objects from one location to another
using appropriate vocabulary and the cardinal points (north, south,
east, west) (2.4.A1a).
Teacher Notes: Transformational geometry is another way to investigate geometric figures by moving every point in a plane figure to a new
location. To help students form images of shapes through different transformations, students can use concrete objects, figures drawn on graph
paper, mirrors or other reflective surfaces, or appropriate technology.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra),
Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the
mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other
indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, with 1.1.K3
referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
2-82
January 31, 2004
▲ – Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
SECOND GRADE
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 4: Geometry From An Algebraic Perspective – The student identifies one or more points on a number
line in a variety of situations.
Second Grade Knowledge Base Indicators
Second Grade Application Indicators
The student...
The student...
1. locates and plots whole numbers from 0 through 1,000 on a
1. solves real-world problems involving counting, adding, and
segment of a number line (horizontal/vertical) (2.4.K1a), e.g.,
subtracting whole numbers from 0 through 1,000 using a segment
using a segment of a number line from 800 to 820 to locate the
of a number line (2.4.A1a) ($), e.g., Adam had collected 894
whole number 805.
marbles. He lost nine marbles. How many does he have now?
2. represents the distance between two whole numbers from 0
Using the number line, Adam shows how he solved the problem.
through 1,000 on a segment of a number line (2.4.K1a).
3. uses a segment of a number line to model addition and
subtraction using whole numbers from 0 through 1,000
(2.4.K1a), e.g., 333 + n = 349 or 333 + 16 = n or 400 – n = 352
or 400 – 48 = n.
Teacher Notes: A number line (a mathematical model) is a diagram that represents numbers with equal distances marked off as points on a
line, and is an example of one-to-one correspondence (a relation). A number line can be used as a visual representation of numbers and
operations. In addition, a number line used horizontally and vertically is a precursor to the coordinate plane; and the distance between two
numbers on a number line is a precursor to absolute value.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra),
Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the
mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other
indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, with 1.1.K3
referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators
in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade
level with a ($). The National Standards in Personal Finance are included in the Appendix.
2-83
January 31, 2004
▲ – Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 4: Data
SECOND GRADE
Standard 4: Data – The student uses concepts and procedures of data analysis in a variety of situations.
Benchmark 1: Probability – The student applies the concepts of probability using concrete objects in a variety of
situations.
Second Grade Knowledge Base Indicators
Second Grade Application Indicators
The student...
The student...
1. recognizes any outcome of a simple event in an experiment or
1. makes a prediction about a simple event in an experiment or
simulation as impossible, possible, certain, likely, or unlikely
simulation; conducts the experiment or simulation including the use
(2.4.K1g) ($).
of concrete objects; records the results in a chart, table, or graph;
2. lists some of the possible outcomes of a simple event in an
and makes an accurate statement about the results (2.4.A1e-f).
experiment or simulation including the use of concrete objects
(2.4.K1g-h).
Teacher Notes: Ideas from probability reinforce concepts in the other Standards, especially Number and Computation and Geometry. Students
need to develop an intuitive concept of chance – whether or not something is unlikely or likely to happen. Probability experiences should be
addressed through the use of concrete objects, coins, and geometric models (spinners or number cubes).
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra),
Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the
mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other
indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3
referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
2-84
January 31, 2004
▲ – Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 4: Data
SECOND GRADE
Standard 4: Data – The student uses concepts and procedures of data analysis in a variety of situations.
Benchmark 2: Statistics – The student collects, organizes, displays, and explains numerical (whole numbers) and
non-numerical data sets including the use of concrete objects in a variety of situations.
Second Grade Knowledge Base Indicators
The student...
1. organizes, displays, and reads numerical (quantitative) and nonnumerical (qualitative) data in a clear, organized, and accurate
manner including a title, labels, categories, and whole number
intervals using these data displays (2.4.K1h) ($):
a. ▲ graphs using concrete objects;
b. ▲ pictographs with a whole symbol or picture representing one,
two, or ten (no partial symbols or pictures);
c. ▲ ■ frequency tables (tally marks);
d. ▲ horizontal and vertical bar graphs;
e. Venn diagrams or other pictorial displays, e.g., glyphs;
f. line plots.
2. collects data using different techniques (observations, interviews, or
surveys) and explains the results (2.4.K1h) ($).
3. identifies the minimum (lowest) and maximum (highest) values in a
whole number data set (2.4.K1a) ($).
4. finds the range for a data set using two-digit whole numbers
(2.4.K1a) ($).
5. finds the mode (most) for a data set using concrete objects that
include (2.4.K1a) ($):
a. quantitative/numerical data (whole numbers through 100);
b. qualitative/non-numerical data (category that occurs most
often).
Second Grade Application Indicators
The student...
3. communicates the results of data collection and answers questions
based on information from (2.4.A1f) ($):
a. graphs using concrete objects,
b. pictographs with a whole symbol or picture representing one (no
partial symbols or pictures),
c. horizontal and vertical bar graphs.
2. determines categories from which data could be gathered (2.4.A1f)
($), e.g., categories could include shoe size, height, favorite candy
bar, or number of pockets in clothing.
3. recognizes that the same data set can be displayed in various
formats including the use of concrete objects (2.4.A1f) ($).
4. recognizes appropriate conclusions from data collected (2.4.A1f) ($).
2-85
January 31, 2004
▲ – Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Graphs (data displays) are pictorial representations of mathematical relationships and are used to tell a story. When a graph is
made, the axes and the scale (numbers running along a side of the graph) are chosen for a reason. The difference between numbers from one grid
line to another is the interval. The interval will depend on the lowest and highest values in the data. Emphasizing the importance of using equalsized pictures or intervals is critical to ensuring that the data display is accurate. Graphs take many forms: bar graphs and pictographs compare
discrete data, frequency tables show how many times a certain piece of data occurs using tally marks to record the data, circle graphs (pie charts)
model parts of a whole, line graphs show change over time, Venn diagrams show relationships between sets of objects, and line plots show
frequency of data on a number line.
The measures of central tendency (averages) of a data set are mean, median, and mode. Conceptual understanding of mean, median, and
mode is developed through the use of concrete objects that represent the data values.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, with 1.1.K3 referring to Standard 1
(Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
2-86
January 31, 2004
▲ – Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
THIRD GRADE
Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 1: Number Sense – The student demonstrates number sense for whole numbers, fractions, decimals,
and money using concrete objects in a variety of situations.
100
Third Grade Knowledge Base Indicators
Third Grade Application Indicators
The student…
1. knows, explains, and represents ($):
a. whole numbers from 0 through 10,000 (2.4.K1a-b)
b. fractions greater than or equal to zero (halves, fourths, thirds,
eighths, tenths, sixteenths) (2.4.K1c) ($);
c. decimals greater than or equal to zero through tenths place
(2.4.K1c).
2. compares and orders:
a. ▲ ■ whole numbers from 0 through 10,000 with and without the
use of concrete objects (2.4.K1a-b) ($);
b. fractions greater than or equal to zero with like denominators
(halves, fourths, thirds, eighths, tenths, sixteenths) using concrete
objects (2.4.K1a,c);
c. decimals greater than or equal to zero through tenths place using
concrete objects (2.4.K1a-c).
3. ▲ knows, explains, and uses equivalent representations including the
use of mathematical models for:
a. addition and subtraction of whole numbers from 0 through 1,000
(2.4.K1a-b) ($), e.g., 144 + 236 =
300 +
80
The student…
1. solves real-world problems using equivalent representations
and concrete objects to ($):
█
██
▌▌▌▌
▌▌▌
▪▪▪▪
▪▪▪▪▪▪
$100
$100
$100
$10
$10
$10
$10
a. compare and order whole numbers from 0 through 5,000
(2.4.A1a-b), e.g., using base ten blocks, represent the total
school attendance for a week; then represent the numbers
using digits and compare and order in different ways;
b. add and subtract whole numbers from 0 through 1,000 and
when used as monetary amounts (2.4.A1a,d) ($), e.g., use real
money to show at least 2 ways to represent $10.42; then
subtract the cost of a book purchases at the school’s book fair
from $10.42 (the amount you have earned and can spend).
2. determines whether or not solutions to real-world problems
that involve the following are reasonable ($).
$10
$10
$10
$10
100
a. whole numbers from 0 through 1,000 (2.4.A1a-b), e.g., a
student says that there are 1,000 students in grade 3 at her
school, is this reasonable?
b. fractions greater than or equal to zero (halves, fourths, thirds,
eighths, tenths, sixteenths) (2.4.A1a,c); e.g., you ate ½ of a
sandwich and a friend ate ¾ of the same sandwich; is this
reasonable?
c. decimals greater than or equal to zero when used as
monetary amounts (2.4.A1d), e.g., a pack of chewing gum
costs what amount - $62
$.75 9¢ $75.00 750¢? Is
this reasonable?;
3-87
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
b. multiplication using the basic facts through the 5s and the
multiplication facts of the 10s (2.4.K1a), e.g., 3 x 2 can be
represented as 4 + 2 or as an array, X X X
3. determines the amount of change owed through $100.00
(2.4.A1d), e.g., school supplies cost $12.37. What was the
amount of change received after giving the clerk $20.00?
To solve, $20.00 – $12.37 = $7.63 (the change).
X X X;
c.
addition and subtraction of money (2.4.K1d) ($), e.g., three half
dollars equals 50¢ + 50¢ + 50¢ or 50¢ + 100¢.
4. ▲N determines the value of mixed coins and bills with a total value of
$50 or less (2.1.K1d) ($).
Teacher Notes: Number sense refers to one’s ability to reason with numbers and to work with numbers in a flexible way. The ability
to compute mentally, to estimate based on understanding of number relationships and magnitudes, and to judge reasonableness of
answers are all involved in number sense.
When we say that someone has good number sense, we mean that he or she possesses a variety of abilities and understandings that
include an awareness of the relationships between numbers, an ability to represent numbers in a variety of ways, a knowledge of the
effects of operations, and an ability to interpret and use numbers in real-world counting and measurement situations. Such a person
predicts with some accuracy the result of an operation and consistently chooses appropriate measurement units. This “friendliness
with numbers” goes far beyond mere memorization of computational algorithms and number facts; it implies an ability to use numbers
flexibly, to choose the most appropriate representation of a number for a given circumstance, and to recognize when operations have
been correctly performed. (Number Sense and Operations: Addenda Series, Grades K-6, NCTM, 1993)
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks
are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can
be used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a)
refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the
Models benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each indicator in the
Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For example, process
models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number
Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked
at three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the
3-88
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Appendix.
3-89
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
THIRD GRADE
Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 2: Number Systems and Their Properties – The student demonstrates an understanding of whole
numbers with a special emphasis on place value and recognizes, uses, and explains the concepts of
properties as they relate to whole numbers, fractions, decimals, and money in a variety of situations.
Third Grade Knowledge Base Indicators
The student…
1. identifies, reads, and writes numbers using numerals and words from
tenths place through ten thousands place (2.4.K1a-b) ($), e.g., sixtyfour thousand, three hundred eighty and five tenths is written in
numerical form as 64,380.5.
2. identifies, models, reads, and writes numbers using expanded form
from tenths place through ten thousands place (2.4.K1b), e.g.,
56,277.3 = (5 x 10,000) + (6 x 1,000) + (2 x 100) + (7 x 10) + (7 x 1) +
(3 x .1) = 50,000 + 6,000 + 200 + 70 + 7 + .3.
3. classifies various subsets of numbers as whole numbers, fractions
(including mixed numbers), or decimals (2.4.K1a-c, 2.4.K1i)
4. identifies the place value of various digits from tenths to one hundred
thousands place (2.4.K1b) ($).
5. identifies any whole number through 1,000 as even or odd (2.4.K1a).
6. uses the concepts of these properties with whole numbers from 0
through 100 and demonstrates their meaning including the use of
concrete objects (2.4.K1a) ($):
a. commutative properties of addition and multiplication, e.g., 7 + 8
= 8 + 7 or 3 x 6 = 6 x 3;
b. zero property of addition (additive identity), e.g., 4 + 0 = 4;
c. property of one for multiplication (multiplicative identity), 1 x 3 = 3;
d. associative property of addition, e.g., (3 + 2) + 4 = 3 + (2 + 4);
e. symmetric property of equality applied to addition and
multiplication, e.g., 100 = 20 + 80 is the same as 20 + 80 = 100
and 3 x 4 = 12 is the same as 12 = 3 x 4;
Third Grade Application Indicators
The student…
1. solves real-world problems with whole numbers from 0 through
100 using place value models, money, and the concepts of these
properties to explain reasoning (2.4.A1a-b,d) ($):
a. commutative property of addition, e.g., a student has a dime, a
nickel, and a quarter to purchase a pencil; he/she totals the
amount of the coins to see whether or not there is enough
money; the student could count the quarter, nickel, and dime
as 25¢ + 5¢ + 10¢ or as 25¢ + 10¢ + 5¢ because adding in
any order does not change the sum;
b. zero property of addition, e.g., a student has 6 marbles in one
pocket and none in the other, so all together there are: 6 + 0 =
6;
c. associative property of addition, e.g., a student has two dimes
and a quarter; there are 2 ways to group the coins to find the
total: 10¢ (dime) + 10¢ (dime) = 20¢, then add the quarter,
20¢ + 25¢ (quarter) = 45¢ or 10¢ (dime) + 25¢ (quarter) = 35¢,
then add the other dime to 35¢ and 35¢ + 10¢ = 45¢ or (D +
D) + Q = D + (D + Q) using coins or money models.
2. performs various computational procedures with whole numbers
from 0 through 100 using the concepts of these properties and
explains how they were used (2.4.A1a-b):
a. commutative property of multiplication, e.g., given 4 x 6, the
student says: I know that 4 x 6 is 24 because I know 6 x 4 is
24 and multiplying in any order gets the same answer;
3-90
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
f. zero property of multiplication, e.g., 9 x 0 = 0 or 0 x 32 = 0.
7. divides whole numbers from 0 through 99,999 into groups of 10,000s;
1,000s; 100s; 10s, and 1s using base ten models (2.4.K1b).
b. zero property of multiplication without computing, e.g., 7 x 3 x
4 x 0 x 5 = □, the student says: I know the answer (product) is
zero because no matter how many factors you have, when
you multiply with a 0, the product is zero;
c. associative property of addition, e.g., 9 + 8 could be solved as
1 + (8 + 8) or (1 + 8) + 8, the student says: I don’t know 9 + 8,
but I know my doubles (8 + 8), so I made the 9 into 1 + 8 and
added 8 + 8 and then added 1 more to make 17.
Teacher Notes: From the Mathematics Dictionary and Handbook (Nichols Schwartz Publishing, 1999), property as a mathematical term means a
characteristic (an attribute) of a number, geometric shape, mathematical operation, equation, or inequality. To give an example:
 Property of a number: 8 is divisible by 2.
 Property of a geometric shape: Each of the four sides of a square is of the same length.
 Property of an operation: Addition is commutative. For all numbers x and y, x + y = y + x.
 Property of an equation: For all numbers a, b, and c, if a = b, then a + c = b + c.
 Property of an inequality: For all numbers a, b, and c, if a > b, then a – c > b – c.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks
are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can
be used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a)
refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the
Models benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each indicator in the
Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For example, process
models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number
Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
3-91
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
THIRD GRADE
Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 3: Estimation – The student uses computational estimation with whole numbers, fractions, and money in
a variety of situations.
Third Grade Knowledge Base Indicators
The student…
1. estimates whole numbers quantities from 0 through 1,000; fractions
(halves, fourths); and monetary amounts through $500 using
various computational methods including mental math, paper and
pencil, concrete objects, and appropriate technology (2.4.K1a-d) ($).
2. uses various estimation strategies to estimate using whole number
quantities from 0 through 1,000 and explains the process used
(2.4.K1a) ($) e.g., 362 rounded to the nearest ten is 360 and 362
rounded to the nearest hundred is 400. Using front-end estimation,
362 is about 300 or 400 depending on the context of the problem.
Using a “nice” number, 362 is about 350 because of the benchmark
number – 350, since 350 is the halfway point between 300 and 400.
3. recognizes and explains the difference between an exact and an
approximate answer (2.4.K1a), e.g., when asked how many
students are in a classroom, an exact answer could be 24.
Whereas, an approximate answer could be 20 since 24 could be
rounded down to the nearest ten (underestimated) or rounded up
to 30 (overestimated).
Third Grade Application Indicators
The student…
1. adjusts original whole number estimate of a real-world problem
using numbers from 0 through 1,000 based on additional
information (a frame of reference) (2.4.A1a) ($), e.g., if given a pint
container and told the number of marbles it has in it, the student
would estimate the number of marbles in a quart container.
2. estimates to check whether or not the result of a real-world problem
using whole numbers from 0 through 1,000 and monetary amounts
through $500 is reasonable and makes predictions based on the
information (2.4.A1a-b,d) ($), e.g., at the movies, you bought
popcorn for $2.35 and a soda for $2.50; and then paid $4.50 for a
ticket. Is it reasonable to say you spent $10? How much will you
need to save to go to the
movies once a week for the next
month?
3. selects a reasonable magnitude from three given quantities based
on a familiar problem situation and explains the reasonableness of
the results (2.4.A1a), e.g., about how many students are in my class
today – 2, 20, 200?
4. determines if a real-world problem with whole numbers from 0
through 1,000 calls for an exact or approximate answer and
performs the appropriate computation using various computational
methods including mental math, paper and pencil, concrete objects,
and appropriate technology (2.4.A1a) ($).
3-92
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Estimate, as a verb, means to make an educated guess based on information in a problem or to give an answer
close to the exact number. Estimation is used when an exact answer is not needed, as in many real-life situations for which “ballpark”
computations are acceptable. Good number sense enables one to estimate a quantity, estimate a measure, or estimate an answer.
Estimation serves as an important companion to computation. It provides a tool for judging the reasonableness of computational
methods including mental math, paper and pencil, concrete objects, and appropriate technology. However, being able to compute
does not automatically lead to an ability to estimate or judge reasonableness of answers. Frequent modeling by the teacher helps
students develop a range of estimation strategies. Students should be encouraged to frequently explain their thinking as they
estimate. As with exact computation, sharing estimation strategies allows students access to others’ thinking and provides
opportunities for class discussion. Identifying the estimation strategy by name is not critical; however, explaining the thinking behind
the strategy to make a valid estimation is important. (Principles and Standards for School Mathematics, NCTM, 2000)
Mental math and estimation are distinct but related mathematical skills. Proficiency in mental math contributes to increased skill in
estimation. In order for students to become more familiar with estimation, teachers should introduce estimation with examples where
rounded or estimated numbers are used. Emphasis should be placed on real-world examples where only estimation is required, e.g.,
About how many hours do you sleep a night? Using the language of estimation is important, so students begin to realize that a variety
of estimates (answers) are possible.
In addition, when students are taught specific estimation strategies, they develop mental math and estimation skills easier. Estimation
strategies include front-end with adjustment, compatible “nice” numbers, clustering, or special numbers.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks
are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can
be used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a)
refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the
Models benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each indicator in the
Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For example, process
models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number
Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked
at three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the
Appendix.
3-93
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
THIRD GRADE
Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 4: Computation – The student models, performs, and explains computation with whole numbers and
money including
the use of concrete objects in a variety of situations.
Third Grade Knowledge Base Indicators
The student…
1. computes with efficiency and accuracy using various computational
methods including mental math, paper and pencil, concrete objects,
and appropriate technology (2.4.K1a) ($).
2. N states and uses with efficiency and accuracy the multiplication facts
through the 5s and the multiplication facts of the 10s and
corresponding division facts (2.4.K1a) ($).
3. skip counts (multiples) by 2s, 3s, 4s, 5s, and 10s (2.4.K1a).
4. N performs and explains these computational procedures:
a. adds and subtracts whole numbers from 0 through 10,000
(2.4.K1a-b);
b. multiplies whole numbers when one factor is 5 or less and the
other factor is a multiple of 10 through 1,000 with or without the
use of concrete objects (2.4.K1a-b), e.g., 400 x 3 = 120 or 70 x 5
= 350;
c. adds and subtracts monetary amounts using dollar and cents
notation through $500.00 (2.4.K1d) ($), e.g., $47.07 + $356.96 =
$404.03.
5. fair shares/measures out (divides) a total amount through 100
concrete objects into equal groups (2.4.K1a-b), e.g., fair sharing 52
pieces of candy with 8 friends resulting in eight groups of 6 with four
pieces left over or measuring out into groups of eight 52 pieces of
candy with four pieces left over.
6. explains the relationship between addition and subtraction (2.4.K1ab) ($).
7. ▲■ N identifies multiplication and division fact families through the 5s
and the multiplication and division fact families of the 10s (2.4.K1a),
e.g., when given 6 x  = 18, the student recognizes the remaining
Third Grade Application Indicators
The student…
1. ▲N solves one-step real-world addition or subtraction problems
with ($):
a. whole numbers from 0 through 10,000 (2.4.A1a-b), e.g., for
the food drive, the school collected 564 cans (cylinders) and
297 boxes (rectangular prisms). How many items did they
collect in all? This problem could be solved with base 10
models: by adding 500 + 200 (700), 60 + 90 (150), and 4 + 7
(11), so 700+ 150 + 11= 861; by adding 564 + 300 (864) and
297 is 3 less than 300, so 864 – 3 = 861; or by using the
traditional algorithm;
b. monetary amounts using dollar and cents notation through
$500.00 (2.4.A1a-b,d), e.g., you are shopping for a new
bicycle; at The Bike Store, the bike you want is $189.69 and at
Sports for All, it is $162.89. How much will you save by buying
the bike at Sports for All?
2. N generates a family of multiplication and division facts through
the 5s (2.4.A1a), e.g., if the student writes 5 x 9 = 45, the
remaining facts generated are: 9 x 5 = 45, 45 ÷ 5 = 9, 45 ÷ 9 = 5.
3-94
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
members of the fact family.
8. reads and writes horizontally, vertically, and with different operational
symbols the same addition, subtraction, multiplication, or division
expression, e.g., 4 • 6 is the same as 4 x 6 or 4(6) or 6 and 10
divided by 2 is the same as 10 ÷ 2 or 10
x4
2.
Teacher Notes: Efficiency and accuracy means that students are able to compute single-digit numbers with fluency. Students increase their
understanding and skill in single-digit addition and subtraction by developing relationships within addition and subtraction combinations and by
counting on for addition and counting up for subtraction and unknown-addend situations. Students learn basic number combinations and develop
strategies for computing that makes sense to them. Through class discussions, students can compare the ease of use and ease of explanation of
various strategies. In some cases, their strategies for computing will be close to conventional algorithms; in other cases, they will be quite different.
Many times, students’ invented approaches are based on a sound understanding of numbers and operations, and these invented approaches often
can be used with efficiency and accuracy. (Principles and Standards for School Mathematics, NCTM, 2000)
The definition of computation is finding the standard representation for a number. For example, 6 + 6, 4 x 3, 17 – 5, and 24 ÷ 2 are all
representations for the standard representation of 12. Mental math is mentally finding the standard representation for a number – calculating in
your head instead of calculating using paper and pencil or technology. One of the main reasons for teaching mental math is to help students
determine if a computed/calculated answer is reasonable; in other words, using mental math to estimate to see if the answer makes sense.
Students develop mental math skills easier when they are taught specific strategies. Mental math strategies include counting on, doubling numbers
(doubles), making ten, and compatible numbers.
Regrouping refers to the reorganization of objects. In computation, regrouping is based on a “partitioning to multiples of ten” strategy. For
example, 46 + 7 could be solved by partitioning 46 into 40 and 6, then 40 + (6 + 7) = 40 + 13 (and then 13 is partitioned into 10 and 3) which then
becomes (40 + 10) + 3 becomes 50 + 3 = 53 or 7 could be partitioned as 4 and 3, then 46 + 4 (bridging through 10) = 50 and 50 + 3 = 53. Before
algorithmic procedures are taught, an understanding of “what happens” must occur. For this to occur, instruction should involve the use of
structured manipulatives. To emphasize the role of the base ten numeration system in algorithms, some form of expanded notation is
recommended. During instruction, each child should have a set of manipulatives to work with rather than sit and watch demonstrations by the
teacher. Some additional computational strategies include doubles plus one or two (i.e., 6 + 8, 6 and 6 are 12, so the answer must be 2 more or
14), compensation (i.e., 9 + 7, if one is taken away from 9, it leaves 8; then that one is given to 7 to make 8, then 8 + 8 = 16), subtracting through
ten (i.e., 13 – 5, 13 take away 3 is 10, then take 2 more away from 10 and that is 8), and nine is one less than ten (i.e., 9 + 6, 10 and 6 are 16, and
1 less than 16 makes 15). (Teaching Mathematics in Grades K-8: Research Based Methods, ed. Thomas R. Post, Allyn and Bacon, 1988)
3-95
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
3-96
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
THIRD GRADE
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 1: Patterns – The student recognizes, describes, extends, develops, and explains relationships in
patterns using concrete objects in a variety of situations.
Third Grade Knowledge Base Indicators
Third Grade Application Indicators
The student…
4. uses concrete objects, drawings, and other representations to work
with types of patterns (2.4.K1a):
e. repeating patterns, e.g., an AB pattern is like 1-2, 1-2, …; an
ABC pattern is like dog-horse-pig, dog-horse-pig, …; an AAB
pattern is like ↑↑→, ↑↑→, …;
f. growing patterns, e.g., 1, 4, 7, 10, …
5. uses these attributes to generate patterns:
c. counting numbers related to number theory (2.4.K1a), e.g.,
evens, odds, or multiples through the 5s;
whole numbers that increase or decrease (2.4.K1a) ($),e.g., 3, 6, 9,
…; 20, 15, 10, …;
geometric shapes including one attribute change (2.4.K1f), e.g., -
The student…
1. generalizes the following patterns using a written description:
a. counting numbers related to number theory (2.4.A1a);
b. whole number patterns (2.4.A1a) ($),
c. patterns using geometric shapes (2.4.A1f),
d. measurement patterns (2.4.A1a),
e. money and time patterns (2.4.A1a,d) ($),
f. patterns using size, shape, color, texture, or movement
(2.4.A1a).
2. ▲ recognizes multiple representations of the same pattern (2.4.A1a)
e.g., the ABC pattern could be represented by clap, snap, stomp, …;
red, green, yellow, …; tricycle, bicycle, unicycle, …; or 3, 2, 1, …
-Δ-▲, --Δ-▲, --Δ-▲,… where the pattern is filled-in
square, square, triangle, filled-in triangle, …; or when using
attribute blocks the change is size only, then shape only, …
such as
…;
measurements (2.4.K1a), e.g., 1 ft, 2 ft, 3 ft, …; 3 lbs, 6 lbs, 9 lbs;
or 2 cups, 4 cups, 6 cups, …;
e. money and time (2.4.K1a,d) ($), e.g., $.25, $.50, $.75, … or
1:05 p.m., 1:10 p.m., 1:15 p.m., …;
f. things related to daily life (2.4.K1a), e.g., water cycle, food
cycle, or life cycle;
3-97
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
g. things related to size, shape, color, texture, or movement
(2.4.K1a), e.g., red-green, red-green, red-green, …; snapping
fingers; clapping hands; stomping feet; or tossing a bean bag
over the head, under the leg, and behind the back (kinesthetic
patterns).
3. identifies, states, and continues a pattern presented in various
formats including numeric (list or table), visual (picture, table, or
graph), verbal (oral description), kinesthetic (action), and written
(2.4.K1a) ($).
4. generates:
a. repeating patterns (2.4.K1a),
b. growing (extending) patterns (2.4.K1a),
c. patterns using function tables (input/output machines, T-tables)
(2.4.K1e).
3-98
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Working with patterns is an important process in the development of mathematical thinking. Patterns can be based on geometric
attributes (shapes, regions, angles); measurement attributes (color, texture, length, weight, volume, number); relational attributes (proportion,
sequence, functions); and affective attributes (values, likes, dislikes, familiarity, heritage, culture). (Learning to Teach Mathematics, Randall J.
Souviney, Macmillan Publishing Company, 1994)
This process (working with patterns) can be used to develop or deepen understandings of important concepts in number theory, whole numbers,
measurement, geometry, probability, and functions. Working with patterns provides opportunities for students to recognize, describe, extend,
develop, and explain.
Number theory is the study of the properties of the counting numbers (positive integers), their relationships, ways to represent them, and patterns
among them. Number theory includes the concepts of odd and even numbers, factors and multiples, primes and composites, and greatest
common factor and least common multiple.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks
are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can
be used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a)
refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the
Models benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each indicator in the
Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For example, process
models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1
(Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
3-99
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
3-100
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
THIRD GRADE
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 2: Variables, Equations, and Inequalities – The student uses symbols and whole numbers to solve
equations including the use of concrete objects in a variety of situations.
Third Grade Knowledge Base Indicators
The student…
1. explains and uses symbols to represent unknown whole number
quantities from 0 through 1,000 (2.4.K1a)
2. finds the sum or difference in one-step equations with ($):
a. whole numbers from 0 through 99 (2.4.K1a) e.g., 89 = 76 + y or
y – 23 = 32;
b. monetary values through a dollar (2.4.K1d), e.g., 25¢ + 10¢ +
5¢ = n.
3. finds the unknown in the multiplication and division fact families
through the 5s and the 10s (2.4.K1a), e.g., 3 •  = 4 • 6.
4. compares two whole numbers from 0 through 1,000 using the
equality and inequality symbols (=, <, >) and their corresponding
meanings (is equal to, is less than, is greater than) (2.4.K1a-b) ($).
Third Grade Application Indicators
The student…
1. represents real-world problems using symbols with one
operation and one unknown that (2.4.A1a) ($):
a. adds or subtracts using whole numbers from 0 through 99, e.g.,
when asked to represent the number of 3rd graders in a school,
students write: 21 + 18 + 19 =  ;
b. multiplies or divides using the basic facts through the 5s and the
basic facts of the 10s, e.g., juice comes in packs of 4. How
many packs are needed for 32 third-graders? Students could
write: 32 ÷ 4 = J.
2. generates one-step equations to solve real-world problems with one
unknown and a whole number solution that (2.4.A1a) ($):
a. adds or subtracts using the basic fact families, e.g., when asked
to generate a simple equation, a student says: I have 5 dogs
and 2 fish. How many pets do I have? This is represented by 5
+ 2 = P and to solve for P, add 5 and 2, P = 7.
b. multiplies or divides using the basic facts through the 5s and the
basic facts of the 10s, e.g., Tom has a sticker book and each
page holds 5 stickers. If the same number of stickers is placed
on each page, the book will hold 30 stickers. How many pages
are in his book? This is represented by 5 x S = 30 or 30 ÷ 5 = S.
3. generates (2.4.A1a) ($):
a. a real-world problem with one operation that matches a given
addition equation or subtraction equation using whole numbers
from 0 through 99, e.g., given the subtraction equation, 69 – G
= 37, the problem could be written: You have 69 guppies and
give away some to a friend and have 37 left. How many
guppies did you give away?
b. a real-world problem with one operation that matches a given
3-101
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
multiplication equation or division equation using basic facts
through the 5s and the basic facts of the 10s, e.g., the problem
could be: I have 25 pictures and glue 5 pictures on each page of
my album. How many pages will I need to use? The equation:
25/5 = ▲.
c. number comparison statements using equality and inequality
symbols (=, <, >) for whole numbers from 0 through 100,
measurement, and money $, e.g.4 ft 4 in > 4 ft 2 in.
Teacher Notes: Understanding the concept of variable is fundamental to algebra. In the early grades, students use various symbols, including
letters and geometric shapes, to represent unknown quantities that do or do not vary. Quantities that are not given and do not vary are often
referred to as unknowns or missing elements when they appear in equations, e.g., 2 x 4 = Δ.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
3-102
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
THIRD GRADE
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 3: Functions – The student recognizes and describes whole number relationships using concrete
objects in a variety of situations.
Third Grade Knowledge Base Indicators
The student…
1. states mathematical relationships between whole numbers from 0
through 200 using various methods including mental math, paper
and pencil, concrete objects, and appropriate technology (2.4.K1a)
($), e.g., every time a quarter is added to the amount; 25¢ is added
to the total.
2. finds the values and determines the rule with one operation
(addition, subtraction) of whole numbers from 0 through 200 using a
horizontal or vertical function table (input/output machine, T-table)
(2.4.K1e), e.g., using this input/output machine, different student
responses might be that the rule is Input minus
Input
Output
10 equals Output, the rule is N – 10,
92
82
or the rule is subtract 10.
156
13
113
?
106
?
N
146
3
103
59
?
?
?
Third Grade Application Indicators
The student…
1. represents and describes mathematical relationships between whole
numbers from 0 through 100 using concrete objects, pictures, written
descriptions, symbols, equations, tables, and graphs (2.4.A1a) ($).
2. finds the rule, states the rule using words, and extends numerical
patterns with whole numbers from 0 through 100 (2.4.A1a,e), e.g., at
school each student must check out three library books. After the
tenth student has checked out, how many total books will have been
checked out? A solution using a function table might be:
Number of Students
Total Number Of Books
1
3
2
6
5
15
10
?
The rule could be that for every student, add three books or multiply
the number of children by three to get the total number of books.
Other solutions might be using a pattern to count by three ten times 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 - or skip count by three ten times.
..
3. ▲ generalizes numerical patterns using whole numbers from 0
through 200 with one operation (addition, subtraction) by stating the
rule using words, e.g., if the sequence is 30, 50, 70, 90, …; in words,
the rule is add twenty to the number before.
4. uses a function table (input/output machine, T-table) to identify and
plot ordered pairs in the first quadrant of a coordinate plane
(2.4.K1a,e).
3-103
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Functions are relationships or rules in which each member of one set is paired with one, and only one, member of another set
(an ordered pair). The concept of function can be introduced using function machines. Any number put in the machine will be changed according to
some rule. A record of inputs and corresponding outputs can be maintained in a two-column format. Function tables, input/output machines, and Ttables may be used interchangeably and serve the same purpose.
Function concepts should be developed from growing patterns. Each term in a number sequence is related to its position in the
sequence – the functional relationship. The pattern – 4, 7, 10, 13, 16, 19, and so on – is an arithmetic sequence with a difference of 3.
The pattern could be described as add 3 meaning that 3 must be added to the previous term to find the next. This pattern is explained
by using the recursive definition for a sequence. The recursive definition for a sequence is a statement or a set of statements that
explains how each successive term in the sequence is obtained from the previous term(s).
In the pattern 1, 4, 9, 16, 25, …, 225; there is no common difference. This sequence is not arithmetic or geometric (no common ratio
between geometric terms). Neither is it a combination of the two; however, there is a pattern and the missing terms between 25 and
225 can be found. To find the term value, square the number of the term. The next missing terms would be 36, 49, 64, 81, 100, 121,
and 144. This pattern is explained by using the explicit formula for a sequence. The explicit formula for a sequence defines a rule for
finding each term in the number sequence related to its position in the sequence. In other words, to find the term value, square the
number of the term – the 5th term is 52, the 8th term is 82, …
Patterns themselves are not explicit or recursive. The RULE for the pattern can be expressed explicitly or recursively and MOST
patterns can be explained using either format especially IF that pattern reflects either an arithmetic sequence or geometric sequence.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks
are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be
used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to
Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models
benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each indicator in the Models
benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For example, process models are
linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and
Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at
three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the
Appendix.
3-104
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
3-105
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
THIRD GRADE
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 4: Models – The student develops and uses mathematical models including the use of concrete objects
to represent and show mathematical relationships in a variety of situations.
Third Grade Knowledge Base Indicators
The student…
1. knows, explains, and uses mathematical models to represent
mathematical concepts, procedures, and relationships.
Mathematical models include:
a. process models (concrete objects, pictures, number lines,
coordinate planes/grids, hundred charts, measurement tools,
multiplication arrays, or division sets) to model computational
procedures and mathematical relationships (1.2.K1, 1.2.K.1a,
1.2.K2 1.2.K3, 1.2.K5-6, 1.3.K1-3, 1.4.K1-3, 1.4.K1a-b, 1.4.K5-7,
2.1.K1, 2.1.K2a, 2.1.K2d-g, 2.1.K3,
2.1.K4a-b, 2.2.K1, 2.2.K2, 2.2.K3-4, 2.3.K1, 2.3.K4, 3.2.K1-4,
3.3.K1, 3.4.K1-3, K.2.K3) ($);
b. place value models (place value mats, hundred charts, base ten
blocks or unifix cubes) to compare, order, and represent
numerical quantities and to model computational procedures
(1.1.K1c, 1.1.K2a, 1.1.K2c, 1.1.K3a, 1.2.K1-4, 1.2.K7, 1.3.K1,
1.4.K4a-b, 1.4.K5-6, 2.2.K4) ($);
c. fraction models (fraction strips or pattern blocks) and decimal
models (base ten blocks or coins) to compare, order, and
represent numerical quantities (1.1.K1b, 1.1.K2b-c, 1.2.K3,
1.3.K1) ($);
d. money models (base ten blocks or coins) to compare, order, and
represent numerical quantities (1.1K3c, 1.1.K4, 1.3.K1, 1.4.K4c,
2.1.K2e, 2.2.K2b) ($);
e. function tables (input/output machines, T-tables) to find
numerical relationships (2.1.K4c, 2.3.K2, 2.3.K4) ($);
Third Grade Application Indicators
The student…
1. recognizes that various mathematical models can be used to
represent the same problem situation. Mathematical models
include:
a. process models (concrete objects, pictures, number lines,
coordinate planes/grids, hundred charts, measurement tools,
multiplication arrays, or division sets) to model computational
procedures and mathematical relationships and to model
problem situations (1.2.A1, 1.2.A2a-b, 1.3.A1-4, 1.4.A1a-b,
1.4.A2, 2.1.A1a-b, 2.1.A1d-f, 2.1.A2, 2.2.A1-3, 2.2.A3a-c,
2.3.A1-2, 3.2.A1-3, 3.3.A1-2, 3.4.A1, 4.2.A2) ($);
b. place value models (place value mats, hundred charts, base ten
blocks, or unifix cubes) to compare, order, and represent
numerical quantities and to model computational procedures
(1.1.A1a, 1.1.A2a, 1.2.A1-2, 1.3.A2, 1.4.A1a-b) ($);
c. fraction models (fraction strips or pattern blocks) and decimal
models (base ten blocks or coins) to compare, order, and
represent numerical quantities (1.1.A2b) ($);
d. money models (base ten blocks or coins) to compare, order, and
represent numerical quantities (1.1.A1b, 1.1.A2c, 1.2.A1, 1.3.A2,
1.4.A1b, 2.1.A1e, 2.2.A3c) ($);
e. function tables (input/output machines, T-tables) to model
numerical relationships (2.3.A2) ($);
f. two-dimensional geometric models (geoboards, dot paper,
pattern blocks, or tangrams) to model perimeter, area, and
properties of geometric shapes and three-dimensional geometric
models (solids) and real-world objects to compare size and to
model attributes of geometric shapes (2.1.A1c, 3.1.A1-3);
3-106
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
f.
two-dimensional geometric models (geoboards, dot paper,
pattern blocks, or tangrams) to model perimeter, area, and
properties of geometric shapes and three-dimensional geometric
models (solids) and real-world objects to compare size and to
model attributes of geometric shapes (2.1.K2c, 3.1.K1-6, 3.2.K5,
3.3.K2);
g. two-dimensional geometric models (spinners), threedimensional models (number cubes), and process models
(concrete objects) to model probability (4.1.K1-2) ($);
h. graphs using concrete objects, representational objects, or
abstract representations, pictographs, frequency tables,
horizontal and vertical bar graphs, Venn diagrams or other
pictorial displays, line plots, charts, and tables to organize
and display data (2.3.K4, 4.1.K2, 4.2.K1a-d, 4.2.K1f-g,
4.2.K2) ($);
i. Venn diagrams to sort data and show relationships
(1.2.K3).
2. creates a mathematical model to show the relationship
between two or more things, e.g., using pattern blocks, a
whole (1) can be represented as
a
two
three
six
g. two-dimensional geometric models (spinners), threedimensional models (number cubes), and process models
(concrete objects) to model probability (4.1.A1-2) ($);
h. graphs using concrete objects, representational objects, or
abstract representations pictographs, frequency tables,
horizontal and vertical bar graphs, Venn diagrams or other
pictorial displays, line plots, charts and tables to organize and
display data (4.1.A1-2, 4.2.A1a-d, 4.2.A1f-g, 4.2.A3) ($);
i. Venn diagrams to sort data and show relationships.
2. selects a mathematical model that is more useful than other
mathematical models in a given situation.
(1/1) or
(2/2) or
(3/3) or
(6/6).
3-107
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: For assessment purposes, the mathematical modeling process appropriate to the indicator may be included as part of the item
being assessed.
The mathematical modeling process involves:
a. selecting key features and relationships within the real-world situation and representing these concepts in mathematical terms through
some sort of mathematical model,
b. performing manipulations and mathematical procedures within the mathematical model,
c. interpreting the results of the manipulations within the mathematical model,
d. using these results to make inferences about the original real-world situation.
The use of mathematical models is necessary for conceptual understanding. The ways in which mathematical ideas are represented is
fundamental to how students understand and use those ideas. As students begin to use multiple representations of the same situation, e.g.,
comparing the number of boys and girls in the classroom can be represented by lining them up in two different lines. The same situation also can
be represented by pictures of the children (pictograph), a bar graph, or by using two different colors of the same manipulative (unifix cubes or color
tiles). As students work with the different representations of the same situation, they begin to develop an understanding of the advantages and
disadvantages of the various representations/models.
Many mathematical models are listed in this benchmark. The indicator lists some of the mathematical models that could be used to teach a
concept. Each indicator in this benchmark is linked to other indicators in other benchmarks; those indicators are identified in the parentheses. For
example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1
(Number Sense), and Knowledge Indicator 3. In addition, the indicator in the other benchmarks identifies, in parentheses, the Models’ indicator. For
example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models).
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
3-108
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
THIRD GRADE
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 1: Geometric Figures and Their Properties – The student recognizes geometric shapes and investigates
their properties using concrete objects in a variety of situations.
Third Grade Knowledge Base Indicators
The student…
1. recognizes and investigates properties of plane figures (circles,
squares, rectangles, triangles, ellipses, rhombi, octagons) using
concrete objects, drawings, and appropriate technology (2.4.K1f).
2. recognizes, draws, and describes plane figures (circles, squares,
rectangles, triangles, ellipses, rhombi, octagons) (2.4.K1f).
3. ■ recognizes the solids (cubes, rectangular prisms, cylinders, cones,
spheres) (2.4.K1f).
4. ▲ recognizes and describes the square, triangle, rhombus, hexagon,
parallelogram, and trapezoid from a pattern block set (2.4.K1f).
5. recognizes and describes a quadrilateral as any four-sided figure
(2.4.K1f).
6. determines if geometric shapes and real-world objects contain line(s)
of symmetry and draws the line(s) of symmetry if the line(s) exist(s)
(2.4.K1f).
Third Grade Application Indicators
The student…
1. solves real-world problems by applying properties of plane figures
(circles, squares, rectangles, triangles, ellipses) to (2.4.A1f), e.g.,
the teacher asked each student to draw a rectangle. A student
draws a square. Did the student follow directions? Why or why
not?
2. demonstrates how (2.4.A1f):
a. plane figures (circles, squares, rectangles, triangles, ellipses,
rhombi, hexagons, trapezoids) can be combined to make a
new shape;
b. solids (cubes, rectangular prisms, cylinders, cones, spheres)
can be combined to make a new shape.
3. identifies the plane figures (circles, squares, rectangles, triangles,
ellipses, rhombi, hexagons, trapezoids) used to form a composite
figure (2.4.A1f).
3-109
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Geometry is the study of shapes, their properties, and their relationships to other shapes. Symbols and numbers are used to
describe their properties and their relationships to other shapes. The term geometry comes from two Greek words meaning “earth measure.” The
fundamental concepts in geometry are point (no dimension), line (one-dimensional), plane (two-dimensional), and space (three-dimensional). Plane
figures are referred to as two-dimensional and solids are referred to as three-dimensional.
From the Mathematics Dictionary and Handbook (Nichols Schwartz Publishing, 1999), property as a mathematical term means a
characteristic (an attribute) of a number, geometric shape, mathematical operation, equation, or inequality. To give an example:





Property of a number: 8 is divisible by 2.
Property of a geometric shape: Each of the four sides of a square is of the same length.
Property of an operation: Addition is commutative. For all numbers x and y, x + y = y + x.
Property of an equation: For all numbers a, b, and c, if a = b, then a + c = b + c.
Property of an inequality: For all numbers a, b, and c, if a > b, then a – c > b – c.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
Pattern blocks are a collection of six geometric shapes in six colors. Each set contains 250 pieces – 50 green triangles, 25 orange squares, 50
blue rhombi, 50 tan rhombi, 50 red trapezoids, and 25 yellow hexagons. The blue rhombus and the tan rhombus also are parallelograms, and the
orange square is a parallelogram. The blocks are designed so that their sides are all the same length with the exception that the trapezoid has one
side twice as long. This feature allows the blocks to be nested together and encourages the exploration of relationships among the shapes.
Activities with pattern blocks help students explore patterns, functions, fractions, congruence, similarity, symmetry, perimeter, area, and graphing. A
template for a pattern block set can be found in the Appendix.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
3-110
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
THIRD GRADE
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 2: Measurement and Estimation – The student estimates and measures using standard and nonstandard
units of measure with concrete objects in a variety of situations.
Third Grade Knowledge Base Indicators
The student…
1. uses whole number approximations (estimations) for length, width,
weight, volume, temperature, time, and perimeter using standard and
nonstandard units of measure (2.4.K1a) ($).
2. ▲ reads and tells time to the minute using analog and digital clocks
(2.4.K1a).
3. selects, explains the selection of, and uses measurement tools, units
of measure, and degree of accuracy appropriate for a given situation
to measure (2.4.K1a) ($):
a. length width, and height to the nearest half inch, inch, foot, and
yard; and to the nearest whole unit of nonstandard unit;
b. length, width, and height to the nearest centimeter and meter;
c. weight to the nearest whole unit of a nonstandard unit;
d. volume to the nearest cup, pint, quart, and gallon;
e. volume to the nearest liter;
f. temperature to the nearest degree.
4. states (2.4.K1a):
a. the number of hours in a day and days in a year;
b. the number of inches in a foot, inches in a yard, and feet in a
yard;
c. the number of centimeters in a meter;
d. the number of cups in a pint, pints in a quart, and quarts in a
gallon.
5. finds the perimeter of squares, rectangles, and triangles given the
measures of all the sides (2.4.K1f).
Third Grade Application Indicators
The student…
1. solves real-world problems by applying appropriate
measurements:
a. ▲ length to the nearest inch, foot, or yard, e.g., Jill has a
piece of rope that is 36 inches long and Bob has a piece that
is 15 inches long. If they put their pieces together, how long
would the piece of rope be?
b. ▲ length to the nearest centimeter or meter, e.g., a new pencil
is about how many centimeters long?
c. length to the nearest whole unit of a nonstandard unit, e.g.,
how many paper clips long is a hot dog?
d. temperature to the nearest degree, e.g., what would the
temperature outside be if it was a good day for swimming?
e. ▲ number of days in a week, e.g., if school started 37 weeks
ago, how many days of school have passed?
2. estimates to check whether or not measurements or calculations
for length, temperature, and time in real-world problems are
reasonable (2.4.A1a) ($), e.g., after finding the range of
temperature over a two-week period, determine whether or not the
answer is reasonable.
3. adjusts original measurement or estimation for length, weight,
temperature, and time in real-world problems based on additional
information (a frame of reference) (2.4.A1a) ($), e.g., the class
estimates that the class gerbil weighs as much as a box of 24
crayons. The gerbil is placed on one side of the pan balance and
a box of 16 crayons is placed on the other side. The pan balance
barely moves. Should the estimate of the gerbil’s weight be
adjusted?
3-111
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: The term geometry comes from two Greek words meaning “earth measure.” Measurement provides the tools required to apply
geometric concepts in the real-world. Estimation in measurement is defined as making guesses as to the exact measurement of an object without
using any type of measurement tool. Estimation helps students develop a relationship between the different sizes of units of measure. It helps
students develop basic properties of measurement and it gives students a tool to determine whether a given measurement is reasonable.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
3-112
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
THIRD GRADE
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 3: Transformational Geometry – The student recognizes and performs one transformation on simple
shapes or concrete objects in a variety of situations.
Third Grade Knowledge Base Indicators
The student…
1. knows and uses cardinal points (north, south, east, west) and
intermediate points (northeast, southeast, northwest, southwest)
(2.4.K1a).
2. recognizes and performs one transformation (reflection/flip,
rotation/turn, and translation/slide) on a two-dimensional figure
(2.4.K1f).
Third Grade Application Indicators
The student…
1. recognizes real-world transformations (reflection/flip, rotation/turn,
and translation/slide) (2.4.A1a), e.g., tiles in a ceiling, bricks in a
sidewalk, or steps on a playground slide.
2. gives and uses directions to move from one location to another on a
map and follows directions including the use of cardinal and
intermediate points (2.4.A1a).
Teacher Notes: Transformational geometry is another way to investigate geometric figures by moving every point in a plane figure to a new
location. To help students form images of shapes through different transformations, students can use concrete objects, figures drawn on graph
paper, mirrors or other reflective surfaces, or appropriate technology.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
3-113
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
THIRD GRADE
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 4: Geometry From An Algebraic Perspective – The student relates geometric concepts to a number line
and the first quadrant of a coordinate plane in a variety of situations.
Third Grade Knowledge Base Indicators
The student…
1. uses a number line (horizontal/vertical) to model the basic
multiplication facts through the 5s and the multiplication facts of the
10s (2.4.K1a).
2. identifies points on a coordinate plane (coordinate grid) using
(2.4.K1a):
a. two positive whole numbers,
b. a letter and a positive whole number.
3. identifies points as ordered pairs in the first quadrant of a coordinate
plane (coordinate grid) (2.4.K1a).
Third Grade Application Indicators
The student…
1. solves real-world problems using coordinate planes (coordinate
grids) and map grids that have positive whole number and letter
coordinates (2.4.A1a), e.g., identifying locations on a map or giving
and following directions to move from one location to another.
3-114
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: A number line (a mathematical model) is a diagram that represents numbers with equal distances marked off as points on a line,
and is an example of one-to-one correspondence (a relation). A number line can be used as a visual representation of numbers and operations. In
addition, a number line used horizontally and vertically is a precursor to the coordinate plane; and the distance between two numbers on a number
line is a precursor to absolute value.
