Download Math K-12 Curriculum Frameworks - Albemarle County Public Schools

Albemarle County Public Schools
K-12 Mathematics Curriculum
May 2007
Mathematics Vertical Team
May, 2007
This curriculum document represents the collective thinking of numerous individuals who have
dedicated themselves to research and conversation on and about mathematics curriculum and
instruction over the past three academic years (2004-2007).
Mathematics Vertical Team (2006-2007)
Brooke Asher
Melissa Batchelet
Nicole Butt
Catherine Coffman
Jennie Critzer
Alexandra Davis
Katie Haskins
Linda Insalaco
Ann Messina
Sheila Porter
Marlene Robinson
Tres Wells
Cale Elementary School
Hollymead Elementary School
Scottsville Elementary School
Albemarle High School
Walton Middle School
Department of Instruction
Monticello High School
Sutherland Middle School
Red Hill Elementary School
Jack Jouett Middle School
Department of Instruction
Burley Middle School
Other teachers have also served on the Mathematics Vertical Team during the 2004 – 2006 school years
and made significant contributions to this document:
Janina Buechner
India Haun
Vicki Hendrix
Jay Smith
Chuck Witt
Mathematics Vertical Team, May, 2007
Henley Middle School
Department of Instruction, Walton Middle School
Baker-Butler Elementary School
Sutherland Middle School, Greer Elementary School
Western Albemarle High School
1
Table of Contents
Introduction …………………………………………………………………………………………
3
Mathematics Concepts and Enduring Understandings
………………………………..
4
Philosophy ………………………………………………………………………………………..
5
Habits of Mind of a Mathematician
6
………………………………………………………..
Organization of the Mathematics Curriculum Document
Annotated Matrix Graphic
.……………………….
7
……………………………………………………………….
9
Mathematics Curriculum Matrix K-12 Vertical Sample …………………………………..
11
Appendix Guide ……………………………………………………………………………….
32
Appendix
Lifelong Learner Standards
………………………………………………………
Mathematics Curriculum Assessment Model
Bloom’s Taxonomy of Questioning
The Content Strands of Mathematics
………………………………
ii
………………………………………………
ii
……………………………………………
iii
Mapping the Mathematical Enduring Understandings to the Mathematics
Content Strands
Number and Operations …………………………………………………..
Data Analysis and Probability …………………………………………….
Geometry and Measurement ……………………………………………
Patterns and Algebra ………………………………………………..........
Bibliography
……………….………………………………………………………………….
Mathematics Vertical Team, May, 2007
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xi
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Introduction
This document represents the work of the Mathematics Vertical Team which received its mandate to develop a
comprehensive K-12 mathematics curriculum to support the development of the Framework for Quality Learning (FQL).
As stated in the FQL:
The Albemarle County Public Schools’ core purpose is to establish a community of learners and learning,
through rigor, relevance, and relationships one student at a time. The Framework for Quality Learning
guides and supports teachers’ development and implementation of a system for high-quality curricula,
assessment, and instruction as they act on this vision and facilitate all students attaining deep
understanding of the disciplines … By organizing standards around key concepts and understandings of
the discipline, we engage the personal intellect and emotions of the students (Erickson, 2002). When
students explore concepts over time as opposed to facts in isolation, they develop deeper
understanding and are able to transfer knowledge across disciplines and situations.
The Framework for Quality Learning sets rigorous expectations for how students learn, analyze
information, and communicate, leading to increased student engagement, content mastery, and
higher-order thinking. Application of the Framework for Quality Learning advances the Division’s vision:
‘All learners believe in their power to embrace learning, to excel, and to own their future’
(Framework for Quality Learning, 2006).
Supporting the FQL, the mathematics curriculum is rooted in standards-based and concept-centered instruction and
curriculum with connections to the Lifelong-Learner Standards, Virginia’s Mathematics Standards of Learning (SOL), and
the process and content standards of the National Council of Teachers of Mathematics (NCTM) as articulated in
Principals and Standards for School Mathematics. These various standards provide insight into what all students must
know, understand, and be able to do in authentic mathematical contexts.
The Mathematics Concepts and Enduring Understandings are outlined in the Framework for Quality Learning (FQL)
connecting the Mathematics (Discipline Level) Concepts and the Interdisciplinary Concepts to the topics or content area
of inquiry (p. 4).
Mathematics Vertical Team, May, 2007
3
Mathematics Concepts and Enduring Understandings
Interdisciplinary
Concepts
Systems
Mathematics
Concepts
Relationships
Quantifying
Representation
Properties and Models
Models
Analysis and
Evaluation
Enduring Understandings
Relationships among numbers and number systems form the foundations of number
sense and mathematical communication.
Spatial relationships can be described using coordinate geometry and other
representational systems.
Attributes of objects can be measured using processes and quantified units, using
appropriate techniques, tools, and formulas.
Situations and structures can be represented, modeled, and analyzed using
algebraic symbols.
Mathematical models are used to predict and make inferences about data.
Data can be collected, organized, and displayed in purposeful ways.
Change and
Interactions
Patterns
Cause and Effect
Various statistical methods can be used to observe, analyze, predict, and make
inferences about data.
Patterns and relationships among operations are essential to making estimates and
computing fluently.
Patterns, relations, and functions can be recognized and understood
mathematically.
Communication
Reasoning and
Justification
Theory
Change, in various contexts, both quantitative and qualitative, can be identified
and analyzed.
Characteristics, properties, and mathematical arguments about geometric
relationships can be analyzed and developed using logical and spatial reasoning.
Transformations, symmetry, and spatial reasoning can be used to analyze and
model mathematical situations.
Probability and data analysis can be used to make predictions.
Mathematics Vertical Team, May, 2007
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Philosophy
Dewey said the most important role of school is learning. And learning is a consequence of thinking. Today’s society
demands trained and agile thinkers, and today’s students must learn to make meaning for themselves and to solve
problems for which they do not have answers (Costa, 1997). The Life-Long Learner Standards (Appendix i) identified in
the Framework for Quality Learning (FQL), set expectations for how students develop a wide variety of knowledge,
understanding, and skills. It is also important to consider the desired characteristics (values, attitudes, and skills) of a
student and ultimately an adult working within the discipline. For the purposes of this document, we developed the
Habits of Mind of a Mathematician (p. 6) through incorporation of what are commonly considered the process standards
of learning about and doing mathematics.
What would we wish for students to know, understand, and be able to do when they complete K-12 mathematics? Our
ultimate goal would be for a student to exhibit the disposition to understand how to think about and do mathematics
and to apply mathematics to authentic applications and real-world situations. We designed the mathematics curriculum
document to illustrate the spiraling of concepts from kindergarten through twelfth grade as an interpretation of how the
discipline level enduring understandings continue to grow throughout a child’s education to ultimately support the
lifelong-learner skills. The curriculum document is not intended to replace the Virginia Mathematics Standards of
Learning, or to be interpreted as a complete mathematics curriculum. It offers connections among the content
standards, essential questions and understandings, processes and skills, various cognitive levels of assessment, and
vocabulary to guide instruction through the strands of mathematics content. The Mathematics Curriculum Matrix can
generate an understanding of the way the concepts of K-12 mathematics move across the grades and impact
instruction at every level by creating consistency and continuity across the division. The document is a means to provide
equal access to quality mathematics instruction for all students, and create a vision of a continuous, seamless integration
of content and process standards that travel with increasing sophistication through the K-12 mathematics curriculum.
Mathematics Vertical Team, May, 2007
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Habits of Mind of a Mathematician
• Analyze situations in mathematical terms and pose and solve problems based on situations observed
• Select and use various types of reasoning to develop and evaluate mathematical arguments and
proofs
• Organize and consolidate mathematical thinking through precise verbal, written, and graphical
communication
• Understand how mathematical ideas interconnect and build on one another to produce a coherent
whole
• Use representations to model and interpret physical, social, and mathematical phenomena
• Evaluate and use technology appropriately as a tool to support and apply the problem-solving
process
Mathematics Vertical Team, May, 2007
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The Organization of the Mathematics Curriculum Document
The Mathematics Curriculum Document is a system of documents that are organized electronically through the
Mathematics Curriculum Matrix. By the document’s very nature, it is difficult to describe its three dimensional capacity on
two dimensional paper. The beauty of the model, however, is its ability to flexibly meet the specific needs of individual
teachers by linking concepts and standards in a way that allows variation depending on the lens of inquiry. The
organization of the model is through the umbrella of the K-12 Mathematics (Discipline Level) Enduring Understandings.
These enduring understandings were developed by using the national and state mathematics content and process
standards and the Framework for Quality Learning. The concepts and ideas were also informed by the ideas of various
education researchers connected to the work of Albemarle County (i.e. Erickson, Wiggins and McTighe, Antonetti,
DuFour, etc).
In the Mathematics Curriculum Document, the Mathematics Enduring Understandings connect to the content standards
of mathematics curriculum commonly identified as: Number and Operations, Data and Probability, Geometry and
Measurement, and Patterns and Algebra (Appendix iii). Through these content strands of mathematics, connections are
made with the Virginia Mathematics Standards of Learning (SOL), and the National Mathematics Standards as
articulated by the National Council of Teachers of Mathematics (NCTM) in Principles and Standards for School
Mathematics. Each of the K-12 Mathematics Enduring Understandings is then translated into curriculum, assessment, and
instruction for various stages of a child’s development, through grade band essential understandings and ultimately to
the grade specific curriculum and assessment. We have mapped these Mathematics Enduring Understandings, the
Mathematics Concepts, and Interdisciplinary Concepts (p.4) to the specific content strands (Appendix iii) of the
mathematics curriculum.
This electronic structure will provide the teacher with the ability to research by concept or content standards
(mathematics strand or SOL); by grade level or grade band; and perhaps by vocabulary. The format will facilitate
making connections across the grade levels and the important overarching concepts and ideas within and eventually
beyond the discipline of mathematics. For every K-12 Mathematics Enduring Understanding each grade level is
represented in the matrix document, as is every Virginia Mathematics SOL.
The following description of each component of the Mathematics Curriculum Matrix is followed by a graphic that
articulates the physical placement of these components as they occur in all grade levels for each Mathematics Enduring
Understanding (p.4).
Mathematics Vertical Team, May, 2007
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The Mathematics Curriculum Matrix is structured into five key components. These components are organized under the
umbrella of each of the Mathematics Enduring Understandings (p.4). The structure enables the teacher to view the
connections not only among the broad Discipline Level Concepts and the Interdisciplinary Level Concepts, but also the
connections through the continuum of K-12 mathematics content.
The five key components are:
1. The grade band expectation
This statement gives a more specific interpretation of the K-12 Mathematics (Discipline Level) Enduring
Understanding and focus for the grade band instruction.
2. The Essential Understanding at the specific grade level
Essential understandings help teachers focus the content on what a student needs to know, understand, and
do by making connections between the topics and the enduring understanding as it spirals through the
grade levels with increasing sophistication.
3. Assessment
The assessment section provides samples within the hierarchy of Bloom’s Taxonomy of the Cognitive Domain
to provide teachers with a better understanding of the different levels of challenge required to meet the
intent of a particular standard.
4. Vocabulary
The vocabulary section provides common terminology that promotes consistency of mathematical
language and facilitates accurate communication as a student moves through the grades.
5. Virginia Mathematics Standards of Learning Connections
The Virginia Mathematics Standards of Learning at every grade level are mapped to the Mathematics
Curriculum Matrix. There are also supporting Standards of Learning that may occur at different grade levels
or in different content areas. If there is no direct link to the Standards of Learning, the required processes and
skills for the grade level are referenced.
In the graphic below, these five key components are numbered respectively under the umbrella of the K-12 Mathematics
Enduring Understandings. While not named as a key component, three other organizational tags occur in the document:
grade level, mathematics strand, and concept number within that particular strand, and are labeled accordingly in the
graphic.
