Download Unit One Organizer: “Dealing with Data”

Unit 3: “Rational Reasoning”
6/20/2008 8:01:00 AM
Unit Overview
6/20/2008 8:01:00 AM
OVERVIEW:
 In this unit students will explore positive and negative numbers, extending
many previously learned concepts to include the use of all
 rational numbers. Algorithms for computing with rational numbers should be
investigated, discovered and formalized by students using models, diagrams,
manipulatives, and patterns.
 Students will:
 • investigate the use of positive and negative numbers in real contexts; • plot
positive and negative numbers in the coordinate plane;
 • order positive and negative rational numbers and plot them on a number line;
• use absolute value to explore the relationship between a number and its
additive inverse;
 • develop algorithms for computing with positive and negative rational
numbers; • simplify and evaluate algebraic expressions involving positive and
negative rational numbers;
 • extend properties of real numbers to include all rational numbers; • solve oneand two-step linear equations in one variable; and,
 • solve problems by defining a variable, writing and solving an equation, and
interpreting the solution of the equation in the context of the original problem.
 To assure that this unit is taught with the appropriate emphasis, depth and
rigor, it is important that the task listed under “Evidence of
 Learning” be reviewed early in the planning process. A variety of resources
should be utilized to supplement, but not completely replace, the textbook.
Textbooks not only provide much needed content information, but excellent
learning activities as well. The
tasks in these units illustrate the type of learning activities that should be utilized
from a variety of sources.
Reference:
Georgia Performance Standards Framework for Mathematics Grade 7.
(2006, September 20, 2006). Retrieved January 15, 2008 from
http://public.doe.k12.ga.us/DMGetDocument.aspx/7th%20Grade%20Unit
%203%20Rational%20Reasoning%202-1907.pdf?p=6CC6799F8C1371F6F46BEA13164EAA7A1945391FE9C84C82B48CA
7E45D5A791C&Type=D
Number of Weeks to Teach Lesson
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5 weeks according to Georgia Performance Standards Curriculum Map,
Atlanta Public Schools Teaching Plans and Georgia Performance
Standards Framework for Mathematics
GPS Standards to be covered
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 KEY STANDARDS:
 M7N1. Students will understand the meaning of positive and negative
rational numbers and use them in computation.
 a. Find the absolute value of a number and understand it as a distance from zero
on a number line. b. Compare and order rational numbers, including repeating
decimals.
 c. Add, subtract, multiply, and divide positive and negative rational numbers. d.
Solve problems using rational numbers.
 M7A1. Students will represent and evaluate quantities using algebraic
expressions.
a. Translate verbal phrases to algebraic expressions.
 b. Simplify and evaluate algebraic expressions, using commutative, associative,
and distributive properties as appropriate.
 c. Add and subtract linear expressions. M7A2. Students will understand and
apply linear equations in one variable.
 a. Given a problem, define a variable, write an equation, solve the equation, and
interpret the solution. b. Use the addition and multiplication properties of
equality to solve one- and two-step linear equations
 M7A3. Students will understand relationships between two variables.
a. Plot points on a coordinate plane.