A coordinate plane (coordinate grid) consists of a horizontal number line called the x-axis and a vertical number line called the y-axis. These two
lines intersect at a point called the origin. The x-axis and the y-axis divide the plane into four sections called quadrants. Any point on the coordinate
plane can be named with two numbers called coordinates. The first number is the x-coordinate. The second number is the y-coordinate. Since the
pair is always named in order (first x, then y), it is called an ordered pair.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
3-115
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 4: Data
THIRD GRADE
Standard 4: Data – The student uses concepts and procedures of data analysis in a variety of situations.
Benchmark 1: Probability – The student applies the concepts of probability to draw conclusions and to make
predictions and decisions including the use of concrete objects in a variety of situations.
Third Grade Knowledge Base Indicators
The student…
1. recognizes any outcome of a simple event in an experiment or
simulation as impossible, possible, certain, likely, unlikely, or
equally likely (2.4.K1g) ($).
2. ▲ ■ lists some of the possible outcomes of a simple event in an
experiment or simulation including the use of concrete objects
(2.4.K1g-h).
Third Grade Application Indicators
The student…
1. makes predictions about a simple event in an experiment or
simulation; conducts the experiment or simulation including the use
of concrete objects; records the results in a chart, table, or graph;
and uses the results to draw conclusions about the event (2.4.A1gh).
2. compares what should happen (theoretical probability/expected
results) with what did happen (experimental probability/empirical
results) in an experiment or simulation with a simple event (2.4.A1g).
3-116
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Ideas from probability reinforce concepts in the other Standards, especially Number and Computation and Geometry. Students
need to develop an intuitive concept of chance – whether or not something is unlikely or likely to happen. Probability experiences should be
addressed through the use of concrete objects, coins, and geometric models (spinners, number cubes, or dartboards). Probabilities are ratios,
expressed as fractions, decimals, or percents, determined by considering results or outcomes of experiments. Some examples of uses of
probability in every day life include: There is a 50% chance of rain today. What is the probability that the team will win every game?
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
3-117
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 4: Data
THIRD GRADE
Standard 4: Data – The student uses concepts and procedures of data analysis in a variety of situations.
Benchmark 2: Statistics – The student collects, organizes, displays, explains, and interprets numerical (whole
numbers) and non-numerical data sets including the use of concrete objects in a variety of situations.
Third Grade Knowledge Base Indicators
The student…
1. organizes, displays, and reads numerical (quantitative) and nonnumerical (qualitative) data in a clear, organized, and accurate
manner including a title, labels, categories, and whole number
intervals using these data displays (2.4.K1h) ($):
a. graphs using concrete objects;
b. pictographs with a whole symbol or picture representing one,
two, five, ten, twenty-five, or one-hundred (no partial symbols
or pictures);
c. frequency tables (tally marks);
d. horizontal and vertical bar graphs;
e. Venn diagrams or other pictorial displays, e.g., glyphs;
f. line plots;
g. charts and tables.
2. collects data using different techniques (observations, polls,
surveys, or interviews) and explains the results (2.4.K1h) ($).
3. ▲ finds these statistical measures of a data set with less than ten
data points using whole numbers from 0 through 1,000 (2.4.K1a) ($):
a. minimum and maximum data values,
b. range,
c. mode (uni-modal only),
d. median when data set has an odd number of data points.
Third Grade Application Indicators
The student…
1. interprets and uses data to make reasonable inferences and
predictions, answer questions, and make decisions from these data
displays (2.4.A1h) ($):
a. graphs using concrete objects;
b. pictographs with a whole symbol or picture representing one,
two, five, ten, twenty-five, or one-hundred (no partial symbols or
pictures);
c. frequency tables (tally marks);
d. horizontal and vertical bar graphs;
e. Venn diagrams or other pictorial displays;
f. line plots;
g. charts and tables.
2. uses these statistical measures with a data set of less than ten data
points using whole numbers from 0 through 1,000 to make
reasonable inferences and predictions, answer questions, and make
decisions (2.4.A1a) ($):
a. minimum and maximum data values,
b. range,
c. mode,
d. median when data set has an odd number of data points.
3. recognizes that the same data set can be displayed in various
formats including the use of concrete objects (2.4.A1h) ($).
3-118
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Graphs (data displays) are pictorial representations of mathematical relationships, are used to tell a story, and are an important
part of statistics. When a graph is made, the axes and the scale (numbers running along a side of the graph) are chosen for a reason. The
difference between numbers from one grid line to another is the interval. The interval will depend on the lowest and highest values in the data set.
Emphasizing the importance of using equal-sized pictures or intervals is critical to ensuring that the data display is accurate.
Graphs take many forms:
– bar graphs and pictographs compare discrete data,
– frequency tables show how many times a certain piece of data occurs,
– circle graphs (pie charts) model parts of a whole,
– line graphs show change over time,
– Venn diagrams show relationships between sets of objects, and
– line plots show frequency of data on a number line.
An important aspect of data is its center. The measures of central tendency (averages) of a data set are mean, median, and mode. Conceptual
understanding of mean, median, and mode is developed through the use of concrete objects that represent the data values.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
3-119
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
Standard 1:
FOURTH GRADE
Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 1:
Number Sense – The student demonstrates number sense for whole numbers, fractions (including
mixed numbers), decimals, and money including the use of concrete objects in a variety of situations.
Fourth Grade Knowledge Base Indicators
The student…
1. knows, explains, and uses equivalent representations for ($):
a. whole numbers from 0 through 100,000 (2.4.K1a-b);
b. fractions greater than or equal to zero (halves, fourths, thirds,
eighths, tenths, twelfths, sixteenths, hundredths) including mixed
numbers (2.4.K1c);
c. decimals greater than or equal to zero through hundredths place
and when used as monetary amounts (2.4.K1c-d) ($), e.g., 7¢ =
$.07 = 7/100 of a dollar or a hundreds grid with 7 sections
colored or .1 = 1/10 =  .
2. compares and orders:
a. whole numbers from 0 through 100,000 (2.4.K1a-b) ($);
b. fractions greater than or equal to zero (halves, fourths, thirds,
eighths, tenths, twelfths, sixteenths, hundredths) including mixed
numbers with a special emphasis on concrete objects (2.4.K1c);
c. decimals greater than or equal to zero through hundredths place
and when used as monetary amounts (2.4.K1c-d) ($).
Fourth Grade Application Indicators
The student…
1. solves real-world problems using equivalent representations and
concrete objects to ($):
a. compare and order whole numbers from 0 through 100,000
(2.4.A1a-b); e.g., using base ten blocks, represent the
attendance at the circus over a three day stay; then represent
the numbers using digits and compare and order in different
ways;
b. add and subtract whole numbers from 0 through 10,000 and
decimals when used as monetary amounts (2.4.A1a-d), e.g.,
use real money to show at least 2 ways to represent $142.78,
then subtract the cost of a pair of tennis shoes;
c. multiply a one-digit whole number by a two-digit whole number
(2.4.A1a-b), e.g., use base ten blocks to represent 24 x 5 to find
the total number of hours in 5 days, or use repeated addition
24 + 24 + 24 + 24 + 25 to solve, or use the algorithm.
2. determines whether or not solutions to real-world problems that
involve the following are reasonable ($):
a. whole numbers from 0 through 10,000 (2.4.A1a-b), e.g., a
student says that there are 1,000 students in grade 4 at her
school, is this reasonable?
b. fractions greater than or equal to zero (halves, fourths, thirds,
eighths, tenths, sixteenths) (2.4.A1c), e.g., you ate ½ of a
sandwich and a friend ate ¾ of the same sandwich; is this
reasonable?
c. decimals greater than or equal to zero when used as monetary
amounts (2.4.A1c-d), e.g., a pack of chewing gum costs what
amount - $62 $.75 9¢ 75.00 750¢? Is this reasonable?
4-120
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Number sense refers to one’s ability to reason with numbers and to work with numbers in a flexible way. The ability to compute
mentally, to estimate based on understanding of number relationships and magnitudes, and to judge reasonableness of answers are all involved in
number sense.
When we say that someone has good number sense, we mean that he or she possesses a variety of abilities and understandings that include an
awareness of the relationships between numbers, an ability to represent numbers in a variety of ways, a knowledge of the effects of operations,
and an ability to interpret and use numbers in real-world counting and measurement situations. Such a person predicts with some accuracy the
result of an operation and consistently chooses appropriate measurement units. This “friendliness with numbers” goes far beyond mere
memorization of computational algorithms and number facts; it implies an ability to use numbers flexibly, to choose the most appropriate
representation of a number for a given circumstance, and to recognize when operations have been correctly performed. (Number Sense and
Operations: Addenda Series, Grades K-6, NCTM, 1993)
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
4-121
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
FOURTH GRADE
Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 2: Number Systems and Their Properties – The student demonstrates an understanding of whole
numbers with a special emphasis on place value; recognizes, uses, and explains the concepts of
properties as they relate to whole numbers; and extends these properties to fractions (including
mixed numbers), decimals, and money.
Fourth Grade Knowledge Base Indicators
Fourth Grade Application Indicators
4-122
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
The student…
1. ▲ identifies, models, reads, and writes numbers using numerals,
words, and expanded notation from hundredths place through onehundred thousands place (2.4.K1a-b) ($), e.g., four hundred sixty-two
thousand, two hundred eighty-four and fifty hundredths = 462,284.50
or 462,284.50 = (4 x 100,000) + (6 x 10,000) + (2 x 1,000) + (2 x 100)
+ (8 x 10) + (4 x 1) + (5 x .1) + (0 x .01) = 400,000 + 60,000 + 2,000 +
200 + 80 + 4 + .5 + .00.
2. classifies various subsets of numbers as whole numbers, fractions
(including mixed numbers), or decimals (2.4.K1b-c, 2.4.K1i).
3. identifies the place value of various digits from hundredths place
through one hundred thousands place (2.4.K1b) ($).
4. identifies any whole number as even or odd (2.4.K1a).
5. uses the concepts of these properties with the whole number system
and demonstrates their meaning including the use of concrete objects
(2.4.K1a) ($):
a. ▲ commutative properties of addition and multiplication, e.g.,
12 + 18 = 18 + 12 and 8 x 9 = 9 x 8;
b. ▲ zero property of addition (additive identity) and property of one
for multiplication (multiplicative identity), e.g., 24 + 0 = 24 and
75 x 1 = 75;
c. ▲ associative properties of addition and multiplication, e.g.,
4 + (2 + 3) = (4 + 2) + 3 and 2 x (3 x 4) = (2 x 3) x 4;
d. ▲ symmetric property of equality applied to addition and
multiplication, e.g., 100 = 20 + 80 is the same as 20 + 80 = 100
and 21 = 7 x 3 is the same as 3 x 7 = 21;
e. zero property of multiplication, e.g., 9 x 0 = 0 or 0 x 112 = 0;
The student…
1.
solves real-world problems with whole numbers from 0 through
10,000 using place value models; money; and the concepts of
these properties to explain reasoning (2.4.A1a-b,d) ($):
a. commutative properties of addition and multiplication, e.g., a
student has a $5, a $10, and a $20 bill; a student totals the
amount to see how much can be spent shopping for school
supplies. The student says: Because you can add in any
order, I can rearrange the money and count $20, $10, and $5
for $20 + $10 + $5. Another student has 4 $5 bills. The
student is asked the amount. The student says: I don’t know 4
x 5 but I know 5 x4 is $20, since multiplication can be done in
any order.
b. zero property of addition, e.g., a student has 6 marbles in one
pocket and none in the other pocket. How many marbles
altogether?
c. property of one for multiplication, e.g., there are 24 students in
our class, each student should have one math book; so I
compute 24 x 1 = 24. Multiplying times 1 does not change the
product because it is one group of 24.
d. associative properties of addition and multiplication, e.g., a
student has two dimes and a quarter. Using coins or money
models, there are at least 2 ways to group the coins to find the
total. One way is 10¢ (dime) + 10¢ (dime) = 20¢, then add the
quarter, so 20¢ + 25¢ (quarter) = 45¢. Another way 10¢
(dime) + 25¢ (quarter) = 35¢, then add the other dime to 35¢
so 35¢ + 10¢ = 45¢ This models that (D + D) + Q =
D + (D + Q).
4-123
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
f.
distributive property, e.g., 6(7 + 3) = (6 • 7) + (6 • 3).
e. zero property of multiplication, e.g., in science, you are
observing a snail. The snail does not move over a 4-hour
period. To figure its total movement, you say 4 x 0 = 0.
2. performs various computational procedures with whole numbers
from 0 through 10,000 using the concepts of the following
properties; extends the properties to fractions (halves, fourths,
thirds, eighths, tenths, sixteenths) including mixed numbers, and
decimals through hundredths place; and explains how the
properties were used (2.4.A1a-c):
a. commutative property of addition and multiplication, e.g., 5 + 6
= 6 + 5, the student says: I know that 5 + 6 = 11 and adding in
any order still gets the answer, so 6 + 5 is the same as 5 + 6.
4 x 6 = 6 x 4, the student says: I know that 4 x 6 = 24 and
multiplying in any order still gets the answer, so 4 x 6 is the
same as 6 x 4.
b. zero property of multiplication without computing, e.g., 158 x 0
= 0; the student says: I know the answer (product) is zero
because no matter how many factors you have, when you
multiply with a 0, the product is zero.
c. associative property of addition, e.g., 9 + 8 could be solved as
1 + (8 + 8) or (1 + 8) + 8, the student says: I don’t know 9 + 8,
but I know my doubles of 8 + 8, so I made the 9 into 1 + 8 and
then added 1 more to make 17.
3. states the reason for using whole numbers, fractions, mixed
numbers, or decimals when solving a given real-world problem
(2.4.A1a-d).
4-124
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: From the Mathematics Dictionary and Handbook (Nichols Schwartz Publishing, 1999), property as a mathematical term means a
characteristic (an attribute) of a number, geometric shape, mathematical operation, equation, or inequality. To give an example:
Property of a number: 8 is divisible by 2.
Property of a geometric shape: Each of the four sides of a square is of the same length.
Property of an operation: Addition is commutative. For all numbers x and y, x + y = y + x.
Property of an equation: For all numbers a, b, and c, if a = b, then a + c = b + c.
Property of an inequality: For all numbers a, b, and c, if a > b, then a – c > b – c.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
4-125
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
Standard 1:
FOURTH GRADE
Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 3: Estimation – The student uses computational estimation with whole numbers, fractions (including
mixed numbers) and money in a variety of situations.
Fourth Grade Knowledge Base Indicators
The student…
1. estimates whole number quantities from 0 through 10,000;
fractions (halves, fourths, thirds); and monetary amounts through
$1,000 using various computational methods including mental
math, paper and pencil, concrete materials, and appropriate
technology (2.4.K1a-d) ($).
2. uses various estimation strategies and explains how they are used
when estimating whole numbers quantities from 0 through 10,000;
fractions [(halves, fourths, thirds) including mixed numbers)]; and
monetary amounts through $1,000 (2.4.K1a-d) ($).
3. recognizes and explains the difference between an exact and an
approximate answer (2.4.K1a), e.g., when asked how many desks
are in the room, the student gives an estimate of about 30 and
then counts the desks and indicates an exact answer is 28 desks.
4. selects from an appropriate range of estimation strategies and
determines if the estimate is an overestimate or underestimate,
(2.4.K1a).
Fourth Grade Application Indicators
The student…
1. adjusts original whole number estimates of a real-world problem using
numbers from 0 through 10,000 based on additional information (a
frame of reference) (2.4.A1a) ($), e.g., if given a small jar and told the
number of pieces of candy it has in it, the student would adjust his/her
original estimate of the number of pieces of candy in a larger jar.
2. estimates to check whether or not the result of a real-world problem
using whole numbers from 0 through 10,000, fractions (including
mixed numbers), and monetary amounts is reasonable and makes
predictions based on the information (2.4.A1a-d) ($), e.g., at the
movies, you bought popcorn for $2.35, a soda for $2.50, and paid
$4.50 for the ticket. Is it reasonable to say you spent $10? How much
will you need to save to go to the movies once a week for the next
month?
3. selects a reasonable magnitude from three given quantities based on
a familiar problem situation and explains the reasonableness of
selection (2.4.A1a), e.g., about how many new pencils will fit in your
pencil box? Is it about 25, about 50, or about 100? The answer will
depend on the size of your pencil box.
4. determines if a real-world problem calls for an exact or approximate
answer and performs the appropriate computation using various
computational methods including mental math, paper and pencil,
concrete objects, and appropriate technology (2.4.A1a) ($).
4-126
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Estimate, as a verb, means to make an educated guess based on information in a problem or to give an answer close to the
exact number. Estimation is used when an exact answer is not needed, as in many real-life situations for which “ballpark” computations are
acceptable. Good number sense enables one to estimate a quantity, estimate a measure, or estimate an answer.
Estimation serves as an important companion to computation. It provides a tool for judging the reasonableness of computational methods
including mental math, paper and pencil, concrete objects, and appropriate technology. However, being able to compute does not automatically
lead to an ability to estimate or judge reasonableness of answers. Frequent modeling by the teacher helps students develop a range of estimation
strategies. Students should be encouraged to frequently explain their thinking as they estimate. As with exact computation, sharing estimation
strategies allows students access to others’ thinking and provides opportunities for class discussion. Identifying the estimation strategy by name is
not critical; however, explaining the thinking behind the strategy to make a valid estimation is important. (Principles and Standards for School
Mathematics, NCTM, 2000).
Mental math and estimation are distinct but related mathematical skills. Proficiency in mental math contributes to increased skill in estimation. In
order for students to become more familiar with estimation, teachers should introduce estimation with examples where rounded or estimated
numbers are used. Emphasis should be placed on real-world examples where only estimation is required, e.g., About how many hours do you
sleep a night? Using the language of estimation is important, so students begin to realize that a variety of estimates (answers) are possible.
In addition, when students are taught specific estimation strategies, they develop mental math and estimation skills easier. Estimation strategies
include front-end with adjustment, compatible “nice” numbers, clustering, or special numbers.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
4-127
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
Standard 1:
FOURTH GRADE
Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 4: Computation – The student models, performs, and explains computation with whole numbers,
fractions, and money including the use of concrete objects in a variety of situations.
Fourth Grade Knowledge Base Indicators
The student…
1. computes with efficiency and accuracy using various computational
methods including mental math, paper and pencil, concrete
materials, and appropriate technology (2.4.K1a) ($).
2. N states and uses with efficiency and accuracy multiplication facts
from 1 x 1 through 12 x 12 and corresponding division facts
(2.4.K1a) ($).
3. N performs and explains these computational procedures ($):
a. adds and subtracts whole numbers from 0 through 100,000 and
when used as monetary amounts (2.4.K1a-b,d);
b. multiplies through a three-digit whole number by a two-
digit whole number (2.4.K1a-b);
c.
multiplies whole dollar monetary amounts (through three-digits)
by a one- or two-digit whole number (2.4.K1d), e.g., $45 x 16;
d. multiplies monetary amounts less than $100.00 by whole
numbers less than ten (2.4.K1d), e.g., $14.12 x 7;
e. divides through a two-digit whole number by a one-digit whole
number with a one-digit whole number quotient with or without a
remainder (2.4.K1a-b), e.g., 47 ÷ 5 = 9 r 2;
f. adds and subtracts fractions greater than or equal to zero with
like denominators (2.4.K1c);
g. figures correct change through $20.00 (2.4.K1d).
4. identifies multiplication and division fact families (2.4.K1a).
5. reads and writes horizontally, vertically, and with different
operational symbols the same addition, subtraction, multiplication,
or division expression, e.g., 6 • 4 is the same as 6 x 4 is the same
as 4 and 6(4) or 10 divided by 2 is the same as 10 ÷ 2 or 10.
x
6
2
Fourth Grade Application Indicators
The student…
1. ▲N solves one- and two-step real-world problems with one or two
operations using these computational procedures ($):
a. adds and subtracts whole numbers from 0 through 10,000 and
when used as monetary amounts (2.4.A1a-b,d), e.g., Lee buys
a bicycle for $139, a helmet for $29, and a reflector for $6. He
paid for it with a $200 check from his grandparents. How much
will he have left from the $200 check?
b. multiplies through a two-digit whole number by a two-digit whole
number (2.4.A1a-b), e.g., at school, there are 22 students in
each classroom. If there are 24 classes, how many students are
in the classrooms?
c. multiplies whole dollar monetary amounts (up through threedigit) by a one- or two-digit whole number (2.4.A1a-b,d), e.g.,
112 third and fourth graders are planning a field trip. The cost
per student is $9.00. How much will the trip cost?
d. multiplies monetary amounts less than $100 by whole numbers
less than ten (2.4.A1a-d), e.g., at the book fair, a student buys 8
books on animals for $2.69 each. How much did the student
pay for the books?
e. ■ figures correct change through $20.00 (2.4.A1a-d), e.g.,
buying a 65¢ drink, paying for it with a $1 bill, and then figuring
the amount of change.
2. generates a family of multiplication and division facts given one
equation/fact (2.4.A1b), e.g., given 8 x 9 = 72, the other facts are
9 x 8 = 72, 72 ÷ 8 = 9, and 72 ÷ 9 = 8.
4-128
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
6. ▲N shows the relationship between these operations with the basic
fact families (addition facts with sums from 0 through 20 and
corresponding subtraction facts, multiplication facts from 1 x 1
through 12 x 12 and corresponding division facts) including the use of
mathematical models (2.4.K1a) ($):
a. addition and subtraction,
b. addition and multiplication,
c. multiplication and division,
d. subtraction and division.
7. finds factors and multiples of whole numbers from 1 through 100
(2.4.K1a).
4-129
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Efficiency and accuracy means that students are able to compute single-digit numbers with fluency. Students increase their
understanding and skill in single-digit addition and subtraction by developing relationships within addition and subtraction combinations and by
counting on for addition and counting up for subtraction and unknown-addend situations. Students learn basic number combinations and develop
strategies for computing that makes sense to them. Through class discussions, students can compare the ease of use and ease of explanation of
various strategies. In some cases, their strategies for computing will be close to conventional algorithms; in other cases, they will be quite different.
Many times, students’ invented approaches are based on a sound understanding of numbers and operations, and these invented approaches often
can be used with efficiency and accuracy. (Principles and Standards for School Mathematics, NCTM, 2000)
The definition of computation is finding the standard representation for a number. For example, 6 + 6, 4 x 3, 17 – 5, and 24 ÷ 2 are all
representations for the standard representation of 12. Mental math is mentally finding the standard representation for a number – calculating in
your head instead of calculating using paper and pencil or technology. One of the main reasons for teaching mental math is to help students
determine if a computed/calculated answer is reasonable; in other words, using mental math to estimate to see if the answer makes sense.
Students develop mental math skills easier when they are taught specific strategies. Mental math strategies include counting on, doubling,
repeated doubling, halving, making tens, dividing with tens, thinking money, and using compatible “nice” numbers.
Regrouping refers to the reorganization of objects. In computation, regrouping is based on a “partitioning to multiples of ten” strategy. For
example, 46 + 7 could be solved by partitioning 46 into 40 and 6, then 40 + (6 + 7) = 40 + 13 (and then 13 is partitioned into 10 and 3) which then
becomes (40 + 10) + 3 becomes 50 + 3 = 53 or 7 could be partitioned as 4 and 3, then 46 + 4 (bridging through 10) = 50 and 50 + 3 = 53. Before
algorithmic procedures are taught, an understanding of “what happens” must occur. For this to occur, instruction should involve the use of
structured manipulatives. To emphasize the role of the base ten numeration system in algorithms, some form of expanded notation is
recommended. During instruction, each child should have a set of manipulatives to work with rather than sit and watch demonstrations by the
teacher. Some additional computational strategies include doubles plus one or two (i.e., 6 + 8, 6 and 6 are 12, so the answer must be 2 more or
14), compensation (i.e., 9 + 7, if one is taken away from 9, it leaves 8; then that one is given to 7 to make 8, then 8 + 8 = 16), subtracting through
ten (i.e., 13 – 5, 13 take away 3 is 10, then take 2 more away from 10 and that is 8), and nine is one less than ten (i.e., 9 + 6, 10 and 6 are 16, and
1 less than 16 makes 15). (Teaching Mathematics in Grades K-8: Research Based Methods, ed. Thomas R. Post, Allyn and Bacon, 1988)
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
4-130
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
FOURTH GRADE
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 1:
Patterns – The student recognizes, describes, extends, develops, and explains relationships in
patterns using concrete objects in a variety of situations.
Fourth Grade Knowledge Base Indicators
The student…
1. uses concrete objects, drawings, and other representations to work
with types of patterns(2.4.K1a):
a. repeating patterns, e.g., an AB pattern is like 1-2, 1-2, …; an
ABC pattern is like dog-horse-pig, dog-horse-pig, …; an AAB
pattern is like ↑↑→, ↑↑→, …;
b. growing patterns e.g., 2, 5, 11, 20, …
2. uses these attributes to generate patterns:
a. counting numbers related to number theory (2.4.K1a), e.g.,
multiples and factors through 12 or multiplying by 10, 100, or
1,000;
b. whole numbers that increase or decrease (2.4.K1a) ($) , e.g.,
20, 15, 10, …;
c. geometric shapes including one or two attributes changes
(2.4.K1f), e.g.,
Fourth Grade Application Indicators
The student…
1. generalizes these patterns using a written description:
a. counting numbers related to number theory (2.4.A1a),
b. whole number patterns (2.4.A1a) ($),
c. patterns using geometric shapes (2.4.A1f),
d. measurement patterns (2.4.A1a),
e. money and time patterns (2.4.A1a,d) ($),
f. patterns using size, shape, color, texture, or movement
(2.4.A1a).
2. recognizes multiple representations of the same pattern (2.4.A1a),
e.g., skip counting by 5s to 60; whole number multiples of 5 through
60; the multiplication table of 5 given the numerical pattern of 5, 10,
15, …, 60; relating the concept of five minute time intervals to each
of the numerals on a clock giving the pattern of 5, 10, 15, …, 60;
one nickel, two nickels, three nickels, …; the number of fingers on
twelve hands; recognizing that all of these representations are the
same general pattern.
… when the next shape
has one more side; or when both color and shape change at the
same time such as
…,
d. measurements (2.4.K1a), e.g., 3 ft., 6 ft., 9 ft., …;
e. money and time (2.4.K1a,d) ($), e.g., $.25, $.50, $.75, … or 1:05
p.m., 1:10 p.m., 1:15 p.m., …;
f. things related to daily life (2.4.K1a), e.g., water cycle, food cycle,
or life cycle;
g. things related to size, shape, color, texture, or movement
(2.4.K1a), e.g., rough, smooth, rough, smooth, rough, smooth, or
4-131
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
clapping hands (kinesthetic patterns).
3. identifies, states and continues a pattern presented in visual various
formats including numeric (list or table), visual (picture, table, or
graph), verbal (oral description), kinesthetic (action), and written
(2.4.K1a) ($).
4. generates:
a. a pattern (repeating, growing) (2.4.K1a);
a pattern using a function table (input/output machines, T-tables)
(2.4.K1e).
Teacher Notes: Working with patterns is an important process in the development of mathematical thinking. Patterns can be based on geometric
attributes (shapes, regions, angles); measurement attributes (color, texture, length, weight, volume, number); relational attributes (proportion,
sequence, functions); and affective attributes (values, likes, dislikes, familiarity, heritage, culture). (Learning to Teach Mathematics, Randall J.
Souviney, Macmillan Publishing Company, 1994)
This process (working with patterns) can be used to develop or deepen understandings of important concepts in number theory, whole numbers,
measurement, geometry, probability, and functions. Working with patterns provides opportunities for students to recognize, describe, extend,
develop, and explain.
Number theory is the study of the properties of the counting numbers (positive integers), their relationships, ways to represent them, and patterns
among them. Number theory includes the concepts of odd and even numbers, factors and multiples, primes and composites, and greatest
common factor and least common multiple.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
4-132
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
FOURTH GRADE
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 2: Variables, Equations, and Inequalities – The student uses variables, symbols, and whole numbers to
solve equations including the use of concrete objects in a variety of situations.
Fourth Grade Knowledge Base Indicators
The student…
1. explains and uses variables and symbols to represent unknown
whole number quantities from 0 through 1,000 (2.4.K1a).
2. ▲ solves one-step equations using whole numbers with one variable
and a whole number solution that:
a. find the unknown in a multiplication or division equation based
on the multiplication facts from 1 x 1 through 12 x 12 and
corresponding division facts (2.4.K1a), e.g., 60 = 10 x n;
b. find the unknown in a money equation using multiplication and
division based upon the facts and addition and subtraction with
values through $10 (2.4.K1d) ($), e.g., 8 quarters + 10 dimes = y
dollars;
c. find the unknown in a time equation involving whole minutes,
hours, days, and weeks with values through 200 (2.4.K1a), e.g.,
180 minutes = y hours.
3. compares two whole numbers from 0 through 10,000 using the
equality and inequality symbols (=, ≠, <, >) and their corresponding
meanings (is equal to, is not equal to, is less than, is greater than)
(2.4.K1b) ($).
4. reads and writes whole number equations and inequalities using
mathematical vocabulary and notation, e.g., 15 = 3 x 5 is the same
as fifteen equals three times five or 4,564 > 1,000 is the same as
four thousand, five hundred sixty-four is greater than one thousand.
Fourth Grade Application Indicators
The student…
1. represents real-world problems using variables and symbols with
unknown whole number quantities from 0 through 1,000 (2.4.A1a)
($), e.g., How many weeks in twenty-eight days? can be
represented by
n x 7 = 28 or n = 28 ÷ 7.
2. generates one-step equations to solve real-world problems with
one unknown (represented by a variable or symbol) and a whole
number solution that (2.4.A1a) ($):
a. add or subtract whole numbers from 0 through 1,000; e.g.,
Homer, Kansas has 832 nonfiction books in its library. Homer,
Idaho has 652 nonfiction books in its library. How many fewer
books nonfiction books are in Homer, Idaho’s library?
832 - 652 = B;
b. multiply or divide using the basic facts, e.g., Tom has a sticker
book and each page holds 5 stickers. If the same number of
stickers is placed on each page, the book will hold 30 stickers.
How many pages are in his book? This is represented by 5 x S
= 30 or 30 ÷ 5 = S.
3. generates (2.4.A1a) ($):
a. real-world problems with one operation to match a given
addition, subtraction, multiplication, or division equation using
whole numbers through 99, e.g., given 12 x 3 = Y, the student
writes: I was sick for 3 days, when I got back I had 3 pages of
homework. There are 12 problems on each page. How many
total problems must I work?
b. number comparison statements using equality and inequality
symbols (=, <, >) with whole numbers, measurement, and
money, e.g., 1 ft < 15 in or 10 quarters > $2.
4-133
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Understanding the concept of variable is fundamental to algebra. Students use various symbols, including letters and
geometric shapes, to represent unknown quantities that both do and do not vary. Quantities that are not given and do not vary are often referred to
as unknowns or missing elements when they appear in equations, e.g., 2 + 4 = Δ or 3 + s = 7. Various symbols or letters should be used
interchangeably in equations.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
4-134
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
FOURTH GRADE
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 3: Functions – The student recognizes and describes whole number relationships including the use of
concrete objects in a variety of situations.
Fourth Grade Knowledge Base Indicators
The student…
1. states mathematical relationships between whole numbers from 0
through 1,000 using various methods including mental math, paper
and pencil, concrete materials, and appropriate technology (2.4.K1a)
($).
2. ▲ finds the values, determines the rule, and states the rule using
symbolic notation with one operation of whole numbers from 0
through 200 using a horizontal or vertical function table (input/output
machine, T-table) (2.4.K1e), e.g., using the function table, find the
rule, the rule is N • 4.
N
?
1
4
5
20
2
8
3
?
4
?
?
24
3. generalizes numerical patterns using whole numbers from 0 through
200 with one operation by stating the rule using words, e.g., if the
pattern is 46, 68,90, 112, 134, …; in words, the rule is add 22 to the
number before.
4. uses a function table (input/output machine, T-table) to identify, plot,
and label the ordered pairs in the first quadrant of a coordinate plane
(2.4.K1a,e).
Fourth Grade Application Indicators
The student…
1. ▲ represents and describes mathematical relationships between
whole numbers from 0 through 1,000 using concrete objects,
pictures, written descriptions, symbols, equations, tables, and
graphs (2.4.A1a) ($).
2. finds the rule, states the rule, and extends numerical patterns using
real-world applications using whole numbers from 0 through 200
(2.4.A1a,e), e.g., the teacher must order supplies for field day. For
every 12 students, one red rubber ball is needed. If 6 balls are
ordered, how many students will be able to play? A solution using a
function table might be:
Number of
Students
12
24
36
48
60
72
N
Number of Balls
1
2
3
4
5
6
N ÷ 12
The rule is divide the number of students by 12 or for each group of
12 students, another ball is added. Other solutions might be using a
pattern to count by 12 six times – 12, 24, 36, 48, 60, 72 or to skip count
by 12 for each ball ordered.
4-135
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Functions are relationships or rules in which each member of one set is paired with one, and only one, member of another set (an
ordered pair). The concept of function can be introduced using function machines. Any number put in the machine will be changed according to
some rule. A record of inputs and corresponding outputs can be maintained in a two-column format. Function tables, input/output machines, and
T-tables may be used interchangeably and serve the same purpose.
Function concepts should be developed from growing patterns. Each term in a number sequence is related to its position in the sequence – the
functional relationship. The pattern – 4, 7, 10, 13, 16, 19, and so on – is an arithmetic sequence with a difference of 3. The pattern could be
described as add 3 meaning that 3 must be added to the previous term to find the next. This pattern is explained by using the recursive definition
for a sequence. The recursive definition for a sequence is a statement or a set of statements that explains how each successive term in the
sequence is obtained from the previous term(s).
In the pattern 1, 4, 9, 16, 25, …, 225; there is no common difference. This sequence is not arithmetic or geometric (no common ratio between
geometric terms). Neither is it a combination of the two; however, there is a pattern and the missing terms between 25 and 225 can be found. To
find the term value, square the number of the term. The next missing terms would be 36, 49, 64, 81, 100, 121, and 144. This pattern is explained
by using the explicit formula for a sequence. The explicit formula for a sequence defines a rule for finding each term in the number sequence
related to its position in the sequence. In other words, to find the term value, square the number of the term – the 5th term is 52, the 8th term is 82, …
Patterns themselves are not explicit or recursive. The RULE for the pattern can be expressed explicitly or recursively and MOST patterns can be
explained using either format especially IF that pattern reflects either an arithmetic sequence or geometric sequence.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
4-136
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
FOURTH GRADE
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 4: Models – The student develops and uses mathematical models including the use of concrete objects
to
represent and explain mathematical relationships in a variety of situations.
Fourth Grade Knowledge Base Indicators
The student…
1. knows, explains, and uses mathematical models to represent
mathematical concepts, procedures, and relationships. Mathematical
models include:
a. process models (concrete objects, pictures, diagrams, number
lines, hundred charts, measurement tools, multiplication arrays,
division sets, or coordinate planes/grids) to model computational
procedures, mathematical relationships, and equations (1.1.K1a,
1.1.K2a, 1.2.K1, 1.2.K4-5, 1.3.K1-4, 1.4.K1-2, 1.4.K3a-b,
1.4.K3e, 1.4.K4, 1.4.K6-7, 2.1.K1, 2.1.K.1a-b, 2.1.K2d-g, 2.1.K3,
2.1.K4a, 2.2.K1, 2.2.K2a, 2.2.K3-4, 2.3.K1, 2.3.K4, 3.2.K1-4,
3.3.K1-2, 3.4.K1-4, 4.2.K3) ($);
b. place value models (place value mats, hundred charts, base ten
blocks, or unifix cubes) to compare, order, and represent
numerical quantities and to model computational procedures
(1.1.K1a, 1.1.K2a, 1.2.K1-3, 1.3.K1-2, 1.4.K3a-b, 1.4.K3e,
2.2.K4) ($);
c. fraction and mixed number models (fraction strips or pattern
blocks) and decimal models (base ten blocks or coins) to
compare, order, and represent numerical quantities (1.1.K1b-c,
1.1.K2b-c, 1.2.K2, 1.3.K1-2, 1.4.K1f) ($);
d. money models (base ten blocks or coins) to compare, order, and
represent numerical quantities (1.1.K1c, 1.2.K1c, 1.3.K1-2,
1.4.K3a, 1.4.K3a, 1.4.K3c-d, 1.4.K3g, 2.1.K2e, 2.2.K2b) ($);
e. function tables (input/output machines, T-tables) to model
numerical and algebraic relationships (2.1.K4b, 2.3.K2, 2.3.K4,
3.4.K4) ($);
Fourth Grade Application Indicators
The student…
1. recognizes that various mathematical models can be used to
represent the same problem situation. Mathematical models
include:
a. process models (concrete objects, pictures, diagrams, number
lines, coordinate planes/grids, hundred charts, measurement
tools, multiplication arrays, or division sets) to model
computational procedures, mathematical relationships, and
problem situations (1.1.A1, 1.1.A2a, 1.2.A1-3, 1.3.A1-4,
1.4.A1, 2.1.A1a-b, 2.1.A1d-f, 2.1.A2, 2.2.A1-3, 2.3.A1-2,
3.2.A1a-g, 3.2.A2-3, 3.3.A1-2, 3.4.A1-2, 4.2.A2) ($);
b. place value models (place value mats, hundred charts, base
ten blocks, or unfix cubes) to model problem situations (1.1A1,
1.1.A2a, 1.2.A1-3, 1.3.A2, 1.4.A1) ($);
c. fraction and mixed number models (fraction strips or pattern
blocks) and decimal models (base ten blocks or coins) to
compare, order, and represent numerical quantities (1.1.A1b,
1.1.A2b-c, 1.2.A2-3, 1.3.A2, 1.4.A1d-e) ($);
d. money models (base ten blocks or coins) to compare, order,
and represent numerical quantities (1.1.A1b, 1.1.A2c, 1.2.A1,
1.2.A3, 1.3.A1, 1.4.A1a, 1.4.A1c-e, 2.1.A1e) ($);
e. function tables (input/output machines, T-tables) to model
numerical and algebraic relationships (2.3.A2) ($);
f. two-dimensional geometric models (geoboards, dot paper,
pattern blocks, or tangrams) to model perimeter, area, and
properties of geometric shapes and three-dimensional
geometric models (solids) and real-world objects to compare
size and to model properties of geometric shapes (2.1.A1c,
3.1.A1-2, 3.2.A1h, 3.3.A3);
4-137
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
f.
two-dimensional geometric models (geoboards, dot paper,
pattern blocks, or tangrams) to model perimeter, area, and
properties of geometric shapes and three-dimensional geometric
models (solids) and real-world objects to compare size and to
model properties of geometric shapes (2.1.K2c, 2.1.K1e, 3.1.K16, 3.2.K5, 3.3.K3);
g. two-dimensional geometric models (spinners), three-dimensional
models (number cubes), and process models (concrete objects)
to model probability (4.1.K1-3) ($);
h. graphs using concrete objects, pictographs, frequency tables,
horizontal and vertical bar graphs, line graphs, circle graphs,
Venn diagrams, line plots, charts, and tables to organize and
display data (4.1.K2, 4.2.K1-2) ($);
i. Venn diagrams to sort data and show relationships
g. two-dimensional geometric models (spinners), threedimensional geometric models (number cubes), and process
models (concrete objects) to model probability (4.1.A1-3) ($);
h. graphs using concrete objects, pictographs, frequency tables,
horizontal and vertical bar graphs, line graphs, Venn
diagrams, line plots, charts, and tables to organize, display,
explain, and interpret data (4.1.A2, 4.2.A1, 4.2.A3-4) ($);
i. Venn diagrams to sort data and show relationships.
2. selects a mathematical model and explains why some
mathematical models are more useful than other mathematical
models in certain situations.
(1.2.K2).
2. creates a mathematical model to show the relationship between two
or more things, e.g., using pattern blocks, a whole (1) can be
represented as
a
two
three
six
(1/1) or
(2/2) or
(3/3) or
(6
(6/6).
4-138
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: For assessment purposes, the mathematical modeling process appropriate to the indicator may be included as part of the item
being assessed.
The mathematical modeling process involves:
a. selecting key features and relationships within the real-world situation and representing these concepts in mathematical terms through
some sort of mathematical model,
b. performing manipulations and mathematical procedures within the mathematical model,
c. interpreting the results of the manipulations within the mathematical model,
d. using these results to make inferences about the original real-world situation.
The use of mathematical models is necessary for conceptual understanding. The ways in which mathematical ideas are represented is
fundamental to how students understand and use those ideas. As students begin to use multiple representations of the same situation, students
begin to develop an understanding of the advantages and disadvantages of the various representations/models.
Many mathematical models are listed in this benchmark. The indicator lists some of the mathematical models that could be used to teach a
concept. Each indicator in this benchmark is linked to other indicators in other benchmarks; those indicators are identified in the parentheses. For
example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1
(Number Sense), and Knowledge Indicator 3. In addition, the indicator in the other benchmarks identifies, in parentheses, the Models’ indicator.
For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models).
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
4-139
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
FOURTH GRADE
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 1: Geometric Figures and Their Properties – The student recognizes geometric shapes and investigates
their properties including the use of concrete objects in a variety of situations.
Fourth Grade Knowledge Base Indicators
The student…
1. recognizes and investigates properties of plane figures (circles,
squares, rectangles, triangles, ellipses, rhombi, octagons, hexagons,
pentagons) using concrete objects, drawings, and appropriate
technology (2.4.K1f).
2. recognizes, draws, and describes plane figures (circles, squares,
rectangles, triangles, ellipses, rhombi, octagons, hexagons,
pentagons) (2.4.K1f).
3. describes the solids (cubes, rectangular prisms, cylinders, cones,
spheres, triangular prisms) using the terms faces, edges, and vertices
(corners) (2.4.K1f).
4. recognizes and describes the square, triangle, rhombus, hexagon,
parallelogram, and trapezoid from a pattern block set (2.4.K1f).
5. recognizes (2.4.k1f):
a. squares, rectangles, rhombi, parallelograms, trapezoids as
special quadrilaterals;
b. similar and congruent figures;
c. points, lines (intersecting, parallel, perpendicular), line segments,
and rays.
6. determines if geometric shapes and real-world objects contain line(s)
of symmetry and draws the line(s) of symmetry if the line(s) exist(s)
(2.4.K1f).
Fourth Grade Application Indicators
The student…
1. solves real-world problems by applying the properties of (2.4.A1f):
a. plane figures (circles, squares, rectangles, triangles, ellipses,
rhombi, parallelograms, hexagons) and lines of symmetry,
e.g., print your name or the school’s name in all capital letters.
Identify the lines of symmetry in each letter.
b. solids (cubes, rectangular prisms, cylinders, cones, spheres),
e.g., you want to design something to store school supplies.
Which of the solids could you use for storage? Why did you
select that solid?
2. ▲ ■ identifies the plane figures (circles, squares, rectangles,
triangles, ellipses, rhombi, octagons, hexagons, pentagons,
trapezoids) used to form a composite figure (2.4.A1f).
4-140
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Geometry is the study of shapes, their properties, and their relationships to other shapes. Symbols and numbers are used to
describe their properties and their relationships to other shapes. The fundamental concepts in geometry are point (no dimension), line (onedimensional), plane (two-dimensional), and space (three-dimensional). Plane figures are referred to as two-dimensional and solids are referred to
as three-dimensional.
From the Mathematics Dictionary and Handbook (Nichols Schwartz Publishing, 1999), property as a mathematical term means a characteristic (an
attribute) of a number, geometric shape, mathematical operation, equation, or inequality. To give an example:
 Property of a number: 8 is divisible by 2.
 Property of a geometric shape: Each of the four sides of a square is of the same length.
 Property of an operation: Addition is commutative. For all numbers x and y, x + y = y + x.

Property of an equation: For all numbers a, b, and c, if a = b, then a + c = b + c.

Property of an inequality: For all numbers a, b, and c, if a > b, then a – c > b – c.
Pattern blocks are a collection of six geometric shapes in six colors. Each set contains 250 pieces – 50 green triangles, 25 orange squares, 50
blue rhombi, 50 tan rhombi, 50 red trapezoids, and 25 yellow hexagons. The blue rhombus and the tan rhombus also are parallelograms, and the
orange square is a parallelogram. The blocks are designed so that their sides are all the same length with the exception that the trapezoid has one
side twice as long. This feature allows the blocks to be nested together and encourages the exploration of relationships among the shapes.
Activities with pattern blocks help students explore patterns, functions, fractions, congruence, similarity, symmetry, perimeter, area, and graphing.
A pattern block template can be found in the Appendix.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
4-141
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
FOURTH GRADE
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 2: Measurement and Estimation – The student estimates and measures using standard and nonstandard
units of measure including the use of concrete objects in a variety of situations.
Fourth Grade Knowledge Base Indicators
The student…
1. uses whole number approximations (estimations) for length, width,
weight, volume, temperature, time, perimeter, and area using
standard and nonstandard units of measure (2.4.K1a) ($).
2. ▲ selects, explains the selection of, and uses measurement tools,
units of measure, and degree of accuracy appropriate for a given
situation to measure (2.4.K1a) ($):
a. length, width, and height to the nearest fourth of an inch or to
the nearest centimeter;
b. volume to the nearest cup, pint, quart, or gallon; to the nearest
liter; or to the nearest whole unit of a nonstandard unit;
c. weight to the nearest ounce or pound or to the nearest whole
unit of a nonstandard unit of measure;
d. temperature to the nearest degree;
e. time including elapsed time.
3. states:
a. the number of weeks in a year;
b. the number of ounces in a pound;
c. the number of milliliters in a liter, grams in a kilogram, and
meters in a kilometer;
d. the number of items in a dozen.
4. converts (2.4.K1a):
a. within the customary system: inches and feet, feet and yards,
inches and yards, cups and pints, pints and quarts, quarts and
gallons;
b. within the metric system: centimeters and meters.
5. finds(2.4.K1f):
a. the perimeter of two-dimensional figures given the measures of
all the sides.
b. the area of squares and rectangles using concrete objects.
Fourth Grade Application Indicators
The student…
1. solves real-world problems by applying appropriate measurements:
a. length to the nearest fourth of an inch (2.4.A1a), e.g., how much
longer is the math textbook than the science textbook?
b. length to the nearest centimeter (2.4.A1a), e.g., a new pencil is
about how many centimeters long?
c. temperature to the nearest degree (2.4.A1a), e.g., what would
the temperature outside be if it was a good day for sledding?
d. weight to the nearest whole unit (pounds, grams, nonstandard
unit) (2.4.A1a), e.g., Brendan went to the store and bought 2
packages of hamburger for a meatloaf. One of the hamburger
packages weighed 1 lb. and 8 ozs. The other packages
weighed 1 lb. and 7 ozs. What is the combined weight (to the
nearest pound) of the two packages of hamburger?
e. time including elapsed time (2.4.A1a), e.g., Joy went to the mall
at 10:00 a.m. She shopped until 4:15 p.m. How long did she
shop at the mall?
f. months in a year (2.4.A1a), e.g., if it takes 208 weeks to get a
college degree, and Susan has completed one year, how many
more weeks does she have to complete to get her degree?
g. minutes in an hour (2.4.A1a), e.g., Bob has spent 240 minutes
working on a project for Science. How many hours has he
worked on the project?
h. perimeter of squares, rectangles, and triangles (2.4.A1f), e.g., a
triangle has 3 equal sides of 32 inches. What is the perimeter of
the triangle?
2. ▲ estimates to check whether or not measurements and
calculations for length, width, weight, volume, temperature, time, and
perimeter in real-world problems are reasonable (2.4.A1a) ($), e.g.,
which is the most reasonable weight for your scissors – 2 ounces, 2
pounds, 20 ounces, or 20 pounds? A teacher measures one side of
4-142
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
a square desktop at 2 feet. Which of the following perimeters is
reasonable for the desktop – 2 feet, 4 square feet, 6 square feet, or
8 feet?
3. adjusts original measurement or estimation for length, width, weight,
volume, temperature, time, and perimeter in real-world problems
based on additional information (a frame of reference) (2.4.A1a) ($),
e.g., your class has a large jar and a small jar. You estimate it will
take 5 small jars of liquid to fill the large jar. After you pour the
contents of 2 small jars in, the large jar is more that half full. Should
you need to adjust your estimate?
Teacher Notes: The term geometry comes from two Greek words meaning “earth measure.” Measurement provides the tools required to apply
geometric concepts in the real-world. Estimation in measurement is defined as making guesses as to the exact measurement of an object
without using any type of measurement tool. Estimation helps students develop a relationship between the different sizes of units of measure. It
helps students develop basic properties of measurement and it gives students a tool to determine whether a given measurement is reasonable.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
4-143
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
FOURTH GRADE
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 3: Transformational Geometry – The student recognizes and performs one transformation on simple
shapes or concrete objects in a variety of situations.
Fourth Grade Knowledge Base Indicators
The student…
1. describes a transformation using cardinal points or positional
directions (2.4.K1a), e.g., go north three blocks and then west four
blocks or move the triangle three units to the right and two units up.
2. ▲ ■ recognizes, performs, and describes one transformation
(reflection/flip, rotation/turn, translation/slide) on a two-dimensional
figure or concrete object (2.4.K1a).
3. recognizes three-dimensional figures (rectangular prisms, cylinders)
and concrete objects from various perspectives (top, bottom, sides,
corners) (2.4.K1f).
Fourth Grade Application Indicators
The student…
1. recognizes real-world transformations (reflection/flip, rotation/turn,
translation/slide) (2.4.A1a).
2. gives and uses cardinal points or positional directions to move from
one location to another on a map or grid (2.4.A1a).
3. describes the properties of geometric shapes or concrete objects
that stay the same and the properties that change when a
transformation is performed (2.4.A1f).
Teacher Notes: Transformational geometry is another way to investigate geometric figures by moving every point in a plane figure to a new
location. To help students form images of shapes through different transformations, students can use concrete objects, figures drawn on graph
paper, mirrors or other reflective surfaces, or appropriate technology.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
4-144
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
FOURTH GRADE
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 4:
Geometry From An Algebraic Perspective – The student relates geometric concepts to a number line
and the first quadrant of a coordinate plane in a variety of situations.