Mathematics Vertical Team, May, 2007
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Annotated Graphic Example of the Mathematics Curriculum Matrix, Grade 3
K-12 Mathematics Enduring Understanding
Mathematics
Content Strand
Grade Level
1
Grade band
expectation
4
Vocabulary
Assessment Sample
Knowledge/Comprehension
2
Essential Understandings
for this grade
3
Virginia SOL
Supporting Virginia SOL
Assessment Sample
Application/Analysis
Assessment Sample
Synthesis/Evaluation
5
Supporting Skills
and Processes
Concept within the
content strand
Interdisciplinary
Concept
Mathematics Vertical Team, May, 2007
Mathematics Concept
(Discipline Level)
9
The next pages present conceptually a vertical view of one K-12 Mathematics Enduring Understanding as it spirals
through the grades. The example you are viewing, from the FQL document Mathematics Curriculum (p. 4), is specifically:
Interdisciplinary
Concepts
Communication
Mathematics
Concepts
Reasoning and
Justification
Enduring Understandings
Characteristics, properties, and mathematical arguments about geometric
relationships can be analyzed and developed using logical and spatial reasoning.
This particular Mathematics Enduring Understanding is the first of four contained in the Geometry and Measurement
content strand (Appendix iii). Within the content strand of Geometry and Measurement, the grade band expectations of
Concept 1 spiral through the K-12 Mathematics Curriculum as identified below.
Geometry and Measurement, Concept 1
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed and
developed using logical and spatial reasoning.
Grade Band Expectations:
In grade band K-2:
Represent and compare 2- and 3- dimensional shapes through drawings, block construction, dramatizations, and words
In grade band 3-5:
Develop clarity and precision in describing properties of geometric objects and classifying into categories by properties
In grade band 6-8:
Investigate relationships of polygons by drawing, measuring, visualizing, comparing, transforming and classifying geometric objects
In grade band 9-12:
In grades 9-12: Analyze characteristics and properties about geometric shapes and develop mathematical arguments about these shapes in
applied settings in authentic situations
It may be helpful to use the graphic (p. 9) to locate the key components of the Mathematics Curriculum Matrix as you
move through the document. Notice the increasing sophistication of the grade band and grade level expectations that
map to the same discipline level enduring understanding. The vocabulary should also create consistency. We would like
to emphasize that this document is a “working document” that has already had numerous iterations, and therefore
subject to revision and improvement as we share and discuss the work with our colleagues in the division.
Mathematics Vertical Team, May, 2007
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Mathematics Curriculum Matrix
as excerpted from
Geometry and Measurement Content Strand, Concept 1
Grades K-12
Mathematics Vertical Team, May, 2007
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
Grade K
In grade band K-2:
Represent and compare 2- and 3-dimensional shapes through drawings, block construction, dramatizations, and words
Essential Understandings
All students will understand:
•
•
•
Two-dimensional shapes
help them represent and
describe their world.
Geometric figures can be
classified by shape.
Two- and threedimensional shapes can be
combined to make new
shapes.
Assessment Samples – SOL/Blooms
Knowledge/Comprehension Level:
• Identify Shapes: Give students a sheet with a circle, square, triangle,
and rectangle drawn on it. Point to a shape and ask, “What is this?”
Do this for all shapes. If the student cannot name shapes, follow up by
asking the student to “Point to a circle.” Do this for all four shapes.
Application/Analysis Level:
• “Sort by Attributes” (ACAP): Give a set of 10 attribute blocks, all the
same color and thickness. Use 5 different shapes and at least 2
different sizes. Ask the students, “How can you sort these into piles
that belong together?” If they do not begin to sort within a minute say,
“Think about how they are alike or different.” Do not give prompts to
sort by shape or size.
Vocabulary
triangle
square
rectangle
circle
side
corner
larger
smaller
same/equal
Synthesis/Evaluation Level:
• Using the sort created above, ask the student:
1. How are the blocks in this pile alike?
2. How are the blocks in this pile different from the other pile?
SOL: K.11 The student will identify, describe, and draw two-dimensional (plane) geometric figures (circle, triangle, square, and rectangle).
K.13 The student will compare the size (larger, smaller) and shape of plane geometric figures (circle, triangle, square, and rectangle).
Communication
Reasoning and Justification
Concept 1
11
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
Grade 1
In grade band K-2:
Represent and compare 2- and 3-dimensional shapes through drawings, block construction, dramatizations, and words
Essential Understandings
All students will understand:
•
Vocabulary used to
explain their strategies to
sort and/or group plane
geometric figures
Assessment Samples – SOL/Blooms
Knowledge/Comprehension Level:
• Give students a sheet with a triangle, square, rectangle, and circle on
it. Tell students to: color all circles red or color all triangles blue.
Vocabulary
triangle
square
rectangle
Application/Analysis Level:
“Shape Cards” (Quilt Squares and Block Towns, Investigations in Data,
Number and Space) (Attachment A.1)
• Cut out the shape cards and sort them into two or more groups.
• Glue them onto a piece of paper.
• Label each group.
Synthesis/Evaluation Level:
• Build a structure using 3 or 4 Geoblocks. Draw what you made.
Rubric:
Advanced
circle
side
corner
square corner
two-dimensional
three-dimensional
•
Student accurately creates a 2-D representation of
the 3-D shapes used in block structure
Proficient
• Student attempts to show size and thickness in
representation
• Student is able to describe characteristics of the
Geoblocks
Near Proficient
• Student uses 3 or 4 blocks
• Student represents each block in the structure by
drawing only 1 face of that block (2-D)
• Student effectively shows how many blocks are
arranged
Needs Improvement • Student uses incorrect number of blocks
• Student does not distinguish individual block (may
draw outline of whole construction)
• Student is unable to show how blocks are arranged
SOL: 1.16 The student will draw, describe, and sort plane geometric figures (triangle, square, rectangle, and circle) according to number of sides,
corners, and square corners.
Supporting SOL: 1.17 The student will identify and describe objects in his/her environment that depict plane geometric figures (triangle, rectangle, square, and
circle).
Communication
Reasoning and Justification
Concept 1
12
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
Grade 2
In grade band K-2:
Represent and compare 2- and 3-dimensional shapes through drawings, block construction, dramatizations, and words
Essential Understandings
All students will understand:
•
•
•
Differences and
similarities between twoand three-dimensional
shapes.
Three-dimensional figures
can be identified by the
number of edges, faces,
corners, and by the shape
of their faces.
Similarities between
everyday objects and 3dimensional figures
Assessment Samples – SOL/Blooms
Knowledge/Comprehension Level:
• Tell students to number 1 – 3 on a piece of paper. Ask the students to
find an example of a cylinder, a sphere, and a cube in the classroom,
and draw and label what they found.
Vocabulary
solid
rectangular solid
square pyramid
sphere
Application/Analysis Level:
• Give students solid figures and ask them to sort them into groups.
Have students explain their rationalization.
cylinder
cone
face
Synthesis/Evaluation Level:
“Making a Shed” (Balanced Assessment, Elementary Grades Assessment)
(Attachment A.2)
• Give students “Making a Shed”. Ask the students to make a
replication of the shed using solid figures. The students need to
explain why their model matches the 2-dimensional version using
geometric language.
base
edge
corner
plane figure
two-dimensional
three-dimensional
SOL: 2.20
2.22
The student will identify, describe, and sort three-dimensional (solid) concrete figures, including a cube, rectangular solid (prism), square
pyramid, sphere, cylinder, and cone, according to the number and shape of the solid’s faces, edges, and corners
The student will compare and contrast plane and solid geometric shapes (circle/sphere, square/cube, and rectangle/rectangular solid).
Communication
Reasoning and Justification
Concept 1
13
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
Grade 3
In grade band 3-5:
Develop clarity and precision in describing properties of geometric objects and classifying into categories by properties
Essential Understandings
All students will understand:
•
•
•
•
Lines are composed of line
segments and points.
Two-dimensional shapes can be
described by the number of sides
(line segments), corners (angles),
and square corners.
Three-dimensional shapes can be
described by the number of faces,
edges, corners, square corners, and
by the shape of faces.
Relationships between 2- and 3dimensional figures (circle/sphere,
square/cube, triangle/pyramid, and
rectangle/rectangular solid).
Assessment Samples – SOL/Blooms
Knowledge/Comprehension Level:
• Show students a solid figure.
Answer the following questions:
1. What is the name of this figure?
2. What shapes are its faces?
3. How many faces does it have?
4. How many edges does it have?
Application/Analysis Level:
• Give students pipe cleaners and clay.
Build a solid that has 3 rectangular faces and 2 triangular faces.
Draw their figure.
Answer these questions about their figures:
1. What is the name of this figure?
2. How many edges does your figure have?
3. How many corners does your figure have?
4. How many line segments does your figure have?
Synthesis/Evaluation Level:
• Give students pipe cleaners and clay. Ask students, “Given two rectangles,
what kinds of other shapes can you add to make a 3-dimensional figure?
Make as many different solids as you can and label them. How do you
know these are all the solids that can be made?”
Vocabulary
line
point
line segment
angle
face
corner
square corner
edge
solid
cube
cylinder
cone
rectangular solid
sphere
square pyramid
SOL: 3.18 The student will analyze two-dimensional (plane) and three-dimensional (solid) geometric figures (circle, square, rectangle, triangle, cube,
rectangular solid [prism], square pyramid, sphere, cone, and cylinder) and identify relevant properties, including the number of corners,
square corners, edges, and the number and shape of faces, using concrete models.
3.19 The student will identify and draw representations of line segments and angles, using a ruler or straightedge.
Supporting Skills and Processes: Name 2- and 3-dimensional figures based on a picture or a written description. Sort 3-dimensional figures. Identify and draw
lines, points, angles, and line segments.
Communication
Reasoning and Justification
Concept 1
14
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
Grade 4
In grade band 3-5:
Develop clarity and precision in describing properties of geometric objects and classifying into categories by properties
Essential Understandings
All students will understand:
•
•
Points, lines, line segments, and
rays are fundamental components
of noncircular geometric figures.
The shortest distance between two
points on a flat surface is a line
segment.
Assessment Samples – SOL/Blooms
Vocabulary
Knowledge/Comprehension Level:
• Which of the following letters have two parallel line segments?
A
L
N
V
point
line
line segment
Z
ray
Application/Analysis Level:
• Describe how shape A is different from shapes B and C.
angle
two-dimensional figure
plane
•
•
•
Lines in a plane either intersect or
are parallel. Perpendicularity is a
special case of intersection.
Students will identify real-world
situations that illustrate parallel,
intersecting, and perpendicular
lines.
Students will understand that 2dimensional (plane) figures are
unique in their defining properties.
parallel
intersect / intersection
perpendicular
circle
polygon
A
B
C
triangle
quadrilateral
square
Synthesis/Evaluation Level:
• Create a drawing of a parallelogram that is composed of a square
and two congruent right triangles.
Now create another drawing of a parallelogram (also composed of a
square and two congruent, right triangles) which is not similar to the
first drawing you did.
Based on what you know about these polygons, explain why both of
these drawings are parallelograms.
rectangle
parallelogram
rhombus
three-dimensional figure
sphere
cube
Communication
Reasoning and Justification
Concept 1
15
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
Grade 4
rectangular solid (prism)
cylinder
cone
face
edge
vertex
base
SOL: 4.14 The student will investigate and describe the relationships between and among points, lines, line segments, and rays.
4.15 The student will a) identify and draw representations of points, lines, line segments, rays, and angles, using a straightedge or ruler; and
b) describe the path of shortest distance between two points on a flat surface.
4.16 The student will identify and draw representations of lines that illustrate intersection, parallelism, and perpendicularity.
4.17 a) The student will analyze and compare the properties of two-dimensional (plane) geometric figures (circle, square, rectangle, triangle,
parallelogram, and rhombus) and three-dimensional (solid) geometric figures (sphere, cube, and rectangular solid [prism]).