b. Represent, describe, and analyze relations from tables, graphs, and
formulas.
c. Describe how change in one variable affects the other variable.
 M7P2. Students will reason and evaluate mathematical arguments a.
Recognize reasoning and proof as fundamental aspects of mathematics.
 b. Make and investigate mathematical conjectures. c. Develop and evaluate
mathematical arguments and proofs.
d. Select and use various types of reasoning and methods of proof.
Reference:
Georgia Performance Standards Framework for Mathematics Grade 7.
(2006, September 20, 2006). Retrieved January 15, 2008 from
http://public.doe.k12.ga.us/DMGetDocument.aspx/7th%20Grade%20Unit
%203%20Rational%20Reasoning%202-1907.pdf?p=6CC6799F8C1371F6F46BEA13164EAA7A1945391FE9C84C82B48CA
7E45D5A791C&Type=D
Notes
The standards found in both the APS Teaching Plans and the GPS
Framework are the same. The Georgia State Framework also includes
related standards which I have excluded for simplicity sake. Please
decide whether or not the related standards should be included.
In the Atlanta Public Schools Teaching Plans, the GPS standards change
slightly from week to week.
Essential Questions
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ESSENTIAL QUESTIONS:
 When are negative numbers used and why are they important?
 Why is it useful for me to know the absolute value of a number? • What
strategies are most useful in helping me develop algorithms for adding,
subtracting, multiplying, and dividing positive and negative numbers?
• What properties and conventions do I need to understand in order to simplify and
evaluate algebraic expressions?
Reference:
Georgia Performance Standards Framework for Mathematics Grade 7.
(2006, September 20, 2006). Retrieved January 15, 2008 from
http://public.doe.k12.ga.us/DMGetDocument.aspx/7th%20Grade%20Unit
%203%20Rational%20Reasoning%202-1907.pdf?p=6CC6799F8C1371F6F46BEA13164EAA7A1945391FE9C84C82B48CA
7E45D5A791C&Type=D
Notes
*The Atlanta Public Schools Teaching Plans is delineated by week, and so
there are essential questions for each week. In contrast, the GPS
Framework for Mathematics is delineated by unit.
*Essential Questions change slightly week to week.
Enduring Understandings
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ENDURING UNDERSTANDINGS:
 Negative numbers are used to represent quantities that are less than zero such
as temperatures, scores in games or sports, and loss of income in business.
 Absolute value is useful in ordering and graphing positive and negative
numbers. • Computation with positive and negative numbers is often necessary
to determine relationships between quantities.
 • Models, diagrams, manipulatives and patterns are useful in developing and
remembering algorithms for computing with positive and negative numbers.
• Properties of real numbers hold for all rational numbers. • Positive and negative
numbers are often used to solve problems in everyday life.
Reference:
Georgia Performance Standards Framework for Mathematics Grade 7.
(2006, September 20, 2006). Retrieved January 15, 2008 from
http://public.doe.k12.ga.us/DMGetDocument.aspx/7th%20Grade
%20Unit%203%20Rational%20Reasoning%202-1907.pdf?p=6CC6799F8C1371F6F46BEA13164EAA7A1945391FE9C
84C82B48CA7E45D5A791C&Type=D
Assessment
6/20/2008 8:01:00 AM
GPS Framework, Grade 7, Unit 2, Patterns and Relationships,
Culminating Task “A Poster,”
Georgia Online Assessment System (GOAS)
Student Tests available for this unit:
Math, Grade 7, A Test 63
Math, Grade 7, A Test 64
Math, Grade 7, A Test 65
Reference:
Georgia Online Assessment System developed by the Georgia State
Department of Education. (2007). Retrieved from
https://www.georgiaoas.org/servlet/a2
Georgia Performance Standards Framework for Mathematics Grade 7.
(2006, September 20, 2006). Retrieved January 15, 2008 from
http://public.doe.k12.ga.us/DMGetDocument.aspx/7th%20Grade%20Unit
%203%20Rational%20Reasoning%202-1907.pdf?p=6CC6799F8C1371F6F46BEA13164EAA7A1945391FE9C84C82B48CA
7E45D5A791C&Type=D
Literacy GPS Connection
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APS Teaching Plans Unit 2 Literacy GPS Connection
ELA7R2 The student understands and acquires new vocabulary and uses it
correctly in reading and writing.
ELA7RC3 The student acquires new vocabulary in each content area and
uses it correctly.
ELA7RC4 The student establishes a context for information acquired by
reading across subject areas.
Note
The GPS Framework for Mathematics – Grade 7 does not include Literacy
GPS connections.
Vocabulary
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GPS Framework for Mathematics – Grade 7
Distributive property: The sum of two addends multiplied by a number will be the sum of the
product of each addend and the


number. Example: a(b + c) = ab + ac Integers: The set of whole
numbers and their opposites {... -3, -2, -1, 0, 1, 2, 3, ...}
Natural numbers: The set of numbers {1, 2, 3, 4, ...}. Natural numbers can also be called
counting numbers. Negative Numbers: The set of numbers less than zero.

Opposite Numbers: Two different numbers that have the same absolute value. Example: 4
and –4 are opposite numbers because both have an absolute value of 4.

Positive Numbers: The set of numbers greater than zero. Rational Numbers: The set of
numbers that can be written in the form a/b where a and b are integers and b ≠ 0.

Sign: a symbol that indicates whether a number is positive or negative. Example: in –4, the
(–) sign shows this number is read “negative four”.