Fourth Grade Knowledge Base Indicators
The student…
1. uses a number line (horizontal/vertical) to model whole number
multiplication facts from 1 x 1 through 12 x 12 and corresponding
division facts (2.4.K1a).
2. uses points in the first quadrant of a coordinate plane (coordinate
grid) to identify locations (2.4.K1a).
3. ▲ ■ identifies and plots points as whole number ordered pairs in the
first quadrant of a coordinate plane (coordinate grid) (2.4.K1a).
4. organizes whole number data using a T-table and plots the ordered
pairs in the first quadrant of a coordinate plane (coordinate grid)
(2.4.K1a,e).
Fourth Grade Application Indicators
The student…
1. solves real-world problems that involve distance and location using
coordinate planes (coordinate grids) and map grids with positive
whole number and letter coordinates (2.4.A1a), e.g., identifying
locations and giving and following directions to move from one
location to another.
2. solves real-world problems by plotting whole number ordered pairs
in the first quadrant of a coordinate plane (coordinate grid)
(2.4.A1a) ($), e.g., given that each movie ticket cost $5, the student
graphs the number of tickets bought and the total cost of tickets to
attend a movie.
4-145
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: A number line (a mathematical model) is a diagram that represents numbers with equal distances marked off as points on a line,
and is an example of one-to-one correspondence (a relation). A number line can be used as a visual representation of numbers and operations.
In addition, a number line used horizontally and vertically is a precursor to the coordinate plane; and the distance between two numbers on a
number line is a precursor to absolute value.
A coordinate plane (coordinate grid) consists of a horizontal number line called the x-axis and a vertical number line called the y-axis. These two
lines intersect at a point called the origin. The x-axis and the y-axis divide the plane into four sections called quadrants. Any point on the
coordinate plane can be named with two numbers called coordinates. The first number is the x-coordinate. The second number is the ycoordinate. Since the pair is always named in order (first x, then y), it is called an ordered pair.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
4-146
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 4: Data
FOURTH GRADE
Standard 4: Data – The student uses concepts and procedures of data analysis in a variety of situations.
Benchmark 1: Probability – The student applies the concepts of probability to draw conclusions and to make predictions and
decisions including the use of concrete objects in a variety of situations.
Fourth Grade Knowledge Base Indicators
The student…
1. recognizes that the probability of an impossible event is zero and
that the probability of a certain event is one (2.4.K1g) ($).
2. lists all possible outcomes of a simple event in an experiment or
simulation including the use of concrete objects (2.4.K1g-h).
3. recognizes and states the probability of a simple event in an
experiment or simulation (2.4.K1g), e.g., when a coin is flipped, the
probability of landing heads up is ½ and the probability of landing
tails up is ½. This can be read as one out of two or one half.
Fourth Grade Application Indicators
The student…
1. makes predictions about a simple event in an experiment or
simulation; conducts an experiment or simulation including the use
of concrete objects; records the results in a chart, table, or graph;
and uses the results to draw conclusions about the event
(2.4.A1g-h).
2. uses the results from a completed experiment or simulation of a
simple event to make predictions in a variety of real-world problems
(2.4.A1g-h), e.g., the manufacturer of Crunchy Flakes puts a prize
in 20 out of every 100 boxes. What is the probability that a shopper
will find a prize in a box of Crunchy Flakes, if they purchase 10
boxes?
3. compares what should happen (theoretical probability/expected
results) with what did happen (empirical probability/experimental
results) in an experiment or simulation with a simple event
(2.4.A1g).
4-147
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Ideas from probability reinforce concepts in the other Standards, especially Number and Computation and Geometry. Students
need to develop an intuitive concept of chance – whether or not something is unlikely or likely to happen. Probability experiences should be
addressed through the use of concrete objects, coins, and geometric models (spinners, number cubes, or dartboards). Probabilities are ratios,
expressed as fractions, decimals, or percents, determined by considering results or outcomes of experiments. Some examples of uses of
probability in every day life include: There is a 50% chance of rain today. What is the probability that the team will win every game?
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories) Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
4-148
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 4: Data
FOURTH GRADE
Standard 4: Data – The student uses concepts and procedures of data analysis in a variety of situations.
Benchmark 2: Statistics – The student collects, organizes, displays, explains, and interprets numerical (whole numbers) and
non-numerical data sets including the use of concrete objects in a variety of situations.
Fourth Grade Knowledge Base Indicators
The student…
1. ▲ ■ organizes, displays, and reads numerical (quantitative) and
non-numerical (qualitative) data in a clear, organized, and accurate
manner including a title, labels, categories, and whole number
intervals using these data displays (2.4.K1h) ($):
a. graphs using concrete objects, (for testing, does not have to use
concrete objects in items);
b. pictographs with a symbol or picture representing one, two, five,
ten, twenty-five, or one-hundred including partial symbols when
the symbol represents an even amount;
c. frequency tables (tally marks);
d. horizontal and vertical bar graphs;
e. Venn diagrams or other pictorial displays, e.g., glyphs;
f. line plots;
g. charts and tables;
h. line graphs;
i. circle graphs.
2. collects data using different techniques (observations, polls, surveys,
interviews, or random sampling) and explains the results (2.4.K1h)
($).
3. identifies, explains, and calculates or finds these statistical
measures of a data set with less than ten whole number data points
using whole numbers from 0 through 1,000 (2.4.K1a) ($):
a. minimum and maximum values,
b. range,
c. mode,
d. median when data set has an odd number of data points,
e. mean when data set has a whole number mean.
Fourth Grade Application Indicators
The student…
1. interprets and uses data to make reasonable inferences and
predictions, answer questions, and make decisions from these data
displays (2.4.A1h) ($):
a. graphs using concrete objects;
b. pictographs with a symbol or picture representing one, two,
five, ten, twenty-five, or one-hundred including partial symbols
when the symbol represents an even amount;
c. frequency tables (tally marks);
d. horizontal and vertical bar graphs;
e. Venn diagrams or other pictorial displays;
f. line plots;
g. charts and tables;
h. line graphs.
2. ▲ uses these statistical measures of a data set using whole
numbers from 0 through 1,000 with less than ten whole number
data points to make reasonable inferences and predictions, answer
questions, and make decisions (2.4.A1a) ($):
a. minimum and maximum values,
b. range,
c. mode,
d. median when the data set has an odd number of data points,
e. mean when the data set has a whole number mean.
3. recognizes that the same data set can be displayed in various
formats including the use of concrete objects (2.4.A1h) ($).
4. recognizes and explains the effects of scale and interval changes
on graphs of whole number data sets (2.4.A1h).
4-149
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Graphs (data displays) are pictorial representations of mathematical relationships, are used to tell a story, and are an important
part of statistics. When a graph is made, the axes and the scale (numbers running along a side of the graph) are chosen for a reason. The
difference between numbers from one grid line to another is the interval. The interval will depend on the lowest and highest values in the data set.
Emphasizing the importance of using equal-sized pictures or intervals is critical to ensuring that the data display is accurate.
Graphs take many forms:
– bar graphs and pictographs compare discrete data,
– frequency tables show how many times a certain piece of data occurs,
– circle graphs (pie charts) model parts of a whole,
– line graphs show change over time,
– Venn diagrams show relationships between sets of objects, and
– line plots show frequency of data on a number line.
An important aspect of data is its center. The measures of central tendency (averages) of a data set are mean, median, and mode. Conceptual
understanding of mean, median, and mode is developed through the use of concrete objects that represent the data values.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
4-150
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
FIFTH GRADE
Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 1: Number Sense – The student demonstrates number sense for integers, fractions, decimals, and
money in a variety of situations.
Fifth Grade Knowledge Base Indicators
Fifth Grade Application Indicators
The student…
1. ▲N knows, explains, and uses equivalent representations for ($):
a. whole numbers from 0 through 1,000,000 (2.4.K1a-b);
b. fractions greater than or equal to zero (including mixed
numbers) (2.4.K1c);
c. decimals greater than or equal to zero through hundredths
place and when used as monetary amounts (2.4.K1c).
2. compares and orders (2.4.K1a-c) ($) :
a. integers,
b. fractions greater than or equal to zero (including mixed
numbers),
c. decimals greater than or equal to zero through hundredths
place.
3. explains the numerical relationships (relative magnitude) between
whole numbers, fractions greater than or equal to zero (including
mixed numbers), and decimals greater than or equal to zero through
hundredths place (2.4.K1a-c).
4. knows equivalent percents and decimals for one whole, one-half,
one-fourth, three-fourths, and one tenth through nine tenths
(2.4.K1c), e.g., 1 = 100% = 1.0, 3/4 = 75% = .75, 3/10 = 30% = .3.
5. identifies integers and gives real-world problems where integers are
used (2.4.K1a), e.g., making a T-table of the temperature each hour
over a twelve hour period in which the temperature at the beginning
is 10 degrees and then decreases 2 degrees per hour.
The student…
1. solves real-world problems using equivalent representations and
concrete objects to ($):
a. compare and order (2.4.A1a-d) –
i. whole numbers from 0 through 1,000,000; e.g., using base
ten blocks, represent the attendance at the circus over a
three day stay; then represent the numbers using digits and
compare and order in different ways;
ii. fractions greater than or equal to zero (including mixed
numbers), e.g., Frank ate 2 ½ pizzas, Tara ate 9/4 of the
pizza. Frank says he ate more. Is he correct? Use a model
to explain. With drawings and shadings, student shows
amount of pizza eaten by Frank and the amount eaten by
Tara.
iii. decimals greater than or equal to zero to hundredths place,
e.g., uses decimal squares, money (dimes as tenths,
pennies as hundredths), the correct amount of hundred
chart filled in, or a number line to show that .42 is less than
.59.
iv. integers, e.g., plot winter temperature for a very cold region
for a week (use Internet data); represent on a thermometer,
number line, and with integers;
b. add and subtract whole numbers from 0 through 100,000 and
decimals when used as monetary amounts (2.4.A1a,c), e.g.,
use real money to show at least 2 ways to represent $846.00,
then subtract the cost of a new computer setup;
5-151
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
c.
multiply through a two-digit whole number by a two-digit
whole number (2.4.A1a-b), e.g., George charges $23 for
mowing a lawn. How much will he make after he mows 3
lawns? Represent the $23 with money models - 2 $10
bills and 3 $1 bills and repeat that 3 times or represent
the $23 using base ten blocks or 23 x 3 or 23 + 23 + 23.;
d. divide through a four-digit whole number by a two-digit whole
number (2.4.A1a-b), e.g., the Boy Scout troop collected cans
and held bake sales for a year and earned $492.60. The money
will be divided evenly among the 12 troop members to buy new
uniforms. Represent each boy’s share of the money at least 2
ways - traditional division; use 4 hundreds, 9 tens, 2 ones, and
6 dimes to act out the situation; or use base ten blocks to act it
out.
2. determines whether or not solutions to real-world problems that
involve the following are reasonable ($):
a. whole numbers from 0 through 100,000 (2.4.A1a-b), e.g., the
football is placed on your own 10-yard line with 90 yards to go
for a touchdown. After the first down, your team gains 7 yards.
On the second down, your team loses 4 yards. Is it reasonable
for the football to be placed on the 3-yard line for the beginning
of the third down?
b. fractions greater than or equal to zero (including mixed
numbers) (2.4.A1c), e.g., explain if it is reasonable to say that a
dog is ½ boxer, ¼ bulldog, ¼ collie, and ¼ rotweiler;
3. decimals greater than or equal to zero through hundredths place
(2.4.A1c), e.g., five people a ate pizza. Is it reasonable to say that
each person ate .3 of the pizza?
5-152
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Number sense refers to one’s ability to reason with numbers and to work with numbers in a flexible way. The ability to compute
mentally, to estimate based on understanding of number relationships and magnitudes, and to judge reasonableness of answers are all involved in
number sense. When we say that someone has good number sense, we mean that he or she possesses a variety of abilities and understandings
that include an awareness of the relationships between numbers, an ability to represent numbers in a variety of ways, a knowledge of the effects of
operations, and an ability to interpret and use numbers in real-world counting and measurement situations. Such a person predicts with some
accuracy the result of an operation and consistently chooses appropriate measurement units. This “friendliness with numbers” goes far beyond
mere memorization of computational algorithms and number facts; it implies an ability to use numbers flexibly, to choose the most appropriate
representation of a number for a given circumstance, and to recognize when operations have been correctly performed. (Number Sense and
Operations: Addenda Series, Grades K-6, NCTM, 1993)
At this grade level, rational numbers include positive numbers, large numbers (one million), and small numbers (one-hundredth).
Relative magnitude refers to the size relationship one number has with another – is it much larger, much smaller, close, or about the
same? For example, using the numbers 219, 264, and 457, answer questions such as –
 Which two are closest? Why?
 Which is closest to 300? To 250?
 About how far apart are 219 and 500? 5,000?
 If these are ‘big numbers,’ what are small numbers? Numbers about the same? Numbers that make these seem
small?
(Elementary and Middle School Mathematics, John A. Van de Walle, Addison Wesley Longman, Inc., 1998)
Mathematical models such as concrete objects, pictures, diagrams, Venn diagrams, number lines, hundred charts, base ten blocks, or factor trees
are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to
represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2
(Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the
mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other
indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3
referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
5-153
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
FIFTH GRADE
Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 2: Number Systems and Their Properties – The student demonstrates an understanding of the whole
number system; recognizes, uses, and explains the concepts of properties as they relate to the whole number
system; and extends these properties to integers, fractions (including mixed numbers), and decimals.
Fifth Grade Knowledge Base Indicators
The student…
1. classifies subsets of numbers as integers, whole number, fractions
(including mixed numbers), or decimals (2.4.K1a-c, 2.4.K1k).
2. identifies prime and composite numbers from 0 through 50.
3. uses the concepts of these properties with whole numbers,
integers, fractions greater than or equal to zero (including mixed
numbers), and decimals greater than or equal to zero and
demonstrates their meaning including the use of concrete objects
(2.4.K1a) ($):
a. commutative properties of addition and multiplication, e.g.,
43 + 34 = 34 + 43 and 12 x 15 = 15 x 12;
b. associative properties of addition and multiplication, e.g.,
4 + (3 + 5) = (4 + 3) + 5;
c. zero property of addition (additive identity) and property of one
for multiplication (multiplicative identity), e.g., 342 + 0 = 342 and
576 x 1 = 576;
d. symmetric property of equality, e.g., 35 = 11 + 24 is the same
as 11 + 24 = 35;
e. zero property of multiplication, e.g., 438,223 x 0 = 0;
f. distributive property, e.g., 7(3 + 5) = 7(3) + 7(5);
g. substitution property, e.g., if a = 3 and a = b, then b = 3.
4. recognizes Roman Numerals that are used for dates, on clock
faces, and in outlines.
5. recognizes the need for integers, e.g., with temperature, below zero
is negative and above zero is positive; in finances, money in your
pocket is positive and money owed someone is negative.
Fifth Grade Application Indicators
The student…
1. solves real-world problems with whole numbers from 0 through
100,000 and decimals through hundredths using place value
models; money; and the concepts of these properties to explain
reasoning (2.4.A1a-c,e) ($):
a. commutative and associative properties of addition and
multiplication, e.g., lay out a $5, $10 and $20 bills. Ask for the
total of the money. The student says: Because you can add in
any order (commutative) I can rearrange the money and count
$20, $10 and $5 for $20 + $10 + $5 or Lay out 4 $5
bills. The student is asked the amount.
The student says: I don’t know what 4 x
5 is, but I know 5 x 4 is $20 and since
multiplication can be done in any order,
then it is $20.
b. zero property of addition, e.g., have students lay out 6
dimes. Tell them to add zero. How many dimes? 6 + 0
=6
c.
property of one for multiplication, e.g., there are 24 students in
our class. I want one math book per student, so I compute 24 x
1= 24. Multiplying times 1 does not change the product because
it is one group of 24.
d. symmetric property of equality, e.g., Pat knows he has $56. He
has 2 twenty-dollar bills in his wallet. How much does he have
at home in his bank? This can be represented as –
56 = (2 x 20) + ▎ , so ( 2 x 20) + ▎ = 56
5-154
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
56 = 40 + ▎ , so 40 + ▎ = 56
56 = 20 + 20 + 16, so 20 + 20 + 16 = 56.
e. zero property of multiplication, e.g., in science, you are
observing a snail. The snail does not move over a 4-hour
period. To figure its total movement, you say 4 x 0 = 0.
f. distributive property, e.g., Juan has 7 quarters and 7 dimes.
What is the total amount of money he has? 7($.25 + $.10) =
7($.25) + 7($.10).
3. performs various computational procedures with whole numbers
from 0 through 100,000 using the concepts of these properties;
extends these properties to fractions greater than or equal to zero
(including mixed numbers) and decimals greater than or equal to
zero through hundredths place; and explains how the properties
were used (2.4.A1a-c,e):
a. commutative and associative properties of addition and
multiplication, e.g., given 4.2 x 10, the student says: I know that
it is 42 because I know that 10 x 4.2 = 42, since you can
multiply in any order and get the same answer. or The student
says I don’t know what 9 + 8 is, but I know my doubles of 8 + 8,
so I make the 9 into 1 + 8 and after adding 8 and 8, I add 1
more;
b. zero property of addition, e.g., given 47 + 917 + 0, the student
says: I know that the answer is 964 because adding 0 does not
change the answer (sum);
c. property of one for multiplication, e.g., $9.62 x 1. The student
says: I know the product is still $9.62 because multiplication by
one never changes the product. It is like if I had $9.62 in one
pile, I would just have $9.62;
d. symmetric property of equality, e.g., given ▎ = ½ + ¼, the
student says: That is the same as ½ + ¼ because I must
make both sides equal;
e. zero property of multiplication e.g., given .7 x 0, the student
says: I know the answer (product) is zero because no matter
how many factors you have, multiplying by 0, the product is 0;
f. distributive property, e.g., given 4 x 614, the student can explain
that you can solve it (in your head?) by computing 4(600) +
5-155
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
4(10) + 4(4), which is 2,400 + 40 + 4 = 2,444.
4. states the reason for using integers, whole numbers, fractions
(including mixed numbers), or decimals when solving a given realworld problem (2.4.A1a-c) ($).
Teacher Notes: From the Mathematics Dictionary and Handbook (Nichols Schwartz Publishing, 1999), property as a mathematical term means a
characteristic (an attribute) of a number, geometric shape, mathematical operation, equation, or inequality. To give an example:
 Property of a number: 8 is divisible by 2.
 Property of a geometric shape: Each of the four sides of a square is of the same length.
 Property of an operation: Addition is commutative. For all numbers x and y, x + y = y + x.
 Property of an equation: For all numbers a, b, and c, if a = b, then a + c = b + c.
 Property of an inequality: For all numbers a, b, and c, if a > b, then a – c > b – c.
Mathematical models such as concrete objects, pictures, diagrams, Venn diagrams, number lines, hundred charts, base ten blocks, or factor trees
are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to
represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2
(Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the
mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other
indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3
referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
5-156
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
FIFTH GRADE
Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 3: Estimation – The student uses computational estimation with whole numbers, fractions, decimals,
and money in a variety of situations.
Fifth Grade Knowledge Base Indicators
The student:
1. estimates whole numbers quantities from 0 through 100,000;
fractions greater than or equal to zero (including mixed numbers);
decimals greater than or equal to zero through hundredths place;
and monetary amounts to $10,000 using various computational
methods including mental math, paper and pencil, concrete
materials, and appropriate technology (2.4.K1a-c) ($).
2. ▲N uses various estimation strategies to estimate whole number
quantities from 0 through 100,000; fractions greater than or equal to
zero (including mixed numbers); decimals greater than or equal to
zero through hundredths place; and monetary amounts to $10,000
and explains how various strategies are used (2.4.K1a-c) ($).
3. recognizes and explains the difference between an exact and an
approximate answer (2.4.K1a-c).
4. explains the appropriateness of an estimation strategy used and
whether the estimate is greater than (overestimate) or less than
(underestimate) the exact answer (2.4.K1a).
Fifth Grade Application Indicators
The student:
1. adjusts original estimate using whole numbers from 0 through
100,000 of a real-world problem based on additional information (a
frame of reference) (2.4.A1a) ($), e.g., given a large container of
marbles, estimate the quantity of marbles. Then, using a smaller
container filled with marbles, count the number of marbles in the
smaller container and adjust your original estimate.
2. estimates to check whether or not the result of a real-world problem
using whole numbers from 0 through 100,000; fractions greater than
or equal to zero (including mixed numbers); decimals greater than or
equal to zero to tenths place; and monetary amounts to $10,000 is
reasonable and makes predictions based on the information
(2.4.A1a-c) ($), e.g., at your birthday party, you ate 4 ½ pepperoni
pizzas, 3 ¼ cheese pizzas, and 2 ¾ sausage pizzas. On the bill they
charged you for 10 pizzas. Is that reasonable? If pizzas cost $6.99
each, about how much should you save for your next birthday party?
3. selects a reasonable magnitude from given quantities based on a
real-world problem using whole numbers from 0 through 100,000
and explains the reasonableness of selection (2.4.A1a), e.g., about
how many tulips can fit in the flower vase, 2, 10, or 25? The student
chooses ten and explains that the vase at home is a jelly jar and
either two or ten will fit, but ten looks prettier.
4. ▲ ■ determines if a real-world problem calls for an exact or
approximate answer using whole numbers from 0 through 100,000
and performs the appropriate computation using various
computational methods including mental math, paper and pencil,
concrete materials, and appropriate technology (2.4.A1a) ($).
5-157
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Estimate, as a verb, means to make an educated guess based on information in a problem or to give an answer
close to the exact number. Estimation is used when an exact answer is not needed, as in many real-life situations for which “ballpark”
computations are acceptable. Good number sense enables one to estimate a quantity, estimate a measure, or estimate an answer.
Estimation serves as an important companion to computation. It provides a tool for judging the reasonableness of computational
methods including mental math, paper and pencil, concrete objects, and appropriate technology. However, being able to compute
does not automatically lead to an ability to estimate or judge reasonableness of answers. Frequent modeling by the teacher helps
students develop a range of estimation strategies. Students should be encouraged to frequently explain their thinking as they
estimate. As with exact computation, sharing estimation strategies allows students access to others’ thinking and provides
opportunities for class discussion. Identifying the estimation strategy by name is not critical; however, explaining the thinking behind
the strategy to make a valid estimation is important. [For example, an invented approach for adding 37 + 89 could involve arriving at
the solution by adding the tens, adding the ones, and then adding the partial sums to get the total.] (Principles and Standards for
School Mathematics, NCTM, 2000)
Mental math and estimation are distinct but related mathematical skills. Proficiency in mental math contributes to increased skill in
estimation. In order for students to become more familiar with estimation, teachers should introduce estimation with examples where
rounded or estimated numbers are used. Emphasis should be placed on real-world examples where only estimation is required, e.g.,
About how many hours do you sleep a night? Using the language of estimation is important, so students begin to realize that a variety
of estimates (answers) are possible.
In addition, when students are taught specific estimation strategies, they develop mental math and estimation skills easier. Estimation
strategies include front-end with adjustment, compatible “nice” numbers, clustering, or special numbers.
Mathematical models such as concrete objects, pictures, diagrams, Venn diagrams, number lines, hundred charts, base ten blocks,
or factor trees are necessary for conceptual understanding and should be used to explain computational procedures. If a
mathematical model can be used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For
example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the
indicator in the Models benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each
indicator in the Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For
example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation),
Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at
three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
5-158
Personal Finance are indicated at each grade level with aJanuary
($). The
National Standards in Personal Finance are included in the
31, 2004
Appendix.
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
5-159
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
FIFTH GRADE
Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 4: Computation – The student models, performs, and explains computation with whole numbers,
fractions including mixed numbers, and decimals including the use of concrete objects in a variety of situations.
Fifth Grade Knowledge Base Indicators
The student…
1. computes with efficiency and accuracy using various
computational methods including mental math, paper and pencil,
concrete materials, and appropriate technology (2.4.K1a).
2. performs and explains these computational procedures:
a. N divides whole numbers through a 2-digit divisor and a 4-digit
dividend with the remainder as a whole number or a fraction
using paper and pencil (2.4.K1a-b), e.g., 7452 ÷ 24 = 310 r 12
or 310 ½;
b. divides whole numbers beyond a 2-digit divisor and a 4-digit
dividend using appropriate technology (2.4.K1a-b), e.g.,
73,368 ÷ 36 = 2,038;
c. N adds and subtracts decimals from thousands place through
hundredths place (2.4.K1c);
d. N multiplies decimals up to three digits by two digits from
hundreds place through hundredths place (2.4.K1c);
e. N adds and subtracts fractions (like and unlike denominators)
greater than or equal to zero (including mixed numbers)
without regrouping and without expressing answers in
simplest form with special emphasis on manipulatives,
drawings, and models; (2.4.K1c);
f. N multiplies and divides by 10; 100; 1,000; or single-digit
multiples of each (2.4.K1a-b), e.g., 20 • 300 or 4,400 ÷ 500.
3. reads and writes horizontally, vertically, and with different
operational symbols the same addition, subtraction, multiplication,
or division expression, e.g., 6 • 4 is the same as 6 x 4 is the same
as 6(4) and 6 or 10 divided by 2 is the same as 10 ÷ 2 or 10
Fifth Grade Application Indicators
The student…
1. ▲N solves one- and two-step real-world problems using these
computational procedures ($) (For the purpose of assessment, twostep could include any combination of a, b, c, d, e, or f.):
a. adds and subtracts whole numbers from 0 through 100,000
(2.4.A1a-b); e.g., Lee buys a bike for $139, a helmet for $29 and
a reflector for $6. How much of his $200 check from his
grandparents will he have left?
b. multiplies through a four-digit whole number by a two-digit whole
number (2.4.A1a-b), e.g., at the amusement park, Monday’s
attendance was 4,414 people. Tuesday’s attendance was 3,042
people. If the cost per person is $23, how much money was
collected on those days?
c. multiplies monetary amounts up to $1,000 by a one- or two-digit
whole number (2.4.A1c), e.g., what is the cost of 4 items each
priced at $3.49?
d. divides whole numbers through a 2-digit divisor and a 4-digit
dividend with the remainder as a whole number or a fraction
(2.4.A1a-c);
e. adds and subtracts decimals from thousands place through
hundredths place when used as monetary amounts (2.4.A1a-c)
(The set of decimal numbers includes whole numbers.), e.g., at
the track meet, Peter ran the 100 meter dash in 12.3 seconds.
Tanner ran the same race in 12.19 seconds. How much faster
was Tanner?
5-160
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
x4
2.
4. ▲N identifies, explains, and finds the greatest common
factor and least common multiple of two or more whole
numbers through the basic multiplication facts from 1 x 1
through 12 x 12 (2.4.K1d).
f.
■ multiplies and divides by 10; 100; and 1,000 and single digit
multiples of each (10, 20, 30, …; 100, 200, 300, …; 1,000; 2,000;
3,000; …) (2.4.A1a-b), e.g., Matti has 1,590 stamps to place in
her stamp album. 30 stamps fit on a page. What is the minimum
number of pages she needs in her album?
5-161
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Efficiency and accuracy means that students are able to compute single-digit numbers with fluency. Students increase their
understanding and skill in addition, subtraction, multiplication, and division by understanding the relationships between addition and subtraction,
addition and multiplication, multiplication and division, and subtraction and division. Students learn basic number combinations and develop
strategies for computing that makes sense to them. Through class discussions, students can compare the ease of use and ease of explanation of
various strategies. In some cases, their strategies for computing will be close to conventional algorithms; in other cases, they will be quite different.
Many times, students’ invented approaches are based on a sound understanding of numbers and operations, and these invented approaches often
can be used with efficiency and accuracy. [For example, an invented approach for adding 37 + 89 could involve arriving at the solution by adding
the tens, adding the ones, and then adding the partial sums to get the total.] (Principles and Standards for School Mathematics, NCTM, 2000)
The definition of computation is finding the standard representation for a number. For example, 6 + 6, 4 x 3, 17 – 5, and 24 ÷ 2 are all
representations for the standard representation of 12. Mental math is mentally finding the standard representation for a number –
calculating in your head instead of calculating using paper and pencil or technology. One of the main reasons for teaching mental
math is to help students determine if a computed/calculated answer is reasonable; in other words, using mental math to estimate to
see if the answer makes sense. Students develop mental math skills easier when they are taught specific strategies. Mental math
strategies include counting on, doubling, repeated doubling, halving, making tens, dividing with tens, thinking money, and using
compatible “nice” numbers.
Mathematical models such as concrete objects, pictures, diagrams, Venn diagrams, number lines, hundred charts, base ten blocks,
or factor trees are necessary for conceptual understanding and should be used to explain computational procedures. If a
mathematical model can be used to represent the concept, the indicator in the Models benchmark is identified in the parentheses.
For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then,
the indicator in the Models benchmark lists some of the mathematical models that could be used to teach the concept. In addition,
each indicator in the Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For
example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation),
Benchmark 1 (Number Sense), and Knowledge Indicator 3.
Technology is changing mathematics and its uses. The use of technology including calculators and computers is an important part of
growing up in a complex society. It is not only necessary to estimate appropriate answers accurately when required; it is also
important to have a good understanding of the underlying concepts in order to know when to apply the appropriate procedure.
Technology does not replace the need to learn
basic facts, to compute mentally, or to do reasonable paper-and-pencil computation. However, dividing a 5-digit number by a 2-digit number is
appropriate with the exception of dividing by 10, 100, or 1,000 and simple multiples of each.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
5-162
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at
three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the
Appendix.
5-163
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
FIFTH GRADE
Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 1: Patterns – The student recognizes, describes, extends, develops, and explains relationships in
patterns in a variety of situations.
Fifth Grade Knowledge Base Indicators
The student…
6. uses concrete objects, drawings, and other representations to work
with these types of patterns(2.4.K1a):
a. repeating patterns, e.g., 9, 10, 11, 9, 10, 11, …;
b. growing patterns, e.g., 20, 30, 28, 38, 36, … where the rule is
add 10, then subtract 2; or 2, 5, 8, … as an example of an
arithmetic sequence – each term after the first is found by
adding the same number to the preceding term.
7. uses these attributes to generate patterns:
a. counting numbers related to number theory (2.4.K1a), e.g.,
multiples or perfect squares;
b. whole numbers (2.4.K1a) ($), e.g., 10; 100; 1,000; 10,000;
100,000; … (powers of ten);
c. geometric shapes through two attribute changes (2.4.K1g), e.g.,
Fifth Grade Application Indicators
The student…
1. generalizes these patterns using a written description:
a. numerical patterns (2.4.K1a) ($),
b. patterns using geometric shapes through two attribute changes
(2.4.A1a,g),
c. measurement patterns (2.4.A1a),
d. patterns related to daily life (2.4.A1a)
2. recognizes multiple representations of the same pattern (2.4.A1a)
($), e.g., 10; 100; 1,000; …
– represented as 10; 10 x 10; 10 x 10 x 10; …;
– represented as a rod, a flat, a cube, … using base ten blocks; or
– represented by a $10 bill; a $100 bill; a $1,000 bill; ….
… when the next shape
has one more side; or when both the color and the shape
change at the same time;
d. measurements (2.4.K1a), e.g., 3 m, 6 m, 9 m, …;
e. things related to daily life (2.4.K1a), e.g., sports scores,
longitude and latitude, elections, eras, or appropriate topics
across the curriculum;
f. things related to size, shape, color, texture, or movement
(2.4.K1a), e.g., square dancing moves (kinesthetic patterns)
5-164
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
3. identifies, states, and continues a pattern presented in various
formats including numeric (list or table), visual (picture, table, or
graph), verbal (oral description), kinesthetic (action), and written
(2.4.K1a) ($).
4. generates:
a. a pattern (repeating, growing) (2.4.K1a).
b. a pattern using a function table (input/output machines, T-tables)
(2.4.K1g).
Teacher Notes: Working with patterns is an important process in the development of mathematical thinking. Patterns can be based on geometric
attributes (shapes, regions, angles); measurement attributes (color, texture, length, weight, volume, number); relational attributes (proportion,
sequence, functions); and affective attributes (values, likes, dislikes, familiarity, heritage, culture). (Learning to Teach Mathematics, Randall J.
Souviney, Macmillan Publishing Company, 1994)
This process (working with patterns) can be used to develop or deepen understandings of important concepts in number theory, positive rational
numbers, measurement, geometry, probability, and functions. Working with patterns provides opportunities for students to recognize, describe,
extend, develop, and explain.
Number theory is the study of the properties of the counting (natural) numbers, their relationships, ways to represent them, and patterns among
them. Number theory includes the concepts of odd and even numbers, factors and multiples, primes and composites, greatest common factor and
least common multiple, and sequences.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that
could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are
identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and
Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
5-165
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
FIFTH GRADE
Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 2: Variables, Equations, and Inequalities – The student uses variables, symbols, whole numbers, and
algebraic expressions in one variable to solve linear equations in a variety of situations.
Fifth Grade Knowledge Base Indicators
The student…
1. ▲ explains and uses variables and symbols to represent unknown
whole number quantities from 0 through 1,000 and variable
relationships (2.4.K1a)
2. ▲N solves one-step linear equations with one variable and a whole
number solution using addition and subtraction with whole numbers
from 0 through 100 and multiplication with the basic facts (2.4.K1a,e)
($), e.g., 3y = 12, 45 = 17 + q, or r – 42 = 36.
3. explains and uses equality and inequality symbols (=, , <, ≤, >, ≥)
and corresponding meanings (is equal to, is not equal to, is less
than, is less than or equal to, is greater than, is greater than or equal
to) with whole numbers from 0 to 100,000 (2.4.K1a-b) ($).
4. recognizes ratio as a comparison of part-to-part and part-to-whole
relationships (2.4.K1a), e.g., the relationship between the number of
boys and the number of girls (part-to-part) or the relationship
between the number of girls to the total number of students in the
classroom (part-to-whole).
Fifth Grade Application Indicators
The student…
1. represents real-world problems using variables, symbols, and onestep equations with unknown whole number quantities from 0
through 1,000 (2.4.A1a,e) ($); e.g., Your parents say you must read
5 minutes each and every day of the next year. How many minutes
will you read? This is represented by 365 x 5 = M.
2. generates one-step linear equations to solve real-world problems
with whole numbers from 0 through 1,000 with one unknown and a
whole number solution using addition, subtraction, multiplication,
and division (2.4.A1a,e) ($), e.g., Ninety-six items are being shared
with four people. How much does each person receive? becomes
96 ÷ 4 = n.
3. generates (2.4.A1a,e) ($):
a. a real-world problem with one operation to match a given
addition, subtraction, multiplication, or division equation using
whole numbers from 0 through 1,000 (2.4.A1a), e.g., given
95 ÷ 5 = x students write: There are 95 kids at camp who need
to be divided into teams of 5. How many teams will there be?
b. number comparison statements using equality and inequality
symbols (=, <, >) with whole numbers, measurement, and
money e.g., 1 ft < 15 in or 10 quarters > $2.
5-166
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Understanding the concept of variable is fundamental to algebra. Students use various symbols, including letters
and geometric shapes to represent unknown quantities that both do and do not vary. Quantities that are not given and do not vary are
often referred to as unknowns or missing elements when they appear in equations, e.g., 2 + 4 = Δ or 3 + s = 7. Various symbols or
letters should be used interchangeably in equations.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks
are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be
used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to
Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models
benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each indicator in the Models
benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For example, process models are
linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and
Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at
three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the
Appendix.
5-167
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
FIFTH GRADE
Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 3: Functions – The student recognizes, describes, and examines whole number relationships in a
variety of situations.
Fifth Grade Knowledge Base Indicators
The student…
1. states mathematical relationships between whole numbers from 0
through 10,000 using various methods including mental math,
paper and pencil, concrete objects, and appropriate technology
(2.4.K1a) ($).
2. finds the values, determines the rule, and states the rule using
symbolic notation with one operation of whole numbers from 0
through 10,000 using a vertical or horizontal function table
(input/output machine, T-table) (2.4.K1f), e.g., using the function
table, fill in the values and find the rule, the rule is N • 80.
N
?
4
320
9
720
11
880
?
640
2
?
7
?
?
800
3. generalizes numerical patterns using whole numbers from 0
through 5,000 up to two operations by stating the rule using words,
e.g., If the sequence is 2400, 1200, 600, 300, 150, …; in words,
the rule could be split the number in half or divide the previous
number by 2 or if the sequence is 4, 11, 25, 53, 109, …; in words,
the rule could be double the number and add 3 to get the next
number or multiply the number by 2 and add 3.
4. ▲ ■ uses a function table (input/output machine, T-table) to
identify, plot, and label whole number ordered pairs in the first
quadrant of a coordinate plane (2.4.K1a,f).
5. plots and locates points for integers (positive and negative whole
numbers) on a horizontal number line and vertical number line
(2.4.K1a).
6. describes whole number relationships using letters and symbols.
Fifth Grade Application Indicators
The student…
1. represents and describes mathematical relationships between whole
numbers from 0 through 5,000 using written and oral descriptions,
tables, graphs, and symbolic notation (2.4.A1a) ($).
2. finds the rule, states the rule, and extends numerical patterns using
real-world problems with whole numbers from 0 through 5,000
(2.4.A1a,f) ($), e.g., the class sells cookies at lunch recess to raise
money for a field trip. The goal is to sell 3,000 cookies at 25¢ each.
Day
# of Cookies Sold
A student notices that every
4
450
4th day, a new case of
8
900
cookies has to be opened.
12
1350
Each case holds 450
16
1800
cookies. If the class keeps
20
2250
selling cookies at the same
24
2700
rate, how many days will it
28
3150
take to sell 3,000 cookies?
A student’s answer might be: 28 days because that will be 150 over
the goal or on day 27 until 3,000 cookies are sold.
3. translates between verbal, numerical, and graphical representations
including the use of concrete objects to describe mathematical
relationships (2.4.A1a,k), e.g., when the temperature is 20º and then
it drops 2º an hour for 12 hours, the result is a negative number; the
student could model this using a vertical number line.
5-168
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Functions are relationships or rules in which each member of one set is paired with one, and only one, member of another set
(an ordered pair). The concept of function can be introduced using function machines. Any number put in the machine will be changed according
to some rule. A record of inputs and corresponding outputs can be maintained in a two-column format. Function tables, input/output machines,
and T-tables may be used interchangeably and serve the same purpose.
Function concepts should be developed from growing patterns. Each term in a number sequence is related to its position in the
sequence – the functional relationship. The pattern – 4, 7, 10, 13, 16, 19, and so on – is an arithmetic sequence with a difference of
3. The pattern could be described as add 3 meaning that 3 must be added to the previous term to find the next. This pattern is
explained by using the recursive definition for a sequence. The recursive definition for a sequence is a statement or a set of
statements that explains how each successive term in the sequence is obtained from the previous term(s).
In the pattern 1, 4, 9, 16, 25, …, 225; there is no common difference. This sequence is not arithmetic or geometric (no common ratio
between geometric terms). Neither is it a combination of the two; however, there is a pattern and the missing terms between 25 and
225 can be found. To find the term value, square the number of the term. The next missing terms would be 36, 49, 64, 81, 100, 121,
and 144. This pattern is explained by using the explicit formula for a sequence. The explicit formula for a sequence defines a rule for
finding each term in the number sequence related to its position in the sequence. In other words, to find the term value, square the
number of the term – the 5th term is 52, the 8th term is 82, …
Patterns themselves are not explicit or recursive. The RULE for the pattern can be expressed explicitly or recursively and MOST
patterns can be explained using either format especially IF that pattern reflects either an arithmetic sequence or geometric sequence.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks
are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can
be used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a)
refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the
Models benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each indicator in the
Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For example, process
models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number
Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked
at three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the
5-169
Appendix.
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
5-170
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
FIFTH GRADE
Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 4: Models – The student develops and uses mathematical models including the use of concrete objects
to represent and explain mathematical relationships in a variety of situations.
Fifth Grade Knowledge Base Indicators
The student…
1. knows, explains, and uses mathematical models to represent
mathematical concepts, procedures, and relationships.
Mathematical models include:
a. process models (concrete objects, pictures, diagrams, number
lines, hundred charts, measurement tools, multiplication arrays,
division sets, or coordinate planes/grids) to model computational
procedures and mathematical relationships and to solve
equations (1.1.K1a, 1.1K1c, 1.1.K2, 1.1.K3, 1.1.K5, 1.2.K1,
1.2.K3, 1.3.K1-4, 1.4.K1, 1.4.K2a-b, 1.4.K.2f, 2.1.K1, 2.1.K2a-b,
2.1.K2d-h, 2.1.K2, 2.2.K1-4, 2.3.K1, 2.3.K4-5, 3.1.K1-6, 3.2.K14, 3.3.K1-2, 3.4.K1-4, 4.2.K3) ($);
b. place value models (place value mats, hundred charts, base ten
blocks, or unifix cubes) to compare, order, and represent
numerical quantities and to model computational procedures
(1.1.K1a, 1.1.K2, 1.1.K4, 1.2.K1, 1.3.K1-3, 1.4.K2a-b, 1.4.K2f,
2.2.K3) ($);
c. fraction and mixed number models (fraction strips or pattern
blocks) and decimal and money models (base ten blocks or
coins) to compare, order, and represent numerical quantities
(1.1.K1b, 1.1.K2-4, 1.2.K1, 1.3.K1-3, 1.4.K2c-e, 4.1.K4) ($);
d. factor trees to find least common multiple and greatest common
factor (1.2.K2, 1.4.K4);
e. equations and inequalities to model numerical relationships
(2.2.K2) ($);
f. function tables (input/output machines, T-tables) to model
numerical and algebraic relationships (2.1.K1c, 2.1.K1j, 3.1.K18, 3.2.K7-8, 3.3.K1-3) ($);
Fifth Grade Application Indicators
The student…
1. recognizes that various mathematical models can be used to
represent the same problem situation. Mathematical models
include:
a. process models (concrete objects, pictures, diagrams, number
lines, hundred charts, measurement tools, multiplication arrays,
division sets, or coordinate planes/grids) to model computational
procedures, mathematical relationships, and problem situations
and to solve equations (1.1.A1, 1.1.A2a, 1.2.A1-3, 1.3.A1-4,
1.4.A1a-b, 1.4.A1d-f, 2.1.A1a, 2.1.A1c-d, 2.1.A2, 2.2.A1-3,
2.3.A1-3, 3.2.A1a-f, 3.2.A2-4, 3.3.A1, 3.4.A1-2. 4.2.A2) ($);
b. place value models (place value mats, hundred charts, base ten
blocks, or unifix cubes) to model problem situations (1.1.A1,
1.1.A2a, 1.2.A1-3, 1.3.A2, 1.4.A1a-b, 1.4.A1f, 1.4.A3a-e, 2.2.K3)
($);
c. fraction and mixed number models (fraction strips or pattern
blocks) and decimal and money models (base ten blocks or
coins) to compare, order, and represent numerical quantities
(1.1.A1a-b, 1.1.A2b-c, 1.2.A1-3, 1.3.A2, 1.4.A1c-e) ($);
d. factor trees to find least common multiple and greatest common
factor;
e. equations and inequalities to model numerical relationships
(2.1.A1-2, 2.2.A1-3) ($);
f. function tables (input/output machines, T-tables) to model
numerical and algebraic relationships (2.3.A2, 3.2.A1g-h,
3.3.A3) ($);
g. two-dimensional geometric models (geoboards or dot paper) to
model perimeter, area, and properties of geometric shapes and
three-dimensional models (nets or solids) and real-world objects
5-171
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
to compare size and to model volume and properties of
geometric shapes (2.1.A1b, 3.1.A1-2, 3.2.A4, 4.1.A1-3);
g. two-dimensional geometric models (geoboards or dot paper) to
model perimeter, area, and properties of geometric shapes and
three-dimensional models (nets or solids) and real-world objects
to compare size and to model volume and properties of
geometric shapes (2.1.K2c, 2.1.K4b, 3.2.K5, 3.3.K3, 4.1.K2);
h. tree diagrams to organize attributes through three different sets
and determine the number of possible combinations (4.1.K2,
4.2.K1a-d, 4.2.K1f-I; 4.2.K2, 4.2);
i. two- and three-dimensional geometric models (spinners or
number cubes) and process models (concrete objects, pictures,
diagrams, or coins) to model probability (4.1.K1-3, 4.2.K1e,
4.2.K2) ($) ;
j. graphs using concrete objects, pictographs, frequency tables,
bar graphs, line graphs, circle graphs, Venn diagrams, line
plots, charts, tables, and single stem-and-leaf plots to organize
and display data (4.1.K2, 4.2.K1-2) ($) ;
k. Venn diagrams to sort data and show relationships.
2. creates mathematical models to show the relationship between two
or more things, e.g., using trapezoids to represent numerical
quantities –
h. scale drawings to model large and small real-world
objects (3.3.A2);
i. tree diagrams to organize attributes through three different
sets and determine the number of possible combinations;
j. two- and three-dimensional geometric models (spinners or
number cubes) and process models (concrete objects,
pictures, diagrams, or coins) to model probability (4.1.A13, 4.2.A1) ($);
k. graphs using concrete objects, pictographs, frequency
tables, bar graphs, line graphs, circle graphs, Venn
diagrams, line plots, charts, and tables to organize,
display, explain, and interpret data (2.3.A3; 4.1.A1-2,
4.2.A1, 4.2.A3-4) ($);
l. Venn diagrams to sort data and show relationships.
2. selects a mathematical model and explains why some
mathematical models are more useful than other
mathematical models in certain situations.
.5 or 1/2
1 or 1.0
1.5 or 1 1/2
2 or 2.0
5-172
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: For assessment purposes, the mathematical modeling process appropriate to the indicator may be included as part of the item
being assessed. The mathematical modeling process involves:
a. selecting key features and relationships within the real-world situation and representing these concepts in mathematical terms through
some sort of mathematical model,
b. performing manipulations and mathematical procedures within the mathematical model,
c. interpreting the results of the manipulations within the mathematical model,
d. using these results to make inferences about the original real-world situation.
The use of mathematical models is necessary for conceptual understanding. The ways in which mathematical ideas are
represented is fundamental to how students understand and use those ideas. As students begin to use multiple representations of
the same situation, they begin
to develop an understanding of the advantages and disadvantages of the various representations/models.
Many mathematical models are listed in this benchmark. The indicator lists some of the mathematical models that could be used to teach a
5-173
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
concept. Each indicator in this benchmark is linked to other indicators in other benchmarks; those indicators are identified in the parentheses. For
example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1
(Number Sense), and Knowledge Indicator 3. In addition, the indicator in the other benchmarks identifies, in parentheses, the Models’ indicator.
For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models).
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
5-174
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
FIFTH GRADE
Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 1: Geometric Figures and Their Properties – The student recognizes geometric shapes and compares
their properties in a variety of situations.
Fifth Grade Knowledge Base Indicators
The student…
1. recognizes and investigates properties of plane figures and solids
using concrete objects, drawings, and appropriate technology
(2.4.K1g).
2. recognizes and describes (2.4.K1g):
a. regular polygons having up to and including ten sides;
b. similar and congruent figures.
3. ▲ recognizes and describes the solids (cubes, rectangular prisms,
cylinders, cones, spheres, triangular prisms, rectangular pyramids,
triangular pyramids) using the terms faces, edges, and vertices
(corners) (2.4.K1g).
4. determines if geometric shapes and real-world objects contain
line(s) of symmetry and draws the line(s) of symmetry if the line(s)
exist(s) (2.4.K1g).
5. recognizes, draws, and describes (2.4.K1g):
a. points, lines, line segments, and rays;
b. angles as right, obtuse, or acute.
6. recognizes and describes the difference between intersecting,
parallel, and perpendicular lines (2.4.K1g).
7. identifies circumference, radius, and diameter of a circle (2.4.K1g).
Fifth Grade Application Indicators
The student…
1. solves real-world problems by applying the properties of (2.4.A1g):
a. ▲ plane figures (circles, squares, rectangles, triangles, ellipses,
rhombi, parallelograms, hexagons, pentagons) and the line(s) of
symmetry; e.g., twins are having a birthday party. The
rectangular birthday cake is to be cut into two pieces of equal
size and with the same shape. How would the cake be cut?
Would the cut be a line of symmetry? How would you know?
b. solids (cubes, rectangular prisms, cylinders, cones, spheres,
triangular prisms) emphasizing faces, edges, vertices, and
bases; e.g., ribbon is to be glued on all of the edges of a cube.
If one edge measures 5 inches, how much ribbon is needed? If
a letter was placed on each face, how many letters would be
needed?
c. intersecting, parallel, and perpendicular lines; e.g., relate these
terms to maps of city streets, bus routes, or walking paths.
Which street is parallel to the street where the school is located?
2. identifies the plane figures (circles, squares, rectangles, triangles,
ellipses, rhombi, octagons, pentagons, hexagons, trapezoids,
parallelograms) used to form a composite figure (2.4.A1g).
5-175
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Geometry is the study of shapes, their properties, and their relationships to other shapes. Symbols and numbers are used to
describe their properties and their relationships to other shapes. The fundamental concepts in geometry are point (no dimension), line (onedimensional), plane (two-dimensional), and space (three-dimensional). Plane figures are referred to as two-dimensional and solids are referred to as
three-dimensional.
From the Mathematics Dictionary and Handbook (Nichols Schwartz Publishing, 1999), property as a mathematical term means a characteristic (an
attribute) of a number, geometric shape, mathematical operation, equation, or inequality. To give an example:
 Property of a number: 8 is divisible by 2.
 Property of a geometric shape: Each of the four sides of a square is of the same length.
 Property of an operation: Addition is commutative. For all numbers x and y, x + y = y + x.
 Property of an equation: For all numbers a, b, and c, if a = b, then a + c = b + c.
 Property of an inequality: For all numbers a, b, and c, if a > b, then a – c > b – c.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that
could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are
identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and
Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
5-176
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
FIFTH GRADE
Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 2: Measurement and Estimation – The student estimates, measures, and uses measurement formulas in
a variety of situations.
Fifth Grade Knowledge Base Indicators
The student…
1. determines and uses whole number approximations (estimations)
for length, width, weight, volume, temperature, time, perimeter, and
area using standard and nonstandard units of measure (2.4.K1a)
($).
2. selects, explains the selection of, and uses measurement tools,
units of measure, and degree of accuracy appropriate for a given
situation to measure length, width, weight, volume, temperature,
time, perimeter, and area using (2.4.K1a) ($):
a. customary units of measure to the nearest fourth and eighth
inch,
b. metric units of measure to the nearest centimeter,
c. nonstandard units of measure to the nearest whole unit,
d. time including elapsed time.