Communication
Reasoning and Justification
Concept 1
16
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
Grade 5
In grade band 3-5:
Develop clarity and precision in describing properties of geometric objects and classifying into categories by properties
Essential Understandings
All students will understand:
•
•
•
The defining properties and
symmetry of various plane figures
are unique.
Simple plane figures can be
combined to make more
complicated figures; and,
complicated figures can be
subdivided into simple plane
figures.
Solid figures are unique in their
defining properties.
Assessment Samples – SOL/Blooms
Knowledge/Comprehension Level:
1. Choose four 3-dimensional figures from the set of geometric
solids, and list the characteristics for each figure.
Vocabulary
parallel
perpendicular
side
2. Using your ruler, draw a polygon that satisfies these conditions.
A. The polygon has only two right angles.
B. The polygon has only two congruent sides.
C. Label the right angles and the two congruent sides.
angle
vertex
diagonal
Application/Analysis Level:
• Alan says that if a figure has four sides, it must be a rectangle. Gina
does not agree. Draw two figures that show Gina is correct. Explain
how your figure supports Gina’s belief that each four-sided figure is
not necessarily a rectangle.
similar
Synthesis/Evaluation Level:
• “Mystery Rings” (Navigating Through Geometry in Grades 3-5,
NCTM) (Attachment A.5)
triangle
congruent
two-dimensional
polygon
right triangle
acute triangle
obtuse triangle
scalene triangle
isosceles triangle
equilateral triangle
quadrilateral
rectangle
square
parallelogram
kite
Communication
Reasoning and Justification
Concept 1
17
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
Grade 5
trapezoid
rhombus
three-dimensional
face
edge
pyramid
prism
cylinder
cone
cube
SOL: 5.15 The student, using two-dimensional (plane) figures (square, rectangle, triangle, parallelogram, rhombus, kite, and trapezoid) will
a) recognize, identify, describe, and analyze their properties in order to develop definitions of these figures; b) identify and explore
congruent, non-congruent, and similar figures; c) investigate and describe the results of combining and subdividing shapes.
5.16 The student will identify, compare, and analyze properties of three-dimensional (solid) geometric shapes (cylinder, cone, cube, square,
pyramid, and rectangular prism).
Supporting SOL:
5.8 The student will describe and determine the perimeter of a polygon and the area of a square, rectangle, and right triangle, given the appropriate
measures.
5.9 The student will identify and describe the diameter, radius, chord, and circumference of a circle.
5.13 The student will measure and draw right, acute, and obtuse angles and triangles, using appropriate tools.
5.14 The student will classify angles and triangles as right, acute, or obtuse.
Supporting Skills and Processes: Construct models of 2- and 3-dimensional figures using a variety of resources to include technology and manipulatives.
Communication
Reasoning and Justification
Concept 1
18
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
Grade 6
In grade band 6-8:
Investigate relationships of polygons by drawing, measuring, visualizing, comparing, transforming, and classifying geometric objects
Essential Understandings
All students will understand:
•
•
•
An angle is two rays diverging
from a common point.
Names for angles and triangles by
defining referents and
characteristics.
Plane figures are identified,
classified, and described by their
similarities, differences, and
defining properties.
•
The meaning of congruence.
•
The attributes of perpendicular
lines and a bisector.
•
•
Assessment Samples – SOL/Blooms
Vocabulary
Use “Shapes and Designs”, Connected Mathematics (Attachment A.6)
for the following:
vertex
Knowledge/Comprehension Level:
• What name could you give A, H, U, and R?
line
ray
line segment
triangle
Application/Analysis Level:
• What are the features of a polygon? An angle?
• Can you make a distinction between a trapezoid and a
parallelogram? Explain your thinking.
Synthesis/Evaluation Level:
• The figures I, L, and V can be grouped together, but X would not
belong in the group. Explain why.
• The figures U, S, and E can be grouped together, but G would not
belong in the group. Explain why.
obtuse
acute
right
isosceles
equilateral
quadrilaterals
rhombus
trapezoid
How to interpret a picture of a
solid figure from a twodimensional diagram and vice
versa.
parallelogram
square
rectangle
pentagon
The decomposition of a solid
figure into a discrete set of
surfaces.
hexagon
heptagon
octagon
polygon
congruent
Communication
Reasoning and Justification
Concept 1
19
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
Grade 6
bisect
rectangular prism
cylinder
pyramid
two-dimensional
three-dimensional
surface
edge
angle ruler
protractor
degrees
two-dimensional model (net)
SOL: 6.13 The student will a) estimate angle measures, using 45°, 90°, and 180° as referents, and use the appropriate tools to measure the given angles;
and b) measure and draw right, acute, and obtuse angles and triangles.
6.14 The student will identify, classify, and describe the characteristics of plane figures, describing their similarities, differences, and defining
properties.
6.15 The student will determine congruence of segments, angles, and polygons by direct comparison, given their attributes. Examples of
noncongruent and congruent figures will be included.
6.16 The student will construct the perpendicular bisector of a line segment and an angle bisector.
6.17 The student will sketch, construct models of, and classify solid figures (rectangular prism, cone, cylinder, and pyramid).
Communication
Reasoning and Justification
Concept 1
20
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
Grade 7
In grade band 6-8:
Investigate relationships of polygons by drawing, measuring, visualizing, comparing, transforming, and classifying geometric objects
Essential Understandings
All students will understand:
•
•
•
•
•
Quadrilaterals can be classified
according to the attributes of their
sides and/or angles.
A quadrilateral can belong to one
or more subsets of the set of
quadrilaterals and has all of the
defining attributes of the subset.
The meaning of prefixes associated
with the number of sides of a
polygon.
Assessment Samples – SOL/Blooms
Vocabulary
Knowledge/Comprehension Level:
• Which of the following characteristics must be present in order to
classify a quadrilateral as a parallelogram?
a. exactly one pair of equal adjacent sides
b. exactly one pair of parallel sides
c. a diagonal as an axis of symmetry
d. two equal adjacent angles
e. two pairs of parallel sides
quadrilateral
Application/Analysis Level:
“Geometric Shapes” (NCTM Mathematics Assessment Sampler, Grades
6-8) (Attachment A.7)
diagonal
parallelogram
rectangle
square
rhombus
trapezoid
perpendicular
polygons
Similar geometric figures have the
same shape but may have different
sizes.
Five geometric terms are listed in alphabetical order: equilateral
triangle, rhombus, right isosceles triangle, square, trapezoid.
• Which one of the geometric terms listed above most accurately
describes shape A? Explain how you know that is your answer.
• Repeat the question above for shapes B, C, and D.
How ratios and proportions can be
used to determine the length of
something that cannot be measured
directly.
Synthesis/Evaluation Level
• Luis claimed that a figure must be a square if it is a parallelogram
and has all its sides the same length. Is Luis correct? Defend your
decision with a written argument.
pentagon
hexagon
heptagon
octagon
nonagon
decagon
sides
triangle
proportion
similar
congruent
regular polygon
corresponding angles
Communication
Reasoning and Justification
Concept 1
21
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
Grade 7
corresponding sides
angle ruler
protractor
degrees
straight angle
two-dimensional model (net)
SOL: 7.9 The student will compare and contrast the following quadrilaterals: parallelogram, rectangle, square, rhombus, and trapezoid. Deductive
reasoning and inference will be used to classify quadrilaterals.
7.10 The student will identify and draw the following polygons: pentagon, hexagon, heptagon, octagon, nonagon, and decagon.
7.11 The student will determine if geometric figures, quadrilaterals and triangles, are similar and write proportions to express the relationships between
corresponding parts of similar figures.
Communication
Reasoning and Justification
Concept 1
22
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
Grade 8
In grade band 6-8:
Investigate relationships of polygons by drawing, measuring, visualizing, comparing, transforming, and classifying geometric objects
Essential Understandings
All students will understand:
•
•
Assessment Samples – SOL/Blooms
Use the following figure to answer the questions below.
vertical
supplementary
Pairs of angles are named by their
defining attributes.
A three-dimensional object can be
represented as a two-dimensional
model that represents views of the
object from different perspectives.
Vocabulary
3
1
3
1
complementary
Building Base Plan
1
front
Knowledge/Comprehension Level:
• Using your cubes, make a model of the building shown in the base
plan above.
Application/Analysis Level:
• A set of building plans includes the base outline, the front view, and
the right view. Sketch a full set of building plans on grid paper for a
new building of your design with the same base outline as the
building represented above.
angle
measure
draw
intersecting
protractors
angle ruler
straight angle
degrees
three-dimensional figures
top
side
bottom
Synthesis/Evaluation Level:
• Explain how your new building can have the same set of building
plans as the original building even though your new building has a
different base plan.
isometric
geometric solids
two-dimensional model (net)
polyhedron
polygon
SOL: 8.6 The student will verify by measuring and describe the relationships among vertical angles, supplementary angles, and complementary angles
and will measure and draw angles of less than 360°.
8.9 The student will construct a three-dimensional model, given the top, side, and/or bottom views.
Communication
Reasoning and Justification
Concept 1
23
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
Grades 9-12
In grade band 9-12:
Analyze characteristics and properties of geometric shapes and develop mathematical arguments about these shapes in applied settings in
authentic situations
Essential Understandings
All students will understand:
•
Intersecting lines form angles
with measurable
characteristics, relationships
and properties.
Assessment Samples – SOL/Blooms
A clothing designer must carefully measure segments and angles to create a
pattern. The figure below is part of a clothing design pattern. All
measurements are in centimeters.
Vocabulary
parallel
perpendicular
point
line
plane
•
•
•
•
•
The interior and exterior
angles of polygons have
measurable characteristics,
relationships, and patterns.
Two- and three-dimensional
shapes can be constructed with
a compass, straight-edge,
patty-paper, and various
technology applications.
Congruency and similarity in
polygons have real-life
application in a variety of areas
including art, architecture, and
sciences.
Congruency and similarity can
be demonstrated using logical
reasoning.
Polygons have defining
characteristics with their sides,
angles and diagonals.
angle
polygon
congruent
similar
Knowledge/Comprehension Level:
• Based on the given information, what do you know about angles B and E.?
Why?
Application/Analysis Level:
• What do you know about triangle ABC and triangle DEC. Justify by using
a two column proof.
• Find the measurement of CD. Support your answer using geometric
concepts.
• Find the measurement of FE, FC, and BC. Justify.
Synthesis/Evaluation Level:
• Determine the perimeter of triangle ABC and triangle DEC. How is this
relevant to our clothing designer?
• If the design is to be repeated twice, what is the total area of the fabric
needed?
side
diagonal
Pythagorean Theorem
triangle
arc
secant
chord
tangent
circle
diameter
radius
perimeter
area
Communication
Reasoning and Justification
Concept 1
24
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
•
•
The Pythagorean Theorem is
essential for solving triangle
problems.
Grades 9-12
The regular tetrahedron shown in figure(a) has grown a regular tetrahedron
(with a side half as long) on each of its faces in figure (b). Figure (c) shows
the new solid (b) in the process of growing tetrahedra on its faces in the same
way.
Many relationships exist
between and among angles,
arcs, secants, chords, and
tangents of a circle.
•
Three-dimensional figures are
a part of everyday life.
•
Modeling is an overall part of
design for products and
structures.
•
Perimeter, area, surface area
and volume are all measurable
characteristics of 2- and 3dimensional shapes.
•
Formulas can be derived to
generalize perimeter, area,
surface area and volume.
Communication
surface area
volume
quadrilateral
iterations
tetrahedron/tetrahedra
Knowledge/Comprehension Level:
•
Complete the tables below.
Reasoning and Justification
Concept 1
25
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
Grades 9-12
Application/Analysis Level:
• Determine the total surface area of the 20th iteration?
• Determine the total surface area of the nth iteration as
•
•
•
n → ∞?
th
Determine the total volume of the 20 iteration?
Determine the total volume of the nth iteration as n → ∞ ?
Compare the total surface area and the total volume as n →
∞.
Synthesis/Evaluation Level:
• Create a similar model using a cube, and then construct a table similar to
the tetrahedron.