Whole numbers: The set of all natural numbers and the number zero.
You may visit www.intermath-uga.gatech.edu and click on dictionary to see definitions and
specific examples of terms and symbols

used in this seventh grade GPS unit.
Variable: A symbol (often a letter) that represents a number.
Proportion: An equation that states two ratios are equal.
Ratio: A comparison of two quantities that share a fixed, multiplicative
relationship.
Rational Number: A number that can be written as a/b where a and b
are integers, but b is not equal to 0.
Equation: A mathematical sentence that contains an equal sign.
Algebraic Expression: A mathematical phrase involving at least one
variable and sometimes numbers
APS Teaching Plans Unit 2 Week 1 Vocabulary
Discrete data: a variable with a limited number of values (e.g.),
gender (male/female), college class
(freshman/sophomore/junior/senior)
Continuous data: data that can take on any value (subject to the
resolution of the measuring process).
Variable: A symbol (often a letter) that represents a number.
Proportion: An equation that states two ratios are equal.
Independent Variable: a variable that is manipulated, measured, or
selected by the researcher as an antecedent condition to an observed
behavior. In a hypothesized cause-and-effect relationship, the
independent variable is the cause and the dependent variable is the
outcome or effect.
Dependent Variable: a variable that is not under the experimenter's
control -- the data. It is the variable that is observed and measured in
response to the independent variable.
Narrative: (a message that tells the particulars of an act or occurrence or
course of events; presented in writing or drama or cinema or as a radio or
television program) "his narrative was interesting"; "Disney's stories
entertain adults as well as children"
APS Teaching Plans Unit 2 Week 2 Vocabulary
Variable: A symbol (often a letter) that represents a number.
Proportion: An equation that states two ratios are equal.
Equation: A mathematical sentence that contains an equals sign.
Algebraic Expression: A mathematical phrase involving at least one
variable and sometimes numbers and operation symbols.
APS Teaching Plans Unit 2 Week 3 Vocabulary
Variable: A symbol (often a letter) that represents a number.
Proportion: An equation that states two ratios are equal.
Proportion: An equation that states two ratios are equal.
Ratio: A comparison of two quantities that have the same unit of
measure.
Algebraic Expression: A mathematical phrase involving at least one
variable and sometimes numbers and operation symbols.
APS Teaching Plans Unit 2 Week 4 Vocabulary
Variable: A symbol (often a letter) that represents a number.
Proportion: An equation that states two ratios are equal.
Ratio: A comparison of two quantities that have the same unit of
measure.
Algebraic Expression: A mathematical phrase involving at least one
variable and sometimes numbers and operation symbols.
References
Organization of Terms produced by Bob Hall of the University of Texas.
(2007). Retrieved September 25, 2007 from
http://bobhall.tamu.edu/FiniteMath/Module8/Introduction.html
The Glossary of Global Productivity Solutions. (2006). Retrieved
September 25, 2007 from
http://www.gpsqtc.com/library/abc.shtml
The Interactive Mathematics Dictionary produced by University of Georgia
Mathematics Education Program. (2007). Retrieved September 25, 2007
from www.intermath-uga.gatech.edu
Miller, G. (2006). Wordnet. Retrieved September 25, 2007 from
http://wordnet.princeton.edu/
Notes
 APS Teaching Plans include vocabulary but not definitions.
 GPS Framework for Mathematics use definitions from www.intermathuga.gatech.edu and so I have defined remaining terms from this site as
well.
 The four vocabulary words in red did not appear on the
www.intermath-uga.gatech.edu or the Holt Mathematics course
Multilingual Glossary. I used the Bob Hall of the University of Texas
statistics site, the global site for the purple and Princeton for
narrative.
Problem Solving
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APS Teaching Plan Unit 2 Week 1 Problem Solving
Solve non-routine problems involving the Find a Pattern strategy from
Holt Mathematics Course 2, Problem Solving Handbook
Solve multi-step routine problems
APS Teaching Plan Unit 2 Week 2 Problem Solving
Solve non-routine problems involving the Make a Table strategy from Holt
Mathematics Course 2, Problem Solving Handbook
Solve multi-step routine problems
APS Teaching Plan Unit 2 Week 3 Problem Solving
Solve non-routine problems involving the Solve a Simpler Problem
strategy from Holt Mathematics Course 2, Problem Solving Handbook,
Solve multi-step routine problems
APS Teaching Plan Unit 2 Week 4 Problem Solving
Solve routine and non-routine problems using the Guess and Check
strategy
Notes:
GPS Framework for Mathematics – Grade 7 does not include a Problem
Solving section.
Focus Lessons
Unit 2
Unit 2
Unit 2
Unit 2
Week
Week
Week
Week
1
2
3
4
Note:
See Focus Lessons on APS Teaching Plans
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Technology Connections
6/20/2008 8:01:00 AM
UNITEDSTREAMING VIDEOS
GPS Addressed this unit: M7A1a-c, M7A2a-b, M7A3a-c, M7D1
GPS Addressed in the video: M7A1.b