3. states the number of feet and yards in a mile (2.4.K1a).
4. converts (2.4.K1a):
a. ▲ ■ within the customary system: inches and feet, feet and
yards, inches and yards, cups and pints, pints and quarts,
quarts and gallons, pounds and ounces;
b. within the metric system: centimeters and meters, meters and
kilometers, milliliters and liters, grams and kilograms.
5. knows and uses perimeter and area formulas for squares and
rectangles (2.4.K1g).
Fifth Grade Application Indicators
The student…
1. solves real-world problems by applying appropriate measurements
and measurement formulas ($):
a. ▲ length to the nearest eighth of an inch or to the nearest
centimeter (2.4.A1a), e.g., in science, we are studying butterflies.
What is the wingspan of each of the butterflies studied to the
nearest eighth of an inch?
b. temperature to the nearest degree (2.4.A1a), e.g., what would the
temperature be if it was a good day for swimming?
c. ▲ weight to the nearest whole unit (pounds, grams, nonstandard
units) (2.4.A1a), e.g., if you bought 200 bricks (each one weighed
5 pounds), how much would the whole load of bricks weigh?
d. time including elapsed time (2.4.A1a), e.g., Bob left Wichita at
10:45 a.m. He arrived in Kansas city at 1:30. How long did it take
Bob to travel to Kansas City?
e. hours in a day, days in a week, and days and weeks in a year
(2.4.A1a), e.g., John spent 59 days in New York City. How many
weeks did he stay in New York City?
f. ▲ months in a year and minutes in an hour (2.4.A1a), e.g., it took
Susan 180 minutes to complete her homework assignment. How
many hours did she spend doing homework?
g. ▲ perimeter of squares, rectangles, and triangles (2.4.A1g), e.g.,
Mark wants to put up a fence up in his rectangle-shaped back
yard. If his yard measures 18 feet by 36 feet, how many feet of
fence will he need to go around his yard?
h. ▲ area of squares and rectangles (2.4.A1g), e.g., a farmer's
square-shaped field is 35 feet on each side. How many square
feet does he have to plow?
5-177
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
2. solves real-world problems that involve conversions within the same
measurement system: inches and feet, feet and yards, inches and
yards, cups and pints, pints and quarts, quarts and gallons,
centimeters and meters (2.4.A1a), e.g., you estimate that each
person will chew 6 inches of bubblegum tape. If each package has 9
feet of bubblegum tape, how many people will get gum from that
package?
3. estimates to check whether or not measurements or calculations for
length, weight, temperature, time, perimeter, and area in real-world
problems are reasonable (2.4.A1a) ($), e.g. is it reasonable to say you
need 30 mL of water to fill a fish tank or would you need 30 L of water
to fill the fish tank?
4. adjusts original measurement or estimation for length, width, weight,
volume, temperature, time, and perimeter in real-world problems
based on additional information (a frame of reference) (2.4.A1a,g) ($),
e.g., after estimating the outside temperature to be 75º F, you find out
that yesterday’s high temperature at 3 p.m. was 62º. Should you
adjust your estimate? Why or why not?
Teacher Notes: The term geometry comes from two Greek words meaning “earth measure.” Measurement provides the tools required to apply
geometric concepts in the real-world. Estimation in measurement is defined as making guesses as to the exact measurement of an object without
using any type of measurement tool. Estimation helps students develop a relationship between the different sizes of units of measure. It helps
students develop basic properties of measurement and it gives students a tool to determine whether a given measurement is reasonable.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that
could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are
identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and
Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
5-178
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
FIFTH GRADE
Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 3: Transformational Geometry – The student recognizes and performs transformations on geometric
shapes including the use of concrete objects in a variety of situations.
Fifth Grade Knowledge Base Indicators
Fifth Grade Application Indicators
The student…
The student…
1. recognizes and performs through two transformations (reflection,
1. describes and draws a two-dimensional figure after performing one
rotation, translation) on a two-dimensional figure (2.4.K1a).
transformation (reflection, rotation, translation) (2.4.A1a).
2. recognizes when an object is reduced or enlarged (2.4.K1a).
2. makes scale drawings of two-dimensional figures using a simple
3. ▲ recognizes three-dimensional figures (rectangular prisms,
scale and grid paper (2.4.A1h), e.g., using the scale 1 cm = 3 m,
cylinders, cones, spheres, triangular prisms, rectangular pyramids)
the student makes a scale drawing of the classroom.
from various perspectives (top, bottom, side, corners) (2.4.K1g).
Teacher Notes: Transformational geometry is another way to investigate geometric figures by moving every point in a plane figure to a new
location. To help students form images of shapes through different transformations, students can use concrete objects, figures drawn on graph
paper, mirrors or other reflective surfaces, or appropriate technology.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories – Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
5-179
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
FIFTH GRADE
Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 4: Geometry From An Algebraic Perspective – The student relates geometric concepts to a number line
and the first quadrant of a coordinate plane in a variety of situations.
Fifth Grade Knowledge Base Indicators
The student…
1. locates and plots points on a number line (vertical/horizontal) using
integers (positive and negative whole numbers) (2.4.K1a).
2. explains mathematical relationships between whole numbers,
fractions, and decimals and where they appear on a number line
(2.4.K1a).
3. identifies and plots points as ordered pairs in the first quadrant of a
coordinate plane (coordinate grid) (2.4.K1a).
4. organizes whole number data using a T-table and plots the ordered
pairs in the first quadrant of a coordinate plane (coordinate grid)
(2.4.K1a,f).
Fifth Grade Application Indicators
The student…
1. solves real-world problems that involve distance and location using
coordinate planes (coordinate grids) and map grids with positive
whole number and letter coordinates (2.4.A1a), e.g., identifying
locations and giving and following directions to move from one
location to another.
2. solves real-world problems by plotting ordered pairs in the first
quadrant of a coordinate plane (coordinate grid) (2.4.A1a) ($) , e.g.,
graph daily the cumulative number of recess minutes in a 5-day
school week.
5-180
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: A number line (a mathematical model) is a diagram that represents numbers with equal distances marked off as points on a line,
and is an example of one-to-one correspondence (a relation). A number line can be used as a visual representation of numbers and operations. In
addition, a number line used horizontally and vertically is a precursor to the coordinate plane; and the distance between two numbers on a number
line is a precursor to absolute value.
A coordinate plane (coordinate grid) consists of a horizontal number line called the x-axis and a vertical number line called the y-axis. These two
lines intersect at a point called the origin. The x-axis and the y-axis divide the plane into four sections called quadrants. Any point on the coordinate
plane can be named with two numbers called coordinates. The first number is the x-coordinate. The second number is the y-coordinate. Since the
pair is always named in order (first x, then y), it is called an ordered pair.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that
could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are
identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and
Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
5-181
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 4: Data
FIFTH GRADE
Data – The student uses concepts and procedures of data analysis in a variety of situations.
Benchmark 1: Probability – The student applies the concepts of probability to draw conclusions and to make
predictions and decisions including the use of concrete objects in a variety of situations.
Fifth Grade Knowledge Base Indicators
The student…
1. recognizes that all probabilities range from zero (impossible) through
one (certain) (2.4.K1i) ($).
2. lists all possible outcomes of a simple event in an experiment or
simulation in an organized manner including the use of concrete
objects (2.4.K1g-j).
3. recognizes a simple event in an experiment or simulation where the
probabilities of all outcomes are equal (2.4.K1i).
4. represents the probability of a simple event in an experiment or
simulation using fractions (2.4.K1c).
Fifth Grade Application Indicators
The student...
1. ■ conducts an experiment or simulation with a simple event
including the use of concrete materials; records the results in a
chart, table, or graph; uses the results to draw conclusions about
the event; and makes predictions about future events (2.4.A1j-k).
2. uses the results from a completed experiment or simulation of a
simple event to make predictions in a variety of real-world situations
(2.4.A1j-k), e.g., the manufacturer of Crunchy Flakes puts a prize in
20 out of every 100 boxes. What is the probability that a shopper
will find a prize in a box of Crunchy Flakes, if they purchase 10
boxes?
3. compares what should happen (theoretical probability/expected
results) with what did happen (empirical probability/experimental
results) in an experiment or simulation with a simple event
(2.4.A1j).
5-182
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Ideas from probability reinforce concepts in the other Standards, especially Number and Computation and Geometry. Students
need to develop an intuitive concept of chance – whether or not something is unlikely or likely to happen. Probability experiences should be
addressed through the use of concrete objects, coins, and geometric models (spinners, number cubes, or dartboards). Probabilities are ratios,
expressed as fractions, decimals, or percents, determined by considering results or outcomes of experiments. Some examples of uses of
probability in every day life include: There is a 50% chance of rain today. What is the probability that the team will win every game?
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
5-183
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 4: Data
FIFTH GRADE
Data – The student uses concepts and procedures of data analysis in a variety of situations.
Benchmark 2: Statistics – The student collects, organizes, displays, explains, and interprets numerical (rational
numbers) and non-numerical data sets in a variety of situations with a special emphasis on measures
of central tendency.
Fifth Grade Knowledge Base Indicators
The student…
1. organizes, displays, and reads numerical (quantitative) and nonnumerical (qualitative) data in a clear, organized, and accurate
manner including a title, labels, categories, and whole number and
decimal intervals using these data displays (2.4.K1j) ($):
a. graphs using concrete objects,
b. pictographs,
c. frequency tables,
d. bar and line graphs,
e. Venn diagrams and other pictorial displays, e.g., glyphs,
f. line plots,
g. charts and tables,
h. circle graphs,
i. single stem-and-leaf plots.
2. collects data using different techniques (observations, polls, tallying,
interviews, surveys, or random sampling) and explains the results
(2.4.K1j) ($).
3. ▲ identifies, explains, and calculates or finds these statistical
measures of a whole number data set of up to twenty whole number
data points from 0 through 1,000 (2.4.K1a) ($):
a. minimum and maximum values,
b. range,
c. mode (no-, uni-, bi-),
d. median (including answers expressed as a decimal or a fraction
without reducing to simplest form),
e. mean (including answers expressed as a decimal or a fraction
without reducing to simplest form).
Fifth Grade Application Indicators
The student…
1. ▲ interprets and uses data to make reasonable inferences,
predictions, and decisions, and to develop convincing arguments
from these data displays (2.4.A1k) ($):
a. graphs using concrete materials,
b. pictographs,
c. frequency tables,
d. bar and line graphs,
e. Venn diagrams and other pictorial displays,
f. line plots,
g. charts and tables,
h. circle graphs.
2. uses these statistical measures of a whole number data set to make
reasonable inferences and predictions, answer questions, and make
decisions (2.4.A1a) ($):
a. minimum and maximum values,
b. range,
c. mode,
d. median,
e. mean when the data set has a whole number mean.
3. recognizes that the same data set can be displayed in various
formats and discusses why a particular format may be more
appropriate than another (2.4.A1k) ($).
4. recognizes and explains the effects of scale and interval changes
on graphs of whole number data sets (2.4.A1k).
5-184
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Graphs (data displays) are pictorial representations of mathematical relationships, are used to tell a story, and are an important
part of statistics. When a graph is made, the axes and the scale (numbers running along a side of the graph) are chosen for a reason. The
difference between numbers from one grid line to another is the interval. The interval will depend on the lowest and highest values in the data set.
Emphasizing the importance of using equal-sized pictures or intervals is critical to ensuring that the data display is accurate.
Graphs take many forms:
– bar graphs and pictographs compare discrete data,
– frequency tables show how many times a certain piece of data occurs,
– circle graphs (pie charts) model parts of a whole,
– line graphs show change over time,
– Venn diagrams show relationships between sets of objects,
– line plots show frequency of data on a number line, and
– single stem-and-leaf plots (closely related to line plots except that the number line is usually vertical and digits are used rather than x’s) show
frequency distribution by arranging numbers (stems) on the left side of a vertical line with numbers (leaves) on the right side.
An important aspect of data is its center. The measures of central tendency (averages) of a data set are mean, median, and mode. Conceptual
understanding of mean, median, and mode is developed through the use of concrete objects that represent the data values.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
5-185
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
SIXTH GRADE
Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 1: Number Sense – The student demonstrates number sense for rational numbers and simple algebraic
expressions in one variable in a variety of situations.
Sixth Grade Knowledge Base Indicators
Sixth Grade Application Indicators
The student…
1. knows, explains, and uses equivalent representations for rational
numbers expressed as fractions, terminating decimals, and percents;
positive rational number bases with whole number exponents; time;
and money (2.4.K1a-c) ($).
2. ▲ compares and orders (2.4.K1a-c) ($):
a. integers;
b. fractions greater than or equal to zero,
c. decimals greater than or equal to zero through thousandths
place.
3. explains the relative magnitude between whole numbers, fractions
greater than or equal to zero, and decimals greater than or equal to
zero (2.4.K1a-c).
4. ▲ N knows and explains numerical relationships between percents,
decimals, and fractions between 0 and 1 (2.4.K1a,c), e.g.,
recognizing that percent means out of a 100, so 60% means 60 out of
100, 60% as a decimal is .60, and 60% as a fraction is 60/100.
5. uses equivalent representations for the same simple algebraic
expression with understood coefficients of 1 (2.4.K1a), e.g., when
students are developing their own formula for the perimeter of a
square, they combine s + s + s + s to make 4s.
The student…
1. generates and/or solves real-world problems using
equivalent representations of (2.4.A1a-c) ($):
a. integers, e.g., the basketball team made 15 out of 25
free throws this season. Express their free throw
shooting as a fraction and as a decimal.
b. fractions greater than or equal to zero, e.g., the
basketball team made 15 out of 25 free throws this
season, express their free throw shooting as a fraction.
c. decimals greater than or equal to zero through
thousandths place (2.4.1a), e.g., the basketball team
made 15 out of 25 free throws this season, express their
free throw shooting as a decimal.
2. determines whether or not solutions to real-world problems that
involve the following are reasonable ($).
a. integers (2.4.A1a), e.g., the football is placed on your own 10yard line with 90 yards to go for a touchdown. After the first
down, your team gains 7 yards. On the second down, your
team loses 4 yards; and on the third down your team gains 2
yards. Is it reasonable for the football to be placed on the 5
yard line for the beginning of the fourth down? Why or why
not?
b. fractions greater than or equal to zero (2.4.A1c), e.g., Gary,
Tom, and their parents are selling greeting cards. Gary
receives 1/3 of the profit and Tom receives 1/4 of the profit. Is
it reasonable that together they received 2/7 of the profits?
6-186
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Why or why not?
c.
decimals greater than or equal to zero through thousandths
place (2.4.A1c), e.g., the beginning bank balance is $250.40
A deposit of $175, a withdrawal of $198, and a $2 service
charge are made. The checkbook balance reads $127.40. Is
this a reasonable balance? Why or why not?
6-187
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Number sense refers to one’s ability to reason with numbers and to work with numbers in a flexible way. The ability to compute
mentally, to estimate based on understanding of number relationships and magnitudes, and to judge reasonableness of answers are all involved in
number sense.
When we say that someone has good number sense, we mean that he or she possesses a variety of abilities and understandings that include an
awareness of the relationships between numbers, an ability to represent numbers in a variety of ways, a knowledge of the effects of operations,
and an ability to interpret and use numbers in real-world counting and measurement situations. Such a person predicts with some accuracy the
result of an operation and consistently chooses appropriate measurement units. This “friendliness with numbers” goes far beyond mere
memorization of computational algorithms and number facts; it implies an ability to use numbers flexibly, to choose the most appropriate
representation of a number for a given circumstance, and to recognize when operations have been correctly performed. (Number Sense and
Operations: Addenda Series, Grades K-6, NCTM, 1993)
At this grade level, rational numbers include positive and negative numbers, large numbers (one million), and small numbers (onethousandth). Relative magnitude refers to the size relationship one number has with another – is it much larger, much smaller, close,
or about the same? For example, using the numbers 219, 264, and 457, answer questions such as –
 Which two are closest? Why?
 Which is closest to 300? To 250?
 About how far apart are 219 and 500? 5,000?
 If these are ‘big numbers,’ what are small numbers? Numbers about the same? Numbers that make these seem
small?
(Elementary and Middle School Mathematics, John A. Van de Walle, Addison Wesley Longman, Inc., 1998)
Mathematical models such as concrete objects, pictures, diagrams, Venn diagrams, number lines, hundred charts, base ten blocks, or factor trees
are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to
represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2
(Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the
mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other
indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3
referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories) Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
Standard 1: Number and Computation
SIXTH GRADE
6-188
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 2: Number Systems and Their Properties – The student demonstrates an understanding of the rational
number system and the irrational number pi; recognizes, uses, and describes their properties; and
extends these properties to algebraic expressions in one variable.
Sixth Grade Knowledge Base Indicators
The student…
1. classifies subsets of the rational number system as counting (natural)
numbers, whole numbers, integers, fractions (including mixed
numbers), or decimals (2.4.K1a,c,k).
2. identifies prime and composite numbers and explains their meaning
(2.4.K1d).
3. uses and describes these properties with the rational number system
and demonstrates their meaning including the use of concrete
objects (2.4.K1a) ($):
a. commutative and associative properties of addition and
multiplication (commutative – changing the order of the numbers
does not change the solution; associative – changing the
grouping of the numbers does not change the solution);
b. identity properties for addition and multiplication (additive identity
– zero added to any number is equal to that number;
multiplicative identity – one multiplied by any number is equal to
that number);
c. symmetric property of equality, e.g., 24 x 72 = 1,728 is the same
as 1,728 = 24 x 72;
d. zero property of multiplication (any number multiplied by zero is
zero);
e. distributive property (distributing multiplication or division over
addition or subtraction), e.g., 26(9 + 15) = 26(9) + 26(15);
f. substitution property (one name of a number can be substituted
for another name of the same number), e.g., if a = 3 and a + 2 =
b, then 3 + 2 = b;
g. addition property of equality (adding the same number to each
side of an equation results in an equivalent equation – an
equation with the same solution), e.g., if a = b, then a + 3 = b + 3;
Sixth Grade Application Indicators
The student…
1. generates and/or solves real-world problems with rational numbers
using the concepts of these properties to explain reasoning
(2.4.A1a-c,e) ($):
a. commutative and associative properties for addition and
multiplication, e.g., at a delivery stop, Sylvia pulls out a
flat of eggs. The flat has 5 columns and 6 rows of eggs.
Show two ways to find the number of eggs: 5 • 6 = 30 or
6 • 5 = 30.
b. additive and multiplicative identities, e.g., the outside
temperature was T degrees during the day. The
temperature rose 5 degrees and by the next morning it
had dropped 5 degrees.
c. symmetric property of equality, e.g., Sam took a $15
check to the bank and received a $10 bill and a $5 bill.
Later Sam took a $10 bill and a $5 bill to the bank and
received a check for $15. $15 = $10 + $5 is the same as
$10 + $5 = $15
d. distributive property, e.g., trim is used around the outside edges
of a bulletin board with dimensions 3 ft by 5 ft. Show two
different ways to solve this problem: 2(3 + 5) = 16 or 2 • 3 + 2 •
5 = 6 + 10 = 16. Then explain why the answers are the same.
e. substitution property, e.g., V = IR [Ohm’s Law: voltage (V) =
current (I) x resistance (R)] If the current is 5 amps (I = 5) and
the resistance is 4 ohms (R = 4), what is the voltage?
f. addition property of equality, e.g., Bob and Sue each read the
same number of books. During the week, they each read 5
more books. Compare the number of books each read:
6-189
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
b= number of books Bob read, s= number of books Sue read,
so b + 5 = s + 5..
h. multiplication property of equality (for any equation, if the same
number is multiplied to each side of that equation, then the new
statement describes an equation equivalent to the original), e.g.,
if a= b, then a x 7 = b x 7;
additive inverse property (every number has a value known as its
additive inverse and when the original number is added to that
additive inverse, the answer is zero), e.g., +5 + (-5) = 0.
4. recognizes and explains the need for integers, e.g., with
g. multiplication property of equality, e.g., Jane watches television
half as much as Tom. Jane watches T.V. for 3 hours. How long
does Tom watch television? Let T = number of hours Tom
watches TV. 3 = ½T, so 2•3 + 2•½T.
h. additive inverse property, e.g., at the shopping mall, you are at
ground level when you take the elevator down 5 floors. Describe
how to get to ground level.
2. analyzes and evaluates the advantages and disadvantages of using
6-190
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
temperature, below zero is negative and above zero is positive; in
finances, money in your pocket is positive and money owed
someone is negative.
5. recognizes that the irrational number pi can be represented by an
approximate rational value, e.g., 22/7 or 3.14.
integers, whole numbers, fractions (including mixed numbers),
decimals, or the irrational number pi and its rational approximations
in solving a given real-world problem (2.4.A1a-c) ($), e.g., in the store
everything is 50% off. When calculating the discount, which
representation of 50% would you use and why?
Teacher Notes: From the Mathematics Dictionary and Handbook (Nichols Schwartz Publishing, 1999), property as a mathematical term means a
characteristic (an attribute) of a number, geometric shape, mathematical operation, equation, or inequality. To give an example:
 Property of a number: 8 is divisible by 2.
 Property of a geometric shape: Each of the four sides of a square is of the same length.
 Property of an operation: Addition is commutative. For all numbers x and y, x + y = y + x.
 Property of an equation: For all numbers a, b, and c, if a = b, then a + c = b + c.
 Property of an inequality: For all numbers a, b, and c, if a > b, then a – c > b – c.
Mathematical models such as concrete objects, pictures, diagrams, Venn diagrams, number lines, hundred charts, base ten blocks, or factor trees
are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to
represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2
(Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the
mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other
indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3
referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
6-191
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
SIXTH GRADE
Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 3: Estimation – The student uses computational estimation with rational numbers and the irrational
number pi in a variety of situations.
Sixth Grade Knowledge Base Indicators
The student…
1. estimates quantities with combinations of rational numbers and/or
the irrational number pi using various computational methods
including mental math, paper and pencil, concrete objects, and/or
appropriate technology (2.4.K1a-c) ($).
2. uses various estimation strategies and explains how they were
used to estimate rational number quantities or the irrational
number pi (2.4.K1a-c) ($)
3. recognizes and explains the difference between an exact and an
approximate answer (2.4.K1a-c).
4. determines the appropriateness of an estimation strategy used
and whether the estimate is greater than (overestimate) or less
than (underestimate) the exact answer and its potential impact on
the result (2.4.K1a).
Sixth Grade Application Indicators
The student…
1. adjusts original rational number estimate of a real-world problem
based on additional information (a frame of reference) (2.4.A1a) ($),
e.g., given a large container of marbles, estimate the quantity of
marbles. Then, using a smaller container filled with marbles, count the
number of marbles in the smaller container and adjust your original
estimate.
2. ▲N estimates to check whether or not the result of a real-world
problem using rational numbers is reasonable and makes predictions
based on the information (2.4.A1a) ($), e.g., a class of 28 students
has a goal of reading 1,000 books during the school year. If each
student reads 13 books each month, will the class reach their goal?
3. selects a reasonable magnitude from given quantities based on a
real-world problem and explains the reasonableness of the selection
(2.4.A1a), e.g., length of a classroom in meters – 1-3 meters, 5-8
meters, 10-15 meters.
4. determines if a real-world problem calls for an exact or approximate
answer and performs the appropriate computation using various
computational methods including mental math, paper and pencil,
concrete objects, or appropriate technology (2.4.A1a) ($), e.g., Kathy
buys items at the grocery store priced at: $32.56, $12.83, $6.99, 5 for
$12.49 each. She has $120 with her to pay for the groceries. To
decide if she can pay for her items, does she need an exact or an
approximate answer?
6-192
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Estimate, as a verb, means to make an educated guess based on information in a problem or to give an answer
close to the exact number. Estimation is used when an exact answer is not needed, as in many real-life situations for which “ballpark”
computations are acceptable. Good number sense enables one to estimate a quantity, estimate a measure, or estimate an answer.
Estimation serves as an important companion to computation. It provides a tool for judging the reasonableness of computational
methods including mental math, paper and pencil, concrete objects, and appropriate technology. However, being able to compute
does not automatically lead to an ability to estimate or judge reasonableness of answers. Frequent modeling by the teacher helps
students develop a range of estimation strategies. Students should be encouraged to frequently explain their thinking as they
estimate. As with exact computation, sharing estimation strategies allows students access to others’ thinking and provides
opportunities for class discussion. Identifying the estimation strategy by name is not critical; however, explaining the thinking behind
the strategy to make a valid estimation is important. (Principles and Standards for School Mathematics, NCTM, 2000)
Mental math and estimation are distinct but related mathematical skills. Proficiency in mental math contributes to increased skill in
estimation. In order for students to become more familiar with estimation, teachers should introduce estimation with examples where
rounded or estimated numbers are used. Emphasis should be placed on real-world examples where only estimation is required, e.g.,
About how many hours do you sleep a night? Using the language of estimation is important, so students begin to realize that a variety
of estimates (answers) are possible.
In addition, when students are taught specific estimation strategies, they develop mental math and estimation skills easier. Estimation
strategies include front-end with adjustment, compatible “nice” (friendly) numbers, clustering, special numbers, or truncation.
Mathematical models such as concrete objects, pictures, diagrams, Venn diagrams, number lines, hundred charts, base ten blocks,
or factor trees are necessary for conceptual understanding and should be used to explain computational procedures. If a
mathematical model can be used to represent the concept, the indicator in the Models benchmark is identified in the parentheses.
For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then,
the indicator in the Models benchmark lists some of the mathematical models that could be used to teach the concept. In addition,
each indicator in the Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For
example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation),
Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked
at three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the
Appendix.
6-193
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
6-194
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
SIXTH GRADE
Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 4: Computation – The student models, performs, and explains computation with positive rational
numbers and integers in a variety of situations.
Sixth Grade Knowledge Base Indicators
Sixth Grade Application Indicators
The student…
The student…
1. computes with efficiency and accuracy using various
computational methods including mental math, paper
and pencil, concrete objects, and appropriate technology
(2.4.K1a).
1. generates and/or solves one- and two-step real-world problems with
rational numbers using these computational procedures ($):
a. division with whole numbers (2.4.A1b), e.g., the perimeter of a
square is 128 feet. What is the length of its side?
b. ▲ ■ addition, subtraction, multiplication, and division of
decimals through hundredths place (2.4.A1a-c), e.g., on a
recent trip, Marion drove 25.8 miles from Allen to Barber, then
15.2 miles from Barber to Chase, then 14.9 miles from Chase to
Douglas. When Marion had completed half of her drive from
Allen to Douglas how many miles did she drive?
c. addition, subtraction, and multiplication of fractions (including
mixed numbers) (2.4.A1c), e.g., the student council is having a
contest between classes. On the average, each student takes
3 1/3 minutes for the relay. How much time is needed for a
class of 24 to run the relay?
2. performs and explains these computational procedures:
a. ▲N divides whole numbers through a two-digit divisor and a
four-digit dividend and expresses the remainder as a whole
number, fraction, or decimal (2.4.K1a-b), e.g., 7452 ÷ 24 =
310 r 12, 310 12/24, 310 ½, or 310.5;
b. N adds and subtracts decimals from millions place through
thousandths place (2.4.K1c);
c. N multiplies and divides a four-digit number by a two-digit
number using numbers from thousands place through
hundredths place (2.4.K1a-b), e.g., 4,350 ÷ 1.2 = 3,625;
d. N multiplies and divides using numbers from thousands place
through thousandths place by 10; 100; 1,000; .1; .01; .001; or
single-digit multiples of each (2.4.K1a-c); e.g., 54.2 ÷ .002 or
54.3 x 300;
e. N adds integers (2.4.K1a); e.g., +6 + –7 = –1
f. ▲N adds, subtracts, and multiplies fractions (including mixed
numbers) expressing answers in simplest form (2.4.K1c); e.g.,
5¼ • 1/3 = 21/4 • 1/3 = 7/4 or 1¾
g. N finds the root of perfect whole number squares (2.4.K1a);
h. N uses basic order of operations (multiplication and division in
order from left to right, then addition and subtraction in order
from left to right) with whole numbers;
6-195
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
i.
adds, subtracts multiplies, and divides rational numbers using
concrete objects.
3. recognizes, describes, and uses different representations to express
the same computational procedures, e.g., 3/4 = 3 ÷ 4 = 4 3 .
4. identifies, explains, and finds the prime factorization of whole
numbers (2.4.K1d).
5. finds prime factors, greatest common factor, multiples, and the least
common multiple (2.4.K1d).
6. finds a whole number percent (between 0 and 100) of a whole
number (2.4.K1a,c) ($), e.g., 12% of 40 is what number?
6-196
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
6-197
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Efficiency and accuracy means that students are able to compute single-digit numbers with fluency. Students
increase their understanding and skill in single-digit addition and subtraction by developing relationships within addition and
subtraction combinations [and by developing relationships with multiplication and division combinations]. Students learn basic
number combinations and develop strategies for computing that makes sense to them. Through class discussions, students can
compare the ease of use and ease of explanation of various strategies. In some cases, their strategies for computing will be close to
conventional algorithms; in other cases, they will be quite different. Many times, students’ invented approaches are based on a sound
understanding of numbers and operations, and these invented approaches often can be used with efficiency and accuracy.
(Principles and Standards for School Mathematics, NCTM, 2000)
The definition of computation is finding the standard representation for a number. For example, 6 + 6, 4 x 3, 17 – 5, and 24 ÷ 2 are all
representations for the standard representation of 12. Mental math is mentally finding the standard representation for a number – calculating in
your head instead of calculating using paper and pencil or technology. One of the main reasons for teaching mental math is to help students
determine if a computed/calculated answer is reasonable; in other words, using mental math to estimate to see if the answer makes sense.
Students develop mental math skills easier when they are taught specific strategies. Mental math strategies include counting on, doubling,
repeated doubling, halving, making tens, multiplying by powers of tens, dividing with tens, thinking money, and using compatible “nice” numbers.
Mathematical models such as concrete objects, pictures, diagrams, Venn diagrams, number lines, hundred charts, base ten blocks,
or factor trees are necessary for conceptual understanding and should be used to explain computational procedures. If a
mathematical model can be used to represent the concept, the indicator in the Models benchmark is identified in the parentheses.
For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then,
the indicator in the Models benchmark lists some of the mathematical models that could be used to teach the concept. In addition,
each indicator in the Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For
example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation),
Benchmark 1 (Number Sense), and Knowledge Indicator 3.
Technology is changing mathematics and its uses. The use of technology including calculators and computers is an important part of
growing up in a complex society. It is not only necessary to estimate appropriate answers accurately when required, but also it is also
important to have a good understanding of the underlying concepts in order to know when to apply the appropriate procedure.
Technology does not replace the need to learn basic facts, to compute mentally, or to do reasonable paper-and-pencil computation.
However, dividing a 5-digit number by a 2-digit number is appropriate with the exception of dividing by 10, 100, or 1,000 and simple
multiples of each.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked
at three grade levels, the indicators in the Kansas Curricular Standards
for Mathematics that correlate with the National Standards in
6-198
31, 2004
Personal Finance are indicated at each grade level with aJanuary
($). The
National Standards in Personal Finance are included in the
▲– Assessed Indicator on the Objective Assessment
Appendix.
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
6-199
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
SIXTH GRADE
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 1: Patterns – The student recognizes, describes, extends, develops, and explains the general rule of a
pattern in variety of situations.
Sixth Grade Knowledge Base Indicators
Sixth Grade Application Indicators
6-200
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
The student…
1. recognizes the same general pattern presented in different
representations [numeric (list or table), visual (picture, table,
or graph), and written] (2.4.A1a,k), e.g., you are selling
cookies by the box. Each box costs $3. You have $2 to begin
your sales. This can be written as a pattern that begins with 2
and adds three each time, as a table or
X
Y
0
2
graph.
Money earned selling cookies
2
3
4
8
11
14
14
money earned
The student…
1. identifies, states, and continues a pattern presented in various
formats including numeric (list or table), visual (picture, table, or
graph), verbal (oral description), kinesthetic (action), and written
using these attributes include:
a. counting numbers including perfect squares, and factors and
multiples (number theory) (2.4.K1a);
b. positive rational numbers limited to two operations (addition,
subtraction, multiplication, division) including arithmetic
sequences (a sequence of numbers in which the difference of
two consecutive numbers is the same) (2.4.K1a);
c. geometric figures through two attribute changes (2.4.K1g);
d. measurements (2.4.K1a);
e. things related to daily life (2.4.K1a) ($), e.g., time (a full moon
every 28 days), tide, calendar, traffic, or appropriate topics
across the curriculum.
2. generates a pattern (repeating, growing) (2.4.K1a).
3. extends a pattern when given a rule of one or two simultaneous
operational changes (addition, subtraction, multiplication, division)
between consecutive terms (2.4.K1a), e.g., find the next three
numbers in a pattern that starts with 3, where you double and add 1
to get the next number; the next three numbers are 7, 15, and 31.
4. ▲ states the rule to find the next number of a pattern with one
operational change (addition, subtraction, multiplication, division) to
move between consecutive terms (2.4.K1a), e.g., given 4, 8, and 16,
double the number to get the next term, multiply the term by 2 to get
the next term, or add the number to itself for the next term.
12
10
8
6
4
2
0
0
1
2
3
4
boxes of cookies sold
2. recognizes multiple representations of the same pattern (2.4.A1a)
($), e.g., 1, 10; 100; 1,000; 10,000…
– represented as 1; 10; 10 x 10; 10 x 10 x 10; 10 x 10 x 10 x 10; …;
– represented as 100; 101; 102; 103; 104; …;
– represented as a unit; a rod; a flat; a cube; … using base ten
blocks; or
– represented as a $1 bill; a $10 bill; a $100 bill ; a $1,000 bill; ….
6-201
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: A fundamental component in the development of classification, number, and problem solving skills is inventing, discovering, and
describing patterns. Patterns pervade all of mathematics and much of nature. All patterns are either repeating or growing or a variation of either or
both. Translating a pattern from one medium to another to find two patterns that are alike, even though they are made with different materials, is
important so students can see the relationships that are critical to repeating patterns. With growing patterns, students not only extend patterns, but
also look for a generalization or an algebraic relationship that will tell them what the pattern will be at any point along the way.
Working with patterns is an important process in the development of mathematical thinking. Patterns can be based on geometric attributes
(shapes, regions, angles); measurement attributes (color, texture, length, weight, volume, number); relational attributes (proportion, sequence,
functions); and affective attributes (values, likes, dislikes, familiarity, heritage, culture). (Learning to Teach Mathematics, Randall J. Souviney,
Macmillan Publishing Company, 1994)
This process (working with patterns) can be used to develop or deepen understandings of important concepts in number theory, rational numbers,
measurement, geometry, probability, and functions. Working with patterns provides opportunities for students to recognize, describe, extend,
develop, and explain.
Number theory is the study of the properties of the counting numbers (positive integers), their relationships, ways to represent them, and patterns
among them. Number theory includes the concepts of odd and even numbers, factors and multiples, primes and composites, greatest common
factor and least common multiple, and sequences.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that
could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are
identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and
Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
6-202
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
SIXTH GRADE
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 2: Variables, Equations, and Inequalities – The student uses variables, symbols, positive rational
numbers, and algebraic expressions in one variable to solve linear equations and inequalities in a
variety of situations.
Sixth Grade Knowledge Base Indicators
Sixth Grade Application Indicators
The student…
1. explains and uses variables and/or symbols to represent unknown
quantities and variable relationships (2.4.K1a), e.g., x < 2.
2. uses equivalent representations for the same simple algebraic
expression with understood coefficients of 1 (2.4.K1a), e.g., when
students are developing their own formula for the perimeter of a
square they combine s + s + s + s to make 4s.
3. solves (2.4.K1a,e) ($):
a. one-step linear equations (addition, subtraction, multiplication,
division) with one variable and whole number solutions, e.g.,
2 x = 8 or x + 7 = 12
b. one-step linear inequalities(addition, subtraction) in one variable
with whole numbers, e.g., x – 5 < 12;
4. explains and uses equality and inequality symbols (=, , <, ≤, >, ≥)
and corresponding meanings (is equal to, is not equal to, is less
than, is less than or equal to, is greater than, is greater than or equal
to) to represent mathematical relationships with positive rational
numbers (2.4.K1a-b) ($).
5. knows and uses the relationship between ratios, proportions, and
percents and finds the missing term in simple proportions where the
missing term is a whole number (2.4.K1a,c), e.g., 1 = x, 2 = 4,
1=
x_.
2 4 3 x
4
100
6. finds the value of algebraic expressions using whole numbers
(2.4.Ka), e.g., If x =3, then 5x = 5(3).
The student…
1. represents real-world problems using variables and symbols to
(2.4.A1a,e) ($):
a. write algebraic or numerical expressions or one-step equations
(addition, subtraction, multiplication, division) with whole number
solutions, e.g., John has three times as much money as his
sister. If M is the amount of money his sister has, what is the
expression that represents the amount of money that John has?
The expression would be written as 3M.
b. ▲ ■ write and/or solve one-step equations (addition, subtraction,
multiplication, and division), e.g., a player scored three more
points today than yesterday. Today, the player scored 17
points. How many points were scored yesterday? Write an
equation to represent this problem. Let Y = number of points
scored yesterday. The equation would be written as y + 3 = 17.
The answer is y = 14.
4. generates real-world problems that represent simple expressions or
one-step linear equations (addition, subtraction, multiplication,
division) with whole number solutions (2.2.A1a,e), ($) e.g., write a
problem situation that represents the expression x + 10. The
problem could be: How old will a person be ten years from now if x
represents the person’s current age?
5. explains the mathematical reasoning that was used to solve a realworld problem using a one-step equation (addition, subtraction,
multiplication, division) (2.2.A1a,e) ($), e.g., use the equation form
y + 3 = 17. Solve by subtracting 3 from both sides to get y = 14.
6-203
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Understanding the concept of variable is fundamental to algebra. Students use various symbols, including letters
and geometric shapes to represent unknown quantities that both do and do not vary. Quantities that are not given and do not vary are
often referred to as unknowns or missing elements when they appear in equations, e.g., 2 + 4 = Δ or 3 • s = 15 where a triangle and s
are used as variables. Various symbols or letters should be used interchangeably in equations.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks
are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be
used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to
Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models
benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each indicator in the Models
benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For example, process models are
linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and
Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at
three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the
Appendix.
6-204
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
SIXTH GRADE
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 3: Functions – The student recognizes, describes, and analyzes linear relationships in a variety of
situations.
Sixth Grade Knowledge Base Indicators
Sixth Grade Application Indicators
The student…
The student…
1. recognizes linear relationships using various methods including
mental math, paper and pencil, concrete objects, and graphing
utilities or appropriate technology (2.4.K1a).
2. finds the values and determines the rule with one operation using a
function table (input/output machine, T-table) (2.4.K1f).
3. generalizes numerical patterns up to two operations by stating the
rule using words (2.4.K1a), e.g., If the sequence is 2400, 1200, 600,
300, 150, …, what is the rule? In words, the rule could be split the
previous number in half or divide the previous number before by 2.
4. uses a given function table (input/output machine, T-table) to
identify, plot, and label the ordered pairs using the four quadrants of
a coordinate plane (2.4.K1a,f).
1. represents a variety of mathematical relationships using written and
oral descriptions of the rule, tables, graphs, and when possible,
symbolic notation (2.4.A1f,k) ($), e.g., linear patterns and graphs can
be used to represent time and distance situations. Pretend you are
in a car traveling from home at 50 miles per hour. Then, represent
the nth term. 50n meaning 50 times the number of hours traveling
equals the distance away from home.
Time | Distance
0 | 0
1 | 50
2 | 100
.
| .
.
| .
.
| .
n | 50n
2. interprets and describes the mathematical relationships of
numerical, tabular, and graphical representations (2.4.Alf,k).
6-205
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Functions are relationships or rules in which each member of one set is paired with one, and only one, member of
another set (an ordered pair). The concept of function can be introduced using function machines. Any number put in the machine will
be changed according to some rule. A record of inputs and corresponding outputs can be maintained in a two-column format.
Function tables, input/output machines, and T-tables may be used interchangeably and serve the same purpose.
Function concepts should be developed from growing patterns. Each term in a number sequence is related to its position in the
sequence – the functional relationship. The pattern – 4, 7, 10, 13, 16, 19, and so on – is an arithmetic sequence with a difference of 3.
The pattern could be described as add 3 meaning that 3 must be added to the previous term to find the next. This pattern is explained
by using the recursive definition for a sequence. The recursive definition for a sequence is a statement or a set of statements that
explains how each successive term in the sequence is obtained from the previous term(s).
6-206
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
In the pattern 1, 4, 9, 16, 25,.., 225; there is no common difference. This sequence is not arithmetic or geometric (no common ratio
between geometric terms). Neither is it a combination of the two; however, there is a pattern and the missing terms between 25 and
225 can be found. To find the term value, square the number of the term. The next missing terms would be 36, 49, 64, 81, 100, 121,
and 144. This pattern is explained by using the explicit formula for a sequence. The explicit formula for a sequence defines a rule for
finding each term in the number sequence related to its position in the sequence. In other words, to find the term value, square the
number of the term – the 5th term is 52, the 8th term is 82, …
Patterns themselves are not explicit or recursive. The RULE for the pattern can be expressed explicitly or recursively and MOST
patterns can be explained using either format especially IF that pattern reflects either an arithmetic sequence or geometric sequence.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks
are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be
used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to
Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models
benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each indicator in the Models
benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For example, process models are
linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and
Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at
three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the
Appendix.
6-207
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
SIXTH GRADE
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 4: Models – The student generates and uses mathematical models to represent and justify mathematical
relationships in a variety of situations.
Sixth Grade Knowledge Base Indicators
Sixth Grade Application Indicators
The student…
1. knows, explains, and uses mathematical models to represent
mathematical concepts, procedures, and relationships.
Mathematical models include:
a. process models (concrete objects, pictures, diagrams, number
lines, hundred charts, measurement tools, multiplication arrays,
division sets, or coordinate planes/grids) to model
computational procedures and mathematical relationships and
to solve equations (1.1.K1-5, 1.2.K1, 1.3.K1-4, 1.4.K1, 1.4.K2a,
1.4.K2c-e, 1.4.K2g, 1.4.K2i, 1.4.K6, 2.1.K1a-b, 2.1.K1d-e,
2.1.K2-4, 2.2.K1-6, 2.3.K1, 2.3.K3-4, 3.2.K1-4, 3.2.K8, 3.3.K14, 3.4.K1-3, 4.2.K4) ($);
b. place value models (place value mats, hundred charts, base
ten blocks, or unifix cubes) to compare, order, and represent
numerical quantities and to model computational procedures
(1.1.K1-4, 1.2.K1, 1.3.K1-3, 1.4.K2b, 1.4.K2c-d, 2.2K4) ($);
c. fraction and mixed number models (fraction strips or pattern
blocks) and decimal and money models (base ten blocks or
coins) to compare, order, and represent numerical quantities
(1.1.K1-4, 1.2.K1, 1.3.K1-3, 1.4.K2b, 1.4.K2d, 1.4.K2f, 1.4.K6,
2.2.K5, 4.1.K4, 4.2.K4) ($):
The student…
1. recognizes that various mathematical models can be used to
represent the same problem situation. Mathematical models include:
a. process models (concrete objects, pictures, diagrams, number
lines, hundred charts, measurement tools, multiplication arrays,
division sets, or coordinate planes/grids) to model computational
procedures and mathematical relationships, to represent problem
situations, and to solve equations (1.1.A1, 1.1.A1a, 1.2.A1-2,
1.3.A1-4, 1.4.A1a-b, 2.1.A1-2, 2.1.A1-3, 3.2.A1a, 3.2.A1c,
3.2.A2, 3.3.A1-2, 3.4.A1-2, 4.2.A1) ($);
b. place value models (place value mats, hundred charts, base ten
blocks, or unifix cubes) to model problem situations (1.1.A1,
1.2.A1-2, 2.2.A3) ($);
c. fraction and mixed number models (fraction strips or pattern
blocks) and decimal and money models (base ten blocks or
coins) to compare, order, and represent numerical quantities
(1.1.A1, 1.1.A2b-c, 1.2.A1-2, 1.4.A1b-c) ($);
d. factor trees to find least common multiple and greatest common
factor;
e. equations and inequalities to model numerical relationships
(2.2.A1-3) ($);
f. function tables (input/output machines, T-tables) to model
numerical and algebraic relationships (2.3.A1-2) ($);
g. two-dimensional geometric models (geoboards or dot paper) to
model perimeter, area, and properties of geometric shapes and
three-dimensional geometric models (nets or solids) and realworld objects to model volume and to identify attributes (faces,
edges, vertices, bases) of geometric shapes (3.1.A1-3. 3.2.A1b,
3.4.A2);
d. factor trees to find least common multiple and greatest
common factor (1.4.K4-5);
e. equations and inequalities to model numerical relationships
(2.2.K3,) ($);
f. function tables (input/output machines, T-tables) to model
numerical and algebraic relationships (2.3.K2, 2.3.K4) ($);
6-208
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
g. two-dimensional geometric models (geoboards or dot paper) to
model perimeter, area, and properties of geometric shapes and
three-dimensional geometric models (nets or solids) and realworld objects to model volume and to identify attributes (faces,
edges, vertices, bases) of geometric shapes (2.1.K1c, 3.1.K15, 3.1.K7-10, 3.2.K7, 3.3.K1-4);
h. tree diagrams to organize attributes and determine the number
of possible combinations (4.1.K2);
i. graphs using concrete objects, two- and three-dimensional
geometric models (spinners or number cubes) and process
models (concrete objects, pictures, diagrams, or coins) to
model probability (4.1.K1-4) ($).
j. frequency tables, bar graphs, line graphs, circle graphs, Venn
diagrams, line plots, charts, tables, single stem-and-leaf plots,
and scatter plots to organize and display data (4.2.K1-3) ($);
k. Venn diagrams to sort data and to show relationships (1.2.K1).
2. uses one or more mathematical models to show the relationship
between two or more things.
h. scale drawings to model large and small real-world objects
(3.4.A2);
i. tree diagrams to organize attributes and determine the number
of possible combinations;
j. two- and three-dimensional geometric models (spinners or
number cubes) and process models (concrete objects, pictures,
diagrams, or coins) to model probability (4.1.A1-3) ($);
k. graphs using concrete objects, frequency tables, bar graphs, line
graphs, circle graphs, Venn diagrams, line plots, charts, tables,
and single stem-and-leaf plots to organize, display, explain, and
interpret data (2.1.A1, 2.3.A1-2, 4.1.A1-2, 4.2.A1-3) ($);
l. Venn diagrams to sort data and to show relationships.
2. selects a mathematical model and justifies why some
mathematical models are more accurate than other
mathematical models in certain situations.
6-209
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: For assessment purposes, the mathematical modeling process appropriate to the indicator may be included as part of the item
being assessed.
The mathematical modeling process involves:
a. selecting key features and relationships within the real-world situation and representing these concepts in mathematical terms through
some sort of mathematical model,
b. performing manipulations and mathematical procedures within the mathematical model,
c. interpreting the results of the manipulations within the mathematical model,
d. using these results to make inferences about the original real-world situation.
The use of mathematical models is necessary for conceptual understanding. The ways in which mathematical ideas are
represented is fundamental to how students understand and use those ideas. As students begin to use multiple representations of
the same situation, they begin to develop an understanding of the advantages and disadvantages of various representations/models.
Many mathematical models are listed in this benchmark. The indicator lists some of the mathematical models that could be used to teach a
concept. Each indicator in this benchmark is linked to other indicators in other benchmarks; those indicators are identified in the parentheses. For
example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1
(Number Sense), and Knowledge Indicator 3. In addition, the indicator in the other benchmarks identifies, in parentheses, the Models’ indicator.
For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models).
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
6-210
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
SIXTH GRADE
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 1: Geometric Figures and Their Properties – The student recognizes geometric figures and compares
their properties in a variety of situations.
Sixth Grade Knowledge Base Indicators
The student…
1. recognizes and compares properties of plane figures and solids
using concrete objects, constructions, drawings, and appropriate
technology (2.4.K1g).
2. recognizes and names regular and irregular polygons through 10
sides including all special types of quadrilaterals: squares,
rectangles, parallelograms, rhombi, trapezoids, kites (2.4.K1g).
3. names and describes the solids [prisms (rectangular and
triangular), cylinders, cones, spheres, and pyramids (rectangular
and triangular)] using the terms faces, edges, vertices, and bases
(2.4.K1g).
4. recognizes all existing lines of symmetry in two-dimensional figures
(2.4.K1g).
5. recognizes and describes the attributes of similar and congruent
figures (2.4.K1g).
6. recognizes and uses symbols for angle (find symbol for), line(↔),
line segment (―), ray (→), parallel (||), and perpendicular ().
7. ▲ classifies (2.4.K1g):
a. angles as right, obtuse, acute, or straight;
b. triangles as right, obtuse, acute, scalene, isosceles, or
equilateral.
8. identifies and defines circumference, radius, and diameter of circles
and semicircles.
9. recognize that the sum of the angles of a triangle equals 180°
(2.4.K1g).
10. determines the radius or diameter of a circle given one or the other.
Sixth Grade Application Indicators
The student…
2. solves real-world problems by applying the properties of (2.4.A1g):
a. plane figures (regular polygons through 10 sides, circles, and
semicircles) and the line(s) of symmetry, e.g., twins are having a
birthday party. The rectangular birthday cake is to be cut into
two equal sizes of the same shape. How would you cut the
cake?
b. solids (cubes, rectangular prisms, cylinders, cones, spheres,
triangular prisms) emphasizing faces, edges, vertices, and
bases, e.g., lace is to be glued on all of the edges of a cube. If
one edge measures 34 cm, how much lace is needed?
c. intersecting, parallel, and perpendicular lines, e.g., railroad
tracks form what type of lines? Two roads are perpendicular,
what is the angle between them?
2. decomposes geometric figures made from (2.4.A1g):
a. regular and irregular polygons through 10 sides, circles, and
semicircles, e.g., draw a picture of a house (rectangular base)
with a roof (triangle) and a chimney on the side of the roof
(trapezoid). Identify the three geometrical figures.
b. nets (two-dimensional shapes that can be folded into threedimensional figures), e.g., the cardboard net that becomes a
shoebox.
3. composes geometric figures made from (2.4.A1g):
a. regular and irregular polygons through 10 sides, circles,
and semicircles;
b. nets (two-dimensional shapes that can be folded into threedimensional figures).