SOL: G.5 The student will a) investigate and identify congruence and similarity relationships between triangles; and b) prove two triangles are
congruent or similar, given information in the form of a figure or statement, using algebraic and coordinate as well as deductive proofs.
G.8: The student will a) investigate and identify properties of quadrilaterals involving opposite sides and angles, consecutive sides and angles, and
diagonals; b) prove these properties of quadrilaterals, using algebraic and coordinate methods as well as deductive reasoning; and c) use
properties of quadrilaterals to solve practical problems.
Supporting SOL:
G.1 The student will construct and judge the validity of a logical argument consisting of a set of premises and a conclusion. This will include a) identifying
the converse, inverse, and contrapositive of a conditional statement; b) translating a short verbal argument into symbolic form; c) using Venn
diagrams to represent set relationships; and d) using deductive reasoning, including the law of syllogism.
G.4 The student will use the relationships between angles formed by two lines cut by a transversal to determine if two lines are parallel and verify, using
algebraic and coordinate methods as well as deductive proofs.
G.6 The student, given information concerning the lengths of sides and/or measures of angles, will apply the triangle inequality properties to
determine whether a triangle exists and to order sides and angles. These concepts will be considered in the context of practical situations.
Supporting Skills and Processes: Knowledge and understanding of the properties and relationships among basic two- and three-dimensional shapes; use of
deductive reasoning to establish or refute conjectures.
Communication
Reasoning and Justification
Concept 1
26
Geometry and Measurement Strand
Grade 1
Assessment Attachment A.1
Shape Cards
Cut out the shape cards. Sort them into two or more groups and glue them onto a piece of
paper. Label each group.
Rubric:
Advanced
Proficient
Nearing
Proficient
Needs
Improvement
•
•
•
•
•
•
Student sorts into 3 or more categories
Student sorts by attributes such as number of sides, straight/curved sides, and number of corn
Student is able to flexibly regroup the shapes
Student uses all shapes and sorts them into distinct categories
Student accurately labels each category
Student may be unable to use all 9 shapes but is able to sort some shapes into distinct groups
•
•
•
Student is unable to use all 9 shapes
Student is unable to sort into distinct groups
Student is unable to label or identify the attribute
Communication
Reasoning and Justification
Concept 1
27
Geometry and Measurement Strand
Grade 2
Making a Shed
Communication
Assessment Attachment A.2
Reasoning and Justification
Concept 1
28
Geometry and Measurement Strand
Grade 5
Communication
Assessment Attachment A.5
Reasoning and Justification
Concept 1
29
Geometry and Measurement Strand
Grade 6
Shapes and Designs
Assessment Attachment A.6
Knowledge/Comprehension Level
• What name could you give A, H, U, and R?
Application/Analysis Level
• What are the features of a polygon? An angle?
• Can you make a distinction between a trapezoid and a parallelogram? Explain your thinking.
Synthesis/Evaluation Level
• The figures I, L, and V can be grouped together, but X would not belong in the group. Explain why.
• The figures U, S, and E can be grouped together, but G would not belong in the group. Explain why.
Communication
Reasoning and Justification
Concept 1
30
Geometry and Measurement Strand
Grade 7
Geometric Shapes
Communication
Assessment Attachment A.7
Reasoning and Justification
Concept 1
31
Appendix
Mathematics Vertical Team, May, 2007
Appendix Guide
Lifelong-Learner Standards
………………………………….……………….…………………..
Assessment in the Mathematics Curriculum Matrix
…………………………………………
ii
………………………………………………..
ii
………………………………………………………….
iii
Bloom’s Taxonomy of the Cognitive Domain
The Content Strands of Mathematics
i
Mapping the Mathematics Enduring Understandings to the Mathematics Content Strands
Number and Operations …………………………………………….. ……………………
Data Analysis and Probability ………………………………………. ……………………
Geometry and Measurement …………………………………………………………….
Patterns and Algebra ………………………………………………....……………………
v
vi
ix
xi
Bibliography
xiii
……………………………………………………………………………………..….
Mathematics Vertical Team, May, 2007
Lifelong-Learner Standards
The Division has identified 12 Lifelong-Learner Standards that set expectations for how students develop a wide variety
of knowledge, understanding, and skills. These standards articulate the necessary components of lifelong learning that
allow all students to succeed as members of a global community and in a global economy. The Lifelong-Learner
Standards are overarching process-based standards, not discrete fact-based standards that can be addressed in a
single lesson or even a single unit. These standards demand attention over time and across all disciplines (FQL, 2006).
Lifelong-Learner Standards
1. Plan and conduct research;
2. Gather, organize, and analyze data, evaluate processes and products, and draw conclusions;
3. Think analytically, critically, and creatively to pursue new ideas, acquire new knowledge, and make decisions;
4. Understand and apply principles of logic and reasoning; develop, evaluate, and defend arguments;
5. Seek, recognize and understand systems, patterns, themes, and interactions;
6. Apply and adapt a variety of appropriate strategies to solve new and increasingly complex problems;
7. Acquire and use precise language to clearly communicate ideas, knowledge, and processes;
8. Explore and express ideas and opinions using multiple media, the arts, and technology;
9. Demonstrate ethical behavior and respect for diversity through daily actions and decision making;
10. Participate fully in civic life, and act on democratic ideals within the context of community and global
interdependence;
11. Understand and follow a physically active lifestyle that promotes good health and wellness; and,
12. Apply habits of mind and metacognitive strategies to plan, monitor, and evaluate one’s own work.
Mathematics Vertical Team, May, 2007
i
Assessment in the Mathematics Curriculum Matrix
The assessment section of the Mathematics Curriculum Matrix provides samples within the hierarchy of Bloom’s
Taxonomy of the Cognitive Domain to provide teachers with a better understanding of the different levels of challenge
required to meet the intent of a particular standard. These assessment samples are intended as examples to help a
teacher focus on the level of questioning and performance needed for a student to gain deep understanding of a
particular standard. There is not a specific assessment provided for all six levels of Bloom’s Taxonomy, but the
assessment examples have been placed in three tiers of the domain to represent low level (Knowledge and
Comprehension), middle level (Application and Analysis), and high level (Synthesis and Evaluation)cognitive demands.
Bloom’s Taxonomy of the Cognitive Domain
Knowledge:
Students recall information; students exhibit memory of previously learned material by recalling facts, terms, basic
concepts, and answers.
Comprehension:
Students recognize what they know in context; students identify relationships between pieces of information;
students demonstrate understanding of facts and ideas by organizing, comparing, translating, interpreting, giving
descriptions, and stating main ideas.
Application:
Students use what they know and comprehend in the performance of a skill; students solve problems applied to new
situations by using acquired knowledge, facts, techniques, and rules in new ways.
Analysis:
Students draw conclusions from new data, making interpretations based on familiar patterns in what they know and
comprehend; students examine and break information into parts by identifying motives or causes; students make
inferences and find evidence to support generalizations.
Synthesis:
Students create a new work that demonstrates their ability to apply their knowledge, comprehension, and analysis of
information in a student-generated product; students compile information together in a different way by combining
elements in a new pattern or proposing alternative solutions based on the application of knowledge and
understanding.
Evaluate:
Students develop, argue and defend opinions based on what they know and comprehend after making an analysis;
students present and defend opinions by making judgments about information; students validate ideas or quality of
work based on a set of criteria.
Mathematics Vertical Team, May, 2007
ii
The Content Strands of Mathematics*
*Most of the ideas and definitions in the “Content Strands of Mathematics” portion of this document have been quoted or paraphrased from
The Principles and Standards for School Mathematics (NCTM, 2000).
Number and Operations
The Number and Operations Strand involves deep and fundamental understanding of, and proficiency with counting,
numbers, and arithmetic, as well as an understanding of number systems and their structures. The concepts and
algorithms of arithmetic are part of number and operations, as are the properties and characteristics of the classes of
numbers that form the beginnings of number theory. Central to these ideas is the development of number sense,
understanding numbers, ways of representing numbers, relationships among numbers, and number systems.
Data Analysis and Probability
Students should learn how to collect data, organize their own or other’s data, and display the data in graphs and
charts that will be useful in answering their questions. This strand also includes learning some methods for analyzing
data and making inferences and conclusions from data. The basic concepts and applications of probability are also
addressed, with an emphasis on the way that probability and data analysis are related. To understand the
fundamentals of statistical ideas, students must work directly with data. This emphasis allows students to meet new
ideas and procedures as they progress through the grades, and make a number of important connections among
ideas and procedures from number, algebra, measurement, and geometry. Work in data analysis and probability
offers a natural way for students to connect mathematics with other content areas and with experiences in their daily
lives.
Geometry and Measurement
Through the study of geometry, students will learn about geometric shapes and structures and how to analyze their
characteristics and relationships. Spatial visualization-building and manipulating mental representations of two- and
three-dimensional objects and perceiving an object from the different perspectives is an important aspect of
geometric thinking. Geometry is a natural place for the development of students’ reasoning and justification skills,
culminating in the work with proof in the secondary grades. Geometric modeling and spatial reasoning offer ways to
interpret and describe physical environments and can be important tools in problem solving.
Measurement is the assignment of a numerical value to an attribute of an object. Understanding what a measurable
attribute is and becoming familiar with the units and processes that are used in measuring attributes is a major
emphasis in this strand. Students should become proficient in using measurement tools, techniques, and formulas in a
range of situations involving measuring and comparing. At more sophisticated levels, measurement involves assigning
Mathematics Vertical Team, May, 2007
iii
a number to a characteristic of a situation, as is done by the consumer price index. The study of measurement offers
the opportunity for learning and applying other mathematics, including number operations, geometric ideas, statistical
concepts, and notions of function. It highlights connections within mathematics and between mathematics and other
content areas.
Patterns and Algebra
Algebra has its historical roots in the study of general methods for solving equations. The Algebra Strand emphasizes
relationships among quantities, including functions, ways of representing mathematical relationships, and the analysis
of change. Functional relationships can be expressed by using symbolic notation, which allows complex
mathematical ideas to be expressed succinctly and allows change to be analyzed efficiently. Understanding change
is fundamental to understanding functions and to understanding many ideas in our world. Today, the methods and
ideas of algebra support mathematical work in many areas, for example, distribution and communication networks,
laws of physics, population models, and statistical results can all be represented in the symbolic language of algebra.
The K-12 Mathematics Enduring Understandings as they connect to each of the Content Strands and the Grade Level
Expectations throughout the Mathematics Curriculum Matrix are outlined below.
Mathematics Vertical Team, May, 2007
iv
Number and Operations
Number and Operations, Concept 1
K-12 Enduring Understanding: Relationships among numbers and number systems form the foundations of number
sense and mathematical communication.
Grade Band Expectations
In grade band K-2:
Develop the ability to deal with numbers mentally and to use number sense to reason with numbers in complex ways
In grade band 3-5:
Develop strategies for judging the relative size of numbers, identify classes of numbers and examine their properties
In grade band 6-8:
Use strategies to build fluency and extend knowledge of the number system
In grade band 9-12:
Develop a full understanding of the system of real numbers and use their understanding to explore new systems
Number and Operations, Concept 2
K-12 Enduring Understanding: Patterns and relationships among operations are essential to making estimates and computing
fluently.
Grade Band Expectations:
In grade band K-2:
Understand the meaning of, and relationship between addition and subtraction to compute fluently
In grade band 3-5:
Understand the meaning of, and relationships between addition and subtraction, and multiplication and division
In grade band 6-8 :
Investigate the properties and obtain computational fluency within the real number system
In grade band 9-12 :
Reason whether a problem calls for an estimate or exact answer, extend understanding of operations to number systems that are
new to students, and select a suitable method of computation
Mathematics Vertical Team, May, 2007
v
Data Analysis and Probability
Data Analysis and Probability, Concept 1
K-12 Enduring Understanding: Data can be collected, organized, and displayed in purposeful ways.