Video Title: Discovering Math: Algebra Solving Linear Equations
Introduction
Reference:
Discovering Math: Algebra (Grades 6-8). Discovery Channel School (2006).
Retrieve January 15, 2008, from unitedstreaming:
http://streaming.discoveryeducation.com/
PODCASTS
Data in Action Podcast, http://georgiastandards.org/math_vc.aspx
Related Materials:
Data in Action PowerPoint Presentation
Centimeter Graph Paper PDF
Reference:
Data in Action. Georgia Department of Education. (2007). Retrieved
September 25, 2007, from http://georgiastandards.org/math_vc.aspx
PROMETHEAN FLIPCHARTS
Algebra
In this lesson students will review basic algebraic problem types (ex:
values of variables, inequalities) used on standardized tests.
http://www.prometheanplanet.com/us/server/show/ConResource.7490
Reference:
Fussillo, S. (2006). The Coordinate Plane. Promethean Planet. Retrieved
on September 26, 2007 from
http://www.prometheanplanet.com/us/server.php?show=conResource.82
23
Strohl, J. (2006). The Coordinate Plane. Promethean Planet. Retrieved on
September 26, 2007 from
http://www.prometheanplanet.com/us/server/show/ConResource.7490
Wagoner, K. (2006). The Coordinate Plane. Promethean Planet. Retrieved
on September 26, 2007 from
http://www.prometheanplanet.com/us/server/show/ConResource.163
WEBQUESTS
Brown, L. Solving Linear Systems of Equations. InstantProjects.Org
Educational Technology Portal and Services. Retrieved on September 26,
2007 from
http://instantprojects.org/webquest/webquest.php?AuthorID=5087
Clark, C. (2003). The Tortise and the Hare: A Webquest developed for
Algebra I or Algebra II on Systems of Equations. Retrieved on September
26, 2007 from http://edtech.tph.wku.edu/~ppetty/cclark.htm
Giguere, S. (1999). Graphing and Applying Linear Equations. Retrieved on
September 26, 2007 from
http://www.gmpdc.org/webquests/sgiguere/linearequations/#Introduction
Womick, A. A Functional Housing Market: Linear Equations. Retrieved on
September 26, 2007 from http://www.spa3.k12.sc.us/house.html
INTERACTIVE LEARNING ACTIVITIES
National Council of Teachers of Mathematics. (2000-2007). Line of Best Fit.
Retrieved on September 26, 2007 from
http://illuminations.nctm.org/ActivityDetail.aspx?ID=146
National Council of Teachers of Mathematics. (2000-2007). Pan Balance Expressions. Retrieved on September 26, 2007 from
http://illuminations.nctm.org/ActivityDetail.aspx?ID=10
National Council of Teachers of Mathematics. (2000-2007). Pan Balance –
Shapes (Fixed Values). Retrieved on September 26, 2007 from
http://illuminations.nctm.org/ActivityDetail.aspx?ID=131
eMath Textbook at www.go.hrw.com
 Homework Help: Course 2 Chapter 4 Patterns and Functions
 http://go.hrw.com/hrw.nd/gohrw_rls1/pKeywordResults?keywor
d=ms7+hwhelp4
 Multilingual Glossary
 http://my.hrw.com/math06_07/nsmedia/tools/glossary/msm/gl
ossary.html
 Chapter Projects: Project Recording Sheet Patterns and Functions
 http://go.hrw.com/resources/go_mt/hm2/cp/7sech04.pdf
 Lab Resources: Chapter 4 Lab Resources
 http://go.hrw.com/hrw.nd/gohrw_rls1/pKeywordResults?keywor
d=ms7+lab4
 Parent Resources
 http://go.hrw.com/hrw.nd/gohrw_rls1/pKeywordResults?keywor
d=mr7+parent
 Career Resources
 http://go.hrw.com/hrw.nd/gohrw_rls1/pKeywordResults?keywor
d=MS7+Career
 Web Extra!: Course 2 Chapter 4
 http://go.hrw.com/hrw.nd/gohrw_rls1/pKeywordResults?keywor
d=ms7+webextra4
 Game Time
 http://go.hrw.com/hrw.nd/gohrw_rls1/pKeywordResults?keywor
d=MS7+Games
 No game for chapter 4
Reference:
Holt, Rinehart and Winston. (2007). Holt Mathematics: Course 2.
Retrieved on September 26, 2007 from
http://www.go.hrw.com/gopages/ma/msm2_07.html
NETS for STUDENT STANDARDS
Leave as is
Sample Tasks
Sample Task 1: Connect the Dots
Sample Task 2: Helicopters and Submarines
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Culminating Tasks
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GPS Framework, Grade 7, Unit 2, Patterns and Relationships,
“Culminating Task: Designing a Poster,”
Evidence of Learning
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GPS Framework for Mathematics – Grade 7 EVIDENCE OF LEARNING:
By the conclusion of this unit, students should be able to demonstrate
the following competencies:
 Distinguish between natural numbers, whole numbers, integers and other
rational numbers. • Find the absolute value of a number.
 Compare two positive and/or negative rational numbers using <, >, or =.
 Order a set of positive and/or negative rational numbers from least to greatest or
greatest to least and plot the numbers on a number line.
 Illustrate computation of positive and negative numbers using models,
diagrams, manipulatives or patterns, including fact families.
 Compute with positive and/or negative rational numbers.
 Use the properties of real numbers (commutative, associative, distributive,
inverse and identity) and the order of operations to simplify and evaluate simple
numeric and algebraic expressions.
 Solve problems involving applications of positive and/or negative rational
numbers.
 The following task represents the level of depth, rigor, and complexity expected
of all 7th grade students. This task or a task of similar depth and rigor should
be used to demonstrate evidence of learning.
Curriculum Map
Note:
Scan this in.
6/20/2008 8:01:00 AM
Concept Map