6-211
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Geometry is the study of shapes, their properties, and their relationships to other shapes. Symbols and numbers are used to
describe their properties and their relationships to other shapes. The fundamental concepts in geometry are point (no dimension), line (onedimensional), plane (two-dimensional), and space (three-dimensional). Plane figures are referred to as two-dimensional. Solids are referred to as
three-dimensional. The base, in terms of geometry, generally refers to the side on which a figure rests. Therefore, depending on the orientation of
the solid, the base changes.
From the Mathematics Dictionary and Handbook (Nichols Schwartz Publishing, 1999), property as a mathematical term means a characteristic (an
attribute) of a number, geometric shape, mathematical operation, equation, or inequality. To give an example:
 Property of a number: 8 is divisible by 2.
 Property of a geometric shape: Each of the four sides of a square is of the same length.
 Property of an operation: Addition is commutative. For all numbers x and y, x + y = y + x.
 Property of an equation: For all numbers a, b, and c, if a = b, then a + c = b + c.
 Property of an inequality: For all numbers a, b, and c, if a > b, then a – c > b – c.
The application of the Knowledge Indicators from the Geometry Benchmark, Geometric Figures and Their Properties are most often applied within
the context of the other Geometry Benchmarks — Measurement and Estimation, Transformational Geometry, and Geometry From an Algebraic
Perspective — rather than in isolation.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
6-212
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
SIXTH GRADE
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 2: Measurement and Estimation – The student estimates, measures, and uses measurement formulas in
a variety of situations.
Sixth Grade Knowledge Base Indicators
The student…
1. determines and uses whole number approximations (estimations) for
length, width, weight, volume, temperature, time, perimeter, and
area using standard and nonstandard units of measure (2.4.K1a) ($).
2. selects, explains the selection of, and uses measurement tools, units
of measure, and level of precision appropriate for a given situation to
find accurate rational number representations for length, weight,
volume, temperature, time, perimeter, area, and angle
measurements (2.4.K1a) ($).
3. converts (2.4.K1a):
a. within the customary system, e.g., converting feet to inches,
inches to feet, gallons to pints, pints to gallons, ounces to
pounds, or pounds to ounces;
b. ▲ within the metric system using the prefixes: kilo, hecto, deka,
deci, centi, and milli; e.g., converting millimeters to meters,
meters to millimeters, liters to kiloliters, kiloliters to liters,
milligrams to grams, or grams to milligrams.
4. uses customary units of measure to the nearest sixteenth of an inch
and metric units of measure to the nearest millimeter (2.4.K1a).
5. recognizes and states perimeter and area formulas for squares,
rectangles, and triangles (2.4.K1g).
a. uses given measurement formulas to find perimeter and area of:
squares and rectangles,
b. figures derived from squares and/or rectangles.
6. describes the composition of the metric system (2.4.K1a):
a. meter, liter, and gram (root measures);
b. kilo, hecto, deka, deci, centi, and milli (prefixes).
7. finds the volume of rectangular prisms using concrete objects
(2.4.K1g).
8. estimates an approximate value of the irrational number pi (2.4.K1a).
Sixth Grade Application Indicators
The student…
1. solves real-world problems by applying these measurement
formulas ($):
a. ▲ perimeter of polygons using the same unit of measurement
(2.4.A1a,g), e.g., measures the length of fence around a yard;
b. ▲ ■ area of squares, rectangles, and triangles using the same
unit of measurement (2.4.A1g), e.g., finds the area of a room for
carpeting;
c. conversions within the metric system (2.4.A1a), e.g., your school
is having a balloon launch. Each student needs 40 centimeters
of string, and there are 42 students. How many meters of string
are needed?
2. estimates to check whether or not measurements and calculations
for length, width, weight, volume, temperature, time, perimeter, and
area in real-world problems are reasonable and adjusts original
measurement or estimation based on additional information (a frame
of reference) (2.4.A1a) ($), e.g., students estimate, in feet, the height
of a bookcase in their classroom. Then a student who is about 5
feet tall stands beside it. The students then adjust the estimate.
6-213
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: The term geometry comes from two Greek words meaning “earth measure.” Measurement provides the tools required to apply
geometric concepts in the real-world. Estimation in measurement is defined as making guesses as to the exact measurement of an object without
using any type of measurement tool. Estimation helps students develop a relationship between the different sizes of units of measure. It helps
students develop basic properties of measurement and it gives students a tool to determine whether a given measurement is reasonable.
Using concrete objects (mathematical models) to conduct experiments to compare the circumference and the radius/diameter of a circle is important
in developing students’ understanding of pi, its approximate value, and the circumference formula. The circumference formula will be emphasized
in the seventh grade.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that
could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are
identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and
Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
.
6-214
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
SIXTH GRADE
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 3: Transformational Geometry – The student recognizes and performs transformations on two- and
three-dimensional geometric figures in a variety of situations.
Sixth Grade Knowledge Base Indicators
The student…
1. ▲ ■ identifies, describes, and performs one or two transformations
(reflection, rotation, translation) on a two-dimensional figure
(2.4.K1a).
2. reduces (contracts/shrinks) and enlarges (magnifies/grows) simple
shapes with simple scale factors (2.4.K1a), e.g., tripling or halving.
3. recognizes three-dimensional figures from various perspectives (top,
bottom, sides, corners) (2.4.K1a).
4. recognizes which figures will tessellate (2.4.K1a).
Sixth Grade Application Indicators
The student…
1. describes a transformation of a given two-dimensional figure that
moves it from its initial placement (preimage) to its final placement
(image) (2.4.A1a).
2. makes a scale drawing of a two-dimensional figure using a simple
scale (2.4.A1a), e.g., using the scale 1 cm = 30 m, the student
makes a scale drawing of the school.
Teacher Notes: Transformational geometry is another way to investigate and interpret geometric figures by moving every point in a plane figure
to a new location. To help students form images of shapes through different transformations, students can use concrete objects, figures drawn on
graph paper, mirrors or other reflective surfaces, or appropriate technology. Some transformations, like translations, reflections, and rotations, do
not change the figure itself. Other transformations like reduction (contraction/shrinking) or enlargement (magnification/growing) change the size of
a figure, but not the shape (congruence vs. similarity).
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and ) and are grouped into four major categories: Income, Spending and Credit, Saving and Investing,
and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in the
Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level with
a ($). The National Standards in Personal Finance are included in the Appendix.
6-215
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
SIXTH GRADE
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 4:
Geometry From An Algebraic Perspective – The student relates geometric concepts to a number line and a
coordinate plane in a variety of situations.
Sixth Grade Knowledge Base Indicators
The student…
1. uses a number line (horizontal/vertical) to order integers and positive
rational numbers (in both fractional and decimal form) (2.4.K1a).
2. organizes integer data using a T-table and plots the ordered pairs in
all four quadrants of a coordinate plane (coordinate grid) (2.4.K1a).
3. ▲ uses all four quadrants of the coordinate plane to (2.4.K1a):
a. identify the ordered pairs of integer values on a given graph;
b. plot the ordered pairs of integer values.
Sixth Grade Application Indicators
The student…
1. represents, generates, and/or solves real-world problems using a
number line with integer values (2.4.A1a) ($), e.g., the difference
between –2 degrees and 10 degrees on a thermometer is 12
degrees (units); similarly, the distance between –2 to +10 on a
number line is 12 units.
2. represents and/or generates real-world problems using a
coordinate plane with integer values to find (2.4.A1a,g-h):
a. the perimeter of squares and rectangles, e.g., Alice made a
scale drawing of her classroom and put it on a coordinate plane
marked off in feet. The rectangular table in the back of the
room was described by the points (8,9), (8,12), (14,12) and
(14,9). Now Alice wants to put a skirting around the outer edge
of the table. Using the drawing, find the amount of skirting she
will need.
b. the area of triangles, squares, and rectangles, e.g., a scale
drawing of a flower garden is found in a book with the
coordinates of the four corners being (9,5), (9,13), (18,13) and
(18,5). The scale is marked off in meters. How many square
meters is the flower garden?
6-216
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: A number line (a mathematical model) is a diagram that represents numbers with equal distances marked off as points on a line,
and is an example of one-to-one correspondence (a relation). A number line can be used as a visual representation of numbers and operations. In
addition, a number line used horizontally and vertically is a precursor to the coordinate plane; and the distance between two numbers on a number
line is a precursor to absolute value.
A coordinate plane (coordinate grid) consists of a horizontal number line called the x-axis and a vertical number line called the y-axis. These two
lines intersect at a point called the origin. The x-axis and the y-axis divide the plane into four sections called quadrants. Any point on the
coordinate plane can be named with two numbers called coordinates. The first number is the x-coordinate. The second number is the ycoordinate. Since the pair is always named in order (first x, then y), it is called an ordered pair.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
6-217
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 4: Data
SIXTH GRADE
Standard 4: Data – The student uses concepts and procedures of data analysis in a variety of situations.
Benchmark 1: Probability – The student applies the concepts of probability to draw conclusions and to make
predictions and decisions including the use of concrete objects in a variety of situations.
Sixth Grade Knowledge Base Indicators
The student…
1. recognizes that all probabilities range from zero (impossible) through
one (certain) and can be written as a fraction, decimal, or a percent
(2.4.K1i) ($) , e.g., when you flip a coin, the probability of the coin
landing on heads (or tails) is ½, .5, or 50%. The probability of
flipping a head on a two-headed coin is 1. The probability of flipping
a tail on a two-headed coin is 0.
2. ▲ ■ lists all possible outcomes of an experiment or simulation with a
compound event composed of two independent events in a clear
and organized way (2.4.K1h-j), e.g., use a tree diagram or list to find
all the possible color combinations of pant and shirt ensembles, if
there are 3 shirts (red, green, blue) and 2 pairs of pants (black and
brown).
3. recognizes whether an outcome in a compound event in an
experiment or simulation is impossible, certain, likely, unlikely, or
equally likely (2.4.K1i).
4. ▲ represents the probability of a simple event in an experiment or
simulation using fractions and decimals (2.4.K1c,i), e.g., the
probability of rolling an even number on a single number cube is
represented by ½ or .5.
Sixth Grade Application Indicators
The student…
1. conducts an experiment or simulation with a compound event
composed of two independent events including the use of concrete
objects; records the results in a chart, table, or graph; and uses the
results to draw conclusions about the events and make predictions
about future events (2.4.A1j-k).
2. analyzes the results of a given experiment or simulation of a
compound event composed of two independent events to draw
conclusions and make predictions in a variety of real-world situations
(2.4.A1j-k), e.g., given the equal likelihood that a customer will order
a pizza with either thick or thin crust, and an equal probability that a
single topping of beef, pepperoni, or sausage will be selected –
1) What is the probability that a pizza ordered will be thin crust with
beef topping?
2) Given sales of 30 pizzas on a Friday night, how many would the
manager expect to be thin crust with beef topping?
3. compares what should happen (theoretical probability/expected
results) with what did happen (empirical probability/experimental
results) in an experiment or simulation with a compound event
composed of two independent events (2.4.A1j).
6-218
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Ideas from probability reinforce concepts in the other Standards, especially Number and Computation and Geometry. Students
need to develop an intuitive concept of chance – whether or not something is unlikely or likely to happen. Probability experiences should be
addressed through the use of concrete objects (process models); spinners, number cubes, or dartboards (geometric models); and coins (money
models). Probabilities are ratios, expressed as fractions, decimals, or percents, determined by considering results or outcomes of experiments.
Some examples of uses of probability in every day life include: There is a 50% chance of rain today. What is the probability that the team will win
every game?
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that
could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are
identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and
Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
6-219
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 4: Data
SIXTH GRADE
Standard 4: Data – The student uses concepts and procedures of data analysis in a variety of situations.
Benchmark 2: Statistics – The student collects, organizes, displays, and explains numerical (rational numbers) and
non-numerical data sets in a variety of situations with a special emphasis on measures of central tendency.
Sixth Grade Knowledge Base Indicators
The student…
1. organizes, displays, and reads quantitative (numerical) and qualitative
(non-numerical) data in a clear, organized, and accurate manner
including a title, labels, categories, and rational number intervals using
these data displays (2.4.K1j) ($):
a. graphs using concrete objects;
b. frequency tables and line plots;
c. bar, line, and circle graphs;
d. Venn diagrams or other pictorial displays;
e. charts and tables;
f. single stem-and-leaf plots;
g. scatter plots;
2. selects and justifies the choice of data collection techniques
(observations, surveys, or interviews) and sampling techniques
(random sampling, samples of convenience, or purposeful sampling)
in a given situation (2.4.K1j).
3. uses sampling to collect data and describe the results (2.4.K1j) ($).
4. determines mean, median, mode, and range for (2.4.K1a,c) ($):
a. a whole number data set,
Sixth Grade Application Indicators
The student…
1. uses data analysis (mean, median, mode, range) of a whole
number data set or a decimal data set with decimals greater than
or equal to zero to make reasonable inferences, predictions, and
decisions and to develop convincing arguments from these data
displays (2.4.A1k) ($):
a. graphs using concrete objects;
b. frequency tables and line plots;
c. bar, line, and circle graphs;
d. Venn diagrams or other pictorial displays;
e. charts and tables;
f. single stem-and-leaf plots.
2. explains advantages and disadvantages of various data displays
for a given data set (2.4.A1k) ($).
3. recognizes and explains the effects of scale and/or interval
changes on graphs of whole number data sets (2.4.A1k).
b. a decimal data set with decimals greater than or equal to
zero.
6-220
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Graphs (data displays) are pictorial representations of mathematical relationships, are used to tell a story, and are an important
part of statistics. When a graph is made, the axes and the scale (numbers running along a side of the graph) are chosen for a reason. The
difference between numbers from one grid line to another is the interval. The interval will depend on the lowest and highest values in the data set.
Emphasizing the importance of using equal-sized pictures or intervals is critical to ensuring that the data display is accurate.
Graphs take many forms:
– bar graphs and pictographs compare discrete data,
– frequency tables show how many times a certain piece of data occurs,
– circle graphs (pie charts) model parts of a whole,
– line graphs show change over time,
– Venn diagrams show relationships between sets of objects,
– line plots show frequency of data on a number line,
– single stem-and-leaf plots (closely related to line plots except that the number line is usually vertical and digits are used rather than x’s) show
frequency distribution by arranging numbers (stems) on the left side of a vertical line with numbers (leaves) on the right side, and
– scatter plots show the relationship between two quantities.
An important aspect of data is its center. The measures of central tendency (averages) of a data set are mean, median, and mode. Conceptual
understanding of mean, median, and mode is developed through the use of concrete objects that represent the data values.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that
could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are
identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and
Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
6-221
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
SEVENTH GRADE
Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 1: Number Sense – The student demonstrates number sense for rational numbers, the irrational number
pi, and simple algebraic expressions in one variable in a variety of situations.
Seventh Grade Knowledge Base Indicators
Seventh Grade Application Indicators
The student…
The student…
1. knows, explains, and uses equivalent representations for rational
numbers and simple algebraic expressions including integers,
fractions, decimals, percents, and ratios; integer bases with whole
number exponents; positive rational numbers written in scientific
notation with positive integer exponents; time; and money (2.4.K1ac) ($), e.g., 253,000 is equivalent to 2.53 x 105 or x + 5x is equivalent
to 6x.
2. compares and orders rational numbers and the irrational number pi
(2.4.K1a) ($).
3. explains the relative magnitude between rational numbers and
between rational numbers and the irrational number pi (2.4.K1a).
4. knows and explains what happens to the product or quotient when
(2.4.K1a):
a. a whole number is multiplied or divided by a rational number
greater than zero and less than one,
b. a whole number is multiplied or divided by a rational number
greater than one,
c. a rational number (excluding zero) is multiplied or divided by
zero.
5. explains and determines the absolute value of rational numbers
(2.4.K1a).
1. generates and/or solves real-world problems using (2.4.A1a) ($):
a. ▲equivalent representations of rational numbers and simple
algebraic expressions, e.g., you are in the mountains. Wilson
Mountain has an altitude of 5.28 x 103 feet. Rush Mountain is
4,300 feet tall. How much higher is Wilson Mountain than Rush
Mountain?
b. fraction and decimal approximations of the irrational number pi,
e.g., Mary measured the distance around her 48-inch diameter
circular table to be 150 inches. Using this information,
approximate pi as a fraction and as a decimal.
2. determines whether or not solutions to real-world problems using
rational numbers, the irrational number pi, and simple algebraic
expressions are reasonable (2.4.A1a) ($), e.g., a sweater that cost
$15 is marked 1/3 off. The cashier charged $12. Is this reasonable?
7-222
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Number sense refers to one’s ability to reason with numbers and to work with numbers in a flexible way. The ability to compute
mentally, to estimate based on understanding of number relationships and magnitudes, and to judge reasonableness of answers are all involved in
number sense.
The student with number sense will look at a problem holistically before confronting the details of the problem. The student will look for
relationships among the numbers and operations and will consider the context in which the question was posed. Students with number sense will
choose or even invent a method that takes advantage of their own understanding of the relationships between numbers and between numbers and
operations, and they will seek the most efficient representation for the given task. Number sense can also be recognized in the students' use of
benchmarks to judge number magnitude (e.g., 2/5 of 49 is less that half of 49), to recognize unreasonable results for calculations, and to employ
non-standard algorithms for mental computation and estimation. (Developing Number Sense: Addenda Series, Grades 5-8, NCTM, 1991)
At this grade level, rational numbers include positive and negative numbers and very large numbers [ten million (10 7)] and very small
numbers [hundred-thousandth (10-5)]. Relative magnitude refers to the size relationship one number has with another – is it much
larger, much smaller, close, or about the same? For example, using the numbers 219, 264, and 457, answer questions such as
 Which two are closest? Why?
 Which is closest to 300? To 250?
 About how far apart are 219 and 500? 5,000?
 If these are ‘big numbers,’ what are small numbers? Numbers about the same? Numbers that make these seem
small?
(Elementary and Middle School Mathematics, John A. Van de Walle, Addison Wesley Longman, Inc., 1998)
Mathematical models such as concrete objects, pictures, diagrams, Venn diagrams, number lines, hundred charts, base ten blocks, or factor trees
are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to
represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2
(Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the
mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other
indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3
referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
7-223
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
SEVENTH GRADE
Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 2: Number Systems and Their Properties – The student demonstrates an understanding of the rational
number system and the irrational number pi; recognizes, uses, and describes their properties; and
extends these properties to algebraic expressions in one variable.
Seventh Grade Knowledge Base Indicators
The student…
4. knows and explains the relationships between natural (counting)
numbers, whole numbers, integers, and rational numbers using
mathematical models (2.4.K1a,k), e.g., number lines or Venn
diagrams.
5. classifies a given rational number as a member of various subsets of
the rational number system (2.4.K1a,k), e.g., - 7 is a rational number
and an integer.
6. names, uses, and describes these properties with the rational number
system and demonstrates their meaning including the use of concrete
objects (2.4.K1a) ($):
a. commutative properties of addition and multiplication (changing
the order of the numbers does not change the solution);
b. associative properties of addition and multiplication (changing the
grouping of the numbers does not change the solution);
c. distributive property [distributing multiplication or division over
addition or subtraction, e.g., 2(4 – 1) = 2(4) – 2(1) = 8 – 2 = 6];
d. substitution property (one name of a number can be substituted
for another name of the same number), e.g., if a = 2, then 3a =
3 x 2 = 6.
7. uses and describes these properties with the rational number system
and demonstrates their meaning including the use of concrete objects
(2.4.K1a) ($):
a. identity properties for addition and multiplication (additive identity
– zero added to any number is equal to that number;
Seventh Grade Application Indicators
The student…
1. generates and/or solves real-world problems with rational numbers
and the irrational number pi using the concepts of these properties
to explain reasoning (2.4.K1a) ($):
a. commutative and associative properties of addition and
multiplication, e.g., at a delivery stop, Sylvia pulls out a flat of
eggs. The flat has 5 columns and 6 rows of eggs. Express
how to find the number of eggs in 2 ways.
b. distributive property, e.g., trim is used around the outside
edges of a bulletin board with dimensions 3 ft by 5 ft. Explain
two different methods of solving this problem.
c. substitution property, e.g., V = IR [Ohm’s Law: voltage (V) =
current (I) x resistance (R)] If the current is 5 amps (I = 5) and
the resistance is 4 ohms (R = 4), what is the voltage?
d. symmetric property of equality, e.g., Sam took a $15 check to
the bank and received a $10 bill and a $5 bill. Later Sam took
a $10 bill and a $5 bill to the bank and received a check for
$15. $15 = $10 + $5 is the same as $10 + $5 = $15.
e. additive and multiplicative identities, e.g., Bob and Sue each
read the same number of books. During the week, they each
read 5 more books. Compare the number of books each read.
Let b=the number of books Bob read and s=the number of
books Sue read, then b+5=s+5
f. zero property of multiplication, e.g., Jenny was thinking of two
numbers. Jenny said that the product of the two numbers was
7-224
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
multiplicative identity – one multiplied by any number is equal to
0. What could you deduct from this statement? Explain your
that number);
reasoning.
b. symmetric property of equality (if 7 + 2x = 9 then 9 = 7 + 2x);
c. zero property of multiplication (any number multiplied by zero is
g. addition and multiplication properties of equality, e.g., the total
zero);
price (P) of a car, including tax (T), is $14, 685. 33. If the tax is
d. addition and multiplication properties of equality
$785.42, what is the sale price of the car (S)?
(adding/multiplying the same number to each side of an equation
h. additive and multiplicative inverse properties, e.g., if 5 candy
results in an equivalent equation);
bars cost $1.00, what does one candy bar cost? Explain your
e. additive and multiplicative inverse properties. (Every number has
reasoning.
a value known as its additive inverse and when the original
2. analyzes and evaluates the advantages and disadvantages of
number is added to that additive inverse, the answer is zero, e.g.,
using integers, whole numbers, fractions (including mixed
+5 + –5 = 0. Every number except 0 has a value known as its
numbers), decimals, or the irrational number pi and its rational
multiplicative inverse and when the original number multiplied by
approximations in solving a given real-world problem (2.4.K1a,
its inverse, the answer will be 1, e.g., 8 x 1/8 =1.)
e.g., in the store everything is 25% off. When calculating the
5. recognizes that the irrational number pi can be represented by
discount, which representation of 25% would you use and why?
approximate rational values, e.g., 22/7 or 3.14.
Teacher Notes: From the Mathematics Dictionary and Handbook (Nichols Schwartz Publishing, 1999), property as a mathematical term means a
characteristic (an attribute) of a number, geometric shape, mathematical operation, equation, or inequality. To give an example:
 Property of a number: 8 is divisible by 2.
 Property of a geometric shape: Each of the four sides of a square is of the same length.
 Property of an operation: Addition is commutative. For all numbers x and y, x + y = y + x.
 Property of an equation: For all numbers a, b, and c, if a = b, then a + c = b + c.
 Property of an inequality: For all numbers a, b, and c, if a > b, then a – c > b – c.
Mathematical models such as concrete objects, pictures, diagrams, Venn diagrams, number lines, hundred charts, base ten blocks, or factor trees
are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to
represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2
(Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the
mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other
indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3
referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
7-225
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
7-226
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
SEVENTH GRADE
Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 3: Estimation – The student uses computational estimation with rational numbers and the irrational
number pi in a variety of situations.
Seventh Grade Knowledge Base Indicators
Seventh Grade Application Indicators
The student…
1. estimates quantities with combinations of rational numbers and/or
the irrational number pi using various computational methods
including mental math, paper and pencil, concrete objects, and/or
appropriate technology (2.4.K1a) ($).
2. N uses various estimation strategies and explains how they were
used to estimate rational number quantities and the irrational
number pi (2.4.K1a) ($).
3. recognizes and explains the difference between an exact and
approximate answer (2.4.K1a).
4. determines the appropriateness of an estimation strategy used and
whether the estimate is greater than (overestimate) or less than
(underestimate) the exact answer and its potential impact on the
result (24.K1a).
5. knows and explains why the fraction (22/7) or decimal (3.14)
representation of the irrational number pi is an approximate value
(2.4.K1c).
The student…
1. adjusts original rational number estimate of a real-world problem
based on additional information (a frame of reference) (2.4.A1a) ($),
e.g., estimate the weight of a bookshelf of books. Then weigh one
book and adjust your estimate.
2. estimates to check whether or not the result of a real-world problem
using rational numbers, the irrational number pi, and/or simple
algebraic expressions is reasonable and makes predictions based
on the information (2.4.A1a), e.g., a goat is staked out in a pasture
with a rope that is 7 feet long. The goat needs 200 square feet of
grass to graze. Does the goat have enough pasture? If not, how
long should the rope be?
3. determines a reasonable range for the estimation of a quantity given
a real-world problem and explains the reasonableness of the range
(2.4.A1a), e.g., how long will it take your teacher to walk two miles?
The range could be 25-35 minutes.
4. determines if a real-world problem calls for an exact or approximate
answer and performs the appropriate computation using various
computational methods including mental math, paper and pencil,
concrete objects, and/or appropriate technology (2.4.A1a) ($), e.g.,
Kathy buys items at the grocery store priced at $32.56, $12.83,
$6.99, and 5 for $12.49 each. She has $120 with her to pay for the
groceries. To decide if she can pay for her items, does she need an
exact or an approximate answer?
7-227
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Estimate, as a verb, means to make an educated guess based on information in a problem or to give an answer
close to the exact number. Estimation is used when an exact answer is not needed, as in many real-life situations for which “ballpark”
computations are acceptable. Good number sense enables one to estimate a quantity, estimate a measure, or estimate an answer.
Estimation serves as an important companion to computation. It provides a tool for judging the reasonableness of computational
methods including mental math, paper and pencil, concrete objects, and appropriate technology. However, being able to compute
does not automatically lead to an ability to estimate or judge reasonableness of answers. Frequent modeling by the teacher helps
students develop a range of estimation strategies. Students should be encouraged to frequently explain their thinking as they
estimate. As with exact computation, sharing estimation strategies allows students access to others’ thinking and provides
opportunities for class discussion. Identifying the estimation strategy by name is not critical; however, explaining the thinking behind
the strategy to make a valid estimation is important. (Principles and Standards for School Mathematics, NCTM, 2000)
Mental math and estimation are distinct but related mathematical skills. Proficiency in mental math contributes to increased skill in
estimation. In order for students to become more familiar with estimation, teachers should introduce estimation with examples where
rounded or estimated numbers are used. Emphasis should be placed on real-world examples where only estimation is required, e.g.,
About how many hours do you sleep a night? Using the language of estimation is important, so students begin to realize that a variety
of estimates (answers) are possible.
In addition, when students are taught specific estimation strategies, they develop mental math and estimation skills easier. Estimation
strategies include front-end with adjustment, compatible “nice” numbers, clustering, special numbers, or truncation.
Mathematical models such as concrete objects, pictures, diagrams, Venn diagrams, number lines, hundred charts, base ten blocks,
or factor trees are necessary for conceptual understanding and should be used to explain computational procedures. If a
mathematical model can be used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For
example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the
indicator in the Models benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each
indicator in the Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For
example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation),
Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at
three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the
Appendix.
7-228
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
7-229
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
SEVENTH GRADE
Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 4: Computation – The student models, performs, and explains computation with rational numbers, the
irrational number pi, and first-degree algebraic expressions in one variable in a variety of situations.
7-230
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Seventh Grade Application Indicators
Seventh Grade Knowledge Base Indicators
The student…
1. computes with efficiency and accuracy using various computational
methods including mental math, paper and pencil, concrete objects,
and appropriate technology (2.4.K1a-c) ($).
2. performs and explains these computational procedures (2.4.K1a):
a. ▲N adds and subtracts decimals from ten millions place through
hundred thousandths place;
b. ▲N multiplies and divides a four-digit number by a two-digit
number using numbers from thousands place through
thousandths place;
c. ▲N multiplies and divides using numbers from thousands place
through thousandths place by 10; 100; 1,000; .1; .01; .001; or
single-digit multiples of each, e.g., 54.2 ÷ .002 or 54.3 x 300;
d. ▲N adds, subtracts, multiplies, and divides fractions and
expresses answers in simplest form;
e. N adds, subtracts, multiplies, and divides integers;
f. N uses order of operations (evaluates within grouping symbols,
evaluates powers to the second or third power, multiplies or
divides in order from left to right, then adds or subtracts in order
from left to right) using whole numbers;
g. simplifies positive rational numbers raised to positive whole
number powers;
h. combines like terms of a first degree algebraic expression.
recognizes, describes, and uses different ways to express computational
procedures, e.g., 5 – 2 = 5 + (–2) or 49 x 23 = (40 x 23) + (9 x 23) or
49 x 23 = (49 x 20) + (49 x 3) or 49 x 23 = (50 x 23) – 23.
finds prime factors, greatest common factor, multiples, and the least
common multiple (2.4.K1d).
The student…
1. generates and/or solves one- and two-step real-world problems
using these computational procedures and mathematical
concepts (2.4.A1a) ($):
a. ■ addition, subtraction, multiplication, and division of rational
numbers with a special emphasis on fractions and expressing
answers in simplest form, e.g., at the candy store, you buy ¾
of a pound of peppermints and ½ of a pound of licorice. The
cost per pound for each kind of candy is $3.00. What is the
total cost of the candy purchased?
b. addition, subtraction, multiplication, and division of rational
numbers with a special emphasis on integers, e.g., the high
temperatures for the week were: 4o, 10o, 1o, 0o, 7o, 3o, and
–5o. What is the mean temperature for the week?
c. first degree algebraic expressions in one variable, e.g., Jenny
rents 3 videos plus $20 of other merchandise. Barb rents 5
videos plus $15 of other merchandise. Represent the total
purchases of Jenny and Barb using V as the price of a video
rental.
d. percentages of rational numbers, e.g., if the sales tax is 5.5%,
what is the sales tax on an item that costs $36?
e. approximation of the irrational number pi, e.g., what is the
approximate diameter of a 400-meter circular track?
5. ▲ finds percentages of rational numbers (2.4.K1a,c) ($), e.g., 12.5% x
$40.25 = n or 150% of 90 is what number? (For the purpose of
assessment, percents will not be between 0 and 1.)
7-231
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Efficiency and accuracy means that students are able to compute single-digit numbers with fluency. Students increase their
understanding and skill in addition, subtraction, multiplication, and division by understanding the relationships between addition and subtraction,
addition and multiplication, multiplication and division, and subtraction and division. Students learn basic number combinations and develop
strategies for computing that makes sense to them. Through class discussions, students can compare the ease of use and ease of explanation of
various strategies. In some cases, their strategies for computing will be close to conventional algorithms; in other cases, they will be quite different.
Many times, students’ invented approaches are based on a sound understanding of numbers and operations, and these invented approaches often
can be used with efficiency and accuracy. (Principles and Standards for School Mathematics, NCTM, 2000)
The definition of computation is finding the standard representation for a number. For example, 6 + 6, 4 x 3, 17 – 5, and 24 ÷ 2 are all
representations for the standard representation of 12. Mental math is mentally finding the standard representation for a number – calculating in
your head instead of calculating using paper and pencil or technology. One of the main reasons for teaching mental math is to help students
determine if a computed/calculated answer is reasonable; in other words, using mental math to estimate to see if the answer makes sense.
Students develop mental math skills easier when they are taught specific strategies. Mental math strategies include counting on, doubling,
repeated doubling, halving, making tens, multiplying by powers of ten, dividing with tens, thinking money, and using compatible “nice” numbers.
Mathematical models such as concrete objects, pictures, diagrams, Venn diagrams, number lines, hundred charts, base ten blocks,
or factor trees are necessary for conceptual understanding and should be used to explain computational procedures. If a
mathematical model can be used to represent the concept, the indicator in the Models benchmark is identified in the parentheses.
For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then,
the indicator in the Models benchmark lists some of the mathematical models that could be used to teach the concept. In addition,
each indicator in the Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For
example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation),
Benchmark 1 (Number Sense), and Knowledge Indicator 3.
Technology is changing mathematics and its uses. The use of technology including calculators and computers is an important part of
growing up in a complex society. It is not only necessary to estimate appropriate answers accurately when required, but also it is also
important to have a good understanding of the underlying concepts in order to know when to apply the appropriate procedure.
Technology does not replace the need to learn basic facts, to compute mentally, or to do reasonable paper-and-pencil computation.
However, dividing a 5-digit number by a 2-digit number is appropriate with the exception of dividing by 10, 100, or 1,000 and simple
multiples of each.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked
at three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the
Appendix.
7-232
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
7-233
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
SEVENTH GRADE
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 1: Patterns – The student recognizes, describes, extends, develops, and explains the general rule of a
pattern in a variety of situations.
Seventh Grade Knowledge Base Indicators
Seventh Grade Application Indicators
The student…
1. identifies, states, and continues a pattern presented in various
formats including numeric (list or table), algebraic (symbolic
notation), visual (picture, table, or graph), verbal (oral description),
kinesthetic (action), and written using these attributes:
a. ▲ counting numbers including perfect squares, cubes, and
factors and multiples (number theory) (2.4.K1a);
b. ▲ positive rational numbers including arithmetic and geometric
sequences (arithmetic: sequence of numbers in which the
difference of two consecutive numbers is the same, geometric: a
sequence of numbers in which each succeeding term is
obtained by multiplying the preceding term by the same number)
(2.4.K1a), e.g., 2, ½, 1/8, 1/32, …;
c. geometric figures (2.4.K1f);
d. measurements (2.4.K1a);
e. things related to daily life (2.4.K1a) ($), e.g., tide, moon cycle, or
temperature.
2. generates a pattern (2.4.K1a).
3. extends a pattern when given a rule of one or two simultaneous
changes (addition, subtraction, multiplication, division) between
consecutive terms (2.4.K1a), e.g., find the next three numbers in a
pattern that starts with 3, where you double and add 1 to get the
next number; the next three numbers are 7, 15, and 31.
4. ▲ ■ states the rule to find the nth term of a pattern with one
operational change (addition or subtraction) between consecutive
terms (2.4.K1a), e.g., given 3, 5, 7, and 9; the nth term is 2n +1. (This
is the explicit rule for the pattern.)
The student…
1. generalizes a pattern by giving the nth term using symbolic notation
(2.4.A1a,f), e.g., given the following, the nth term is 2n.
1 person has 2 eyes
2 people have 4 eyes
3 people have 6 eyes
4 people have 8 eyes
.
.
.
.
.
.
n people have 2n eyes
X
1
2
3
4
.
.
.
n
Y
2
4
6
8
.
.
.
2n
2. recognizes the same general pattern presented in different
representations [numeric (list or table), visual (picture, table, or
graph), and written] (2.4.A1a,f,j-k) ($).
7-234
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
7-235
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: A fundamental component in the development of classification, number, and problem solving skills is inventing,
discovering, and describing patterns. Patterns pervade all of mathematics and much of nature. All patterns are either repeating or
growing or a variation of either or both. Translating a pattern from one medium to another to find two patterns that are alike, even
though they are made with different materials, is important so students can see the relationships that are critical to repeating patterns.
With growing patterns, students not only extend patterns, but also look for a generalization or an algebraic relationship that will tell
them what the pattern will be at any point along the way.
Working with patterns is an important process in the development of mathematical thinking. Patterns can be based on geometric attributes
(shapes, regions, angles); measurement attributes (color, texture, length, weight, volume, number); relational attributes (proportion, sequence,
functions); and affective attributes (values, likes, dislikes, familiarity, heritage, culture). (Learning to Teach Mathematics, Randall J. Souviney,
Macmillan Publishing Company, 1994)
In the pattern that begins with 3, 5, 7, and 9; the explicit rule is 2n +1 and the recursive rule is add 2 to the previous term. Patterns
themselves are not explicit or recursive. The RULE for the pattern can be expressed explicitly or recursively and MOST patterns can
be explained using either format especially IF that pattern reflects either an arithmetic sequence or geometric sequence.
This process (working with patterns) can be used to develop or deepen understandings of important concepts in number theory, rational numbers,
measurement, geometry, probability, and functions. Working with patterns provides opportunities for students to recognize, describe, extend,
develop, and explain.
Number theory is the study of the properties of the counting (natural) numbers, their relationships, ways to represent them, and
patterns among them. Number theory includes the concepts of odd and even numbers, factors and multiples, primes and composites,
greatest common factor and least common multiple, and sequences.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks
are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be
used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to
Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models
benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each indicator in the Models
benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For example, process models are
linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and
Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at
three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
Personal Finance are indicated at each grade level with a ($). 7-236
The National Standards in Personal Finance are included in the
January 31, 2004
Appendix.
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
7-237
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
SEVENTH GRADE
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 2: Variable, Equations, and Inequalities – The student uses variables, symbols, rational numbers, and
simple algebraic expressions in one variable to solve linear equations and inequalities in a variety of
situations.
Seventh Grade Knowledge Base Indicators
Seventh Grade Application Indicators
The student…
1. knows and explains that a variable can represent a single quantity
that changes (2.4.K1a), e.g., daily temperature.
2. knows, explains, and uses equivalent representations for the same
simple algebraic expressions (2.4.K1a), e.g., x + y + 5x is the same
as 6x + y.
3. shows and explains how changes in one variable affects other
variables (2.4.A1a), e.g., changes in diameter affects circumference.
4. explains the difference between an equation and an expression.
5. solves (2.4.K1a,e) ($):
a. one-step linear equations in one variable with positive rational
coefficients and solutions, e.g., 7x = 28 or x + 3 = 7 or x = 5;
4
3
b. two-step linear equations in one variable with counting number
coefficients and constants and positive rational solutions;
c. one-step linear inequalities with counting numbers and one
variable, e.g., 3x > 12.
6. explains and uses the equality and inequality symbols (=, , <, ≤, >, ≥)
and corresponding meanings (is equal to, is not equal to, is less than,
is less than or equal to, is greater than, is greater than or equal to) to
represent mathematical relationships with rational numbers (2.4.K1a)
($).
7. ▲ knows the mathematical relationship between ratios, proportions,
and percents and how to solve for a missing term in a proportion with
positive rational number solutions and monomials (2.4.K1a,c) ($), e.g.,
5=2
The student…
1. ▲ represents real-world problems using variables and symbols to
write linear expressions, one- or two-step equations (2.4.A1e) ($),
e.g., John has three times as much money as his sister. If M is
the amount of money his sister has, what is the equality that
represents the amount of money that John has? To represent the
problem situation, J = 3M could be written.
2. solves real-world problems with one- or two-step linear equations
in one variable with whole number coefficients and constants and
positive rational solutions intuitively and analytically (2.4.A1e) ($),
e.g., Kim has read 5 more than twice the number of pages as
Hank. Kim has read 15 pages. How many pages has Hank read?
To solve analytically, write 2h + 5 = 15. Then find the answer.
3. generates real-world problems that represent one- or two-step
linear equations (2.4.A1e) ($), e.g., given the equation x + 10 = 30,
the problem could be: Two items cost $30.00. If one item costs
$10.00, what is the cost of the other item?
4. explains the mathematical reasoning that was used to solve a realworld problem using a one- or two-step linear equation (2.4.A1e)
($), e.g., Kim has read 5 more than twice the number of pages as
Hank. Kim has read 15 pages. How many pages has Hank read?
To solve, write 2h + 5 = 15. Let h=number of pages Hank read.
Then to find the answer subtract 5 from both sides of the equation.
2h = 10, then divide both sides of the equation by 2, so h = 5.
7-238
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
6 x.
8. ▲ evaluates simple algebraic expressions using positive rational
numbers (2.4.K1c) ($), e.g., if x = 3/2, y = 2, then 5xy + 2 = 5(3/2)(2) +
2 = 17.
Teacher Notes: Understanding the concept of variable is fundamental to algebra. Students use various symbols, including letters
and geometric shapes to represent unknown quantities that both do and do not vary. Quantities that are not given and do not vary
are often referred to as unknowns or missing elements when they appear in equations, e.g., 2 + Δ = 4 or 3 • s = 15 where a triangle
and s are used as variables. Various symbols or letters should be used interchangeably in equations.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks
are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can
be used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a)
refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the
Models benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each indicator in the
Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For example, process
models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number
Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked
at three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the
Appendix.
7-239
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
SEVENTH GRADE
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 3: Functions – The student recognizes, describes, and analyzes constant and linear relationships in a
variety of situations.
Seventh Grade Knowledge Base Indicators
Seventh Grade Application Indicators
The student…
1. recognizes constant and linear relationships using various methods
including mental math, paper and pencil, concrete objects, and
graphing utilities or appropriate technology (2.4.K1a,e-g) ($).
2. finds the values and determines the rule through two operations
using a function table (input/output machine, T-table) (2.4.K1f).
3. demonstrates mathematical relationships using ordered pairs in all
four quadrants of a coordinate plane (2.4.K1g).
4. describes and/or gives examples of mathematical relationships that
remain constant (2.4.K1e-g) ($), e.g., you will get $10.00 to do a job,
no matter how long it takes for you to do it.
The student…
1. represents a variety of constant and linear relationships using
written or oral descriptions of the rule, tables, graphs, and
when possible, symbolic notation (2.4.A13-g,k) ($), e.g., the
relationship between cars and their wheels (written) becomes
a table:
Cars | Wheels
1
| 4
 (1,4)
2
| 8
 (2,8)
10 | 40
 (10,40)
.
| .
.
| .
.
| .
n | 4n
and then the ordered pairs of (1,4), (2,8), (10,40), and (n,4n) can
be graphed.
2. interprets, describes, and analyzes the mathematical relationships
of numerical, tabular, and graphical representations, including
translations between the representations (2.4.A1k) ($).
7-240
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Functions are relationships or rules in which each member of one set is paired with one, and only one, member of another set (an
ordered pair). The concept of function can be introduced using function machines. Any number put in the machine will be changed according to
some rule. A record of inputs and corresponding outputs can be maintained in a two-column format. Function tables, input/output machines, and
T-tables may be used interchangeably and serve the same purpose.
Function concepts should be developed from growing patterns. Each term in a number sequence is related to its position in the
sequence – the functional relationship. The pattern – 4, 7, 10, 13, 16, 19, and so on – is an arithmetic sequence with a difference of
3. The pattern could be described as add 3 meaning that 3 must be added to the previous term to find the next. This pattern is
explained by using the recursive definition for a sequence. The recursive definition for a sequence is a statement or a set of
statements that explains how each successive term in the sequence is obtained from the previous term(s).
In the pattern 1, 4, 9, 16, 25, …, 225; there is no common difference. This sequence is not arithmetic or geometric (no common ratio
between geometric terms). Neither is it a combination of the two; however, there is a pattern and the missing terms between 25 and
225 can be found. To find the term value, square the number of the term. The next missing terms would be 36, 49, 64, 81, 100, 121,
and 144. This pattern is explained by using the explicit formula for a sequence. The explicit formula for a sequence defines a rule for
finding each term in the number sequence related to its position in the sequence. In other words, to find the term value, square the
number of the term – the 5th term is 52, the 8th term is 82, …
Patterns themselves are not explicit or recursive. The RULE for the pattern can be expressed explicitly or recursively and MOST
patterns can be explained using either format especially IF that pattern reflects either an arithmetic sequence or geometric sequence.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks
are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can
be used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a)
refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the
Models benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each indicator in the
Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For example, process
models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number
Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked
at three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the
Appendix.
7-241
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
7-242
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
SEVENTH GRADE
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 4: Models – The student generates and uses mathematical models to represent and justify mathematical
relationships found in a variety of situations.
7-243
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Seventh Grade Knowledge Base Indicators
The student…
1. knows, explains, and uses mathematical models to represent and
explain mathematical concepts, procedures, and relationships.
Mathematical models include:
a. process models (concrete objects, pictures, diagrams, number
lines, hundred charts, measurement tools, multiplication arrays,
division sets, or coordinate grids) to model computational
procedures, algebraic relationships, and mathematical
relationships and to solve equations (1.1.K1-5, 1.2.K1-4, 1.3.K14, 1.4.K1-2, 1.4.K5, 2.1.K1a-b, 2.1.K1e. 2.1.K2-4, 2.2.K1-3,
2.2.K5-6, 2.3.K1, 3.1.K9, 3.2.K1-3, 3.2.K9, 3.3.K1-4, 3.4.K1,
4.2.K4-6) ($);
b. place value models (place value mats, hundred charts, base ten
blocks, or unifix cubes) to compare, order, and represent
numerical quantities and to model computational procedures
(1.1.K1, 1.4.K2) ($);
c. fraction and mixed number models (fraction strips or pattern
blocks) and decimal and money models (base ten blocks or
coins) to compare, order, and represent numerical quantities
(1.1.K1, 1.3.K5, 1,4,K2, 2.2.K7-8, 4.1.K3) ($):
d. factor trees to find least common multiple and greatest common
factor and to model prime factorization (1.4.K4);
e. equations and inequalities to model numerical relationships
(2.2.K5, 2.3.K1, 23.K4) ($);
f. function tables to model numerical and algebraic relationships
(2.3.K1-2, 2.3.K4) ($);
g. coordinate planes to model relationships between ordered pairs
and linear equations (2.3.K1, 2.3.K3-4, 3.4.K2-4) ($);
h. two- and three-dimensional geometric models (geoboards, dot
paper, nets or solids) to model perimeter, area, volume, and
surface area, and properties of two- and three-dimensional
(2.1.K1c, 3.1.K1, 3.1.K3-8, 3.1.K10, 3.2.K1-2, 3.2.K4-8, 3.2.K10);
i. geometric models (spinners, targets, or number cubes),
process models (coins, pictures, or diagrams), and tree
diagrams to model probability (4.1.K1, 4.1.K4) ($);
j. frequency tables, bar graphs, line graphs, circle graphs,
Seventh Grade Application Indicators
The student…
1. recognizes that various mathematical models can be used to
represent the same problem situation. Mathematical models
include:
a. process models (concrete objects, pictures, diagrams,
flowcharts, number lines, hundred charts, measurement tools,
multiplication arrays, division sets, or coordinate grids) to
model computational procedures, algebraic relationships,
mathematical relationships, and problem situations and to
solve equations (1.1.A1, 1.2.A1-2, 1.3.A1-4, 1.4.A1, 2.1.A1-2,
3.1.A1, 3. 2.A1a, 3.2.A1d, 3.2.A1f, 3.2.A2, 3.3.A1. 4.2.A4) ($);
b. place value models (place value mats, hundred charts, base
ten blocks, or unifix cubes) to model problem situations
(1.1.A1a, 1.1.A2a, 1.2.A12, 1.3A1.2, 1.4.A3a-e, 2.2.A3) ($);
c. fraction and mixed number models (fraction strips or pattern
blocks) and decimal and money models (base ten blocks or
coins) to compare, order, and represent numerical quantities
(1.1.A1b, 1.1.A2b, 1.2.A1-2, 1.3.A1-2) ($);
d. factor trees to find least common multiple and greatest
common factor and to model prime factorization (1.4.K5)
e. equations and inequalities to model numerical relationships
(2.2.A1-4, 2.3.A1, 3.2.A1e). ($)
f. function tables to model numerical and algebraic relationships
(2.1.A1-2, 2.3.A1) ($);
g. coordinate planes to model relationships between ordered
pairs and linear equations (2.3.A1, 3.4.A1) ($);
h. two- and three-dimensional geometric models (geoboards, dot
paper, nets or solids) to model perimeter, area, volume, and
surface area, and properties of two- and three-dimensional
models (3.1.A2-3, 3.2.A1b-c, 3.2.A1e, 3.3.A2, 3.4.A1);
i. scale drawings to model large and small real-world objects
(3.3.A3);
j. geometric models (spinners, targets, or number cubes),
process models (coins, pictures, or diagrams), and tree
diagrams to model probability (4.1.A1) ($);
k. frequency tables, bar graphs, line graphs, circle graphs, Venn
7-244
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Venn diagrams, charts, tables, single stem-and-leaf plots,
scatter plots, and box-and-whisker plots to organize and
display data (4.2.K1) ($);
k. Venn diagrams to sort data and show relationships (1.2.K12).
diagrams, charts, tables, single stem-and-leaf plots, scatter
plots, and box-and-whisker plots to describe, interpret, and
analyze data (2.1.A2, 2.3.A1-2, 4.1.A1-2, 4.2.A1-3) ($);
l. Venn diagrams to sort data and show relationships.
2. selects a mathematical model and justifies why some
mathematical models are more accurate than other
mathematical models in certain situations, e.g., recognizes
that change over time is better represented through a line
graph than through a table of ordered pairs.
3. uses the mathematical modeling process to make inferences
about real-world situations when the mathematical model
used to represent the situation is given.
7-245
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: For assessment purposes, the mathematical modeling process appropriate to the indicator may be included as part of the item
being assessed.
The mathematical modeling process involves:
e. selecting key features and relationships within the real-world situation and representing these concepts in mathematical terms through
some sort of mathematical model,
f. performing manipulations and mathematical procedures within the mathematical model,
g. interpreting the results of the manipulations within the mathematical model,
h. using these results to make inferences about the original real-world situation.
The use of mathematical models is necessary for conceptual understanding. The ways in which mathematical ideas are
represented is fundamental to how students understand and use those ideas. As students begin to use multiple representations of
the same situation, they begin to develop an understanding of the advantages and disadvantages of various representations/models.
Many mathematical models are listed in this benchmark. The indicator lists some of the mathematical models that could be used to teach a
concept. Each indicator in this benchmark is linked to other indicators in other benchmarks; those indicators are identified in the parentheses. For
example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1
(Number Sense), and Knowledge Indicator 3. In addition, the indicator in the other benchmarks identifies, in parentheses, the Models’ indicator.
For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models).
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
7-246
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
SEVENTH GRADE
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 1: Geometric Figures and Their Properties – The student recognizes geometric figures and compares
their properties in a variety of situations.
Seventh Grade Knowledge Base Indicators
Seventh Grade Application Indicators
7-247
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
The student…
1. recognizes and compares properties of two- and three-dimensional
figures using concrete objects, constructions, drawings, appropriate
terminology, and appropriate technology (2.4.K1h).
2. classifies regular and irregular polygons having through ten sides as
convex or concave.
3. ▲ identifies angle and side properties of triangles and quadrilaterals
(2.4.K1h):
a. sum of the interior angles of any triangle is 180°;
b. sum of the interior angles of any quadrilateral is 360°;
c. parallelograms have opposite sides that are parallel and
congruent;
d. rectangles have angles of 90°, opposite sides are congruent;
e. rhombi have all sides the same length, opposite angles are
congruent;
f. squares have angles of 90°, all sides congruent;
g. trapezoids have one pair of opposite sides parallel and the other
pair of opposite sides are not parallel.