Grade Band Expectations
In grade band K-2:
Pose questions, gather, and use various methods to represent data about themselves and their surroundings
In grade band 3-5:
Collect, represent, and investigate data and how data collection methods affect the nature of the data set
In grade band 6-8:
Formulate questions, design studies, collect relevant data, and create and use appropriate graphical representations of data
In grade band 9-12:
Know the characteristics and differences of well designed studies and the meaning and types of inferences that can be drawn
from measurement data
Data Analysis and Probability, Concept 2
K-12 Enduring Understanding: Various statistical methods can be used to observe, analyze, predict, and make
inferences about data.
Grade Band Expectations:
In grade band K-2:
Describe parts of the data and a set of data to determine what the data show
In grade band 3-5:
Describe and compare related data sets, their distributions, and measures of center
In grade band 6-8:
Discuss and understand the correspondence between data sets and their graphic representations, and find, use, and interpret
their measures of central tendency
In grade band 9-12:
Graphically display univariate and bivariate data and understand the implications of its characteristics
Mathematics Vertical Team, May, 2007
vi
Data Analysis and Probability, Concept 3
K-12 Enduring Understanding: : Mathematical models are used to predict and make inferences about data.
Grade Band Expectations:
In grade band K-2:
Discuss events related to students’ experiences as likely or unlikely
In grade band 3-5:
Propose and justify conclusions and predictions that are based on data and design studies to further investigate the conclusions
or predictions
In grade band 6-8:
Use observations about differences between two or more samples to make conjectures and develop new questions about the
populations from which they were taken
In grade band 9-12:
Use simulations to explore the variability of sample statistics from a known population and construct sampling distributions
Data Analysis and Probability, Concept 4
K-12 Enduring Understanding: Probability and data analysis can be used to make predictions
Grade Band Expectations:
In grade band K-2:
Develop and evaluate inferences and predictions that are based on data
In grade band 3-5:
Describe and discuss the degree of likelihood of events and predict the probability of outcomes of simple experiments
In grade band 6-8:
Use a basic understanding of probability to make and test conjectures about the results of experiments and simulations
In grade band 9-12:
Understand the concepts of sample space and probability distribution, use simulations, and compute and interpret expected
value
Mathematics Vertical Team, May, 2007
vii
Geometry and Measurement
Geometry and Measurement, Concept 1
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships
can be analyzed and developed using logical and spatial reasoning.
Grade Band Expectations:
In grade band K-2:
Represent and compare 2- and 3-dimensional shapes through drawings, block construction, dramatizations, and words
In grade band 3-5:
Develop clarity and precision in describing properties of geometric objects and classifying into categories by properties
In grade band 6-8:
Investigate relationships of polygons by drawing, measuring, visualizing, comparing, transforming and classifying geometric
objects
In grade band 9-12:
Analyze characteristics and properties of geometric shapes and develop mathematical arguments about these shapes in
applied settings in authentic situations
Geometry and Measurement, Concept 2
K-12 Enduring Understanding: Spatial relationships can be described using coordinate geometry and other
representational systems.
Grade Band Expectations:
In grade band K-2:
Describe, name, and interpret relative positions, directions, and distances in space
In grade band 3-5:
Use a coordinate system to aid in describing location and movement in all four quadrants of the Cartesian plane
In grade band 6-8:
Explore and represent the properties of shapes using coordinate geometry formulas
In grade band 9-12:
Investigate and use Cartesian coordinates and other coordinate systems to analyze geometric situations
Mathematics Vertical Team, May, 2007
viii
Geometry and Measurement, Concept 3
K-12 Enduring Understanding: Transformations, symmetry, and spatial reasoning can be used to analyze and model
mathematical situations.
Grade Band Expectations:
In grade band K-2:
Develop an understanding of transformations and symmetry with shapes
In grade band 3-5:
Identify symmetry and congruency, and describe the results of various transformations of plane figures
In grade band 6-8:
Create and quantify the results of various transformations, including dilation
In grade band 9-12:
a. Understand and represent translations and reflections, rotations, and dilations of objects in the plane by using sketches,
coordinates, vectors, function notation, and matrices
b. Use various representations to help understand the effects of simple transformations and their compositions
Geometry and Measurement, Concept 4
K-12 Enduring Understanding: Attributes of objects can be measured using processes and quantified units, using
appropriate techniques, tools, and formulas.
Grade Band Expectations:
In grade band K-2:
Understand attributes of measurement by directly comparing objects, and selecting an appropriate unit and tools
In grade band 3-5:
Develop and deepen understanding of what it means to measure an object: identify an attribute to be measured, choose an
appropriate unit, and compare that unit to the object being measured
In grade band 6-8:
Become proficient in selecting the appropriate size and type of unit for a given measurement situation, including length, area
and volume
In grade band 9-12:
Apply appropriate technology, and understand the limitations; make reasonable estimates and accurate predictions about
measurement
Mathematics Vertical Team, May, 2007
ix
Patterns and Algebra
Patterns and Algebra, Concept 1
K-12 Enduring Understanding: Patterns, relations, and functions can be recognized and understood mathematically.
Grade Band Expectations:
In grade band K-2:
Recognize, describe, and extend patterns, and sort and classify objects
In grade band 3-5:
Investigate numerical and geometric patterns and express them mathematically in words and symbols
In grade band 6-8:
Investigate patterns that arise when there is a rate of change
In grade band 9-12:
Create and use tabular, symbolic, graphical, and verbal representations and analyze and understand patterns, relations, and
functions
Patterns and Algebra, Concept 2
K-12 Enduring Understanding: Situations and structures can be represented, modeled, and analyzed using algebraic
symbols.
Grade Band Expectations:
In grade band K-2:
Illustrate and model general principles and properties of operations
In grade band 3-5:
Investigate, describe, and represent various rates of change
In grade band 6-8:
Solve problems and understand that relationships among quantities can often be expressed symbolically and represented in
more than one way
In grade band 9-12:
Fluently manipulate algebraic expressions by combining them and re-expressing them in alternative forms, including identification
and selection of relevant features in applications of real-world situations
Mathematics Vertical Team, May, 2007
x
Patterns and Algebra, Concept 3
K-12 Enduring Understanding: Change, in various contexts, both quantitative and qualitative, can be identified and
analyzed.
Grade Band Expectations:
In grade band K-2:
Describe change both qualitatively and quantitatively
In grade band 3-5:
Propose and justify conclusions and predictions that are based on data and design studies to further investigate the conclusions
or predictions
In grade band 6-8:
Use graphs to analyze the nature of changes in quantities in linear relationships
In grade band 9-12:
Approximate and interpret rates of change from graphical and numerical data
Mathematics Vertical Team, May, 2007
xi
Bibliography
Balanced Assessment—Elementary Grades Assessment. (1999). Dale Seymour Publications.
DuFour, Robert; Eaker, R.; & DuFour, Rebecca (Editors). (2005). On Common Ground: The Power of Professional Learning
Communities.
Erickson, L. (2006–-not yet published). Concept-Based Curriculum and Instruction for the Thinking Classroom.
Erickson, L. (1998). Concept-based Curriculum and Instruction: Teaching Beyond the Facts.
Erickson, L. (1994). Stirring the Head, Heart, and Soul: Redefining Curriculum and Instruction.
Lappan, G.; Fey, J.; Fitzgerald, W.; Friel, S.; & Phillips, E. D. (Editors). (2004). Connected Mathematics, shapes and
designs, two-dimensional geometry.
Marzano, R. (2004). Building Background Knowledge for Academic Achievement: Research on What Works in Schools.
Marzano, R. (2003). What Works in Schools: Translating Research into Action. Alexandria, VA: Association for Supervision and
Curriculum Development.
Marzano, R., Pickering, D., & Pollock, J. (2001). Classroom Instruction that Works: Research-based Strategies for Increasing
Student Achievement. Alexandria, VA: Association for Supervision and Curriculum Development.
Marzano, R. (2000). Designing a New Taxonomy of Educational Objectives.
McTighe, J., Seif, E., & Wiggins, G. (2004). You can teach for meaning. Educational Leadership, 62(1), 26-30.
National Council of Teachers of Mathematics. (2005). Mathematics Assessment Sampler, Grades 3-5. Reston, VA.
National Council of Teachers of Mathematics. (2005). Mathematics Assessment Sampler, Grades 6-8. Reston, VA.
National Council of Teachers of Mathematics. (2001). Navigating Through Geometry in Grades 3-5. Reston, VA.
National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA.
Mathematics Vertical Team, May, 2007
xii
Russell, S. J.; Clements, D.; & Sarama, J. (1998). Investigations in Number, Data, and Space, Quilt squares and block towns.
Stiggins, R. J., Arter, J. A., Chappius, J., & Chappius, S.. (2004). Classroom Assessment for Student Learning: Doing it right –
using it well.
Tomlinson, Carol Ann (2001). How to Differentiate Instruction in Mixed-Ability Classrooms, 2nd Edition.
Tomlinson, Carol Ann. (1999) The Differentiated Classroom: Responding to the Needs of All Learners.
Wiggins, G. & McTighe, J. (1998).
Development.
Understanding by Design. Alexandria, VA:
Mathematics Vertical Team, May, 2007
Association for Supervision and Curriculum
xiii
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
Grade K
In grade band K-2:
Represent and compare 2- and 3-dimensional shapes through drawings, block construction, dramatizations, and words
Essential Understandings
All students will understand:
•
•
•
Two-dimensional shapes
help them represent and
describe their world.
Geometric figures can be
classified by shape.
Two- and threedimensional shapes can be
combined to make new
shapes.
Assessment Samples – SOL/Blooms
Knowledge/Comprehension Level:
• Identify Shapes: Give students a sheet with a circle, square, triangle,
and rectangle drawn on it. Point to a shape and ask, “What is this?”
Do this for all shapes. If the student cannot name shapes, follow up by
asking the student to “Point to a circle.” Do this for all four shapes.
Application/Analysis Level:
• “Sort by Attributes” (ACAP): Give a set of 10 attribute blocks, all the
same color and thickness. Use 5 different shapes and at least 2
different sizes. Ask the students, “How can you sort these into piles
that belong together?” If they do not begin to sort within a minute say,
“Think about how they are alike or different.” Do not give prompts to
sort by shape or size.
Vocabulary
triangle
square
rectangle
circle
side
corner
larger
smaller
same/equal
Synthesis/Evaluation Level:
• Using the sort created above, ask the student:
1. How are the blocks in this pile alike?
2. How are the blocks in this pile different from the other pile?
SOL: K.11 The student will identify, describe, and draw two-dimensional (plane) geometric figures (circle, triangle, square, and rectangle).
K.13 The student will compare the size (larger, smaller) and shape of plane geometric figures (circle, triangle, square, and rectangle).
Communication
Reasoning and Justification
Concept 1
11
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
Grade 1
In grade band K-2:
Represent and compare 2- and 3-dimensional shapes through drawings, block construction, dramatizations, and words
Essential Understandings
All students will understand:
•
Vocabulary used to
explain their strategies to
sort and/or group plane
geometric figures
Assessment Samples – SOL/Blooms
Knowledge/Comprehension Level:
• Give students a sheet with a triangle, square, rectangle, and circle on
it. Tell students to: color all circles red or color all triangles blue.
Vocabulary
triangle
square
rectangle
Application/Analysis Level:
“Shape Cards” (Quilt Squares and Block Towns, Investigations in Data,
Number and Space) (Attachment A.1)
• Cut out the shape cards and sort them into two or more groups.
• Glue them onto a piece of paper.
• Label each group.
Synthesis/Evaluation Level:
• Build a structure using 3 or 4 Geoblocks. Draw what you made.