6/20/2008 8:01:00 AM
http://public.doe.k12.ga.us/DMGetDocument.aspx/G7-CMU2.pdf?p=6CC6799F8C1371F6D266E4B28571448133B78FD2FA1A841E3
2775F9A835269A0&Type=D
Parent Letter(s)
6/20/2008 8:01:00 AM
Parent Letter from Math textbook at:
http://georgiastandards.org/DMGetDocument.aspx/grade_7_unit_3_parent_letter.p
df?p=6CC6799F8C1371F65664838EA6545E94BD640990D0E54AD3D0F3C905
75E7A048&Type=D
Bloom’s Taxonomy
6/20/2008 8:01:00 AM
“Patterns and Relationships” for Grade Seven should be at ???
Bloom's Taxonomy of Educational Objectives Cognitive Domain
1. Knowledge (Remembering previously learned material)
Mathematics: Using the Rectangle Activity Sheet, students will describe
communicative rectangles with 90% accuracy.
2. Comprehension (Grasping the meaning of material)
Mathematics: Using the “Fibonacci in Nature Overhead”, students will
compare the Golden Ratio in nature, architecture, and art with 90%
accuracy.
3. Application (Using information in concrete situations)
Mathematics: Using graphic calculators, students will illustrate the
process of plotting points on a coordinate plane with 90% accuracy.
4. Analysis (Breaking down material into parts)
Mathematics: Using computer graphing software, students will examine
the slopes, equations and x-intercept of linear functions data with 90%
accuracy.
5. Synthesis (Putting parts together into a whole)
Mathematics: Using a rubric, students will construct a sample algebra
examination with 90% accuracy.
6. Evaluation (Judging the value of a product for a given purpose, using
definite criteria)
Mathematics: Using the culminating activity, students will debate Jamal’s
mathematical reasoning for purchasing a PlayStation 2 with 90% accuracy.
Reference:
Bloom’s Taxonomy of Educational Objectives. Retrieved on September
26, 2007 from
http://education.calumet.purdue.edu/vockell/edpsybook/Edpsy3/edpsy3
_bloom.htm
Note:
 Adapted from
http://education.calumet.purdue.edu/vockell/edpsybook/Edpsy3/edp
sy3_bloom.htm and GPS standards.