4. identifies and describes (2.4.K1h):
a. the altitude and base of a rectangular prism and triangular prism,
b. the radius and diameter of a cylinder.
5. identifies corresponding parts of similar and congruent triangles and
quadrilaterals (2.4.K1h).
6. uses symbols for right angle within a figure (
), parallel (||),
perpendicular (), and triangle (Δ) to describe geometric
figures(2.4.K1h).
7. classifies triangles as (2.4.K1h):
a. scalene, isosceles, or equilateral;
b. right, acute, obtuse, or equiangular.
The student…
3. solves real-world problems by applying the properties of (2.4.A1a):
d. plane figures (regular and irregular polygons through 10 sides,
circles, and semicircles) and the line(s) of symmetry; e.g., two
guide wires are used to stabilize a tower. The wires with the
ground form an isosceles triangle. The two base angles form
20º angles with the ground. What is the size of the vertex
angle made where the wires meet on the tower?
e. solids (cubes, rectangular prisms, cylinders, cones, spheres,
triangular prisms) emphasizing faces, edges, vertices, and
bases; e.g., lace is to be glued on all of the edges of a cube.
If one edge measures 34 cm, how much lace is needed?
2. decomposes geometric figures made from (2.4.A1h):
c. regular and irregular polygons through 10 sides, circles, and
semicircles;
d. nets (two-dimensional shapes that can be folded into threedimensional figures), e.g., the cardboard net that becomes a
shoebox;
e. prisms, pyramids, cylinders, cones, spheres, and
hemispheres.
3. composes geometric figures made from (2.4.A1h):
c. regular and irregular polygons through 10 sides, circles,
and semicircles;
d. nets (two-dimensional shapes that can be folded into threedimensional figures);
e. prisms, pyramids, cylinders, cones, spheres, and
hemispheres.
8. determines if a triangle can be constructed given sides of three
different lengths(2.4.K1h).
9. generates a pattern for the sum of angles for 3-, 4-, 5-, … n-sides
polygons (2.4.K1a).
10. describes the relationship between the diameter and the
circumference of a circle (2.4.K1h).
7-248
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Geometry is the study of shapes, their properties, and their relationships to other shapes. Symbols and numbers are used to
describe their properties and their relationships to other shapes. The fundamental concepts in geometry are point (no dimension), line (onedimensional), plane (two-dimensional), and space (three-dimensional). Plane figures are referred to as two-dimensional. Solids are referred to as
three-dimensional. The base, in terms of geometry, generally refers to the side on which a figure rests. Therefore, depending on the orientation of
the solid, the base changes.
From the Mathematics Dictionary and Handbook (Nichols Schwartz Publishing, 1999), property as a mathematical term means a characteristic (an
attribute) of a number, geometric shape, mathematical operation, equation, or inequality. To give an example:
 Property of a number: 8 is divisible by 2.
 Property of a geometric shape: Each of the four sides of a square is of the same length.
 Property of an operation: Addition is commutative. For all numbers x and y, x + y = y + x.
 Property of an equation: For all numbers a, b, and c, if a = b, then a + c = b + c.
 Property of an inequality: For all numbers a, b, and c, if a > b, then a – c > b – c.
The application of the Knowledge Indicators from the Geometry Benchmark, Geometric Figures and Their Properties are most often applied within
the context of the other Geometry Benchmarks — Measurement and Estimation, Transformational Geometry, and Geometry From an Algebraic
Perspective — rather than in isolation.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
7-249
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
SEVENTH GRADE
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 2: Measurement and Estimation – The student estimates, measures, and uses measurement formulas in a
variety of situations.
Seventh Grade Knowledge Base Indicators
Seventh Grade Application Indicators
7-250
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
The student…
1. determines and uses rational number approximations (estimations)
for length, width, weight, volume, temperature, time, perimeter, and
area using standard and nonstandard units of measure (2.4.K1a) ($).
2. selects and uses measurement tools, units of measure, and level of
precision appropriate for a given situation to find accurate rational
number representations for length, weight, volume, temperature,
time, perimeter, area, and angle measurements (2.4.K1a) ($).
3. converts within the customary system and within the metric system
(2.4.K1a).
4. ▲ knows and uses perimeter and area formulas for circles, squares,
rectangles, triangles, and parallelograms (2.4.K1h).
5. finds perimeter and area of two-dimensional composite figures of
circles, squares, rectangles, and triangles (2.4.K1h).
6. ▲ uses given measurement formulas to find (2.4.K1h):
a. surface area of cubes,
b. volume of rectangular prisms.
7. finds surface area of rectangular prisms using concrete objects
(2.4.K1h).
8. uses appropriate units to describe rate as a unit of measure
(2.4.K1a), e.g., miles per hour.
9. finds missing angle measurements in triangles and quadrilaterals
(2.4.K1h).
The student…
1. solves real-world problems by ($):
a. converting within the customary and metric systems (2.4.A1a),
e.g., James added 30 grams of sand to his model boat that
weighed 2 kg. With the sand included, what is the total weight
of his boat in kg?
b. finding perimeter and area of circles, squares, rectangles,
triangles, and parallelograms (2.4.A1h), e.g., the dimensions of
a room are 22’ x 12’. What is the total length of wallpaper
border needed?
c. ▲ ■ finding perimeter and area of two-dimensional composite
figures of squares, rectangles, and triangles (2.4.A1h), e.g., the
front of a barn is rectangular in shape with a height of 10 feet
and a width of 48 feet. Above the rectangle is a triangle that is 7
feet high with sides 25 feet long. What is the area of the front of
the barn?
d. using appropriate units to describe rate as a unit of measure
(2.4.A1a), e.g., a person traveled 20 miles in 10 minutes. What
is the rate of travel? The answer could be 2 miles per minute or
120 miles per hour.
e. finding missing angle measurements in triangles and
quadrilaterals (2.4.A1h), e.g., a fenced pasture is a quadrilateral
with angles of 30o, 120o, and 90o degrees. What is the measure
of the fourth angle?
f. applying various measurement techniques (selecting and using
measurement tools, units of measure, and level of precision) to
find accurate rational number representations for length, weight,
volume, temperature, time, perimeter, and area appropriate to a
given situation (2.4.A1a).
7-251
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
2. estimates to check whether or not measurements or calculations for
length, width, weight, volume, temperature, time, perimeter, and
area in real-world problems are reasonable and adjusts original
measurement or estimation based on additional information (a frame
of reference) (2.4.A1a) ($), e.g., students estimate the weight of their
book in grams. Then the weight of their calculator is measured in
grams. Students then adjust their estimate.
Teacher Notes: The term geometry comes from two Greek words meaning “earth measure.” Measurement provides the tools required to apply
geometric concepts in the real-world. Estimation in measurement is defined as making guesses as to the exact measurement of an object without
using any type of measurement tool. Estimation helps students develop a relationship between the different sizes of units of measure. It helps
students develop basic properties of measurement and it gives students a tool to determine whether a given measurement is reasonable.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that
could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are
identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and
Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
7-252
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
SEVENTH GRADE
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 3: Transformational Geometry – The student recognizes and performs transformations on two- and threedimensional geometric figures in a variety of situations.
Seventh Grade Knowledge Base Indicators
Seventh Grade Application Indicators
The student…
1. identifies, describes, and performs single and multiple
transformations [reflection, rotation, translation, reduction
(contraction/shrinking), enlargement (magnification/growing)] on a
two-dimensional figure (2.4.K1a).
2. identifies three-dimensional figures from various perspectives (top,
bottom, sides, corners) (2.4.K1a).
3. draws three-dimensional figures from various perspectives (top,
bottom, sides, corners) (2.4.K1a).
4. generates a tessellation (2.4.K1a).
The student…
1. describes the impact of transformations [reflection, rotation,
translation, reduction (contraction/shrinking), enlargement
(magnification/growing)] on the perimeter and area of squares and
rectangles (2.4.A1a); e.g., when a square is rotated, the perimeter
and area stays the same, however, when the length of the sides of a
square are tripled, the perimeter triples, and the area is 9 times
bigger.
2. investigates congruency and similarity of geometric figures using
transformations (2.4.A1h).
3. ▲ ■ determines the actual dimensions and/or measurements of a
two-dimensional figure represented in a scale drawing (2.4.A1i).
7-253
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Transformational geometry is another way to investigate and interpret geometric figures by moving every point in a plane figure to
a new location. To help students form images of shapes through different transformations, students can use concrete objects, figures drawn on
graph paper, mirrors or other reflective surfaces, or appropriate technology. Some transformations, like translations, reflections, and rotations, do
not change the figure itself. Other transformations like reduction (contraction/shrinking) or enlargement (magnification/growing) change the size of a
figure, but not the shape (congruence vs. similarity).
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that
could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are
identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and
Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
7-254
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
SEVENTH GRADE
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 4: Geometry From An Algebraic Perspective – The student relates geometric concepts to a number line
and a coordinate plane in a variety of situations.
Seventh Grade Knowledge Base Indicators
Seventh Grade Application Indicators
The student…
1. finds the distance between the points on a number line by computing
the absolute value of their difference (2.4.K1a).
2. uses all four quadrants of a coordinate plane to (2.4.K1g):
a. identify in which quadrant or on which axis a point lies when
given the coordinates of a point,
b. plot points,
c. identify points,
d. list through five ordered pairs of a given line.
3. uses a given linear equation with whole number coefficients and
constants and a whole number solution to find the ordered pairs,
organize the ordered pairs using a T-table, and plot the ordered
pairs on the coordinate plane (2.4.K1e-g).
4. examines characteristics of two-dimensional figures on a coordinate
plane using various methods including mental math, paper and
pencil, concrete objects, and graphing utilities or other appropriate
technology (2.4.A1g).
The student…
1. represents and/or generates real-world problems using a coordinate
plane to find (2.4.A1g-h):
a. perimeter of squares and rectangles; e.g., using a coordinate
plane Jack started at the school (1, 2), traveled to the post office
(3 ½, 2), then went by the fire station (3 ½, 3), then visited the
park (1, 3), and finally returned to the school. Determine the
distance Jack traveled.
b. circumference (perimeter) of circles, e.g., Jane jogs on a circular
track with a radius of 100 feet. How far would she jog in one
lap?
c. area of circles, parallelograms, triangles, squares, and
rectangles; e.g., if the sprinkler head is centered at (3, 4) and we
know it reaches point (3, - 2) on the coordinate plane, determine
the area being watered.
7-255
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: A number line (a mathematical model) is a diagram that represents numbers with equal distances marked off as points on a line,
and is an example of one-to-one correspondence (a relation). A number line can be used as a visual representation of numbers and operations. In
addition, a number line used horizontally and vertically is a precursor to the coordinate plane; and the distance between two numbers on a number
line is a precursor to absolute value.
A coordinate plane (coordinate grid) consists of a horizontal number line called the x-axis and a vertical number line called the y-axis. These two
lines intersect at a point called the origin. The x-axis and the y-axis divide the plane into four sections called quadrants. Any point on the coordinate
plane can be named with two numbers called coordinates. The first number is the x-coordinate. The second number is the y-coordinate. Since the
pair is always named in order (first x, then y), it is called an ordered pair.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that
could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are
identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and
Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
7-256
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 4: Data
SEVENTH GRADE
Standard 4: Data – The student uses concepts and procedures of data analysis in a variety of situations.
Benchmark 1: Probability – The student applies the concepts of probability to draw conclusions, generate convincing
arguments, and make predictions and decisions including the use of concrete objects in a
variety of situations.
Seventh Grade Knowledge Base Indicators
Seventh Grade Application Indicators
The student…
1. finds the probability of a compound event composed of two
independent events in an experiment or simulation (2.4.K1i) ($).
2. explains and gives examples of simple or compound events in an
experiment or simulation having probability of zero or one.
3. uses a fraction, decimal, and percent to represent the probability of
(2.4.K1c):
a. a simple event in an experiment or simulation;
b. a compound event composed of two independent events in an
experiment or simulation.
4. finds the probability of a simple event in an experiment or simulation
using geometric models (2.4.K1i), e.g., using spinners or dartboards,
what is the probability of landing on 2? The answer is ¼, .25, or
25%.
The student…
1. conducts an experiment or simulation with a compound event
composed of two independent events including the use of concrete
objects; records the results in a chart, table, or graph; and uses the
results to draw conclusions and make predictions about future
events (2.4.A1j-k).
2. analyzes the results of an experiment or simulation of a compound
event composed of two independent events to draw conclusions,
generate convincing arguments, and make predictions and decisions
in a variety of real-world situations (2.4.A1j-k).
3. compares results of theoretical (expected) probability with empirical
(experimental) probability in an experiment or situation with a
compound event composed of two simple independent events and
understands that the larger the sample size, the greater the
likelihood that the experimental results will equal the theoretical
probability(2.4.A1j).
4. makes predictions based on the theoretical probability of a simple
event in an experiment or simulation (2.4.A1j).
7-257
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Ideas from probability reinforce concepts in the other Standards, especially Number and Computation and Geometry. Students
need to develop an intuitive concept of chance – whether or not something is unlikely or likely to happen. Probability experiences should be
addressed through the use of concrete objects (process models); spinners, number cubes, or dartboards (geometric models); and coins (money
models). Probabilities are ratios, expressed as fractions, decimals, or percents, determined by considering results or outcomes of experiments.
Some examples of uses of probability in every day life include: There is a 50% chance of rain today. What is the probability that the team will win
every game?
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that
could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are
identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and
Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
7-258
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 4: Data
SEVENTH GRADE
Standard 4: Data – The student uses concepts and procedures of data analysis in a variety of situations.
Benchmark 2: Statistics – The student collects, organizes, displays, and explains numerical (rational numbers) and
non-numerical data sets in a variety of situations with a special emphasis on measures of central
tendency.
Seventh Grade Knowledge Base Indicators
Seventh Grade Application Indicators
The student…
1. ▲ organizes, displays, and reads quantitative (numerical) and
qualitative (non-numerical) data in a clear, organized, and accurate
manner including a title, labels, categories, and rational number
intervals using these data displays (2.4.K1j) ($):
a. frequency tables;
b. bar, line, and circle graphs;
c. Venn diagrams or other pictorial displays;
d. charts and tables;
e. stem-and-leaf plots (single);
f. scatter plots;
g. box-and-whiskers plots.
2. selects and justifies the choice of data collection techniques
(observations, surveys, or interviews) and sampling techniques
(random sampling, samples of convenience, or purposeful sampling)
in a given situation.
3. conducts experiments with sampling and describes the results.
4. determines the measures of central tendency (mode, median, mean)
for a rational number data set (2.4.K1a) ($).
5. identifies and determines the range and the quartiles of a rational
number data set (2.4.K1a) ($).
6. identifies potential outliers within a set of data by inspection rather
than formal calculation (2.4.K1a) ($), e.g., consider the data set of 1,
100, 101, 120, 140, and 170; the outlier is 1.
The student…
1. uses data analysis (mean, median, mode, range) of a rational
number data set to make reasonable inferences and predictions, to
analyze decisions, and to develop convincing arguments from these
data displays (2.4.A1k) ($):
a. frequency tables;
b. bar, line, and circle graphs;
c. Venn diagrams or other pictorial displays;
d. charts and tables;
e. stem-and-leaf plots (single);
f. scatter plots;
g. box-and-whiskers plots.
2. explains advantages and disadvantages of various data displays for
a given data set (2.4.A1k) ($).
3. ▲ recognizes and explains (2.4.A1k):
a. ■ misleading representations of data;
b. the effects of scale or interval changes on graphs of data sets.
4. determines and explains the advantages and disadvantages of using
each measure of central tendency and the range to describe a data
set (2.4.A1a) ($).
7-259
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Graphs (data displays) are pictorial representations of mathematical relationships, are used to tell a story, and are an important
part of statistics. When a graph is made, the axes and the scale (numbers running along a side of the graph) are chosen for a reason. The
difference between numbers from one grid line to another is the interval. The interval will depend on the lowest and highest values in the data set.
Emphasizing the importance of using equal-sized pictures or intervals is critical to ensuring that the data display is accurate.
Graphs take many forms:
– bar graphs and pictographs compare discrete data,
– frequency tables show how many times a certain piece of data occurs,
– circle graphs (pie charts) model parts of a whole,
– line graphs show change over time,
– Venn diagrams show relationships between sets of objects,
– line plots show frequency of data on a number line,
– single stem-and-leaf plots (closely related to line plots except that the number line is usually vertical and digits are used rather than x’s) show
frequency distribution by arranging numbers (stems) on the left side of a vertical line with numbers (leaves) on the right side,
– scatter plots show the relationship between two quantities, and
– box-and-whisker plots are visual representations of the five-number summary – the median, the upper and lower quartiles, and the least and
greatest values in the distribution – therefore, the center, the spread, and the overall range are immediately evident by looking at the plot.
Two important aspects of data are its center and its spread. The mean, median, and mode are measures of central tendency (averages) that
describe where data are centered. Each of these measures is a single number that describes the data. However, each does it slightly differently.
The range describes the spread (dispersion) of data. The easiest way to measure spread is the range, the difference between the greatest and the
least values in a data set. Quartiles are boundaries that break the data into fourths.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that
could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are
identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and
Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
7-260
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
EIGHTH GRADE
Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 1: Number Sense – The student demonstrates number sense for real numbers and simple algebraic
expressions in a variety of situations.
Eighth Grade Knowledge Base Indicators
Eighth Grade Application Indicators
The student…
The student…
1. knows, explains, and uses equivalent representations for rational
numbers and simple algebraic expressions including integers,
fractions, decimals, percents, and ratios; rational number bases with
integer exponents; rational numbers written in scientific notation with
integer exponents; time; and money (2.4.K1a) ($).
2. compares and orders rational numbers, the irrational number pi, and
algebraic expressions (2.4.K1a) ($), e.g., which expression is greater
–3n or 3n? It depends on the value of n. If n is positive, 3n is
greater. If n is negative, -3n is greater. If n is zero, they are equal.
3. explains the relative magnitude between rational numbers, the
irrational number pi, and algebraic expressions (2.4.K1a).
4. recognizes and describes irrational numbers (2.4.K1a), e.g., √ 2 is
a non-repeating, non-terminating decimal; or  (pi) is a nonterminating decimal.
5. ▲ knows and explains what happens to the product or quotient
when (2.4.K1a):
a. a positive number is multiplied or divided by a rational number
greater than zero and less than one, e.g., if 24 is divided by 1/3,
will the answer be larger than 24 or smaller than 24? Explain.
b. a positive number is multiplied or divided by a rational number
greater than one, C
c. a nonzero real number is multiplied or divided by zero, (For
purposes of assessment, an explanation of division by zero will
not be expected.)
6. explains and determines the absolute value of real numbers
(2.4.K1a).
1. generates and/or solves real-world problems using equivalent
representations of rational numbers and simple algebraic
expressions (2.4.A1a) ($), e.g., a paper reports a company’s gross
income as $1.2 billion and their total expenses as $30,450,000.
What is the company’s net profit?
2. determines whether or not solutions to real-world problems using
rational numbers, the irrational number pi, and simple algebraic
expressions are reasonable (2.4.A1a) ($), e.g., the city park is
putting a picket fence around their circular rose garden. The
garden has a diameter of 7.5 meters. The planner wants to buy 20
meters of fencing. Is this reasonable?
8-261
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Number sense refers to one’s ability to reason with numbers and to work with numbers in a flexible way. The ability to compute
mentally, to estimate based on understanding of number relationships and magnitudes, and to judge reasonableness of answers are all involved in
number sense.
The student with number sense will look at a problem holistically before confronting the details of the problem. The student will look for
relationships among the numbers and operations and will consider the context in which the question was posed. Students with number sense will
choose or even invent a method that takes advantage of their own understanding of the relationships between numbers and between numbers and
operations, and they will seek the most efficient representation for the given task. Number sense can also be recognized in the students' use of
benchmarks to judge number magnitude (e.g., 2/5 of 49 is less that half of 49), to recognize unreasonable results for calculations, and to employ
non-standard algorithms for mental computation and estimation. (Developing Number Sense: Addenda Series, Grades 5-8, NCTM, 1991)
If you think about division as undoing multiplication, you’ll see why we don’t divide by zero. EXAMPLE: Try to divide 245 ÷ 0. To divide, think of
the related multiplication equation. 245 ÷ 0 = ? asks the same question as ? x 0 = 245. When you think about it this way, you can see that you’re
really stuck because any number times zero is zero! So, mathematicians say that division by zero is undefined. Math On Call 226. Don’t try to
divide by Zero.
At this grade level, real numbers include positive and negative numbers and very large numbers (one billion) and very small numbers
(one-millionth). Relative magnitude refers to the size relationship one number has with another – is it much larger, much smaller,
close, or about the same? For example, using the numbers 219, 264, and 457, answer questions such as:
 Which two are closest? Why?
 Which is closest to 300? To 250?
 About how far apart are 219 and 500? 5,000?
 If these are ‘big numbers,’ what are small numbers? Numbers about the same? Numbers that make these seem
small?
(Elementary and Middle School Mathematics, John A. Van de Walle, Addison Wesley Longman, Inc., 1998)
Mathematical models such as concrete objects, pictures, diagrams, Venn diagrams, number lines, hundred charts, base ten blocks, or factor trees
are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to
represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2
(Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the
mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other
indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3
referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the
Appendix.
8-262
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
8-263
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
EIGHTH GRADE
Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 2: Number Systems and Their Properties – The student demonstrates an understanding of the real
number system; recognizes, applies, and explains their properties; and extends these
properties
to algebraic expressions.
Eighth Grade Knowledge Base Indicators
Eighth Grade Application Indicators
8-264
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
The student…
1. explains and illustrates the relationship between the subsets of the
real number system [natural (counting) numbers, whole numbers,
integers, rational numbers, irrational numbers] using mathematical
models (2.4.K1a), e.g., number lines or Venn diagrams.
2. ▲ identifies all the subsets of the real number system [natural (counting)
numbers, whole numbers, integers, rational numbers, irrational numbers] to
which a given number belongs (2.4.K1l). (For the purpose of assessment,
irrational numbers will not be included.)
3. names, uses, and describes these properties with the rational number system
and demonstrates their meaning including the use of concrete objects
(2.4.K1a) ($):
a. commutative, associative, distributive, and substitution
properties [commutative: a + b = b + a and ab = ba; associative:
a + (b + c) = (a + b) + c and a(bc) = (ab)c; distributive: a(b + c) =
ab + ac; substitution: if a = 2, then 3a = 3 x 2 = 6];
b. identity properties for addition and multiplication and inverse
properties of addition and multiplication (additive identity: a + 0 =
a, multiplicative identity: a • 1 = a, additive inverse: +5 + –5 = 0,
multiplicative inverse: 8 x 1/8 = 1);
c. symmetric property of equality, e.g., 7 + 2 = 9 has the same
meaning as 9 = 7 + 2;
d. addition and multiplication properties of equalities, e.g., if a = b,
then a + c = b + c;
e. addition property of inequalities, e.g., if a > b, then a + c > b + c;
f. zero product property, e.g., if ab = 0, then a = 0 and/or b = 0.
The student…
1. generates and/or solves real-world problems with rational numbers
using the concepts of these properties to explain reasoning
(2.4.A1a) ($):
a. ▲ commutative, associative, distributive, and substitution
properties; e.g., we need to place trim around the outside edges
of a bulletin board with dimensions of 3 ft by 5 ft. Explain two
different methods of solving this problem and why the answers
are equivalent.
b. ▲ identity and inverse properties of addition and multiplication;
e.g., I had $50. I went to the mall and spent $20 in one store,
$25 at a second store and then $5 at the food court. To solve:
[$50 – ($20 + $25 + $5) = $50 - $50 = 0]. Explain your
reasoning.
c. symmetric property of equality; e.g., Sam took a $15 check to
the bank and received a $10 bill and a $5 bill. Later Sam took a
$10 bill and a $5 bill to the bank and received a check for $15.
$15 = $10 + $5 is the same as $10 + $5 = $15
d. addition and multiplication properties of equality; e.g., the total
price (P) of a car, including tax (T), is $14, 685. 33. If the tax is
$785.42, what is the sale price of the car (S)?
e. zero product property, e.g., Jenny was thinking of two numbers.
Jenny said that the product of the two numbers was 0. What
could you deduct from this statement? Explain your reasoning
8-265
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
2. analyzes and evaluates the advantages and disadvantages of
using integers, whole numbers, fractions (including mixed
numbers), or decimals in solving a given real-world problem
(2.4.A1a) ($), e.g., in the store everything is 33 1/3% off. When
calculating the discount, which representation of 33 1/3%
would you use and why?
Teacher Notes: From the Mathematics Dictionary and Handbook (Nichols Schwartz Publishing, 1999), property as a mathematical term means a
characteristic (an attribute) of a number, geometric shape, mathematical operation, equation, or inequality. To give an example:
 Property of a number: 8 is divisible by 2.
 Property of a geometric shape: Each of the four sides of a square is of the same length.
 Property of an operation: Addition is commutative. For all numbers x and y, x + y = y + x.
 Property of an equation: For all numbers a, b, and c, if a = b, then a + c = b + c.
 Property of an inequality: For all numbers a, b, and c, if a > b, then a – c > b – c.
Mathematical models such as concrete objects, pictures, diagrams, Venn diagrams, number lines, hundred charts, base ten blocks, or factor trees
are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to
represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2
(Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the
mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other
indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3
referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
8-266
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
EIGHTH GRADE
Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 3: Estimation – The student uses computational estimation with real numbers in a variety of situations.
Eighth Grade Knowledge Base Indicators
The student…
1. estimates real number quantities using various computational
methods including mental math, paper and pencil, concrete objects,
and/or appropriate technology (2.4.K1a) ($).
2. uses various estimation strategies and explains how they were used
to estimate real number quantities and simple algebraic expressions
(2.4.K1a) ($).
3. knows and explains why a decimal representation of the irrational
number pi is an approximate value (2.4.K1c).
4. knows and explains between which two consecutive integers an
irrational number lies (2.4.K1a).
Eighth Grade Application Indicators
The student…
1. adjusts original rational number estimate of a real-world problem
based on additional information (a frame of reference) (2.4.A1a) ($),
e.g., estimate the height of a building from a picture. In another
picture, a person is standing next to the building. By using the
person as a frame of reference adjust your original estimate.
2. estimates to check whether or not the result of a real-world problem
using rational numbers and/or simple algebraic expressions is
reasonable and makes predictions based on the information
(2.4.A1a) ($), e.g., you have a $4,000 debt on a credit card. You pay
the minimum of $30 per month. Is it reasonable to pay off the debt
in 10 years?
3. determines a reasonable range for the estimation of a quantity given
a real-world problem and explains the reasonableness of the range
(2.4.A1c) ($), e.g., determine the reasonable range for the weight of
a book and explain why this range is reasonable.
4. determines if a real-world problem calls for an exact or approximate
answer and performs the appropriate computation using various
computational methods including mental mathematics, paper and
pencil, concrete objects, and/or appropriate technology (2.4.A1a) ($),
e.g., do you need an exact or an approximate answer when
calculating the area of the walls in a room to determine the number
of rolls of wallpaper needed to paper the room?. An approximation
is appropriate for the area but an exact answer is needed for the
number of roles. What would you do if you were wallpapering 2
rooms?
8-267
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
5. explains the impact of estimation on the result of a real-world
problem (underestimate, overestimate, range of estimates)
(2.4.A1a) ($), e.g., you are estimating the total of three large
purchases ($489, $553, and $92). If you rounded each to the
nearest $10, would your estimate be slightly lower or higher
than the actual amount spent? If you rounded each to the
nearest $100, would your estimate be slightly lower or higher
than the actual amount spent?
8-268
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Estimate, as a verb, means to make an educated guess based on information in a problem or to give an answer
close to the exact number. Estimation is used when an exact answer is not needed, as in many real-life situations for which “ballpark”
computations are acceptable. Good number sense enables one to estimate a quantity, estimate a measure, or estimate an answer.
Estimation serves as an important companion to computation. It provides a tool for judging the reasonableness of computational
methods including mental math, paper and pencil, concrete objects, and appropriate technology. However, being able to compute
does not automatically lead to an ability to estimate or judge reasonableness of answers. Frequent modeling by the teacher helps
students develop a range of estimation strategies. Students should be encouraged to frequently explain their thinking as they
estimate. As with exact computation, sharing estimation strategies allows students access to others’ thinking and provides
opportunities for class discussion. Identifying the estimation strategy by name is not critical; however, explaining the thinking behind
the strategy to make a valid estimation is important. (Principles and Standards for School Mathematics, NCTM, 2000)
Mental math and estimation are distinct but related mathematical skills. Proficiency in mental math contributes to increased skill in
estimation. In order for students to become more familiar with estimation, teachers should introduce estimation with examples where
rounded or estimated numbers are used. Emphasis should be placed on real-world examples where only estimation is required, e.g.,
About how many hours do you sleep a night? Using the language of estimation is important, so students begin to realize that a variety
of estimates (answers) are possible.
In addition, when students are taught specific estimation strategies, they develop mental math and estimation skills easier. Estimation
strategies include front-end with adjustment, compatible “nice” numbers, clustering, special numbers, or truncation.
Mathematical models such as concrete objects, pictures, diagrams, Venn diagrams, number lines, hundred charts, base ten blocks,
or factor trees are necessary for conceptual understanding and should be used to explain computational procedures. If a
mathematical model can be used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For
example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the
indicator in the Models benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each
indicator in the Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For
example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation),
Benchmark 1 (Number Sense), and Knowledge Indicator.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at
three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the
Appendix.
8-269
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
8-270
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
EIGHTH GRADE
Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 4: Computation – The student models, performs, and explains computation with rational numbers, the
irrational number pi, and algebraic expressions in a variety of situations.
Eighth Grade Knowledge Base Indicators
Eighth Grade Application Indicators
The student…
1. computes with efficiency and accuracy using various computational
methods including mental math, paper and pencil, concrete objects,
and appropriate technology (2.4.K1a) ($).
2. performs and explains these computational procedures with rational
numbers (2.4.K1a):
a. ▲N addition, subtraction, multiplication, and division of integers
b. ▲N order of operations (evaluates within grouping symbols,
evaluates powers to the second or third power, multiplies or
divides in order from left to right, then adds or subtracts in order
from left to right);
c. approximation of roots of numbers using calculators;
d. multiplication or division to find:
i. a percent of a number, e.g., what is 0.5% of 10?
ii. percent of increase and decrease, e.g., if two coins are
removed from ten coins, what is the percent of decrease?
iii. percent one number is of another number, e.g., what
percent of 80 is 120?
iv. a number when a percent of the number is given, e.g., 15%
of what number is 30?
e. addition of polynomials, e.g., (3x – 5) + (2x + 8).
f. simplifies algebraic expressions in one variable by combining
like terms or using the distributive property (2.4.K1a), e.g.,
–3(x – 4) is the same as –3x + 12.
The student…
1. ▲ generates and/or solves one- and two-step real-world
problems using computational procedures and mathematical
concepts (2.4.A1a) with ($):
a. ■ rational numbers, e.g., find the height of a triangular
garden given that the area to be covered is 400 square
feet with a base of 12½ feet;
b. the irrational number pi as an approximation, e.g., before
planting, a farmer plows a circular region that has an
approximate area of 7,300 square feet. What is the radius
of the circular region to the nearest tenth of a foot?
c. applications of percents, e.g., sales tax or discounts. (For
the purpose of assessment, percents greater than or equal
to 100% will NOT be used).
3. finds factors and common factors of simple monomial
expressions (2.4.K1d), e.g., given the monomials 10m2n3 and
15a2mn2 some common factors would be 5m, 5mn2, and n2.
8-271
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Efficiency and accuracy means that students are able to compute single-digit numbers with fluency. Students increase their understanding and
skill in addition, subtraction, multiplication, and division by understanding the relationships between addition and subtraction, addition and multiplication,
multiplication and division, and subtraction and division. Students learn basic number combinations and develop strategies for computing that makes sense to
them. Through class discussions, students can compare the ease of use and ease of explanation of various strategies. In some cases, their strategies for computing
will be close to conventional algorithms; in other cases, they will be quite different. Many times, students’ invented approaches are based on a sound
understanding of numbers and operations, and these invented approaches often can be used with efficiency and accuracy. (Principles and Standards for School
Mathematics, NCTM, 2000)
The definition of computation is finding the standard representation for a number. For example, 6 + 6, 4 x 3, 17 – 5, and 24 ÷ 2 are all representations for the
standard representation of 12. Mental math is mentally finding the standard representation for a number – calculating in your head instead of calculating using
paper and pencil or technology. One of the main reasons for teaching mental math is to help students determine if a computed/calculated answer is reasonable; in
other words, using mental math to estimate to see if the answer makes sense. Students develop mental math skills easier when they are taught specific strategies.
Mental math strategies include counting on, doubling, repeated doubling, halving, making tens, multiplying by powers of ten, dividing with tens, finding fractional
parts, thinking money, and using compatible “nice” numbers.
Mathematical models such as concrete objects, pictures, diagrams, Venn diagrams, number lines, hundred charts, base ten blocks,
or factor trees are necessary for conceptual understanding and should be used to explain computational procedures. If a
mathematical model can be used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For
example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the
indicator in the Models benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each
indicator in the Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For
example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation),
Benchmark 1 (Number Sense), and Knowledge Indicator 3.
Technology is changing mathematics and its uses. The use of technology including calculators and computers is an important part of growing up in a complex
society. It is not only necessary to estimate appropriate answers accurately when required, but also it is also important to have a good understanding of the
underlying concepts in order to know when to apply the appropriate procedure. Technology does not replace the need to learn basic facts, to compute mentally, or
to do reasonable paper-and-pencil computation. However, dividing a 5-digit number by a 2-digit number is appropriate with the exception of dividing by 10, 100,
or 1,000 and simple multiples of each.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at
three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the
Appendix.
8-272
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
EIGHTH GRADE
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 1: Patterns – The student recognizes, describes, extends, develops, and explains the general rule
of a pattern from a variety of situations.
Eighth Grade Knowledge Base Indicators
The student…
Eighth Grade Application Indicators
The student…
2. identifies, states, and continues a pattern presented in various
formats including numeric (list or table), algebraic (symbolic
notation), visual (picture, table, or graph), verbal (oral description),
kinesthetic (action), and written using these attributes:
f. counting numbers including perfect squares, cubes, and factors
and multiples with positive rational numbers (number theory)
(2.4.K1a).
g. rational numbers including arithmetic and geometric sequences
(arithmetic: sequence of numbers in which the difference of two
consecutive numbers is the same, geometric: a sequence of
numbers in which each succeeding term is obtained by
multiplying the preceding term by the same number) (2.4.K1a),
e.g., 1, 1, 3, …;
4 2 4
c. geometric figures (2.4.K1h);
d. measurements (2.4.K1a);
e. things related to daily life ($);
f. variables and simple expressions, e.g., 1 – x, 2 – x, 3 – x, 4 – x,
...; or x, x2, x3, ...
2. generates and explains a pattern (2.4.K1a).
3. generates a pattern limited to two operations (addition, subtraction,
multiplication, division, exponents) when given the rule for the nth
term (2.4.K1a), e.g., the nth term is n2+1, find the first 4 terms
beginning with n = 1; the terms are 2, 5, 10, and 17.
4. states the rule to find the nth term of a pattern using explicit symbolic
notation (2.4.K1a), e.g., given 2, 5, 8, 11, …; find the rule for the nth
term, the rule is 3n –1.
1. generalizes numerical patterns using algebra and then translates
between the equation, graph, and table of values resulting from the
generalization (2.4.A1d-e,j) ($), e.g., water is billed at $1.00 per
1,000 gallons, plus a basic fee of $10 per month for being connected
to the water district.
1,000 Gallons | Cost in a given month
Graph these:
| $10 + 1*1.00

(1, 11)
| $10 + 2*.1.00

(2, 12)
|
.
|
.
| 10 + n*1.00

(n, 1.00n + 10)
where C = total cost and G = gallons used,
C = 1.00 G + 10]
Monthly
Water
Bill Charges
(Title/Labels
needed)
Total Cost per Month
1
2
.
.
n
25
20
15
10
5
0
0
5
10
15
Thousands of Gallons Used
8-273
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
5. describes the pattern when given a table of linear values and plots
the ordered pairs on a coordinate plane (2.4.K1f-g), e.g., in the table
below, the pattern could be described as the x-coordinates are
increasing by three, while the y-coordinates are increasing by 6, or
the x is doubled and one is added to find the y.
X
Y
2
5
5
11
8
17
2. recognizes the same general pattern presented in different
representations [numeric (list or table), visual (picture, table, or
graph), and written] (2.4.A1a,j) ($).
11
23
Teacher Notes: A fundamental component in the development of classification, number, and problem solving skills is inventing,
discovering, and describing patterns. Patterns pervade all of mathematics and much of nature. All patterns are either repeating or
growing or a variation of either or both. Translating a pattern from one medium to another to find two patterns that are alike, even
though they are made with different materials, is important so students can see the relationships that are critical to repeating patterns.
With growing patterns, students not only extend patterns, but also look for a generalization or an algebraic relationship that will tell
them what the pattern will be at any point along the way.
Working with patterns is an important process in the development of mathematical thinking. Patterns can be based on geometric attributes
(shapes, regions, angles); measurement attributes (color, texture, length, weight, volume, number); relational attributes (proportion, sequence,
functions); and affective attributes (values, likes, dislikes, familiarity, heritage, culture). (Learning to Teach Mathematics, Randall J. Souviney,
Macmillan Publishing Company, 1994)
In the pattern that begins with 3, 5, 7, and 9; the explicit rule is 2n +1 and the recursive rule is add 2 to the previous term. Patterns themselves are
not explicit or recursive. The RULE for the pattern can be expressed explicitly or recursively and MOST patterns can be explained using either
format especially IF that pattern reflects either an arithmetic sequence or geometric sequence.
This process (working with patterns) can be used to develop or deepen understandings of important concepts in number theory, rational numbers,
measurement, geometry, probability, and functions. Working with patterns provides opportunities for students to recognize, describe, extend,
develop, and explain.
Number theory is the study of the properties of the counting numbers (positive integers), their relationships, ways to represent them, and patterns
among them. Number theory includes the concepts of odd and even numbers, factors and multiples, primes and composites, greatest common
factor and least common multiple, and sequences.
8-274
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that
could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are
identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and
Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
8-275
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
EIGHTH GRADE
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 2: Variable, Equations, and Inequalities – The student uses variables, symbols, real numbers, and
algebraic expressions to solve equations and inequalities in a variety of situations.
Eighth Grade Knowledge Base Indicators
The student…
Eighth Grade Application Indicators
The student…
1. identifies independent and dependent variables within a given
situation.
2. simplifies algebraic expressions in one variable by combining like
terms or using the distributive property (2.4.K1a), e.g., –3(x – 4) is
the same as – 3 x + 12.
3. solves (2.4.K1a,e) ($):
a. ▲ one- and two-step linear equations in one variable with
rational number coefficients and constants intuitively and/or
analytically;
b. one-step linear inequalities in one variable with rational number
coefficients and constants intuitively, analytically, and
graphically;
c. systems of given linear equations with whole number
coefficients and constants graphically.
4. knows and describes the mathematical relationship between ratios,
proportions, and percents and how to solve for a missing monomial
or binomial term in a proportion (2.4.K1c), e.g., 2 = _ 1_
5
x + 2.
5. represents and solves algebraically ($):
a. the number when a percent and a number are given,
b. what percent one number is of another number,
c. percent of increase or decrease, e.g., the price of a loaf of
bread is $2.00. With a coupon, the cost is $1.00. What is the
percent of decrease?
6. evaluates formulas using substitution ($).
1. represents real-world problems using (2.4.A1d) ($):
a. ▲ ■ variables, symbols, expressions, one- or two-step equations
with rational number coefficients and constants, e.g., today John
is 3.25 inches more than half his sister’s height. If J = John’s
height, and S = his sister’s height, then J = 0.5S + 3.25.
b. one-step inequalities with rational number coefficients and
constants, e.g., after Randy paid $38.50 for a watch, he did not
have enough money to by a calculator for $5.50. Represent this
situation with an inequality.
c. systems of linear equations with whole number coefficients and
constants, e.g., two students collected the same amount of
money for a walk-a-thon. One student received $5 per mile and a
donation of $10, while the other student received $2 per mile and
a donation of $20. How many miles did they walk?
2. solves real-world problems with two-step linear equations in one
variable with rational number coefficients and constants and rational
solutions intuitively, analytically, and graphically (2.4.A1e) e.g., Mike
and Albert are friends, but Joe and Albert are not friends. Which of
the following diagrams can be used to describe this situation? (Three
dots labeled J, M, A: there is a line between J and M and line
between M and A, but no line between J and A.)
3. generates real-world problems that represent (2.4.A1d) ($):
a. one- or two-step linear equations, ($), e.g., given the equation 2x
+ 10 = 30, the problem could be I bought two shirts and a pair of
pants for $10. How much was a shirt, if the total bill was $30?
b. one-step linear inequalities, e.g., write a real-world situation that
represents the inequality x + 10 > 30. The problem could be: If
you give me $10, I will have more than $30.
8-276
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
4. explains the mathematical reasoning that was used to solve a
real-world problem using one- or two-step linear equations and
inequalities and discusses the advantages and disadvantages
to various strategies that may have been used to solve the
problem, (2.4.A1d) ($), e.g., given the inequality x + 10 > 30,
subtract the same number from both sides of the inequality or
graph as y1 = x + 10 and y = 30 and find on the graph where
y1 is less than y2. The first method gives an exact answer; the
second method is a visual representation and can be used to
solve more difficult inequalities.
Teacher Notes: Understanding the concept of variable is fundamental to algebra. Students use various symbols, including letters
and geometric shapes to represent unknown quantities that both do and do not vary. Quantities that are not given and do not vary are
often referred to as unknowns or missing elements when they appear in equations, e.g., 2 + Δ = 4 or 3 • s = 15 where a triangle and s
are used as variables. Various symbols or letters should be used interchangeably in equations.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are
necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be
used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to
Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models
benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each indicator in the Models
benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For example, process models are
linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and
Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at
three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the
Appendix.
8-277
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
EIGHTH GRADE
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 3: Functions – The student recognizes, describes, and analyzes constant, linear, and nonlinear
relationships in a variety of situations.
Eighth Grade Knowledge Base Indicators
The student…
1. recognizes and examines constant, linear, and nonlinear
relationships using various methods including mental math, paper
and pencil, concrete objects, and graphing utilities or appropriate
technology (2.4.K1a,e-g) ($).
2. knows and describes the difference between constant, linear, and
nonlinear relationships (2.4.K1g).
3. explains the concepts of slope and x- and y-intercepts of a line
(2.4.K1g).
4. recognizes and identifies the graphs of constant and linear functions
(2.4.K1g) ($).
5. identifies ordered pairs from a graph, and/or plots ordered pairs
using a variety of scales for the x- and y-axis (2.4.K1g).
Eighth Grade Application Indicators
The student…
1. represents a variety of constant and linear relationships using written
or oral descriptions of the rule, tables, graphs, and symbolic notation
(2.4.A1d-f) ($).
2. interprets, describes, and analyzes the mathematical relationships of
numerical, tabular, and graphical representations (2.4.A1j) ($).
3. ▲ translates between the numerical, tabular, graphical, and
symbolic representations of linear relationships with integer
coefficients and constants (2.4.A1a), e.g., a fish tank is being filled
with water with a 2-liter jug. There are already 5 liters of water in the
fish tank. Therefore, you are showing how full the tank is as you
empty 2-liter jugs of water into it. Y = 2x + 5 (symbolic) can be
represented in a table (tabular) –
Y
X 0
5
1
7
2
9
3
11
and as a graph (graphical) –
8-278
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Functions are relationships or rules in which each member of one set is paired with one, and only one, member of
another set (an ordered pair). The concept of function can be introduced using function machines. Any number put in the machine will
be changed according to some rule. A record of inputs and corresponding outputs can be maintained in a two-column format.
Function tables, input/output machines, and T-tables may be used interchangeably and serve the same purpose.
Function concepts should be developed from growing patterns. Each term in a number sequence is related to its position in the
sequence – the functional relationship. The pattern – 4, 7, 10, 13, 16, 19, and so on – is an arithmetic sequence with a difference of 3.
The pattern could be described as add 3 meaning that 3 must be added to the previous term to find the next. This pattern is explained
by using the recursive definition for a sequence. The recursive definition for a sequence is a statement or a set of statements that
explains how each successive term in the sequence is obtained from the previous term(s).
In the pattern 1, 4, 9, 16, 25, …, 225; there is no common difference. This sequence is not arithmetic or geometric (no common ratio
between geometric terms). Neither is it a combination of the two; however, there is a pattern and the missing terms between 25 and
225 can be found. To find the term value, square the number of the term. The next missing terms would be 36, 49, 64, 81, 100, 121,
and 144. This pattern is explained by using the explicit formula for a sequence. The explicit formula for a sequence defines a rule for
finding each term in the number sequence related to its position in the sequence. In other words, to find the term value, square the
number of the term – the 5th term is 52, the 8th term is 82, …
Patterns themselves are not explicit or recursive. The RULE for the pattern can be expressed explicitly or recursively and MOST
patterns can be explained using either format especially IF that pattern reflects either an arithmetic sequence or geometric sequence.
In the Cartesian Coordinate System, the expression x = 3 cannot represent a function. When you graph all points where x = 3 you get
a vertical line, so more than one y-value exists for the x-value x = 3. By definition, a function must have only one output value for any
given input. For more information, consult a reference book under the topic, vertical line test.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks
are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be
used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to
Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models
benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each indicator in the Models
benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For example, process models are
linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and
Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although
8-279 the National Standards in Personal Finance are benchmarked at
three grade levels, the indicators in the Kansas Curricular Standards
for Mathematics that correlate with the National Standards in
January 31, 2004
▲– Assessed
Indicator are
on theindicated
Objective Assessment
Personal
Finance
at each grade level with a ($). The National Standards in Personal Finance are included in the
■ – Assessed Indicator on the Optional Constructed Response Assessment
Appendix.
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
8-280
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
EIGHTH GRADE
Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 4:
Models – The student generates and uses mathematical models to represent and justify mathematical
relationships found in a variety of situations.
Eighth Grade Knowledge Base Indicators
The student…
1. knows, explains, and uses mathematical models to represent and
explain mathematical concepts, procedures, and relationships.
Mathematical models include:
h. process models (concrete objects, pictures, diagrams, number
lines, hundred charts, measurement tools, multiplication arrays,
division sets, or coordinate grids) to model computational
procedures, algebraic relationships, and mathematical
relationships and to solve equations (1.1.K1-6, 1.2.K1, 1.2.K3,
1.3.K1-2, 1.3.K4, 1.4.K1-2, 2.1.K1a-b, 2.1.K1d-e, 2.1.K2-4,
2.2.K2-3, 3.1.K9, 3.2.K1-4, 3.3.K1-4, 3.4.K4, 4.2.K4-5) ($);
i. place value models (place value mats, hundred charts, base ten
blocks, or unifix cubes) to compare, order, and represent
numerical quantities and to model computational procedures ($);
j. fraction and mixed number models (fraction strips or pattern
blocks) and decimal and money models (base ten blocks or
coins) to compare, order, and represent numerical quantities
(1.3.K3, 2.3.K6) ($):
k. factor trees to model least common multiple, greatest common
factor, and prime factorization (1.4.K3);
l. equations and inequalities to model numerical relationships
(2.2.K3, (3.4.K2) ($);
m. function tables to model numerical and algebraic relationships
(2.1.K5, 3.4.K2) ($) ;
n. coordinate planes to model relationships between ordered pairs
and linear equations and inequalities (2.1.K5, 2.3.K1-5, 3.4.K23) ($);
Eighth Grade Application Indicators
The student…
1. recognizes that various mathematical models can be used to
represent the same problem situation. Mathematical models include:
a. process models (concrete objects, pictures, diagrams,
flowcharts, number lines, hundred charts, measurement tools,
multiplication arrays, division sets, or coordinate grids) to model
computational procedures, algebraic relationships,
mathematical relationships, and problem situations and to solve
equations (1.1.A1-2, 1.2.A1-2, 1.3.A1-5, 1.4.A1, 2.1.A1, 3.1.A1,
3.2.A1-2, 3.3.A1, 3.4.A1-2) ($);
b. place value models (place value mats, hundred charts, base ten
blocks, or unifix cubes) to model problem situations ($);
c. fraction and mixed number models (fraction strips or pattern
blocks) and decimal and money models (base ten blocks or
coins) to compare, order, and represent numerical quantities
(3.2.A3) ($);
d. equations and inequalities to model numerical relationships
( 2.1.A2, 2.2.A1-2, 2.3.A1, 3.4.A2) ($);
e. function tables to model numerical and algebraic relationships
(2.1.A2, 2.3.A2, 3.4.A2) ($);
f. coordinate planes to model relationships between ordered pairs
and linear equations and inequalities (2.3.A1 3.4.A2) ($);
g. two- and three-dimensional geometric models (geoboards, dot
paper, nets, or solids) and real-world objects to model perimeter,
area, volume, surface area and properties of two- and threedimensional figures (3.3.A3, 3.4.A2);
h. scale drawings to model large and small real-world objects
(3.1.A1-2, 3.3.A4);
8-281
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
o. two- and three-dimensional geometric models (geoboards, dot
paper, nets, or solids) and real-world objects to model perimeter,
area, volume, surface area, and properties of two-and threedimensional figures (2.1.K1c, 3.1.K1-6, 3.1.K8, 3.1.K10, 3.2.K5,
3.3.K4-5);
p. scale drawings to model large and small real-world objects
(3.3.K3-4);
q. geometric models (spinners, targets, or number cubes),
process models (coins, pictures, or diagrams), and tree
diagrams to model probability (4.1.K1-5) ($);
r. frequency tables, bar graphs, line graphs, circle graphs,
Venn diagrams, charts, tables, single and double stemand-leaf plots, scatter plots, box-and-whisker plots, and
histograms to organize and display data (4.2.K1, 4.2.K6)
($);
s. Venn diagrams to sort data and to show relationships
(1.2.K2).
i.
geometric models (spinners, targets, or number cubes), process
models (coins, pictures, or diagrams), and tree diagrams to
model probability (4.1.A1-4);
j. frequency tables, bar graphs, line graphs, circle graphs, Venn
diagrams, charts, tables, single and double stem-and-leaf plots,
scatter plots, box-and-whisker plots, and histograms to describe,
interpret, and analyze data (2.1.A1-2, 2.3.A2-3, 4.2.A1, 4.2.A3,
4.2.A1-3) ($);
k. Venn diagrams to sort data and to show relationships.