Rubric:
Advanced
circle
side
corner
square corner
two-dimensional
three-dimensional
•
Student accurately creates a 2-D representation of
the 3-D shapes used in block structure
Proficient
• Student attempts to show size and thickness in
representation
• Student is able to describe characteristics of the
Geoblocks
Near Proficient
• Student uses 3 or 4 blocks
• Student represents each block in the structure by
drawing only 1 face of that block (2-D)
• Student effectively shows how many blocks are
arranged
Needs Improvement • Student uses incorrect number of blocks
• Student does not distinguish individual block (may
draw outline of whole construction)
• Student is unable to show how blocks are arranged
SOL: 1.16 The student will draw, describe, and sort plane geometric figures (triangle, square, rectangle, and circle) according to number of sides,
corners, and square corners.
Supporting SOL: 1.17 The student will identify and describe objects in his/her environment that depict plane geometric figures (triangle, rectangle, square, and
circle).
Communication
Reasoning and Justification
Concept 1
12
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
Grade 2
In grade band K-2:
Represent and compare 2- and 3-dimensional shapes through drawings, block construction, dramatizations, and words
Essential Understandings
All students will understand:
•
•
•
Differences and
similarities between twoand three-dimensional
shapes.
Three-dimensional figures
can be identified by the
number of edges, faces,
corners, and by the shape
of their faces.
Similarities between
everyday objects and 3dimensional figures
Assessment Samples – SOL/Blooms
Knowledge/Comprehension Level:
• Tell students to number 1 – 3 on a piece of paper. Ask the students to
find an example of a cylinder, a sphere, and a cube in the classroom,
and draw and label what they found.
Vocabulary
solid
rectangular solid
square pyramid
sphere
Application/Analysis Level:
• Give students solid figures and ask them to sort them into groups.
Have students explain their rationalization.
cylinder
cone
face
Synthesis/Evaluation Level:
“Making a Shed” (Balanced Assessment, Elementary Grades Assessment)
(Attachment A.2)
• Give students “Making a Shed”. Ask the students to make a
replication of the shed using solid figures. The students need to
explain why their model matches the 2-dimensional version using
geometric language.
base
edge
corner
plane figure
two-dimensional
three-dimensional
SOL: 2.20
2.22
The student will identify, describe, and sort three-dimensional (solid) concrete figures, including a cube, rectangular solid (prism), square
pyramid, sphere, cylinder, and cone, according to the number and shape of the solid’s faces, edges, and corners
The student will compare and contrast plane and solid geometric shapes (circle/sphere, square/cube, and rectangle/rectangular solid).
Communication
Reasoning and Justification
Concept 1
13
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
Grade 3
In grade band 3-5:
Develop clarity and precision in describing properties of geometric objects and classifying into categories by properties
Essential Understandings
All students will understand:
•
•
•
•
Lines are composed of line
segments and points.
Two-dimensional shapes can be
described by the number of sides
(line segments), corners (angles),
and square corners.
Three-dimensional shapes can be
described by the number of faces,
edges, corners, square corners, and
by the shape of faces.
Relationships between 2- and 3dimensional figures (circle/sphere,
square/cube, triangle/pyramid, and
rectangle/rectangular solid).
Assessment Samples – SOL/Blooms
Knowledge/Comprehension Level:
• Show students a solid figure.
Answer the following questions:
1. What is the name of this figure?
2. What shapes are its faces?
3. How many faces does it have?
4. How many edges does it have?
Application/Analysis Level:
• Give students pipe cleaners and clay.
Build a solid that has 3 rectangular faces and 2 triangular faces.
Draw their figure.
Answer these questions about their figures:
1. What is the name of this figure?
2. How many edges does your figure have?
3. How many corners does your figure have?
4. How many line segments does your figure have?
Synthesis/Evaluation Level:
• Give students pipe cleaners and clay. Ask students, “Given two rectangles,
what kinds of other shapes can you add to make a 3-dimensional figure?
Make as many different solids as you can and label them. How do you
know these are all the solids that can be made?”
Vocabulary
line
point
line segment
angle
face
corner
square corner
edge
solid
cube
cylinder
cone
rectangular solid
sphere
square pyramid
SOL: 3.18 The student will analyze two-dimensional (plane) and three-dimensional (solid) geometric figures (circle, square, rectangle, triangle, cube,
rectangular solid [prism], square pyramid, sphere, cone, and cylinder) and identify relevant properties, including the number of corners,
square corners, edges, and the number and shape of faces, using concrete models.
3.19 The student will identify and draw representations of line segments and angles, using a ruler or straightedge.
Supporting Skills and Processes: Name 2- and 3-dimensional figures based on a picture or a written description. Sort 3-dimensional figures. Identify and draw
lines, points, angles, and line segments.
Communication
Reasoning and Justification
Concept 1
14
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
Grade 4
In grade band 3-5:
Develop clarity and precision in describing properties of geometric objects and classifying into categories by properties
Essential Understandings
All students will understand:
•
•
Points, lines, line segments, and
rays are fundamental components
of noncircular geometric figures.
The shortest distance between two
points on a flat surface is a line
segment.
Assessment Samples – SOL/Blooms
Vocabulary
Knowledge/Comprehension Level:
• Which of the following letters have two parallel line segments?
A
L
N
V
point
line
line segment
Z
ray
Application/Analysis Level:
• Describe how shape A is different from shapes B and C.
angle
two-dimensional figure
plane
•
•
•
Lines in a plane either intersect or
are parallel. Perpendicularity is a
special case of intersection.
Students will identify real-world
situations that illustrate parallel,
intersecting, and perpendicular
lines.
Students will understand that 2dimensional (plane) figures are
unique in their defining properties.
parallel
intersect / intersection
perpendicular
circle
polygon
A
B
C
triangle
quadrilateral
square
Synthesis/Evaluation Level:
• Create a drawing of a parallelogram that is composed of a square
and two congruent right triangles.
Now create another drawing of a parallelogram (also composed of a
square and two congruent, right triangles) which is not similar to the
first drawing you did.
Based on what you know about these polygons, explain why both of
these drawings are parallelograms.
rectangle
parallelogram
rhombus
three-dimensional figure
sphere
cube
Communication
Reasoning and Justification
Concept 1
15
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
Grade 4
rectangular solid (prism)
cylinder
cone
face
edge
vertex
base
SOL: 4.14 The student will investigate and describe the relationships between and among points, lines, line segments, and rays.
4.15 The student will a) identify and draw representations of points, lines, line segments, rays, and angles, using a straightedge or ruler; and
b) describe the path of shortest distance between two points on a flat surface.
4.16 The student will identify and draw representations of lines that illustrate intersection, parallelism, and perpendicularity.
4.17 a) The student will analyze and compare the properties of two-dimensional (plane) geometric figures (circle, square, rectangle, triangle,
parallelogram, and rhombus) and three-dimensional (solid) geometric figures (sphere, cube, and rectangular solid [prism]).
Communication
Reasoning and Justification
Concept 1
16
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
Grade 5
In grade band 3-5:
Develop clarity and precision in describing properties of geometric objects and classifying into categories by properties
Essential Understandings
All students will understand:
•
•
•
The defining properties and
symmetry of various plane figures
are unique.
Simple plane figures can be
combined to make more
complicated figures; and,
complicated figures can be
subdivided into simple plane
figures.
Solid figures are unique in their
defining properties.
Assessment Samples – SOL/Blooms
Knowledge/Comprehension Level:
1. Choose four 3-dimensional figures from the set of geometric
solids, and list the characteristics for each figure.
Vocabulary
parallel
perpendicular
side
2. Using your ruler, draw a polygon that satisfies these conditions.
A. The polygon has only two right angles.
B. The polygon has only two congruent sides.
C. Label the right angles and the two congruent sides.
angle
vertex
diagonal
Application/Analysis Level:
• Alan says that if a figure has four sides, it must be a rectangle. Gina
does not agree. Draw two figures that show Gina is correct. Explain
how your figure supports Gina’s belief that each four-sided figure is
not necessarily a rectangle.
similar
Synthesis/Evaluation Level:
• “Mystery Rings” (Navigating Through Geometry in Grades 3-5,
NCTM) (Attachment A.5)
triangle
congruent
two-dimensional
polygon
right triangle
acute triangle
obtuse triangle
scalene triangle
isosceles triangle
equilateral triangle
quadrilateral
rectangle
square
parallelogram
kite
Communication
Reasoning and Justification
Concept 1
17
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
Grade 5
trapezoid
rhombus
three-dimensional
face
edge
pyramid
prism
cylinder
cone
cube
SOL: 5.15 The student, using two-dimensional (plane) figures (square, rectangle, triangle, parallelogram, rhombus, kite, and trapezoid) will
a) recognize, identify, describe, and analyze their properties in order to develop definitions of these figures; b) identify and explore
congruent, non-congruent, and similar figures; c) investigate and describe the results of combining and subdividing shapes.
5.16 The student will identify, compare, and analyze properties of three-dimensional (solid) geometric shapes (cylinder, cone, cube, square,
pyramid, and rectangular prism).
Supporting SOL:
5.8 The student will describe and determine the perimeter of a polygon and the area of a square, rectangle, and right triangle, given the appropriate
measures.
5.9 The student will identify and describe the diameter, radius, chord, and circumference of a circle.
5.13 The student will measure and draw right, acute, and obtuse angles and triangles, using appropriate tools.
5.14 The student will classify angles and triangles as right, acute, or obtuse.
Supporting Skills and Processes: Construct models of 2- and 3-dimensional figures using a variety of resources to include technology and manipulatives.
Communication
Reasoning and Justification
Concept 1
18
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
Grade 6
In grade band 6-8:
Investigate relationships of polygons by drawing, measuring, visualizing, comparing, transforming, and classifying geometric objects
Essential Understandings
All students will understand:
•
•
•
An angle is two rays diverging
from a common point.
Names for angles and triangles by
defining referents and
characteristics.
Plane figures are identified,
classified, and described by their
similarities, differences, and
defining properties.
•
The meaning of congruence.
•
The attributes of perpendicular
lines and a bisector.
•
•
Assessment Samples – SOL/Blooms
Vocabulary
Use “Shapes and Designs”, Connected Mathematics (Attachment A.6)
for the following:
vertex
Knowledge/Comprehension Level:
• What name could you give A, H, U, and R?
line
ray
line segment
triangle
Application/Analysis Level:
• What are the features of a polygon? An angle?
• Can you make a distinction between a trapezoid and a
parallelogram? Explain your thinking.
Synthesis/Evaluation Level:
• The figures I, L, and V can be grouped together, but X would not
belong in the group. Explain why.
• The figures U, S, and E can be grouped together, but G would not
belong in the group. Explain why.
obtuse
acute
right
isosceles
equilateral
quadrilaterals
rhombus
trapezoid
How to interpret a picture of a
solid figure from a twodimensional diagram and vice
versa.
parallelogram
square
rectangle
pentagon
The decomposition of a solid
figure into a discrete set of
surfaces.
hexagon
heptagon
octagon
polygon
congruent
Communication
Reasoning and Justification
Concept 1
19
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
Grade 6
bisect
rectangular prism
cylinder
pyramid
two-dimensional
three-dimensional
surface
edge
angle ruler
protractor
degrees
two-dimensional model (net)
SOL: 6.13 The student will a) estimate angle measures, using 45°, 90°, and 180° as referents, and use the appropriate tools to measure the given angles;
and b) measure and draw right, acute, and obtuse angles and triangles.
6.14 The student will identify, classify, and describe the characteristics of plane figures, describing their similarities, differences, and defining
properties.
6.15 The student will determine congruence of segments, angles, and polygons by direct comparison, given their attributes. Examples of
noncongruent and congruent figures will be included.
6.16 The student will construct the perpendicular bisector of a line segment and an angle bisector.
6.17 The student will sketch, construct models of, and classify solid figures (rectangular prism, cone, cylinder, and pyramid).
Communication
Reasoning and Justification
Concept 1
20
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
Grade 7
In grade band 6-8:
Investigate relationships of polygons by drawing, measuring, visualizing, comparing, transforming, and classifying geometric objects
Essential Understandings
All students will understand:
•
•
•
•
•
Quadrilaterals can be classified
according to the attributes of their
sides and/or angles.
A quadrilateral can belong to one
or more subsets of the set of
quadrilaterals and has all of the
defining attributes of the subset.
The meaning of prefixes associated
with the number of sides of a
polygon.
Assessment Samples – SOL/Blooms
Vocabulary
Knowledge/Comprehension Level:
• Which of the following characteristics must be present in order to
classify a quadrilateral as a parallelogram?
a. exactly one pair of equal adjacent sides
b. exactly one pair of parallel sides
c. a diagonal as an axis of symmetry
d. two equal adjacent angles
e. two pairs of parallel sides
quadrilateral
Application/Analysis Level:
“Geometric Shapes” (NCTM Mathematics Assessment Sampler, Grades
6-8) (Attachment A.7)
diagonal
parallelogram
rectangle
square
rhombus
trapezoid
perpendicular
polygons
Similar geometric figures have the
same shape but may have different
sizes.
Five geometric terms are listed in alphabetical order: equilateral
triangle, rhombus, right isosceles triangle, square, trapezoid.
• Which one of the geometric terms listed above most accurately
describes shape A? Explain how you know that is your answer.
• Repeat the question above for shapes B, C, and D.
How ratios and proportions can be
used to determine the length of
something that cannot be measured
directly.
Synthesis/Evaluation Level
• Luis claimed that a figure must be a square if it is a parallelogram
and has all its sides the same length. Is Luis correct? Defend your
decision with a written argument.
pentagon
hexagon
heptagon
octagon
nonagon
decagon
sides
triangle
proportion
similar
congruent
regular polygon
corresponding angles
Communication
Reasoning and Justification
Concept 1
21
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
Grade 7
corresponding sides
angle ruler
protractor
degrees
straight angle
two-dimensional model (net)
SOL: 7.9 The student will compare and contrast the following quadrilaterals: parallelogram, rectangle, square, rhombus, and trapezoid. Deductive
reasoning and inference will be used to classify quadrilaterals.
7.10 The student will identify and draw the following polygons: pentagon, hexagon, heptagon, octagon, nonagon, and decagon.
7.11 The student will determine if geometric figures, quadrilaterals and triangles, are similar and write proportions to express the relationships between
corresponding parts of similar figures.
Communication
Reasoning and Justification
Concept 1
22
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
Grade 8
In grade band 6-8:
Investigate relationships of polygons by drawing, measuring, visualizing, comparing, transforming, and classifying geometric objects
Essential Understandings
All students will understand:
•
•
Assessment Samples – SOL/Blooms
Use the following figure to answer the questions below.
vertical
supplementary
Pairs of angles are named by their
defining attributes.
A three-dimensional object can be
represented as a two-dimensional
model that represents views of the
object from different perspectives.
Vocabulary
3
1
3
1
complementary
Building Base Plan
1
front
Knowledge/Comprehension Level:
• Using your cubes, make a model of the building shown in the base
plan above.
Application/Analysis Level:
• A set of building plans includes the base outline, the front view, and
the right view. Sketch a full set of building plans on grid paper for a
new building of your design with the same base outline as the
building represented above.
angle
measure
draw
intersecting
protractors
angle ruler
straight angle
degrees
three-dimensional figures
top
side
bottom
Synthesis/Evaluation Level:
• Explain how your new building can have the same set of building
plans as the original building even though your new building has a
different base plan.
isometric
geometric solids
two-dimensional model (net)
polyhedron
polygon
SOL: 8.6 The student will verify by measuring and describe the relationships among vertical angles, supplementary angles, and complementary angles
and will measure and draw angles of less than 360°.
8.9 The student will construct a three-dimensional model, given the top, side, and/or bottom views.
Communication
Reasoning and Justification
Concept 1
23
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
Grades 9-12
In grade band 9-12:
Analyze characteristics and properties of geometric shapes and develop mathematical arguments about these shapes in applied settings in
authentic situations
Essential Understandings
All students will understand:
•
Intersecting lines form angles
with measurable
characteristics, relationships
and properties.
Assessment Samples – SOL/Blooms
A clothing designer must carefully measure segments and angles to create a
pattern. The figure below is part of a clothing design pattern. All
measurements are in centimeters.
Vocabulary
parallel
perpendicular
point
line
plane
•
•
•
•
•
The interior and exterior
angles of polygons have
measurable characteristics,
relationships, and patterns.
Two- and three-dimensional
shapes can be constructed with
a compass, straight-edge,
patty-paper, and various
technology applications.
Congruency and similarity in
polygons have real-life
application in a variety of areas
including art, architecture, and
sciences.
Congruency and similarity can
be demonstrated using logical
reasoning.
Polygons have defining
characteristics with their sides,
angles and diagonals.
angle
polygon
congruent
similar
Knowledge/Comprehension Level:
• Based on the given information, what do you know about angles B and E.?
Why?
Application/Analysis Level:
• What do you know about triangle ABC and triangle DEC. Justify by using
a two column proof.
• Find the measurement of CD. Support your answer using geometric
concepts.
• Find the measurement of FE, FC, and BC. Justify.
Synthesis/Evaluation Level:
• Determine the perimeter of triangle ABC and triangle DEC. How is this
relevant to our clothing designer?
• If the design is to be repeated twice, what is the total area of the fabric
needed?
side
diagonal
Pythagorean Theorem
triangle
arc
secant
chord
tangent
circle
diameter
radius
perimeter
area
Communication
Reasoning and Justification
Concept 1
24
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
•
•
The Pythagorean Theorem is
essential for solving triangle
problems.
Grades 9-12
The regular tetrahedron shown in figure(a) has grown a regular tetrahedron
(with a side half as long) on each of its faces in figure (b). Figure (c) shows
the new solid (b) in the process of growing tetrahedra on its faces in the same
way.
Many relationships exist
between and among angles,
arcs, secants, chords, and
tangents of a circle.
•
Three-dimensional figures are
a part of everyday life.
•
Modeling is an overall part of
design for products and
structures.
•
Perimeter, area, surface area
and volume are all measurable
characteristics of 2- and 3dimensional shapes.
•
Formulas can be derived to
generalize perimeter, area,
surface area and volume.
Communication
surface area
volume
quadrilateral
iterations
tetrahedron/tetrahedra
Knowledge/Comprehension Level:
•
Complete the tables below.
Reasoning and Justification
Concept 1
25
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed
and developed using logical and spatial reasoning.
Geometry and Measurement Strand
Grades 9-12
Application/Analysis Level:
• Determine the total surface area of the 20th iteration?
• Determine the total surface area of the nth iteration as
•
•
•
n → ∞?
th
Determine the total volume of the 20 iteration?
Determine the total volume of the nth iteration as n → ∞ ?
Compare the total surface area and the total volume as n →
∞.
Synthesis/Evaluation Level:
• Create a similar model using a cube, and then construct a table similar to
the tetrahedron.
SOL: G.5 The student will a) investigate and identify congruence and similarity relationships between triangles; and b) prove two triangles are
congruent or similar, given information in the form of a figure or statement, using algebraic and coordinate as well as deductive proofs.
G.8: The student will a) investigate and identify properties of quadrilaterals involving opposite sides and angles, consecutive sides and angles, and
diagonals; b) prove these properties of quadrilaterals, using algebraic and coordinate methods as well as deductive reasoning; and c) use
properties of quadrilaterals to solve practical problems.
Supporting SOL:
G.1 The student will construct and judge the validity of a logical argument consisting of a set of premises and a conclusion. This will include a) identifying
the converse, inverse, and contrapositive of a conditional statement; b) translating a short verbal argument into symbolic form; c) using Venn
diagrams to represent set relationships; and d) using deductive reasoning, including the law of syllogism.
G.4 The student will use the relationships between angles formed by two lines cut by a transversal to determine if two lines are parallel and verify, using
algebraic and coordinate methods as well as deductive proofs.
G.6 The student, given information concerning the lengths of sides and/or measures of angles, will apply the triangle inequality properties to
determine whether a triangle exists and to order sides and angles. These concepts will be considered in the context of practical situations.
Supporting Skills and Processes: Knowledge and understanding of the properties and relationships among basic two- and three-dimensional shapes; use of
deductive reasoning to establish or refute conjectures.
Communication
Reasoning and Justification
Concept 1
26
Geometry and Measurement Strand
Grade 1
Assessment Attachment A.1
Shape Cards
Cut out the shape cards. Sort them into two or more groups and glue them onto a piece of
paper. Label each group.
Rubric:
Advanced
Proficient
Nearing
Proficient
Needs
Improvement
•
•
•
•
•
•
Student sorts into 3 or more categories
Student sorts by attributes such as number of sides, straight/curved sides, and number of corn
Student is able to flexibly regroup the shapes
Student uses all shapes and sorts them into distinct categories
Student accurately labels each category
Student may be unable to use all 9 shapes but is able to sort some shapes into distinct groups
•
•
•
Student is unable to use all 9 shapes
Student is unable to sort into distinct groups
Student is unable to label or identify the attribute
Communication
Reasoning and Justification
Concept 1
27
Geometry and Measurement Strand
Grade 2
Making a Shed
Communication
Assessment Attachment A.2
Reasoning and Justification
Concept 1
28
Geometry and Measurement Strand
Grade 5
Communication
Assessment Attachment A.5
Reasoning and Justification
Concept 1
29
Geometry and Measurement Strand
Grade 6
Shapes and Designs
Assessment Attachment A.6
Knowledge/Comprehension Level
• What name could you give A, H, U, and R?
Application/Analysis Level
• What are the features of a polygon? An angle?
• Can you make a distinction between a trapezoid and a parallelogram? Explain your thinking.
Synthesis/Evaluation Level
• The figures I, L, and V can be grouped together, but X would not belong in the group. Explain why.
• The figures U, S, and E can be grouped together, but G would not belong in the group. Explain why.
Communication
Reasoning and Justification
Concept 1
30
Geometry and Measurement Strand
Grade 7
Geometric Shapes
Communication
Assessment Attachment A.7
Reasoning and Justification
Concept 1
31
Appendix
Mathematics Vertical Team, May, 2007
Appendix Guide
Lifelong-Learner Standards
………………………………….……………….…………………..
Assessment in the Mathematics Curriculum Matrix
…………………………………………
ii
………………………………………………..
ii
………………………………………………………….
iii
Bloom’s Taxonomy of the Cognitive Domain
The Content Strands of Mathematics
i
Mapping the Mathematics Enduring Understandings to the Mathematics Content Strands
Number and Operations …………………………………………….. ……………………
Data Analysis and Probability ………………………………………. ……………………
Geometry and Measurement …………………………………………………………….
Patterns and Algebra ………………………………………………....……………………
v
vi
ix
xi
Bibliography
xiii
……………………………………………………………………………………..….
Mathematics Vertical Team, May, 2007
Lifelong-Learner Standards
The Division has identified 12 Lifelong-Learner Standards that set expectations for how students develop a wide variety
of knowledge, understanding, and skills. These standards articulate the necessary components of lifelong learning that
allow all students to succeed as members of a global community and in a global economy. The Lifelong-Learner
Standards are overarching process-based standards, not discrete fact-based standards that can be addressed in a
single lesson or even a single unit. These standards demand attention over time and across all disciplines (FQL, 2006).