2. ▲ determines if a given graphical, algebraic, or geometric model is
an accurate representation of a given real-world situation ($).
3. uses the mathematical modeling process to analyze and make
inferences about real-world situations ($).
8-282
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: For assessment purposes, the mathematical modeling process appropriate to the indicator may be included as part of the item
being assessed.
The mathematical modeling process involves:
i. selecting key features and relationships within the real-world situation and representing these concepts in mathematical terms through
some sort of mathematical model,
j. performing manipulations and mathematical procedures within the mathematical model,
k. interpreting the results of the manipulations within the mathematical model,
l. using these results to make inferences about the original real-world situation.
The use of mathematical models is necessary for conceptual understanding. The ways in which mathematical ideas are represented
is fundamental to how students understand and use those ideas. As students begin to use multiple representations of the same
situation, they begin to develop an understanding of the advantages and disadvantages of various representations/models.
Many mathematical models are listed in this benchmark. The indicator lists some of the mathematical models that could be used to teach a
concept. Each indicator in this benchmark is linked to other indicators in other benchmarks; those indicators are identified in the parentheses. For
example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1
(Number Sense), and Knowledge Indicator 3. In addition, the indicator in the other benchmarks identifies, in parentheses, the Models’ indicator. For
example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models).
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
8-283
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
EIGHTH GRADE
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 1: Geometric Figures and Their Properties – The student recognizes geometric figures and compares
their properties in a variety of situations.
Eighth Grade Knowledge Base Indicators
The student…
1. recognizes and compares properties of two- and three-dimensional
figures using concrete objects, constructions, drawings, appropriate
terminology, and appropriate technology (2.4.K1h).
2. discusses properties of triangles and quadrilaterals related to
(2.4.K1h):
h. sum of the interior angles of any triangle is 180°;
i. sum of the interior angles of any quadrilateral is 360°;
a. parallelograms have opposite sides that are parallel and
congruent, opposite angles are congruent;
b. rectangles have angles of 90°, sides may or may not be equal;
c. rhombi have all sides equal in length, angles may or may not be
equal;
d. squares have angles of 90°, all sides congruent;
e. trapezoids have one pair of opposite sides parallel and the other
pair of opposite sides are not parallel;
f. kites have two distinct pairs of adjacent congruent sides.
3. recognizes and describes the rotational symmetries and line
symmetries that exist in two-dimensional figures (2.4.K1h), e.g.,
draw a picture with a line of symmetry in it. Explain why it is a line of
symmetry.
4. recognizes and uses properties of corresponding parts of similar and
congruent triangles and quadrilaterals to find side or angle measures
using standard notation for similarity (~) and congruence (  )
(2.4.K1h).
5. knows and describes Triangle Inequality Theorem to determine if a
triangle exists (2.4.K1h).
Eighth Grade Application Indicators
The student…
1. solves real-world problems by (2.4.A1a):
a. ▲ ■ using the properties of corresponding parts of similar and
congruent figures, e.g., scale drawings, map reading,
proportions, or indirect measurements.
b. applying the Pythagorean Theorem, e.g., indirect
measurements, map reading/distance, or diagonals.
8-284
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
6. ▲ uses the Pythagorean theorem to (2.4.K1h):
a. determine if a triangle is a right triangle,
b. find a missing side of a right triangle where the lengths of
all three sides are whole numbers.
7. recognizes and compares the concepts of a point, line, and plane.
8. describes the intersection of plane figures, e.g., two circles could
intersect at no point, one point, two points, or all points.
9. describes and explains angle relationships:
a. when two lines intersect including vertical and supplementary
angles;
b. when formed by parallel lines cut by a transversal including
corresponding, alternate interior, and alternate exterior angles.
10. recognizes and describes arcs and semicircles as parts of a circle
and uses the standard notation for arc () and circle () (2.4.K1h).
8-285
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Geometry is the study of shapes, their properties, and their relationships to other shapes. Symbols and numbers are used to
describe their properties and their relationships to other shapes. The fundamental concepts in geometry are point (no dimension), line (onedimensional), plane (two-dimensional), and space (three-dimensional). Plane figures are referred to as two-dimensional. Solids are referred to as
three-dimensional. The base, in terms of geometry, generally refers to the side on which a figure rests. Therefore, depending on the orientation of
the solid, the base changes.
From the Mathematics Dictionary and Handbook (Nichols Schwartz Publishing, 1999), property as a mathematical term means a characteristic (an
attribute) of a number, geometric shape, mathematical operation, equation, or inequality. To give an example:
 Property of a number: 8 is divisible by 2.
 Property of a geometric shape: Each of the four sides of a square is of the same length.
 Property of an operation: Addition is commutative. For all numbers x and y, x + y = y + x.
 Property of an equation: For all numbers a, b, and c, if a = b, then a + c = b + c.
 Property of an inequality: For all numbers a, b, and c, if a > b, then a – c > b – c.
The application of the Knowledge Indicators from the Geometry Benchmark, Geometric Figures and Their Properties are most often
applied within the context of the other Geometry Benchmarks - Measurement and Estimation, Transformational Geometry, and
Geometry From an Algebraic Perspective - rather than in isolation.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks
are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be
used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to
Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models
benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each indicator in the Models
benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For example, process models are
linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and
Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance;
benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending
and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at
three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in
Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the
Appendix.
8-286
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
EIGHTH GRADE
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 2: Measurement and Estimation – The student estimates, measures, and uses geometric formulas in a
variety of situations.
Eighth Grade Knowledge Base Indicators
Eighth Grade Application Indicators
The student…
The student…
1. determines and uses rational number approximations (estimations)
for length, width, weight, volume, temperature, time, perimeter, area,
and surface area using standard and nonstandard units of measure
(2.4.K1a) ($).
2. selects and uses measurement tools, units of measure, and level of
precision appropriate for a given situation to find accurate real
number representations for length, weight, volume, temperature,
time, perimeter, area, surface area, and angle measurements
(2.4.K1a) ($).
3. converts within the customary system and within the metric system.
4. estimates the measure of a concrete object in one system given the
measure of that object in another system and the approximate
conversion factor (2.4.K1a), e.g., a mile is about 2.2 kilometers; how
far is 2 miles?
5. uses given measurement formulas to find (2.4.K1h):
a. area of parallelograms and trapezoids;
b. surface area of rectangular prisms, triangular prisms, and
cylinders;
c. volume of rectangular prisms, triangular prisms, and cylinders.
6. recognizes how ratios and proportions can be used to measure
inaccessible objects (2.4.K1c), e.g., using shadows to measure the
height of a flagpole.
7. calculates rates of change, e.g., speed or population growth.
1. solves real-world problems (2.4.A1a) by ($):
a. converting within the customary and the metric systems, e.g.,
James added 30 grams of sand to his model boat that weighed
2 kg before it sank. With the sand included, what is the total
weight of his boat?
b. finding perimeter and area of circles, squares, rectangles,
triangles, parallelograms, and trapezoids; e.g., Jane jogs on a
circular track with a radius of 100 feet. How far would she jog
in one lap?
c. finding the volume and surface area of rectangular prisms, e.g.,
how much paint would be needed to cover a box with
dimensions of 3 feet by 4 feet by 5 feet?
2. estimates to check whether or not measurements or calculations for
length, weight, volume, temperature, time, perimeter, area, and
surface area in real world problems are reasonable and adjusts
original measurement or estimation based on additional information
(a frame of reference) (2.4.A1a) ($), e.g., to check your calculation
in finding the area of the floor in the kitchen; you count how many
foot-square tiles there are on the floor.
3. uses ratio and proportion to measure inaccessible objects
(2.4.A1c), e.g., using the length of a shadow to measure the height
of a flagpole.
8-287
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: The term geometry comes from two Greek words meaning “earth measure.” Measurement provides the tools required to apply
geometric concepts in the real-world. Estimation in measurement is defined as making guesses as to the exact measurement of an object without
using any type of measurement tool. Estimation helps students develop a relationship between the different sizes of units of measure. It helps
students develop basic properties of measurement and it gives students a tool to determine whether a given measurement is reasonable.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
8-288
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
EIGHTH GRADE
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 3: Transformational Geometry – The student recognizes and applies transformations on geometric
figures in a variety of situations.
Eighth Grade Knowledge Base Indicators
The student…
1. identifies, describes, and performs single and multiple
transformations [reflection, rotation, translation, reduction
(contraction/shrinking), enlargement (magnification/growing)] on a
two-dimensional figure (2.4.K1a).
2. describes a reflection of a given two-dimensional figure that moves it
from its initial placement (preimage) to its final placement (image) in
the coordinate plane over the x- and y-axis (2.4.K1a,i).
3. draws (2.4.K1a):
a. three-dimensional figures from a variety of perspectives (top,
bottom, sides, corners);
b. a scale drawing of a two-dimensional figure;
c. a two-dimensional drawing of a three-dimensional figure.
4. determines where and how an object or a shape can be tessellated
using single or multiple transformations (2.4.K1a).
Eighth Grade Application Indicators
The student…
1. generalizes the impact of transformations on the area and perimeter
of any two-dimensional geometric figure (2.4.A1a), e.g., enlarging
by a factor of three triples the perimeter (circumference) and
multiplies the area by a factor of nine.
2. describes and draws a two-dimensional figure after undergoing two
specified transformations without using a concrete object.
3. investigates congruency, similarity, and symmetry of geometric
figures using transformations (2.4.A1g).
4. uses a scale drawing to determine the actual dimensions and/or
measurements of a two-dimensional figure represented in a scale
drawing (2.4.A1h).
8-289
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Transformational geometry is another way to investigate and interpret geometric figures by moving every point in a plane figure
to a new location. To help students form images of shapes through different transformations, students can use concrete objects, figures drawn on
graph paper, mirrors or other reflective surfaces, or appropriate technology. Some transformations, like translations, reflections, and rotations, do
not change the figure itself. Other transformations, like reduction (contraction/shrinking) or enlargement (magnification/growing), change the size of
a figure, but not the shape (congruence vs. similarity).
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that
could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are
identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and
Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
8-290
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
EIGHTH GRADE
Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 4: Geometry from an Algebraic Perspective – The student uses an algebraic perspective to examine the
geometry of two-dimensional figures in a variety of situations.
Eighth Grade Knowledge Base Indicators
Eighth Grade Application Indicators
The student…
1. uses the coordinate plane to (2.4.K1a):
a. ▲ list several ordered pairs on the graph of a line and find the
slope of the line;
b. ▲ recognize that ordered pairs that lie on the graph of an
equation are solutions to that equation;
c. ▲ recognize that points that do not lie on the graph of an
equation are not solutions to that equation;
d. ▲ determine the length of a side of a figure drawn on a
coordinate plane with vertices having the same x- or ycoordinates;
e. solve simple systems of linear equations.
2. uses a given linear equation with integer coefficients and constants
and an integer solution to find the ordered pairs, organizes the
ordered pairs using a T-table, and plots the ordered pairs on a
coordinate plane (2.4.K1e-g).
3. examines characteristics of two-dimensional figures on a coordinate
plane using various methods including mental math, paper and
pencil, concrete objects, and graphing utilities or other appropriate
technology (2.4.A1g).
The student…
1. represents, generates, and/or solves distance problems (including
the use of the Pythagorean theorem, but not necessarily the
distance formula) (2.4.A1a), e.g., a student lives five miles west and
three miles north of school and another student lives 4 miles south
and 7 miles east of school. What is the shortest distance between
the students’ homes (as the crow flies)?
2. translates between the written, numeric, algebraic, and geometric
representations of a real-world problem (2.4.A1a,d-g), e.g., given a
situation: make a T-table, define the algebraic relationship, and
graph the ordered pairs. The T-table can be represented as –
as an algebraic relationship – 2 χ =
X 0
1
2
3
5,
Y
5
7
9
11
and as a graph (graphical)
8-291
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: A number line (a mathematical model) is a diagram that represents numbers with equal distances marked off as points on a line,
and is an example of one-to-one correspondence (a relation). A number line can be used as a visual representation of numbers and operations. In
addition, a number line used horizontally and vertically is a precursor to the coordinate plane; and the distance between two numbers on a number
line is a precursor to absolute value.
A coordinate plane (coordinate grid) consists of a horizontal number line called the x-axis and a vertical number line called the y-axis. These two
lines intersect at a point called the origin. The x-axis and the y-axis divide the plane into four sections called quadrants. Any point on the coordinate
plane can be named with two numbers called coordinates. The first number is the x-coordinate. The second number is the y-coordinate. Since the
pair is always named in order (first x, then y), it is called an ordered pair.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that
could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are
identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and
Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
8-292
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 4: Data
EIGHTH GRADE
Standard 4: Data – The student uses concepts and procedures of data analysis in a variety of situations.
Benchmark 1: Probability – The student applies the concepts of probability to draw conclusions, generate
convincing arguments, and make predictions and decisions including the use of concrete objects in a
variety of situations.
Eighth Grade Knowledge Base Indicators
The student…
Eighth Grade Application Indicators
1. knows and explains the difference between independent and
dependent events in an experiment, simulation, or situation (2.4.K1j)
($).
2. identifies situations with independent or dependent events in an
experiment, simulation, or situation (2.4.K1j), e.g., there are three
marbles in a bag. If you draw one marble and give it to your brother,
and another marble and give it to your sister, are these independent
events or dependent events?
3. ▲ finds the probability of a compound event composed of two
independent events in an experiment, simulation, or situation
(2.4.K1j), e.g., what is the probability of getting two heads, if you
toss a dime and a quarter?
4. finds the probability of simple and/or compound events using
geometric models (spinners or dartboards) (2.4.K1j).
5. finds the odds of a desired outcome in an experiment or simulation
and expresses the answer as a ratio (2/3 or 2:3 or 2 to 3) (2.4.K1j).
6. describes the difference between probability and odds.
The student…
1. conducts an experiment or simulation with independent or
dependent events including the use of concrete objects; records the
results in a chart, table, or graph; and uses the results to draw
conclusions and make predictions about future events (2.4.A1i-j).
2. analyzes the results of an experiment or simulation of two
independent events to generate convincing arguments, draw
conclusions, and make predictions and decisions in a variety of realworld situations (2.4.A1i-j).
3. compares theoretical probability (expected results) with empirical
probability (experimental results) in an experiment or simulation with
a compound event composed of two independent events and
understands that the larger the sample size, the greater the
likelihood that the experimental results will equal the theoretical
probability (2.4.A1i).
4. makes predictions based on the theoretical probability of (2.4.A1a,i):
a. ▲ ■ a simple event in an experiment or simulation,
b. compound events composed of two independent events in an
experiment or simulation.
8-293
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Ideas from probability reinforce concepts in the other Standards, especially Number and Computation and Geometry. Students
need to develop an intuitive concept of chance – whether or not something is unlikely or likely to happen. Probability experiences should be
addressed through the use of concrete objects (process models); spinners, number cubes, or dartboards (geometric models); and coins (money
models). Probabilities are ratios, expressed as fractions, decimals, or percents, determined by considering results or outcomes of experiments.
Some examples of uses of probability in every day life include: There is a 50% chance of rain today. What is the probability that the team will win
every game?
Odds is a ratio of favorable to unfavorable outcomes whereas probability is a ratio of favorable outcomes to all possible outcomes. Probability can
be written as a fraction, a decimal, and a percent, whereas odds can only be written as a ratio, e.g., 2/3, 2:3, or 2 to 3.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks
are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be
used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to
Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models
benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each indicator in the Models
benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For example, process models are
linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and
Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
8-294
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 4: Data
EIGHTH GRADE
Standard 4: Data – The student uses concepts and procedures of data analysis in a variety of situations.
Benchmark 2: Statistics – The student collects, organizes, displays, explains, and interprets numerical (rational) and
non-numerical data sets in a variety of situations.
Eighth Grade Knowledge Base Indicators
The student…
1. organizes, displays and reads quantitative (numerical) and
qualitative (non-numerical) data in a clear, organized, and accurate
manner including a title, labels, categories, and rational number
intervals using these data displays (2.4.K1k) ($):
a. frequency tables;
b. bar, line, and circle graphs;
c. Venn diagrams or other pictorial displays;
d. charts and tables;
e. stem-and-leaf plots (single and double);
f. scatter plots;
g. box-and-whiskers plots;
h. histograms.
2. recognizes valid and invalid data collection and sampling
techniques.
3. ▲ determines and explains the measures of central tendency
(mode, median, mean) for a rational number data set (2.4.K1a).
4. determines and explains the range, quartiles, and interquartile range
for a rational number data set (2.4.K1a).
5. explains the effects of outliers on the median, mean, and range of a
rational number data set (2.4.K1a).
6. makes a scatter plot and draws a line that approximately represents
the data, determines whether a correlation exists, and if that
correlation is positive, negative, or that no correlation exists
(2.4.K1k).
Eighth Grade Application Indicators
The student…
1. uses data analysis (mean, median, mode, range) in real-world
problems with rational number data sets to compare and contrast
two sets of data, to make accurate inferences and predictions, to
analyze decisions, and to develop convincing arguments from these
data displays (2.4.A1j) ($):
a. frequency tables;
b. bar, line, and circle graphs;
c. Venn diagrams or other pictorial displays;
d. charts and tables;
e. stem-and-leaf plots (single and double);
f. scatter plots;
g. box-and-whiskers plots;
h. histograms.
5. explains advantages and disadvantages of various data collection
techniques (observations, surveys, or interviews), and sampling
techniques (random sampling, samples of convenience, biased
sampling, or purposeful sampling) in a given situation (2.4.A1j) ($).
6. recognizes and explains (2.4.A1j):
a. misleading representations of data;
b. the effects of scale or interval changes on graphs of data sets.
4. recognizes faulty arguments and common errors in data analysis.
8-295
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Graphs (data displays) are pictorial representations of mathematical relationships, are used to tell a story, and are an important
part of statistics. When a graph is made, the axes and the scale (numbers running along a side of the graph) are chosen for a reason. The
difference between numbers from one grid line to another is the interval. The interval will depend on the lowest and highest values in the data set.
Emphasizing the importance of using equal-sized pictures or intervals is critical to ensuring that the data display is accurate.
Graphs take many forms:
– bar graphs and pictographs compare discrete data,
– frequency tables show how many times a certain piece of data occurs,
– circle graphs (pie charts) model parts of a whole,
– line graphs show change over time,
– Venn diagrams show relationships among sets of objects,
– line plots show frequency of data on a number line,
– single stem-and-leaf plots (closely related to line plots except that the number line is usually vertical and digits are used rather than x’s) show
frequency distribution by arranging numbers (stems) on the left side of a vertical line with numbers (leaves) on the right side,
– scatter plots show the relationship between two quantities,
– box-and-whisker plots are visual representations of the five-number summary – the median, the upper and lower quartiles, and the least and
greatest values in the distribution – therefore, the center, the spread, and the overall range are immediately evident by looking at the plot, and
– histograms (closely related to stem-and-leaf plots) describe how data falls into different ranges.
Two important aspects of data are its center and its spread. The mean, median, and mode are measures of central tendency (averages) that
describe where data are centered. Each of these measures is a single number that describes the data. However, each does it slightly differently.
The range describes the spread (dispersion) of data. The easiest way to measure spread is the range, the difference between the greatest and the
least values in a data set. Quartiles are boundaries that break the data into fourths.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that
could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are
identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and
Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
8-296
January 31, 2004
▲– Assessed Indicator on the Objective Assessment
■ – Assessed Indicator on the Optional Constructed Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
NINTH AND TENTH GRADES
Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 1: Number Sense – The student demonstrates number sense for real numbers and algebraic expressions
in a variety of situations.
Ninth and Tenth Grades Knowledge Base Indicators
The student…
1. knows, explains, and uses equivalent representations for real
numbers and algebraic expressions including integers, fractions,
decimals, percents, ratios; rational number bases with integer
exponents; rational numbers written in scientific notation; absolute
value; time; and money (2.4.K1a) ($), e.g., –4/2 = (–2); a(-2) b(3) = b3/a2.
2. compares and orders real numbers and/or algebraic expressions and
explains the relative magnitude between them (2.4.K1a) ($), e.g., will
(5n)2 always, sometimes, or never be larger than 5n? The student
might respond with (5n)2 is greater than 5n if n > 1 and (5n)2 is
smaller than 5 if o < n < 1.
3. knows and explains what happens to the product or quotient when a
real number is multiplied or divided by (2.4.K1a):
a. a rational number greater than zero and less than one,
b. a rational number greater than one,
c. a rational number less than zero.
Ninth and Tenth Grades Application Indicators
The student…
1. generates and/or solves real-world problems using equivalent
representations of real numbers and algebraic expressions
(2.4.A1a) ($), e.g., a math classroom needs 30 books and 15
calculators. If B represents the cost of a book and C represents
the cost of a calculator, generate two different expressions to
represent the cost of books and calculators for 9 math classrooms.
2. determines whether or not solutions to real-world problems using
real numbers and algebraic expressions are reasonable (2.4.A1a)
($), e.g., in January, a business gave its employees a 10% raise.
The following year, due to the sluggish economy, the employees
decided to take a 10% reduction in their salary. Is it reasonable to
say they are now making the same wage they made prior to the
10% raise?
9/10-297
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Number sense refers to one’s ability to reason with numbers and to work with numbers in a flexible way. The ability to compute
mentally, to estimate based on understanding of number relationships and magnitudes, and to judge reasonableness of answers are all involved in
number sense.
At this grade level, real numbers include positive and negative numbers and very large numbers (one billion) and very small numbers (onebillionth). Relative magnitude refers to the size relationship one number has with another – is it much larger, much smaller, close, or about the
same? For example, using the numbers 219, 264, and 457, answer questions such as




Which two are closest? Why?
Which is closest to 300? To 250?
About how far apart are 219 and 500? 5,000?
If these are ‘big numbers,’ what are small numbers? Numbers about the same? Numbers that make these seem
small?
(Elementary and Middle School Mathematics, John A. Van de Walle, Addison Wesley Longman, Inc., 1998)
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
9/10-298
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
NINTH AND TENTH GRADES
Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 2: Number Systems and Their Properties – The student demonstrates an understanding of the real
number system; recognizes, applies, and explains their properties, and extends these properties to
algebraic expressions.
Ninth and Tenth Grades Knowledge Base Indicators
The student…
1. explains and illustrates the relationship between the subsets of the
real number system [natural (counting) numbers, whole numbers,
integers, rational numbers, irrational numbers] using mathematical
models (2.4.K1a), e.g., number lines or Venn diagrams.
2. identifies all the subsets of the real number system [natural
(counting) numbers, whole numbers, integers, rational numbers,
irrational numbers] to which a given number belongs (2.4.K1m).
3. ▲ names, uses, and describes these properties with the real
number system and demonstrates their meaning including the use of
concrete objects (2.4.K1a) ($):
a. commutative (a + b = b + a and ab = ba), associative [a + (b + c)
= (a + b) + c and a(bc) = (ab)c], distributive [a (b + c) = ab + ac],
and substitution properties (if a = 2, then 3a = 3 x 2 = 6);
b. identity properties for addition and multiplication and inverse
properties of addition and multiplication (additive identity: a + 0 =
a, multiplicative identity: a • 1 = a, additive inverse: +5 + –5 = 0,
multiplicative inverse: 8 x 1/8 = 1);
c. symmetric property of equality (if a = b, then b = a);
d. addition and multiplication properties of equality (if a = b, then a
+ c = b + c and if a = b, then ac = bc) and inequalities (if a > b,
then a + c > b + c and if a > b, and c > 0 then ac > bc);
e. zero product property (if ab = 0, then a = 0 and/or b = 0).
4. uses and describes these properties with the real number system
(2.4.K1a) ($):
a. transitive property (if a = b and b = c, then a = c),
Ninth and Tenth Grades Application Indicators
The student…
1. generates and/or solves real-world problems with real numbers
using the concepts of these properties to explain reasoning
(2.4.A1a) ($):
a. commutative, associative, distributive, and substitution
properties, e.g., the chorus is sponsoring a trip to an amusement
park. They need to purchase 15 adult tickets at $6 each and 15
student tickets at $4 each. How much money will the chorus
need for tickets? Solve this problem two ways.
b. identity and inverse properties of addition and multiplication,
e.g., the purchase price (P) of a series EE Savings Bond is
found by the formula ½ F = P where F is the face value of the
bond. Use the formula to find the face value of a savings bond
purchased for $500.
c. symmetric property of equality, e.g., Sam took a $15 check to
the bank and received a $10 bill and a $5 bill. Later Sam took a
$10 bill and a $5 bill to the bank and received a check for $15. $
addition and multiplication properties of equality, e.g., the total
price for the purchase of three shirts in $62.54 including tax. If
the tax is $3.89, what is the cost of one shirt, if all shirts cost the
same?
d. addition and multiplication properties of equality, e.g., the total
price for the purchase of three shirts is $62.54 including tax. If
the tax is $3.89, what is the cost of one shirt?
T = 3s + t
$62.54 = 3s + $3.89 - $3.89
9/10-299
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
b. reflexive property (a = a).
$62.54 - $3.89 = 3s
$58.65 = 3s
$58.65 = 3s = 3s ÷ 3
$19.55 = s
e. zero product property, e.g., Jenny was thinking of two numbers.
Jenny said that the product of the two numbers was 0. What
could you deduct from this statement? Explain your reasoning.
2. analyzes and evaluates the advantages and disadvantages of using
integers, whole numbers, fractions (including mixed numbers),
decimals or irrational numbers and their rational approximations in
solving a given real-world problem (2.4.A1a) ($), e.g., a store sells
CDs for $12.99 each. Knowing that the sales tax is 7%, Marie
estimates the cost of a CD plus tax to be $14.30. She selects nine
CDs. The clerk tells Marie her bill is $157.18. How can Marie
explain to the clerk she has been overcharged?
Teacher Notes: From the Mathematics Dictionary and Handbook (Nichols Schwartz Publishing, 1999), property as a mathematical term means a
characteristic (an attribute) of a number, geometric shape, mathematical operation, equation, or inequality. To give an example:
 Property of a number: 8 is divisible by 2.
 Property of a geometric shape: Each of the four sides of a square is of the same length.
 Property of an operation: Addition is commutative. For all numbers x and y, x + y = y + x.
 Property of an equation: For all numbers a, b, and c, if a = b, then a + c = b + c.
 Property of an inequality: For all numbers a, b, and c, if a > b, then a – c > b – c.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that
could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are
identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and
Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
9/10-300
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
NINTH AND TENTH GRADES
Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 3: Estimation – The student uses computational estimation with real numbers in a variety of situations.
Ninth and Tenth Grades Knowledge Base Indicators
The student…
1. estimates real number quantities using various
computational methods including mental math, paper and
pencil, concrete objects, and/or appropriate technology
(2.4.K1a) ($).
2. uses various estimation strategies and explains how they were
used to estimate real number quantities and algebraic expressions
(2.4.K1a) ($).
3. knows and explains why a decimal representation of an irrational
number is an approximate value(2.4.K1a).
4. knows and explains between which two consecutive integers an
irrational number lies (2.4.K1a).
Ninth and Tenth Grades Application Indicators
The student…
1. ▲ adjusts original rational number estimate of a real-world problem
based on additional information (a frame of reference) (2.4.A1a) ($),
e.g., estimate how long it takes to walk from here to there; time how
long it takes to take five steps and adjust your estimate.
2. estimates to check whether or not the result of a real-world problem
using real numbers and/or algebraic expressions is reasonable and
makes predictions based on the information (2.4.A1a) ($), e.g., if you
have a $4,000 debt on a credit card and the minimum of $30 is paid
per month, is it reasonable to pay off the debt in 10 years?
3. determines if a real-world problem calls for an exact or approximate
answer and performs the appropriate computation using various
computational strategies including mental math, paper and pencil,
concrete objects, and/or appropriate technology (2.4.A1a) ($), e.g., do
you need an exact or an approximate answer in calculating the area
of the walls to determine the number of rolls of wallpaper needed to
paper a room? What would you do if you were wallpapering 2
rooms?
4. explains the impact of estimation on the result of a real-world problem
(underestimate, overestimate, range of estimates) (2.4.A1a) ($), e.g.,
if the weight of 25 pieces of paper was measured as 530.6 grams,
what would the weight of 2,000 pieces of paper equal to the nearest
gram? If the student were to estimate the weight of one piece of
paper as about 20 grams and then multiply this by 2,000 rather than
multiply the weight of 25 pieces of paper by 80; the answer would
differ by about 2,400 grams. In general, multiplying or dividing by a
rounded number will cause greater discrepancies than rounding after
multiplying or dividing.
9/10-301
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Estimate, as a verb, means to make an educated guess based on information in a problem or to give an answer close to the exact
number. Estimation is used when an exact answer is not needed, as in many real-life situations for which “ballpark” computations are acceptable.
Good number sense enables one to estimate a quantity, estimate a measure, or estimate an answer.
Estimation serves as an important companion to computation. It provides a tool for judging the reasonableness of computational methods including
mental math, paper and pencil, concrete objects, and appropriate technology. However, being able to compute does not automatically lead to an
ability to estimate or judge reasonableness of answers. Frequent modeling by the teacher helps students develop a range of estimation strategies.
Students should be encouraged to frequently explain their thinking as they estimate. As with exact computation, sharing estimation strategies allows
students access to others’ thinking and provides opportunities for class discussion. Identifying the estimation strategy by name is not critical;
however, explaining the thinking behind the strategy to make a valid estimation is important. (Principles and Standards for School Mathematics,
NCTM, 2000)
Mental math and estimation are distinct but related mathematical skills. Proficiency in mental math contributes to increased skill in estimation. In
order for students to become more familiar with estimation, teachers should introduce estimation with examples where rounded or estimated
numbers are used. Emphasis should be placed on real-world examples where only estimation is required, e.g., About how many hours do you sleep
a night? Using the language of estimation is important, so students begin to realize that a variety of estimates (answers) are possible. In addition,
when students are taught specific estimation strategies, they develop mental math and estimation skills easier. Estimation strategies include frontend with adjustment, compatible “nice” numbers, clustering, special numbers, truncation, or simulation.
Mathematical models such as concrete objects, pictures, diagrams, Venn diagrams, number lines, hundred charts, base ten blocks, or factor trees
are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to
represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra),
Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical
models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1
(Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and Investing,
and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in the Kansas
Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level with a ($). The
National Standards in Personal Finance are included in the Appendix.
9/10-302
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 1: Number and Computation
NINTH AND TENTH GRADES
Number and Computation – The student uses numerical and computational concepts and procedures in a
variety of situations.
Benchmark 4: Computation – The student models, performs, and explains computation with real numbers and
polynomials in a variety of situations.
Ninth and Tenth Grades Knowledge Base Indicators
The student…
1. computes with efficiency and accuracy using various computational
methods including mental math, paper and pencil, concrete objects,
and appropriate technology (2.4.K1a) ($).
2. performs and explains these computational procedures (2.4.K1a):
a. N addition, subtraction, multiplication, and division using the
order of operations
b. multiplication or division to find ($):
i. a percent of a number, e.g., what is 0.5% of 10?
ii. percent of increase and decrease, e.g., a college raises its
tuition form $1,320 per year to $1,425 per year. What
percent is the change in tuition?
iii. percent one number is of another number, e.g., 89 is what
percent of 82?
iv. a number when a percent of the number is given, e.g., 80 is
32% of what number?
c. manipulation of variable quantities within an equation or
inequality (2.4.K1d), e.g., 5x – 3y = 20 could be written as 5x –
20 = 3y or 5x(2x + 3) = 8 could be written as 8/(5x) = 2x + 3;
d. simplification of radical expressions (without rationalizing
denominators) including square roots of perfect square
monomials and cube roots of perfect cubic monomials;
e. simplification or evaluation of real numbers and algebraic
monomial expressions raised to a whole number power and
algebraic binomial expressions squared or cubed;
f. simplification of products and quotients of real number and
algebraic monomial expressions using the properties of
Ninth and Tenth Grades Application Indicators
The student…
1. generates and/or solves multi-step real-world problems with real
numbers and algebraic expressions using computational procedures
(addition, subtraction, multiplication, division, roots, and powers
excluding logarithms), and mathematical concepts with ($):
a. ▲ applications from business, chemistry, and physics that
involve addition, subtraction, multiplication, division, squares,
and square roots when the formulae are given as part of the
problem and variables are defined (2.4.A1a) ($), e.g., given F =
ma, where F = force in newtons, m = mass in kilograms, a =
acceleration in meters per second squared. Find the
acceleration if a force of 20 newtons is applied to a mass of 3
kilograms.
b. ▲ volume and surface area given the measurement formulas of
rectangular solids and cylinders (2.4.A1f), e.g., a silo has a
diameter of 8 feet and a height of 20 feet. How many cubic feet
of grain can it store?
c. probabilities (2.4.A1h), e.g., if the probability of getting a
defective light bulb is 2%, and you buy 150 light bulbs, how
many would you expect to be defective?
d. ▲ ■ application of percents (2.4.A1a), e.g., given the formula
A = P(1+
r
n
)nt, A = amount, P= principal, r = annual interest, n =
number of compounding periods per year, t = number of years.
If $1,000 is placed in a savings account with a 6% annual
interest rate and is compounded semiannually, how much
money will be in the account at the end of 2 years?
9/10-303
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
exponents;
g. matrix addition ($), e.g., when computing (with one operation) a
building’s expenses (data) monthly, a matrix is created to
include each of the different expenses; then at the end of the
year, each type of expense for the building is totaled;
h. scalar-matrix multiplication ($), e.g., if a matrix is created with
everyone’s salary in it, and everyone gets a 10% raise in pay;
to find the new salary, the matrix would be multiplied by 1.1.
3. finds prime factors, greatest common factor, multiples, and the least
common multiple of algebraic expressions (2.4.K1b).
e. simple exponential growth and decay (excluding logarithms)
and economics (2.4.A1a) ($), e.g., a population of cells doubles
every 20 years. If there are 20 cells to start with, how long will it
take for there to be more than 150 cells? or If the radiation level
is now 400 and it decays by ½ or its half-life is 8 hours, how
long will it take for the radiation level to be below an acceptable
level of 5?
9/10-304
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Efficiency and accuracy means that students are able to compute with fluency. Students increase their understanding and skill
in addition, subtraction, multiplication, and division by understanding the relationships between addition and subtraction, addition and
multiplication, multiplication and division, and subtraction and division. Students learn basic number combinations and develop strategies for
computing that makes sense to them. Through class discussions, students can compare the ease of use and ease of explanation of various
strategies. In some cases, their strategies for computing will be close to conventional algorithms; in other cases, they will be quite different. Many
times, students’ invented approaches are based on a sound understanding of numbers and operations, and these invented approaches often can
be used with efficiency and accuracy. (Principles and Standards for School Mathematics, NCTM, 2000)
The definition of computation is finding the standard representation for a number. For example, 6 + 6, 4 x 3, 17 – 5, and 24 ÷ 2 are all
representations for the standard representation of 12. Mental math is mentally finding the standard representation for a number – calculating in
your head instead of calculating using paper and pencil or technology. One of the main reasons for teaching mental math is to help students
determine if a computed/calculated answer is reasonable; in other words, using mental math to estimate to see if the answer makes sense.
Students develop mental math skills easier when they are taught specific strategies. Mental math strategies include counting on, doubling,
repeated doubling, halving, making tens, multiplying by powers of tens, dividing with tens, finding fractional parts, thinking money, and using
compatible “nice” numbers.
Mathematical models such as concrete objects, pictures, diagrams, Venn diagrams, number lines, or factor trees are necessary for conceptual
understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the concept, the
indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models),
and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that could be
used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are
identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number
and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
Technology is changing mathematics and its uses. The use of technology including calculators and computers is an important part of growing up
in a complex society. It is not only necessary to estimate appropriate answers accurately when required, but also it is also important to have a
good understanding of the underlying concepts in order to know when to apply the appropriate procedure. Technology does not replace the need
to learn basic facts, to compute mentally, or to do reasonable paper-and-pencil computation. However, dividing a 5-digit number by a 2-digit
number is appropriate with the exception of dividing by 10, 100, or 1,000 and simple multiples of each.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
9/10-305
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
NINTH AND TENTH GRADES
Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 1: Patterns – The student recognizes, describes, extends, develops, and explains the general rule of a
pattern in a variety of situations.
Ninth and Tenth Grades Knowledge Base Indicators
The student…
1. identifies, states, and continues the following patterns using various
formats including numeric (list or table), algebraic (symbolic
notation), visual (picture, table, or graph), verbal (oral description),
kinesthetic (action), and written
a. arithmetic and geometric sequences using real numbers and/or
exponents (2.4.K1a); e.g., radioactive half-lives;
b. patterns using geometric figures (2.4.K1h);
c. algebraic patterns including consecutive number patterns or
equations of functions, e.g., n, n + 1, n + 2, ... or f(n) = 2n – 1
(2.4.K1c,e);
d. special patterns (2.4.K1a), e.g., Pascal’s triangle and the
Fibonacci sequence.
2. generates and explains a pattern (2.4.K1f).
3. classify sequences as arithmetic, geometric, or neither.
4. defines (2.4.K1a):
a. a recursive or explicit formula for arithmetic sequences and finds
any particular term,
b. a recursive or explicit formula for geometric sequences and finds
any particular term.
Ninth and Tenth Grades Application Indicators
The student…
1. recognizes the same general pattern presented in different
representations [numeric (list or table), visual (picture, table, or
graph), and written] (2.4.A1i) ($).
2. solves real-world problems with arithmetic or geometric sequences
by using the explicit equation of the sequence (2.4.K1c) ($), e.g., an
arithmetic sequence: A brick wall is 3 feet high and the owners want
to build it higher. If the builders can lay 2 feet every hour, how long
will it take to raise it to a height of 20 feet? or a geometric sequence:
A savings program can double your money every 12 years. If you
place $100 in the program, how many years will it take to have over
$1,000?
9/10-306
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: A fundamental component in the development of classification, number, and problem solving skills is inventing, discovering, and
describing patterns. Patterns pervade all of mathematics and much of nature. All patterns are either repeating or growing or a variation of either or
both. Translating a pattern from one medium to another to find two patterns that are alike, even though they are made with different materials, is
important so students can see the relationships that are critical to repeating patterns. With growing patterns, students not only extend patterns, but
also look for a generalization or an algebraic relationship that will tell them what the pattern will be at any point along the way.
Working with patterns is an important process in the development of mathematical thinking. Patterns can be based on geometric attributes
(shapes, regions, angles); measurement attributes (color, texture, length, weight, volume, number); relational attributes (proportion, sequence,
functions); and affective attributes (values, likes, dislikes, familiarity, heritage, culture). (Learning to Teach Mathematics, Randall J. Souviney,
Macmillan Publishing Company, 1994)
In the pattern that begins with 3, 5, 7, and 9; the explicit rule is 2n +1 and the recursive rule is add 2 to the previous term. Patterns themselves are
not explicit or recursive. The RULE for the pattern can be expressed explicitly or recursively and MOST patterns can be explained using either
format especially IF that pattern reflects either an arithmetic sequence or geometric sequence.
This process (working with patterns) can be used to develop or deepen understandings of important concepts in number theory, rational numbers,
measurement, geometry, probability, and functions. Working with patterns provides opportunities for students to recognize, describe, extend,
develop, and explain.
Number theory is the study of the properties of the counting numbers (positive integers), their relationships, ways to represent them, and patterns
among them. Number theory includes the concepts of odd and even numbers, factors and multiples, primes and composites, and greatest common
factor and least common multiple, and sequences.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that
could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are
identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and
Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
9/10-307
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
9/10-308
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
NINTH AND TENTH GRADES
Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 2: Variables, Equations, and Inequalities – The student uses variables, symbols, real numbers, and
algebraic expressions to solve equations and inequalities in variety of situations.
Ninth and Tenth Grades Knowledge Base Indicators
Ninth and Tenth Grades Application Indicators
The student…
1. knows and explains the use of variables as parameters for a specific
variable situation (2.4.K1f), e.g., the m and b in y = mx + b or the h,
k, and r in (x – h)2 + (y – k)2 = r2.
2. manipulates variable quantities within an equation or inequality
(2.4.K1e), e.g., 5x – 3y = 20 could be written as 5x – 20 = 3y or
5x(2x + 3) = 8 could be written as 8/(5x) = 2x + 3.
3. solves (2.4.K1d) ($):
a. N linear equations and inequalities both analytically and
graphically;
b. quadratic equations with integer solutions (may be solved by
trial and error, graphing, quadratic formula, or factoring);
c. ▲N systems of linear equations with two unknowns using
integer coefficients and constants;
d. radical equations with no more than one inverse operation
around the radical expression;
e. equations where the solution to a rational equation can be
simplified as a linear equation with a nonzero denominator, e.g.,
__3__ = __5__.
(x + 2) (x – 3)
f. equations and inequalities with absolute value quantities
containing one variable with a special emphasis on using a
number line and the concept of absolute value.
g. exponential equations with the same base without the aid of a
calculator or computer, e.g., 3x + 2 = 35.
The student…
1. represents real-world problems using variables, symbols,
expressions, equations, inequalities, and simple systems of linear
equations (2.4.A1c-e) ($).
2. represents and/or solves real-world problems with (2.4.A1c) ($):
a. ▲N linear equations and inequalities both analytically and
graphically, e.g., tickets for a school play are $5 for adults and
$3 for students. You need to sell at least $65 in tickets. Give
an inequality and a graph that represents this situation and
three possible solutions.
b. quadratic equations with integer solutions (may be solved by
trial and error, graphing, quadratic formula, or factoring), e.g., a
fence is to be built onto an existing fence. The three sides will
be built with 2,000 meters of fencing. To maximize the
rectangular area, what should be the dimensions of the fence?
c. systems of linear equations with two unknowns, e.g., when
comparing two cellular telephone plans, Plan A costs $10 per
month and $.10 per minute and Plan B costs $12 per month and
$.07 per minute. The problem is represented by Plan A = .10x +
10 and Plan B = .07x + 12 where x is the number of minutes.
d. radical equations with no more than one inverse operation
around the radical expression, e.g., a square rug with an area of
200 square feet is 4 feet shorter than a room. What is the
length of the room?
e. a rational equation where the solution can be simplified as a
linear equation with a nonzero denominator, e.g., John is 2 feet
taller than Fred. John’s shadow is 6 feet in length and Fred’s
shadow is 4 feet in length. How tall is Fred?
9/10-309
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
3. explains the mathematical reasoning that was used to solve a realworld problem using equations and inequalities and analyzes the
advantages and disadvantages of various strategies that may have
been used to solve the problem (2.4.A1c).
Teacher Notes: One of the most powerful ways we have of expressing the regularities found in mathematics is with variables. Variables enable
us to use mathematical symbolism as a tool to think and help better understand mathematical ideas in the same way physical objects and drawings
are used. But if variables are to be included with these other “thinker toys,” it is important to help students develop an understanding of the various
ways they are used. A variable is a symbol that can stand for any one of a set of numbers or other objects. Although correct, this simple-sounding
definition has a variety of interpretations, depending on how the variables are used.
Meanings of variables change with the way they are used. Usiskin (The ideas of Algebra, K-12, pp. 8-19, NCEM, 1988) identified three uses of
variables that are commonly encountered in school mathematics:
1. As a specific unknown. In the early grades, this is the use found in equations such as 8 +
= 12. Later, we see exercises such as this: If
3x + 2 = 4x – 1, solve for x.
2. As a pattern generalizer. Variables are used in statements that are true for all numbers. For example, a x b – b x a for all real numbers.
Formulas such as A = L x W also show a pattern.
3. As quantities that vary in joint variation. Joint variation occurs when change in one variable determines a change in another. In y = 3x + 5,
as x changes or varies, so does y. Formulas are also an example of joint variation. In C = 2πr, the radius, changes, so does C, the
circumference. (Elementary and Middle School Mathematics, John A. Van de Walle, Addison Weslen Longman, Inc., 1988)
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
Standard 2: Algebra
NINTH AND TENTH GRADES
9/10-310
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 3: Functions – The student analyzes functions in a variety of situations.
Ninth and Tenth Grades Application Indicators
The student…
1. translates between the numerical, graphical, and symbolic
representations of functions (2.4.A1c-e) ($).
2. ▲ ■ interprets the meaning of the x- and y- intercepts, slope, and/or
points on and off the line on a graph in the context of a real-world
situation (2.4.A1e) ($), e.g., the graph below represents a tank full of
water being emptied. What does the y-intercept represent? What
does the x-intercept represent? What is the rate at which it is
emptying? What does the point (2, 25) represent in this situation?
What does the point (2,30) represent in this situation?
The Water Tank
Gallons
Ninth and Tenth Grades Knowledge Base Indicators
The student…
1. evaluates and analyzes functions using various methods including
mental math, paper and pencil, concrete objects, and graphing
utilities or other appropriate technology (2.4.K1a,d-f).
2. matches equations and graphs of constant and linear functions and
quadratic functions limited to y = ax2 + c (2.4.K1d,f).
3. determines whether a graph, list of ordered pairs, table of values, or
rule represents a function (2.4.K1e-f).
4. determines x- and y-intercepts and maximum and minimum values
of the portion of the graph that is shown on a coordinate plane
(2.4.K1f).
5. identifies domain and range of:
a. relationships given the graph or table (2.4.K1e-f),
b. linear, constant, and quadratic functions given the equation(s)
(2.4.K1d).
6. ▲ recognizes how changes in the constant and/or slope within a
linear function changes the appearance of a graph (2.4.K1f) ($).
7. uses function notation.
8. evaluates function(s) given a specific domain ($).
9. describes the difference between independent and dependent
variables and identifies independent and dependent variables ($).
Hours
3. analyzes (2.4.A1c-e):
a. the effects of parameter changes (scale changes or restricted
domains) on the appearance of a function’s graph,
b. how changes in the constants and/or slope within a linear
function affects the appearance of a graph,
c. how changes in the constants and/or coefficients within a
quadratic function in the form of y = ax2 + c affects the
appearance of a graph.
9/10-311
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Functions are relationships or rules in which each member of one set is paired with one, and only one, member of another set (an
ordered pair). The concept of function can be introduced using function machines. Any number put in the machine will be changed according to
some rule. A record of inputs and corresponding outputs can be maintained in a two-column format. Function tables, input/output machines, and Ttables may be used interchangeably and serve the same purpose.
Function concepts should be developed from growing patterns. Each term in a number sequence is related to its position in the sequence – the
functional relationship. The pattern – 4, 7, 10, 13, 16, 19, and so on – is an arithmetic sequence with a difference of 3. The pattern could be
described as add 3 meaning that 3 must be added to the previous term to find the next. This pattern is explained by using the recursive definition for
a sequence. The recursive definition for a sequence is a statement or a set of statements that explains how each successive term in the sequence
is obtained from the previous term(s).
In the pattern 1, 4, 9, 16, 25, …, 225; there is no common difference. This sequence is not arithmetic or geometric (no common ratio between
geometric terms). Neither is it a combination of the two; however, there is a pattern and the missing terms between 25 and 225 can be found. To
find the term value, square the number of the term. The next missing terms would be 36, 49, 64, 81, 100, 121, and 144. This pattern is explained
by using the explicit formula for a sequence. The explicit formula for a sequence defines a rule for finding each term in the number sequence related
to its position in the sequence. In other words, to find the term value, square the number of the term – the 5th term is 52, the 8th term is 82, …
Patterns themselves are not explicit or recursive. The RULE for the pattern can be expressed explicitly or recursively and MOST patterns can be
explained using either format especially IF that pattern reflects either an arithmetic sequence or geometric sequence.
In the Cartesian Coordinate System, the expression x = 3 cannot represent a function. When you graph all points where x = 3 you get a vertical
line, so more than one y-value exists for the x-value x = 3. By definition, a function must have only one output value for any given input. For more
information, consult a reference book under the topic, vertical line test.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary for
conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the concept,
the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4
(Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that
could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are
identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and
Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
9/10-312
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 2: Algebra
NINTH AND TENTH GRADES
Algebra – The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 4: Models – The student develops and uses mathematical models to represent and justify mathematical
relationships found in a variety of situations involving tenth grade knowledge and skills.
Ninth and Tenth Grades Knowledge Base Indicators
The student…
1. knows, explains, and uses mathematical models to represent and
explain mathematical concepts, procedures, and relationships.
Mathematical models include:
a. process models (concrete objects, pictures, diagrams, number
lines, hundred charts, measurement tools, multiplication arrays,
division sets, or coordinate grids) to model computational
procedures, algebraic relationships, and mathematical
relationships and to solve equations (1.1.K1-3, 1.2.K1, 1.2.K3-4,
1.3.K1-4, 1.4.K1, 1.4.K2a-b, 2.1.K1a, 2.1.K1d, 2.1.K2, 2.2.K4,
2.3.K1, 3.2.K1-3, 3.2.K6, 3.3.K1-4, 4.2.K3-4) ($);
b. factor trees to model least common multiple, greatest common
factor, and prime factorization (1.4.K3);
c. algebraic expressions to model relationships between two
successive numbers in a sequence or other numerical patterns
(2.1.K1c);
d. equations and inequalities to model numerical and geometric
relationships (1.4.K2c, 2.2.K3, 2.3.K1-2, 3.2.K7) ($);
e. function tables to model numerical and algebraic relationships
(2.1.K1c, 2.2.K2, 2.3.K1, 2.3.K3, 2.3.K5) ($);
f. coordinate planes to model relationships between ordered pairs
and equations and inequalities and linear and quadratic
functions (2.2.K1, 2.3.K1-6, 3.4.K1-8) ($);
g. constructions to model geometric theorems and properties
(3.1.K2, 3.1.K6);
h. two- and three-dimensional geometric models (geoboards, dot
paper, coordinate plane, nets, or solids) and real-world objects
to model perimeter, area, volume, and surface area, properties
of two- and three-dimensional figures, and isometric views of
three-dimensional figures (2.1.K1b, 3.1.K1-8, 3.2.K1, 3.2.K4-5,
3.3.K1-4);
Ninth and Tenth Grades Application Indicators
The student…
1. recognizes that various mathematical models can be used to
represent the same problem situation. Mathematical models
include:
a. process models (concrete objects, pictures, diagrams,
flowcharts, number lines, hundred charts, measurement tools,
multiplication arrays, division sets, or coordinate grids) to model
computational procedures, algebraic relationships,
mathematical relationships, and problem situations and to solve
equations (1.1.K1, 1.2.A1-2, 1.3.A1-4, 1.4.A1a, 1.4A1d-e,
3.1.A1-3, 3.2.A1-3, 3.3.A2, 3.3.A4, 3.4.A2, 4.2.A1a-b) ($);
b. algebraic expressions to model relationships between two
successive numbers in a sequence or other numerical patterns;
c. equations and inequalities to model numerical and geometric
relationships (2.1.A2, 2.2.A1-3, 2.3.A1) ($);
d. function tables to model numerical and algebraic relationships
(2.3.A1, 2.3.A3, 3.4.A2) ($);
e. coordinate planes to model relationships between ordered pairs
and equations and inequalities and linear and quadratic
functions (2.2.A1, 2.3.A1-3, 3.4.A1-2, 3.4.A4) ($);
f. two- and three-dimensional geometric models (geoboards, dot
paper, coordinate plane, nets, or solids) and real-world objects
to model perimeter, area, volume, and surface area, properties
of two- and three-dimensional figures and isometric views of
three-dimensional figures (3.3.A1, 4.2.A1c);
g. scale drawings to model large and small real-world objects
(3.3.A3, 3.4.A3);
h. geometric models (spinners, targets, or number cubes),
process models (coins, pictures, or diagrams), and tree
diagrams to model probability (1.4.A1c, 4.2.A1, 4.2.A3);
9/10-313
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
i.
j.
k.
scale drawings to model large and small real-world objects;
i. frequency tables, bar graphs, line graphs, circle graphs, Venn
Pascal’s Triangle to model binomial expansion and probability;
diagrams, charts, tables, single and double stem-and-leaf plots,
geometric models (spinners, targets, or number cubes),
scatter plots, box-and-whisker plots, histograms, and matrices
process models (concrete objects, pictures, diagrams, or coins),
to describe, interpret, and analyze data (2.1.A1, 4.1.A1, 4.1.A3and tree diagrams to model probability (4.1.K1-3);
4, 4.1.A6, 4.2.A1) ($);
l. frequency tables, bar graphs, line graphs, circle graphs, Venn
j. Venn diagrams to sort data and show relationships.
diagrams, charts, tables, single and double stem-and-leaf plots,
2. uses the mathematical modeling process to analyze and make
scatter plots, box-and-whisker plots, histograms, and matrices to
inferences about real-world situations ($).
organize and display data (4.2.K1, 4.2.K5-6) ($);
m. Venn diagrams to sort data and show relationships (1.2.K2).
Teacher Notes: For assessment purposes, the mathematical modeling process appropriate to the indicator may be included as part of the item
being assessed. The mathematical modeling process involves:
a. selecting key features and relationships within the real-world situation and representing these concepts in mathematical terms through
some sort of mathematical model,
b. performing manipulations and mathematical procedures within the mathematical model,
c. interpreting the results of the manipulations within the mathematical model,
d. using these results to make inferences about the original real-world situation.
The use of mathematical models is necessary for conceptual understanding. The ways in which mathematical ideas are represented is
fundamental to how students understand and use those ideas. As students begin to use multiple representations of the same situation, they begin
to develop an understanding of the advantages and disadvantages of various representations/models.
Many mathematical models are listed in this benchmark. The indicator lists some of the mathematical models that could be used to teach a
concept. Each indicator in this benchmark is linked to other indicators in other benchmarks; those indicators are identified in the parentheses. For
example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1
(Number Sense), and Knowledge Indicator 3. In addition, the indicator in the other benchmarks identifies, in parentheses, the Models’ indicator.
For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models).
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
9/10-314
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
NINTH AND TENTH GRADES
Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 1: Geometric Figures and Their Properties – The student recognizes geometric figures and compares
and justifies their properties of geometric figures in a variety of situations.
Ninth and Tenth Grades Knowledge Base Indicators
The student…
1. recognizes and compares properties of two-and three-dimensional
figures using concrete objects, constructions, drawings, appropriate
terminology, and appropriate technology (2.4.K1h).
2. discusses properties of regular polygons related to (2.4.K1g-h):
a. angle measures,
b. diagonals.
3. recognizes and describes the symmetries (point, line, plane) that
exist in three-dimensional figures (2.4.K1h).
4. recognizes that similar figures have congruent angles, and their
corresponding sides are proportional (2.4.K1h).
5. uses the Pythagorean Theorem to (2.4.K1h):
a. determine if a triangle is a right triangle,
b. find a missing side of a right triangle.
6. recognizes and describes (2.4.K1g-h):
a. congruence of triangles using: Side-Side-Side (SSS), AngleSide-Angle (ASA), Side-Angle-Side (SAS), and Angle-AngleSide (AAS);
Ninth and Tenth Grades Application Indicators
The student…
1. solves real-world problems by (2.4.A1a):
a. using the properties of corresponding parts of similar and
congruent figures, e.g., scale drawings, map reading, or
proportions;
b. ▲ ■ applying the Pythagorean Theorem, e.g., when checking for
square corners on concrete forms for a foundation, determine if
a right angle is formed by using the Pythagorean Theorem;
c. using properties of parallel lines, e.g., street intersections.
2. uses deductive reasoning to justify the relationships between the
sides of 30°-60°-90° and 45°-45°-90° triangles using the ratios of
sides of similar triangles (2.4.A1a).
3. understands the concepts of and develops a formal or informal proof
through understanding of the difference between a statement
verified by proof (theorem) and a statement supported by examples
(2.4.A1a).
b. the ratios of the sides in special right triangles: 30°-60°-90°
and 45°-45°-90°.
7. recognizes, describes, and compares the relationships of the angles
formed when parallel lines are cut by a transversal (2.4.K1h).
8. recognizes and identifies parts of a circle: arcs, chords, sectors of
circles, secant and tangent lines, central and inscribed angles
(2.4.K1h).
9/10-315
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Geometry is the study of points, lines, angles, planes, and shapes and their relationships. Symbols and numbers are used to
describe the properties of these points, lines, angles, planes, and shapes. The term geometry comes from two Greek words meaning “earth
measure.” The fundamental concepts in geometry are point (no dimension), line (one-dimensional), plane (two-dimensional), and space (threedimensional). Plane figures are referred to as two-dimensional and solids are referred to as three-dimensional.
From the Mathematics Dictionary and Handbook (Nichols Schwartz Publishing, 1999), property as a mathematical term means a
characteristic (an attribute) of a number, geometric shape, mathematical operation, equation, or inequality. To give an example:





Property of a number: 8 is divisible by 2.
Property of a geometric shape: Each of the four sides of a square is of the same length.
Property of an operation: Addition is commutative. For all numbers x and y, x + y = y + x.
Property of an equation: For all numbers a, b, and c, if a = b, then a + c = b + c.
Property of an inequality: For all numbers a, b, and c, if a > b, then a – c > b – c.
The application of the Knowledge Indicators from the Geometry Benchmark, Geometric Figures and Their Properties are most often applied within
the context of the other Geometry Benchmarks — Measurement and Estimation, Transformational Geometry, and Geometry From an Algebraic
Perspective — rather than in isolation.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
9/10-316
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
NINTH AND TENTH GRADES
Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 2: Measurement and Estimation – The student estimates, measures and uses geometric formulas in a
variety of situations.
Ninth and Tenth Grades Knowledge Base Indicators
The student…
1. determines and uses real number approximations (estimations) for
length, width, weight, volume, temperature, time, distance,
perimeter, area, surface area, and angle measurement using
standard and nonstandard units of measure (2.4.K1a) ($).
2. selects and uses measurement tools, units of measure, and level of
precision appropriate for a given situation to find accurate real
number representations for length, weight, volume, temperature,
time, distance, area, surface area, mass, midpoint, and angle
measurements (2.4.K1a) ($).
3. approximates conversions between customary and metric systems
given the conversion unit or formula (2.4.K1a).
4. states, recognizes, and applies formulas for (2.4.K1h) ($):
a. perimeter and area of squares, rectangle, and triangles;
b. circumference and area of circles; volume of rectangular solids.
5. uses given measurement formulas to find perimeter, area, volume,
and surface area of two- and three-dimensional figures (regular and
irregular) (2.4.K1h).
6. recognizes and applies properties of corresponding parts of similar
and congruent figures to find measurements of missing sides
(2.4.K1a).
7. knows, explains, and uses ratios and proportions to describe rates of
change (2.4.K1d) ($), e.g., miles per gallon, meters per second,
calories per ounce, or rise over run.
Ninth and Tenth Grades Application Indicators
The student…
1. solves real-world problems by (2.4.A1a) ($):
a. converting within the customary and the metric systems,
e.g., Marti and Ginger are making a huge batch of cookies
and so they are multiplying their favorite recipe quite a few
times. They find that they need 45 tablespoons of liquid.
To the nearest ¼ of a cup, how many cups would be
needed?
b. finding the perimeter and the area of circles, squares,
rectangles, triangles, parallelograms, and trapezoids, e.g., a
track is made up of a rectangle with dimensions 100 meters by
50 meters with semicircles at each end (having a diameter of 50
meters). What is the distance of one lap around the inside lane
of the track?
c. finding the volume and the surface area of rectangular solids
and cylinders, e.g., a car engine has 6 cylinders. Each cylinder
has a height of 8.4 cm and a diameter of 8.8 cm. What is the
total volume of the cylinders?
d. using the Pythagorean theorem, e.g., a baseball diamond is a
square with 90 feet between each base. What is the
approximate distance from home plate to second base?
e. using rates of change, e.g., the equation w = –52 + 1.6t can be
used to approximate the wind chill temperatures for a wind
speed of 40 mph. Find the wind chill temperature (w) when the
actual temperature (t) is 32 degrees. What part of the equation
represents the rate of change?
2. estimates to check whether or not measurements or calculations for
length, weight, volume, temperature, time, distance, perimeter, area,
surface area, and angle measurement in real-world problems are
9/10-317
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
reasonable and adjusts original measurement or estimation based
on additional information (a frame of reference) (2.4.A1a) ($).
3. uses indirect measurements to measure inaccessible objects
(2.4.A1a), e.g., you are standing next to the railroad tracks and a
train passes. The number of cars in the train can be determined if
you know how long it takes for one car to pass and the length of time
the whole train takes to pass you.
Teacher Notes: The term geometry comes from two Greek words meaning “earth measure.” Measurement provides the tools required to apply
geometric concepts in the real-world. Estimation in measurement is defined as making guesses as to the exact measurement of an object without
using any type of measurement tool. Estimation helps students develop a relationship between the different sizes of units of measure. It helps
students develop basic properties of measurement and it gives students a tool to determine if a given measurement is reasonable.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that
could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are
identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and
Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
9/10-318
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
NINTH AND TENTH GRADES
Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 3: Transformational Geometry – The student recognizes and applies transformations on two- and threedimensional figures in a variety of situations.
Ninth and Tenth Grades Knowledge Base Indicators
The student…
1. describes and performs single and multiple transformations
[refection, rotation, translation, reduction (contraction/shrinking),
enlargement (magnification/growing)] on two- and three-dimensional
figures (2.4.K1a).
2. recognizes a three-dimensional figure created by rotating a simple
two-dimensional figure around a fixed line (2.4.K1a), e.g., a
rectangle rotated about one of its edges generates a cylinder; an
isosceles triangle rotated about a fixed line that runs from the vertex
to the midpoint of its base generates a cone.
3. generates a two-dimensional representation of a three-dimensional
figure (2.4.K1a).
4. determines where and how an object or a shape can be tessellated
using single or multiple transformations and creates a tessellation
(2.4.K1a).
Ninth and Tenth Grades Application Indicators
The student…
1. ▲ analyzes the impact of transformations on the perimeter and area
of circles, rectangles, and triangles and volume of rectangular prisms
and cylinders (2.4.A1f), e.g., reducing by a factor of ½ multiplies an
area by a factor of ¼ and multiplies the volume by a factor of 1/8,
whereas, rotating a geometric figure does not change perimeter or
area.
2. describes and draws a simple three-dimensional shape after
undergoing one specified transformation without using concrete
objects to perform the transformation (2.4.A1a).
3. uses a variety of scales to view and analyze two- and threedimensional figures (2.4.A1g).
4. analyzes and explains transformations using such things as
sketches and coordinate systems (2.4.A1a).
9/10-319
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Transformational geometry is another way to investigate and interpret geometric figures by moving every point in a plane figure
to a new location. To help students form images of shapes through different transformations, students can use concrete objects, figures drawn on
graph paper, mirrors or other reflective surfaces, or appropriate technology. Some transformations, like translations, reflections, and rotations, do
not change the figure itself. Other transformations, like reduction (contraction/shrinking) or enlargement (magnification/growing), change the size of
a figure, but not the shape (congruence vs. similarity).
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that
could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are
identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and
Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
9/10-320
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 3: Geometry
NINTH AND TENTH GRADES
Geometry – The student uses geometric concepts and procedures in a variety of situations.
Benchmark 4: Geometry from an Algebraic Perspective – The student uses an algebraic perspective to analyze the
geometry of two- and three-dimensional figures in a variety of situations.
Ninth and Tenth Grades Knowledge Base Indicators
The student…
1. recognizes and examines two- and three-dimensional figures and
their attributes including the graphs of functions on a coordinate
plane using various methods including mental math, paper and
pencil, concrete objects, and graphing utilities or other appropriate
technology (2.4.K1f).
2. determines if a given point lies on the graph of a given line or
parabola without graphing and justifies the answer (2.4.K1f).
3. calculates the slope of a line from a list of ordered pairs on the line
and explains how the graph of the line is related to its slope
(2.4.K1f).
4. ▲ finds and explains the relationship between the slopes of parallel
and perpendicular lines (2.4.K1f), e.g., the equation of a line 2x + 3y
= 12. The slope of this line is ‫־‬2/3. What is the slope of a line
perpendicular to this line?
5. uses the Pythagorean Theorem to find distance (may use the
distance formula) (2.4.K1f).
6. ▲ recognizes the equation of a line and transforms the equation into
slope-intercept form in order to identify the slope and y-intercept and
uses this information to graph the line (2.4.K1f).
7. recognizes the equation y = ax2 + c as a parabola; represents and
identifies characteristics of the parabola including opens upward or
opens downward, steepness (wide/narrow), the vertex, maximum
and minimum values, and line of symmetry; and sketches the graph
of the parabola (2.4.K1f).
Ninth and Tenth Grades Application Indicators
The student…
1. represents, generates, and/or solves real-world problems that
involve distance and two-dimensional geometric figures including
parabolas in the form ax2 + c (2.4.A1e), e.g., compare the heights of
2 different objects whose paths are represented h1(t) = 3t² + 1 and
h2(t) = ½t² + 4 (where h represents the height in feet and t
represents elapsed time in seconds) after 5 seconds.
2. translates between the written, numeric, algebraic, and geometric
representations of a real-world problem (2.4.A1a-e) ($), e.g., given a
situation, write a function rule, make a T-table of the algebraic
relationship, and graph the order pairs.
3. recognizes and explains the effects of scale changes on the
appearance of the graph of an equation involving a line or parabola
(2.4.A1g).
4. analyzes how changes in the constants and/or leading coefficients
within the equation of a line or parabola affects the appearance of
the graph of the equation (2.4.A1e).
9/10-321
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
8. explains the relationship between the solution(s) to systems of
equations and systems of inequalities in two unknowns and their
corresponding graphs (2.4.K1f), e.g., for equations, the lines
intersect in either one point, no points, or infinite points; and for
inequalities, all points in double-shaded areas are solutions for both
inequalities.
Teacher Notes: A coordinate plane (coordinate grid) consists of a horizontal number line called the x-axis and a vertical number line called the yaxis. These two lines intersect at a point called the origin. The x-axis and the y-axis divide the plane into four sections called quadrants. Any point
on the coordinate plane can be named with two numbers called coordinates. The first number is the x-coordinate. The second number is the ycoordinate. Since the pair is always named in order (first x, then y), it is called an ordered pair.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that
could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are
identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and
Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
9/10-322
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 4: Data
NINTH AND TENTH GRADES
Data – The student uses concepts and procedures of data analysis in a variety of situations.
Benchmark 1: Probability – The student applies probability theory to draw conclusions, generate convincing
arguments, make predictions and decisions, and analyze decisions including the use of concrete
objects in a variety of situations.
Ninth and Tenth Grades Knowledge Base Indicators
The student…
1. finds the probability of two independent events in an experiment,
simulation, or situation (2.4.K1k) ($) .
2. finds the conditional probability of two dependent events in an
experiment, simulation, or situation (2.4.K1k).
3. ▲ explains the relationship between probability and odds and
computes one given the other (2.4.K1a,k).
Ninth and Tenth Grades Application Indicators
The student…
1. conducts an experiment or simulation with two dependent events;
records the results in charts, tables, or graphs; and uses the results
to generate convincing arguments, draw conclusions and make
predictions (2.4.A1h-i).
2. uses theoretical or empirical probability of a simple or compound
event composed of two or more simple, independent events to
make predictions and analyze decisions about real-world situations
including:
a. work in economics, quality control, genetics, meteorology, and
other areas of science (2.4.A1a);
b. games (2.4.A1a);
c. situations involving geometric models, e.g., spinners or
dartboards (2.4.A1f).
3. compares theoretical probability (expected results) with empirical
probability (experimental results) of two independent and/or
dependent events and understands that the larger the sample size,
the greater the likelihood that experimental results will match
theoretical probability (2.4.A1h).
4. uses conditional probabilities of two dependent events in an
experiment, simulation, or situation to make predictions and
analyze decisions.
9/10-323
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Teacher Notes: Ideas from probability reinforce concepts in the other Standards, especially Number and Computation and Geometry. Students
need to develop an intuitive concept of chance – whether or not something is unlikely or likely to happen. Probability experiences should be
addressed through the use of concrete objects (process models); spinners, number cubes, or dartboards (geometric models); and coins (money
models). Probabilities are ratios, expressed as fractions, decimals, or percents, determined by considering results or outcomes of experiments.
Some examples of uses of probability in every day life include: There is a 50% chance of rain today. What is the probability that the team will win
every game?
Whereas probability is a ratio of favorable outcomes to all possible outcomes, odds is a ratio of favorable to unfavorable outcomes. If
the odds of rolling an even number on a number cube is 1:1 (1/1 or 1 to 1), then the probability is 1/2 (.5 or 50%).
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models
that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those
indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard
1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
9/10-324
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Standard 4: Data
NINTH AND TENTH GRADES
Data – The student uses concepts and procedures of data analysis in a variety of situations.
Benchmark 2: Statistics – The student collects, organizes, displays, explains, and interprets numerical (rational) and
non-numerical data sets in a variety of situations.
Ninth and Tenth Grades Knowledge Base Indicators
The student…
1. organizes, displays, and reads quantitative (numerical) and
qualitative (non-numerical) data in a clear, organized, and accurate
manner including a title, labels, categories, and rational number
intervals using these data displays (2.4.K1l):
a. frequency tables and line plots;
b. bar, line, and circle graphs;
c. Venn diagrams or other pictorial displays;
d. charts and tables;
e. stem-and-leaf plots (single and double);
f. scatter plots;
g. box-and-whiskers plots;
h. histograms.
2. explains how the reader’s bias, measurement errors, and display
distortions can affect the interpretation of data.
3. calculates and explains the meaning of range, quartiles and
interquartile range for a real number data set (2.4.K1a).
4. ▲ explains the effects of outliers on the measures of central
tendency (mean, median, mode) and range and interquartile range
of a real number data set (2.4.K1a).
5. ▲ approximates a line of best fit given a scatter plot and makes
predictions using the graph or the equation of that line (2.4.K1k).
6. compares and contrasts the dispersion of two given sets of data in
terms of range and the shape of the distribution including (2.4.K1k):
a. symmetrical (including normal),
b. skew (left or right),
c. bimodal,
d. uniform (rectangular).
Ninth and Tenth Grades Application Indicators
The student…
1. ▲ uses data analysis (mean, median, mode, range, quartile,
interquartile range) in real-world problems with rational number data
sets to compare and contrast two sets of data, to make accurate
inferences and predictions, to analyze decisions, and to develop
convincing arguments from these data displays (2.4.A1i) ($):
a. ■ frequency tables and line plots;
b. bar, line, and circle graphs;
c. Venn diagrams or other pictorial displays;
d. charts and tables;
e. stem-and-leaf plots (single and double);
f. scatter plots
g. box-and-whiskers plots;
h. histograms.
2. determines and describes appropriate data collection techniques
(observations, surveys, or interviews) and sampling techniques
(random sampling, samples of convenience, biased sampling,
census of total population, or purposeful sampling) in a given
situation.
3. uses changes in scales, intervals, and categories to help support a
particular interpretation of the data (2.4.A1i).
4. determines and explains the advantages and disadvantages of using
each measure of central tendency and the range to describe a data
set (2.4.K1i).
5. analyzes the effects of:
a. outliers on the mean, median, and range of a real number data
set;
b. changes within a real number data set on mean, median, mode,
range, quartiles, and interquartile range.
9/10-325
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
–
–
6. approximates a line of best fit given a scatter plot, makes
predictions, and analyzes decisions using the equation of that line
(2.4.A1i).
Teacher Notes: Graphs (data displays) are pictorial representations of mathematical relationships and are an important part of statistics. When a
graph is made, the axes and the scale (numbers running along a side of the graph) are chosen for a reason. The difference between numbers from
one grid line to another is the interval. The interval will depend on the lowest and highest values in the data. Emphasizing the importance of using
equal-sized pictures or intervals is critical to ensuring that the data is displayed accurately.
Graphs take many forms:
– bar graphs compare discrete data;
–
frequency tables show how many times a certain piece of data occurs;
–
circle graphs (pie charts) model parts of a whole;
–
line graphs show change over time;
–
Venn diagrams show relationships between sets of objects;
–
line plots show frequency of data on a number line;
stem-and-leaf plots (closely related to line plots) show frequency distribution by arranging numbers (stems) on the left side of a vertical line
(single stem-and-leaf) with numbers (leaves) on the right side;
–
scatter plots show the relationship between two quantities;
box-and-whisker plots are visual representations of the five-number summary – the median, the upper and lower quartiles, and the least and
greatest values in the distribution – therefore, the center, the spread, and the overall range are immediately evident by looking at the plot
and;
– histograms (closely related to stem-and-leaf plots) describe how data falls into different ranges.
Two important aspects of data are its center and its spread. The mean, median, and mode are measures of central tendency (averages) that
describe where data are centered. Each of these measures is a single number that describes the data. However, each does it slightly differently.
The range describes the spread (dispersion) of data. The easiest way to measure spread is the range, the difference between the greatest and the
least values in a data set. Quartiles are boundaries that break the data into fourths. The interquartile range is the difference between the upper
quartile and the lower quartile and is considered a measure of dispersion.
Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary
for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the
concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark
4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that
could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are
identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and
Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.
9/10-326
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are
provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and
Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in
the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level
with a ($). The National Standards in Personal Finance are included in the Appendix.
9/10-327
January 31, 2004
▲– Assessed Indicator
■ – Assessed Indicator on the Optional Response Assessment
N – Noncalculator
($) – Financial Literacy
THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.
Glossary
A
absolute value – a number’s distance from zero on the number line (The value is always positive.)
acute angle – an angle that measures less than 90
addend – one of a set of numbers to be added
addition problem –
8 (addend)
+ 4 (addend)
12 (sum)
additive identity- 0 is the additive identity, because for any real number a: a + 0 = a
additive inverse – the opposite of a number; when a number is added to its additive inverse, the sum is 0. The additive inverse of –7 is: +7
because –7 + +7 = 0
algebra – a strand of mathematics in which variables are used to express general rules about numbers, number relationships, and operations
algebraic expression – a group of numbers, symbols, and variables that express an operation or a series of operations (3x is an expression for
three times some number.)
algorithm – a systematic way for carrying out computations
altitude- a line segment which measures the height of a figure.
angle – a figure formed by two rays having a common endpoint
approximation – an amount or estimate that is nearly exact
area – the measure of the region inside a closed plane figure, measured in square units
arithmetic sequence – a sequence of numbers in which the difference of two consecutive numbers is the same
array – a rectangular arrangement of objects with equal amounts in each column and row
Associative Property – for any real numbers a, b, and c: (a + b) + c = a + (b + c); and for any real numbers a, b, and c: (ab)c = a(bc)
Kansas Curricular Standards for Mathematics
January 2004
328
attribute – a characteristic of a shape or set of data
axis – one of the reference lines (perpendicular number lines) on a coordinate graph
B
balance (bank) – the amount of money in a bank account
balance (scale) – a tool used for comparing the mass or weight of objects
bar graph – a way of organizing data in horizontal or vertical bars
base (polygon) – the side of a polygon that contains one end of the altitude OR a special side of a two-dimensional object.
base (solid figure) – a special face of a three-dimensional object
benchmark number – a number used in making comparisons or judgments
binomial – the sum or difference of two monomials
bisect – to divide into two equal parts
box-and-whisker plot – a representation of a five-number summary that displays the high and low values, the median, and the quartiles of a set
of data, but does not display any other specific values (In the graph, a rectangle (“box”) represents the middle 50% of the data set and the line
segments (“whiskers”) at both ends of the rectangle represent the remainder of the data.)
C
calendar – a tool to keep track of the date
capacity – a measurement of the amount that an object can hold
cardinal number – a whole number that answers the question “How many objects in a set?”
Cartesian plane – a two-dimensional coordinate grid
Celsius – a temperature scale where 0 is the freezing point of water, and 100 is the boiling point of water
center (of a circle) – the point inside a circle that is the same distance from every point on the circle
centimeter -- a metric unit of length equal to 1/100 of a meter
Kansas Curricular Standards for Mathematics
January 2004
329
central tendency – a number somewhere in the middle of the data set, or a number with a lot of data clustered around it (Mean, median, and
mode are measures of central tendency (averages).)
certain (event) – an event with a probability of one
chart – a sheet of information arranged in lists, pictures, tables, or diagrams
chord – a line segment connecting any two points on a circle
circle – a closed plane figure with every point the same distance from the center
circumference – the perimeter of a circle, the distance around the outside of the circle
clustering – an estimation strategy which groups numbers with relatively the same estimation value (7,225 + 6,734 + 7,123 + 6,642 to 7,000 +
7,000 + 7,000 + 7,000)
coefficient – in algebra, the numerical factor of a term (In 3x, the coefficient of x is 3.)
coins – metal money
Commutative Property – for any real numbers a and b: a + b = b + a; and for any real numbers a and b: ab = ba
compass – a tool for drawing circles
compatible numbers – numbers that easily “fit together” in computation and are easy to manipulate mentally
composite figure – a figure made up of several different geometric shapes
composite number – a number that has more than itself and 1 as factors (6 has factors of 1, 2, 3, and 6.)
compound event – an event composed of at least two simple events
compound interest – interest paid on the accumulated amount, including previous interest
compute – to find a numerical result, usually by adding, subtracting, multiplying, or dividing
cone – a three-dimensional figure with a circular base and a vertex (like an ice cream cone)
congruency – having the same size and shape
Kansas Curricular Standards for Mathematics
January 2004
330
congruent – figures that are exact “duplicates” of each other; having the same size and shape
coordinate grid/plane – a two-dimensional system in which the coordinates of every point can be expressed as an ordered pair that describes
the distance of that point from the x-axis and y-axis (Cartesian plane)
coordinates – a pair of numbers that give the location of a point on a grid/plane or graph
cube – a three-dimensional figure having six congruent, square faces
cube number – a number raised to the third power
cup – a unit of liquid measure; 8 ounces = 1 cup, 2 cups = 1 pint
customary system of measurement – a system of measurement commonly used in the United States (inches, cups, pounds, etc.)
cylinder – a three-dimensional figure with parallel, congruent circular bases
D
data – information, especially numerical, usually organized for analysis
data displays – representations such as frequency tables, histograms, line plots, bar graphs, line graphs, circle graphs, Venn diagrams, charts,
and tables
decimal – a name for a fractional number expressed with a decimal point such as .27
decompose (figures) – breaking apart a composite figure into the basic shapes from which it was made
denominator – the number or expression below the line in fraction, the equal parts in the whole
dependent event – an event whose outcome is affected by the outcome of another event
dependent variable – in a function, a variable whose value is determined by the value of the related independent variable
diagonal – a line segment that joins two non-consecutive vertices of a polygon
diameter – a line segment that has its endpoints on a circle (chord) and passes through the center of the circle
difference – the result of subtracting one number from another
digit – one of the ten numerals in our numeration system: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9
Kansas Curricular Standards for Mathematics
January 2004
331
digital root – the result of adding digits in a number until only one digit remains
discount – a reduction applied to a given price
discrete – discontinuous, made up of distinct parts
Distributive Property – for all real numbers a, b, and c: a(b + c) = ab + ac and a(b - c) = ab - ac
dividend (financial) - payment to stockholders, usually quarterly, that represents their part of the company’s profits
dividend (mathematical) – the number that is being divided by another number
divisibility – a number n is divisible by a number k if the quotient n/k is a whole number
divisibility rules – .Rules for determining which numbers are divisible by a given number
Some common divisibility rules include:
Two: A number is divisible by 2 if the ones digit is 0,2,4,6, or 8
Three: A number is divisible by 3 if the sum of the digits is divisible by 3
Four: A number is divisible by 4 if the number formed by the last two digits is divisible by 4
Five: A number is divisible by 5 if the last digit is 0 or 5
Six:
A number is divisible by 6 if the number is divisible by 2 and 3
Seven: No easy rule exists
Eight: A number is divisible by 8 if the number formed by the last 3 digits is divisible by 8
Nine: A number is divisible by 9 if the digital root is 9
division - the operation of making equal groups to find how many in each group or how many groups
division problem –
12 (quotient)
(divisor)
4 48 (dividend)
48 ÷ 12 = 4
48/12 = 4
divisor – the number by which the dividend is divided
double – twice as much
Kansas Curricular Standards for Mathematics
January 2004
332
E
ellipse – a closed plane curve generated by a point moving in such a way that the sums of its distances from two fixed points is a constant
empirical results – results based on experiments
equation – a mathematical sentence that states that two expressions are equal; 2 + 3 = 1 + 4
equilateral – all sides congruent
equivalent – numbers or variable expressions that have the same value (5 is equivalent to 2 + 3)
estimation strategies –
Front-end with Adjustment
$1.26
4.79 Add whole number part (left most digit.)
.99 1 + 4 + 0 + 1 + 2 = 8
1.37 Adjust by looking at decimal parts.
2.58
.26 + .79 Estimate is 1
.99 Estimate is 1
.37 + .58 Estimate is 1
3
Therefore, the estimate is 8 + 3 = 11.
Rounding
$4.78
2.93 Round number to nearest whole number.
1.25
+ 3.12 5 + 3 + 1 + 3
Therefore, the estimate is 12.
Special Numbers
Examples of special numbers are 1; 10; 100; 1,000; 1/2; 0.
Estimate by rounding to one of the special numbers.
67 x 102
5,421/9.87
4/7+12/13
4,805 x 11/21
Clustering
Kansas Curricular Standards for Mathematics
January 2004
Estimate 67 x 100 or 6,700
Estimate 5,421/10 or 542.1
Estimate 1 + ½ or 1½
Estimate 4800 x ½ or 2,400
333
Clustering can be used as an estimation strategy when you have a group of numbers that cluster around a
common value.
7225
6734 This set of numbers clusters around 7,000.
6829 A good estimate would be 7000 x 6 or 42,000.
7295
7101
+ 6642
Compatible “nice” Numbers
Compatible (nice) numbers are numbers that can easily “fit together” and are easy to manipulate mentally. Take
a global look at all numbers, look for numbers that can be paired for easy computation.
27
49
38
56
81
+ 65
27 + 81 = 100
49 + 56 = 100
38 + 65 = 100
Therefore, the estimate is 300.
Truncation
Truncation is the process of ignoring all digits to the right of a chosen place value. An example of using truncation
as an estimation technique is saying gasoline sells for $1.13 per gallon when the actual price is $1.139 ($1.139).
evaluate (an algebraic expression) – substitute the values given for the variables and solve using the order of operations
even number – a number that can be divided by 2 with no remainder (2, 4, 6, 8, 10, ...)
expanded form – the way to write numbers that shows the place values of each digit
explicit rule for a pattern – statement that explains how each term in the number sequence is related to its position in the sequence
exponent – a numeral telling how many times a number is to be multiplied by itself
expression – a variable or combination of variables, numbers, and symbols that represents a mathematical relationship
F
face - a plane figure that serves as one side of a solid figure; a flat surface on a solid figure
fact family – a group of addition/subtraction or multiplication/division facts that use the same set of numbers; 3+5=8, 5+3=8, 8-3=5, 8-5=3
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factor (of a number) – a number that divides evenly into a given number
factor (in multiplication) – one of two or more numbers that are multiplied to get an answer (in 3 x 4 = 12, both the 3 and the 4 are known as
factors of 12)
Fahrenheit – a temperature scale where the freezing point of water is 32 and the boiling point of water is 212
first quadrant – the upper right fourth of a plane
flip – a mirror-image of a figure on the opposite side of a line; ( the size and shape of the figure stay the same); also called reflection
foot – a customary unit of length; 12 inches = 1 foot
formulate – to express as a formula; to work out and state in an orderly, exact way
fraction – a part of a whole; the name for a fractional number written in the form a/b (4/9)
numerator – the number or expression above the line in fraction (number of equal parts being described)
4 (numerator)
7 (denominator)
denominator – the number or expression below the line in fraction (total number of equal parts)
improper fraction – a fraction with a numerator larger than a denominator (7/3)
proper fraction – a fraction with a numerator smaller than a denominator (3/7)
mixed number – writing a number greater than one with an integer part and a fractional part
lowest terms – a fraction written so that the numerator and denominator have 1 as the
only common factor (4/7 is in lowest terms but 8/14 is not since 2 will go into both the
numerator and the denominator, i.e., 2 is a common factor.)
reduce or simplify – to write a fraction in lowest terms depending on the context, simplest form does not always mean a mixed number
frame of reference – adjusting estimation based on additional information
frequency – the number of times an event occurs
front-end with adjustment – rounding to the nearest whole number such as ($4.96 + $5.88 to $5.00 + $6.00)
function – a special set of ordered pairs; one output value for each input value
function table – a table that assigns exactly one output value for each input value
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G
gallon – a customary unit of liquid measure; 4 quarts = 1 gallon
geometric sequence – a sequence of terms in which the ratio of two succeeding terms is the same
geometry – a strand of mathematics dealing with figures and their parts
glyph – a simple pictorial shorthand used to display data and details of a subject
gram - a metric unit of mass
graphical displays – (list below and define/also in document)
bar graph
circle graph
line plot
histogram
stem and leaf
greater than – more than, the symbol used is >
greatest common factor – the greatest number that is a common factor of two or more numbers (6 is the GCF of 12 and 18 because it is the
largest number that goes into both 12 and 18.)
grid – a set of horizontal and vertical lines spaced uniformly used to graph points
gross income – income before any expenses and deductions are subtracted
growing pattern – a sequence in which each element is larger than the previous one.
H
hexagon – a six-sided plane figure
histogram – a graph in which the labels for the bars are numerical intervals
I
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impossible event – an event with a probability of zero
improper fraction – a fraction with a numerator larger than a denominator
inch – a customary unit of length; 1/12 of a foot
independent events – two or more events in which the outcome of the one event does not affect the outcome of the other events
inequality (algebraic) – an algebraic statement that a quantity is greater than or less than another quantity, such as 7t + 5 < 40
inequality (numerical) -- a numerical statement that a quantity is greater than or less than another quantity; 7 + 5 < 40
integer – any member of the set of whole numbers and their opposites (... -4, -3, -2, -1, 0, 1, 2, 3, 4 ...)
interest – an amount of money paid for the use of money
interquartile range – the difference between the upper (third) and lower (first) quartiles in a set of data
intersecting lines – 2 or more lines that cross each other at one or more points
intersection – the point at which two lines meet
irrational number – a real number that is not rational; A number that can be written as a non-repeating infinite decimal
isosceles (triangle) – at least two congruent sides
isosceles (trapezoid) – non-parallel opposite sides are congruent
K
kilometer – a metric unit of length equal to 1000 meters
kilogram – a metric unit of mass equal to 1000 grams
kite – a quadrilateral in which there are two pairs of consecutive (adjacent) congruent sides
L
least common multiple – the smallest number that is a common multiple of two or more numbers (24 is the LCM of 6 and 8 because it is the
smallest number that both 6 and 8 will go into evenly)
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length – the distance from one point to another; one dimension of a 2- or 3-dimensional figure
less than – fewer than, the symbol used is <
line – a set of points that form a straight path extending infinitely in both directions
linear equation – an equation that results in a straight line when graphed
linear function – a function of the form f(x) = ax + b where a and b are constants
linear inequality – an inequality whose boundary in the coordinate plane is a line
liter -- a metric unit of volume
M
mass – a measurement that tells how much matter an object has
matrix – a rectangular array of data
mean – one of the measures of central tendency (average); the sum of numbers in a set of data divided by the number of pieces of data
measures of central tendency – measures of central tendency (average) include mean, median, mode
median – one measure of central tendency (average); the middle number in a set of numbers, determined by arranging them in order from lowest
to highest and counting to the middle; in the case of an even number data set, add the two middle numbers and divide by two
meter – a metric unit of length
meter stick – a measurement tool used for measuring meters
metric units – a system of measurement based on the decimal system (meters, centimeters, etc.)
mile – a unit of length (5,280 feet = 1 mile)
milliliter -- a metric unit of volume equal to 1/1000 of a liter
milligram – a metric unit of mass equal to 1/1000 of a gram
millimeter – a metric unit of length equal to 1/1000 of a meter
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minuend – the name given to the number from which another number is to be subtracted
mixed number – writing a number greater than one with an integer part and a fractional part
mode – a measure of central tendency (average); the number found most frequently in a set of data; there can be no mode, one mode, or more
than one mode
model – a representation of a situation that includes process models, place value models, fraction and mixed number models, decimal and money
models, two- and three-dimensional models, equations, inequalities, and graphs; models can be mathematical representations of situations, or
non-mathematical models of mathematical situations
money – anything that is generally accepted as a medium of exchange to buy goods and services; serves as a standard of value; and has a store
of value; gold, silver, shells, tobacco, beads, and paper have all been used as money
monomial – an expression that can be a constant, a variable, or a product of a constant and one or more variables such as 5 (a constant), x (a
variable), -3z (a product of a constant and a variable), 6xy (a product of a constant and more than one variable), or x ^3 ( a variable to a power)
multiple – the product of a given whole number and any other whole number
multiplicand – a name given to the number that is being multiplied by another number
multiplication problem –
5 (multiplicand or factor)
x 4 (multiplier or factor)
20 (product)
5 X 4 = 20
5 · 4 = 20
5(4) = 20
Multiplication Property of One – for any real number a: a • 1 =1; 1 • a = 1
Multiplication Property of Zero – for any real number a: a • 0 =0; 0 • a = 0
Multiplicative Identity – 1 is the multiplicative identity, because for any real number a: a • 1 = a
Multiplicative Inverse – the reciprocal of a number (When a number is multiplied by its multiplicative inverse, the product is always one.) The
multiplicative inverse of 4 is ¼, because 4 · ¼ = 1
Multiplicative Property of Equality – for any real numbers a and b: if a = b, then ac = bc
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multiplier – the number by which the multiplicand is being multiplied
N
neighbor – the whole number that is adjacent to another whole number (The neighbors of 6 are 5 and 7.)
net – a pattern to be cut and folded without overlapping to make a solid shape
net income – amount left over after the expenses and deductions have been subtracted from the total
nice numbers – numbers that are easy to think about and work with (100, 500, and 750 are easier to work with than 94, 517, and 762; other
multiplies of 100 are very “nice” numbers as are multiples of 10.)
nonstandard unit – measurement unit other than customary or metric, such as paperclip, penny, thumbnail
number – a mathematical idea contained in a set; a number can be named in word form (six), standard form (22), or expanded form [(3 x 1,000) +
(0 x 100) + (4 x 10) + (6 x 1)]
numeral – the symbolic representation of a number
numerator – the number above the line in fraction; the number of equal parts being described
O
obtuse – an angle that measures more than 90 and less than 180
octagon – an eight-sided plane figure
odds – the ratio of favorable to unfavorable outcomes, whereas probability is a ratio of favorable outcomes to all possible outcomes. If the odds of
rolling an even number on a number cube is 1:1 (1/1 or 1 to 1), then the probability is ½ (.5 or 50%)
odds (against an event) – the ratio of the number of ways an event cannot occur to the number of ways that an event can occur
odds (for an event) – the ratio of the number of ways an event can occur to the number of ways that an event cannot occur
ordered pair – a pair of numbers that give the coordinates of points on a grid in this order (horizontal coordinate, vertical coordinate)
order of operations – the agreed-upon order of evaluating numerical expressions (First, work inside parentheses, then do powers. Then do
multiplications or divisions, from left to right. Then do additions or subtractions, from left to right.)
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ordinal number – a number used to name the position in an ordered arrangement
ounce – a measurement of weight;16 ounces = 1 pound
outcome – a possible result of an experiment
outlier – a piece of numerical data that is much smaller or larger than the rest of the data in a set.
P
parabola –the graph of a quadratic equation of the form y= ax^2+bx+c
parallel lines – lines in the same plane that do not intersect
parallelogram – a four-sided plane figure with parallel opposite sides
pattern – a repeated sequence, either repeating or growing
pattern blocks – a collection of six geometric shapes in color (green triangle, orange square, blue rhombus, tan rhombus, red trapezoid, yellow
hexagon), also see Appendix 7 for a pattern block template
pentagon – a five-sided plane figure
percent – a comparison of a number with 100 (43 compared to 100 is 43%.)
perimeter – the measurement of the distance around the outside of a figure
perpendicular – lines in the same plane that intersect at right angles
pi – the constant ratio of the diameter of a circle to its circumference
pictograph – a visual display of data that uses pictures to represent amounts
pint – a customary unit of liquid measure; 2 cups = 1 pint
place value – the value assigned to a digit due to its position in a numeral
plot – to place points on a coordinate grid/plane
point – a location in space represented by a dot
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poll – a survey of public opinion concerning a particular subject
polygon – a simple closed plane figure having line segments as sides
polynomial – a sum and/or difference of monomials
pound – a customary measurement of weight;16 ounces =1 pound
precision of measurement – the smallest unit of measurement used in measuring
prediction – an educated guess of the number of occurrences in a probability situation; something that is guessed in advance, based on known
facts.
prime factorization – an expression showing a whole number as the product of prime factors (24 = 4 x 6 is not a prime factorization since 4 can
be broken down into 2 x 2 and 6 can be broken into 2 x 3. The correct prime factorization of 24 is 2 x 2 x 2 x 3.)
prime number – a positive integer that can be divided by only 1 and itself (2, 3, 5, 7, 11...)
probability (of an event) – the likelihood that an event will occur, the ratio of the number of favorable outcomes to the number of equally likely
possible outcomes; a number from 0 to 1 that measures the likelihood that an event will occur.
product – the answer in a multiplication problem
profit – income minus expenses, usually stated in dollars. (Extent to which people or organizations are better off at the end of a period than at the
beginning of that period.)
proper fraction – a fraction with a numerator smaller than a denominator
property – a characteristic of a number, geometric figure, mathematical operation, equation, or inequality
proportion – a statement of equality between two ratios (For example, 3:6 = 1:2, read: 3 is to 6 as 1 is to 2, can also be written as two equal
fractions 3/6 = 1/2.)
protractor – a tool for measuring and drawing angles
purposeful sampling – a process used to select a sample; sampling from a deliberately limited population
pyramid – a three-dimensional figure whose base is a polygon and all other faces are triangles that share a common vertex.
Pythagorean Theorem – in a right triangle, the square of the length of the hypotenuse = the sum of the squares of the legs
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Q
quadrant – one of the four regions of the coordinate plane formed by the x-axis and y-axis
quadratic equation – a polynomial equation with a variable to the second degree (power of 2) of the form y=ax 2 + bx + c where a cannot be zero
quadratic function – a function whose value is given by a quadratic polynomial
quadrilateral – a four-sided polygon
quart – a customary unit of liquid measure; 2 pints = 1 quart
quarter – a coin with a value of $.25, one-fourth of something
quotient – the answer in a division problem
R
radius – a line segment having one endpoint in the center of the circle and another on the circle
range – in a collection of numbers, the difference between the maximum value and the minimum value;, the length of an interval
ratio – a comparison of two numbers that can be expressed as a fraction (3/4), or with a colon (3:4), or in words (3 to 4)
rational number – a number than can be written as the ratio of two integers
ray – a portion of a line extending from one endpoint in one direction indefinitely
real number system – the combined set of rational and irrational numbers
real numbers – rational and irrational numbers
rectangle – a four-sided plane figure with 90° angles and opposite sides are parallel
recursive rule for a pattern– statement or a set of statements that explains how each successive term in the sequence is obtained from the
previous term(s)
reflection – a mirror-image of a figure on the opposite side of a line; (the size and shape of the figure stay the same); also called flip
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regrouping – the process of renaming one place value to another; The number 1 in 10 represents ten ones
relative magnitude – the size relationship one number has with another number (is it much larger, much smaller, close, or about the same)
remainder – the number left over when things are divided into equal shares
repeating pattern – a pattern made up of repeating a segment, e.g., 235235235235235…
revenue – business’ receipts from the sale or rental of goods or services; money from other sources such as dividends and interest
rhombus – a quadrilateral that has four congruent sides
right angle – an angle that measures 90
right triangle – a triangle that has a 90angle
Roman numerals – symbols used in the ancient Roman number system
rotation – turning a figure about a point, which serves as the center of rotation; (does not change size or shape); also called turn
rounding – approximating a number to a specific place value so it is easier to work with; round 678 to the nearest ten: 680
ruler – a measuring tool used to determine length
S
sales tax rate – tax specified as a percent of the price of an item
scalar – a single number used with multiplication of a given data such as a vector or a matrix
scale – an arrangement of numbers in some order at uniform intervals
scalene triangle – a triangle having no sides equal
segment – a part of a line defined by two endpoints
similarity – the property of geometric figures having angles of the same size
simple event – an outcome which cannot be broken down further; rolling a number cube and getting 6 is a simple event, rolling two number
cubes and getting a sum of six is a compound event
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simulation – a model of an experiment that might be impractical to carry out
simultaneous change – two or more changes occurring at the same time
slide – a transformation that slides a figure a given distance in a given direction; (shape and size do not change); also called a translation
slope – the slant of a line given as the ratio of the rate of change in y (rise) with respect to a change in x (run)
sphere – a three-dimensional figure formed by a set of points equidistant from a center point; a round ball
square – a quadrilateral with four right angles and four congruent sides
square number – the number of dots in a square array; a number that is the result of multiplying an integer by itself
standard form – the numeric representation of written or pictorial forms of a number concept, e.g., one hundred thirty-two is represented by 132
standard unit of measure – measurement unit such as customary or metric
stem-and-leaf plot – a method of organizing data from least to greatest using the digits of the greatest place value to group data; shows the
greatest, least, and median values in a set of data
subtraction problem –
28 (minuend)
- 7 (subtrahend)
21 (difference)
28 – 7 = 21
subtrahend – the number or the term to be subtracted
sum – the result of adding two or more numbers
Symmetric Property – for any real numbers a and b, if a = b, then b = a
Symmetry (line of) – a line that divides a figure into two congruent halves that are mirror images of each other; e.g. diameter of a circle, diagonal
of a square
systems of linear equations – two or more related linear equations for which you can seek a common solution; The system may have no
common solutions, one common solution, or many common solutions.
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systems of linear inequalities - two or more related linear inequalities for which you can seek a common solution; The system may have no
common solutions, one common solution, or many common solutions
T
T-chart/T-table – a chart/table of ordered pairs; input-output table; function table
temperature – the measurement of hotness or coldness
term – a number, variable, product or quotient in an expression
tessellate – to arrange an area in a repeating geometric pattern
thermometer – a measurement tool used to measure temperature in Fahrenheit or Celsius
three-dimensional – a figure in space having height, length, and depth
ton (customary) – a customary unit of weight equal to 2,000 pounds
ton (metric) -- a metric unit of mass equal to 1000 kilograms
transformation – a movement or change of a figure in two- or three-dimensional space. Standard transformations include rotations, reflections,
and translations (do not change size of a figure) and reduction and enlargement (change size of a figure)
transitive – a law of equality which says if a = b, and b = c, then a = c
translation – a transformation that moves every point on a figure a given distance in a given direction; (shape and size do not change); also called
a slide
transversal – a line that intersects two or more other lines
trapezoid – a quadrilateral with exactly one pair of parallel sides
tree diagram – a connected, branching graph used to diagram probabilities or factors
triangle – a polygon with three sides
truncate – to make numbers with many digits easier to read and use by ignoring all digits to the right of a chosen place
turn - rotating a figure about a point, which serves as the center of rotation; (does not change size or shape); also called rotation
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two-dimensional – a plane figure having only the dimensions of width and length
U
unit – a precisely fixed quantity used to measure; one’s place
unit price – cost for a small unit of measure, such as an ounce or pound, used to compare the costs of a product in different-sized packages
V
variable – a symbol that represents an unstated numerical value in a number sentence; In the equation 6 + a = 10, “a” is the variable which
stands for 4 and a^2 + b^2 = C^2 where the variable’s values depend on the specific situation
vector – a line segment that has direction and distance
Venn diagram – a drawing with circles that shows relationships among sets of data
vertex – a common point at which two line segments, lines, or rays meet
volume – the measure of space enclosed by a figure
W
weight – the measure of how heavy an object is
whole number – the set of numbers {0, 1, 2, 3, ...}
width – the distance from one point to another; one dimension of a 2- or 3-dimensional figure
withdrawal – subtracting money from an account
X
x-axis – the horizontal axis in a coordinate graph
x-intercept -- the value of x at the point where a line or graph of a function intersects the x-axis
Y
y-axis – the vertical axis in a coordinate graph
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yard – a customary unit of length;3 feet = 1 yard
yardstick – a measurement tool used to measure yards, feet, and inches
y-intercept – the value of y at the point where a line or graph of a function intersects the y-axis
Z
zero product property – for any real numbers a and b, if ab = 0, then a = 0 or b = 0
Zero Property – for any real number a, a · 0 = 0 and 0 · a = 0
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