Lifelong-Learner Standards
1. Plan and conduct research;
2. Gather, organize, and analyze data, evaluate processes and products, and draw conclusions;
3. Think analytically, critically, and creatively to pursue new ideas, acquire new knowledge, and make decisions;
4. Understand and apply principles of logic and reasoning; develop, evaluate, and defend arguments;
5. Seek, recognize and understand systems, patterns, themes, and interactions;
6. Apply and adapt a variety of appropriate strategies to solve new and increasingly complex problems;
7. Acquire and use precise language to clearly communicate ideas, knowledge, and processes;
8. Explore and express ideas and opinions using multiple media, the arts, and technology;
9. Demonstrate ethical behavior and respect for diversity through daily actions and decision making;
10. Participate fully in civic life, and act on democratic ideals within the context of community and global
interdependence;
11. Understand and follow a physically active lifestyle that promotes good health and wellness; and,
12. Apply habits of mind and metacognitive strategies to plan, monitor, and evaluate one’s own work.
Mathematics Vertical Team, May, 2007
i
Assessment in the Mathematics Curriculum Matrix
The assessment section of the Mathematics Curriculum Matrix provides samples within the hierarchy of Bloom’s
Taxonomy of the Cognitive Domain to provide teachers with a better understanding of the different levels of challenge
required to meet the intent of a particular standard. These assessment samples are intended as examples to help a
teacher focus on the level of questioning and performance needed for a student to gain deep understanding of a
particular standard. There is not a specific assessment provided for all six levels of Bloom’s Taxonomy, but the
assessment examples have been placed in three tiers of the domain to represent low level (Knowledge and
Comprehension), middle level (Application and Analysis), and high level (Synthesis and Evaluation)cognitive demands.
Bloom’s Taxonomy of the Cognitive Domain
Knowledge:
Students recall information; students exhibit memory of previously learned material by recalling facts, terms, basic
concepts, and answers.
Comprehension:
Students recognize what they know in context; students identify relationships between pieces of information;
students demonstrate understanding of facts and ideas by organizing, comparing, translating, interpreting, giving
descriptions, and stating main ideas.
Application:
Students use what they know and comprehend in the performance of a skill; students solve problems applied to new
situations by using acquired knowledge, facts, techniques, and rules in new ways.
Analysis:
Students draw conclusions from new data, making interpretations based on familiar patterns in what they know and
comprehend; students examine and break information into parts by identifying motives or causes; students make
inferences and find evidence to support generalizations.
Synthesis:
Students create a new work that demonstrates their ability to apply their knowledge, comprehension, and analysis of
information in a student-generated product; students compile information together in a different way by combining
elements in a new pattern or proposing alternative solutions based on the application of knowledge and
understanding.
Evaluate:
Students develop, argue and defend opinions based on what they know and comprehend after making an analysis;
students present and defend opinions by making judgments about information; students validate ideas or quality of
work based on a set of criteria.
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The Content Strands of Mathematics*
*Most of the ideas and definitions in the “Content Strands of Mathematics” portion of this document have been quoted or paraphrased from
The Principles and Standards for School Mathematics (NCTM, 2000).
Number and Operations
The Number and Operations Strand involves deep and fundamental understanding of, and proficiency with counting,
numbers, and arithmetic, as well as an understanding of number systems and their structures. The concepts and
algorithms of arithmetic are part of number and operations, as are the properties and characteristics of the classes of
numbers that form the beginnings of number theory. Central to these ideas is the development of number sense,
understanding numbers, ways of representing numbers, relationships among numbers, and number systems.
Data Analysis and Probability
Students should learn how to collect data, organize their own or other’s data, and display the data in graphs and
charts that will be useful in answering their questions. This strand also includes learning some methods for analyzing
data and making inferences and conclusions from data. The basic concepts and applications of probability are also
addressed, with an emphasis on the way that probability and data analysis are related. To understand the
fundamentals of statistical ideas, students must work directly with data. This emphasis allows students to meet new
ideas and procedures as they progress through the grades, and make a number of important connections among
ideas and procedures from number, algebra, measurement, and geometry. Work in data analysis and probability
offers a natural way for students to connect mathematics with other content areas and with experiences in their daily
lives.
Geometry and Measurement
Through the study of geometry, students will learn about geometric shapes and structures and how to analyze their
characteristics and relationships. Spatial visualization-building and manipulating mental representations of two- and
three-dimensional objects and perceiving an object from the different perspectives is an important aspect of
geometric thinking. Geometry is a natural place for the development of students’ reasoning and justification skills,
culminating in the work with proof in the secondary grades. Geometric modeling and spatial reasoning offer ways to
interpret and describe physical environments and can be important tools in problem solving.
Measurement is the assignment of a numerical value to an attribute of an object. Understanding what a measurable
attribute is and becoming familiar with the units and processes that are used in measuring attributes is a major
emphasis in this strand. Students should become proficient in using measurement tools, techniques, and formulas in a
range of situations involving measuring and comparing. At more sophisticated levels, measurement involves assigning
Mathematics Vertical Team, May, 2007
iii
a number to a characteristic of a situation, as is done by the consumer price index. The study of measurement offers
the opportunity for learning and applying other mathematics, including number operations, geometric ideas, statistical
concepts, and notions of function. It highlights connections within mathematics and between mathematics and other
content areas.
Patterns and Algebra
Algebra has its historical roots in the study of general methods for solving equations. The Algebra Strand emphasizes
relationships among quantities, including functions, ways of representing mathematical relationships, and the analysis
of change. Functional relationships can be expressed by using symbolic notation, which allows complex
mathematical ideas to be expressed succinctly and allows change to be analyzed efficiently. Understanding change
is fundamental to understanding functions and to understanding many ideas in our world. Today, the methods and
ideas of algebra support mathematical work in many areas, for example, distribution and communication networks,
laws of physics, population models, and statistical results can all be represented in the symbolic language of algebra.
The K-12 Mathematics Enduring Understandings as they connect to each of the Content Strands and the Grade Level
Expectations throughout the Mathematics Curriculum Matrix are outlined below.
Mathematics Vertical Team, May, 2007
iv
Number and Operations
Number and Operations, Concept 1
K-12 Enduring Understanding: Relationships among numbers and number systems form the foundations of number
sense and mathematical communication.
Grade Band Expectations
In grade band K-2:
Develop the ability to deal with numbers mentally and to use number sense to reason with numbers in complex ways
In grade band 3-5:
Develop strategies for judging the relative size of numbers, identify classes of numbers and examine their properties
In grade band 6-8:
Use strategies to build fluency and extend knowledge of the number system
In grade band 9-12:
Develop a full understanding of the system of real numbers and use their understanding to explore new systems
Number and Operations, Concept 2
K-12 Enduring Understanding: Patterns and relationships among operations are essential to making estimates and computing
fluently.
Grade Band Expectations:
In grade band K-2:
Understand the meaning of, and relationship between addition and subtraction to compute fluently
In grade band 3-5:
Understand the meaning of, and relationships between addition and subtraction, and multiplication and division
In grade band 6-8 :
Investigate the properties and obtain computational fluency within the real number system
In grade band 9-12 :
Reason whether a problem calls for an estimate or exact answer, extend understanding of operations to number systems that are
new to students, and select a suitable method of computation
Mathematics Vertical Team, May, 2007
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Data Analysis and Probability
Data Analysis and Probability, Concept 1
K-12 Enduring Understanding: Data can be collected, organized, and displayed in purposeful ways.
Grade Band Expectations
In grade band K-2:
Pose questions, gather, and use various methods to represent data about themselves and their surroundings
In grade band 3-5:
Collect, represent, and investigate data and how data collection methods affect the nature of the data set
In grade band 6-8:
Formulate questions, design studies, collect relevant data, and create and use appropriate graphical representations of data
In grade band 9-12:
Know the characteristics and differences of well designed studies and the meaning and types of inferences that can be drawn
from measurement data
Data Analysis and Probability, Concept 2
K-12 Enduring Understanding: Various statistical methods can be used to observe, analyze, predict, and make
inferences about data.
Grade Band Expectations:
In grade band K-2:
Describe parts of the data and a set of data to determine what the data show
In grade band 3-5:
Describe and compare related data sets, their distributions, and measures of center
In grade band 6-8:
Discuss and understand the correspondence between data sets and their graphic representations, and find, use, and interpret
their measures of central tendency
In grade band 9-12:
Graphically display univariate and bivariate data and understand the implications of its characteristics
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Data Analysis and Probability, Concept 3
K-12 Enduring Understanding: : Mathematical models are used to predict and make inferences about data.
Grade Band Expectations:
In grade band K-2:
Discuss events related to students’ experiences as likely or unlikely
In grade band 3-5:
Propose and justify conclusions and predictions that are based on data and design studies to further investigate the conclusions
or predictions
In grade band 6-8:
Use observations about differences between two or more samples to make conjectures and develop new questions about the
populations from which they were taken
In grade band 9-12:
Use simulations to explore the variability of sample statistics from a known population and construct sampling distributions
Data Analysis and Probability, Concept 4
K-12 Enduring Understanding: Probability and data analysis can be used to make predictions
Grade Band Expectations:
In grade band K-2:
Develop and evaluate inferences and predictions that are based on data
In grade band 3-5:
Describe and discuss the degree of likelihood of events and predict the probability of outcomes of simple experiments
In grade band 6-8:
Use a basic understanding of probability to make and test conjectures about the results of experiments and simulations
In grade band 9-12:
Understand the concepts of sample space and probability distribution, use simulations, and compute and interpret expected
value
Mathematics Vertical Team, May, 2007
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Geometry and Measurement
Geometry and Measurement, Concept 1
K-12 Enduring Understanding: Characteristics, properties, and mathematical arguments about geometric relationships
can be analyzed and developed using logical and spatial reasoning.
Grade Band Expectations:
In grade band K-2:
Represent and compare 2- and 3-dimensional shapes through drawings, block construction, dramatizations, and words
In grade band 3-5:
Develop clarity and precision in describing properties of geometric objects and classifying into categories by properties
In grade band 6-8:
Investigate relationships of polygons by drawing, measuring, visualizing, comparing, transforming and classifying geometric
objects
In grade band 9-12:
Analyze characteristics and properties of geometric shapes and develop mathematical arguments about these shapes in
applied settings in authentic situations
Geometry and Measurement, Concept 2
K-12 Enduring Understanding: Spatial relationships can be described using coordinate geometry and other
representational systems.
Grade Band Expectations:
In grade band K-2:
Describe, name, and interpret relative positions, directions, and distances in space
In grade band 3-5:
Use a coordinate system to aid in describing location and movement in all four quadrants of the Cartesian plane
In grade band 6-8:
Explore and represent the properties of shapes using coordinate geometry formulas
In grade band 9-12:
Investigate and use Cartesian coordinates and other coordinate systems to analyze geometric situations
Mathematics Vertical Team, May, 2007
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Geometry and Measurement, Concept 3
K-12 Enduring Understanding: Transformations, symmetry, and spatial reasoning can be used to analyze and model
mathematical situations.
Grade Band Expectations:
In grade band K-2:
Develop an understanding of transformations and symmetry with shapes
In grade band 3-5:
Identify symmetry and congruency, and describe the results of various transformations of plane figures
In grade band 6-8:
Create and quantify the results of various transformations, including dilation
In grade band 9-12:
a. Understand and represent translations and reflections, rotations, and dilations of objects in the plane by using sketches,
coordinates, vectors, function notation, and matrices
b. Use various representations to help understand the effects of simple transformations and their compositions
Geometry and Measurement, Concept 4
K-12 Enduring Understanding: Attributes of objects can be measured using processes and quantified units, using
appropriate techniques, tools, and formulas.
Grade Band Expectations:
In grade band K-2:
Understand attributes of measurement by directly comparing objects, and selecting an appropriate unit and tools
In grade band 3-5:
Develop and deepen understanding of what it means to measure an object: identify an attribute to be measured, choose an
appropriate unit, and compare that unit to the object being measured
In grade band 6-8:
Become proficient in selecting the appropriate size and type of unit for a given measurement situation, including length, area
and volume
In grade band 9-12:
Apply appropriate technology, and understand the limitations; make reasonable estimates and accurate predictions about
measurement
Mathematics Vertical Team, May, 2007
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Patterns and Algebra
Patterns and Algebra, Concept 1
K-12 Enduring Understanding: Patterns, relations, and functions can be recognized and understood mathematically.
Grade Band Expectations:
In grade band K-2:
Recognize, describe, and extend patterns, and sort and classify objects
In grade band 3-5:
Investigate numerical and geometric patterns and express them mathematically in words and symbols
In grade band 6-8:
Investigate patterns that arise when there is a rate of change
In grade band 9-12:
Create and use tabular, symbolic, graphical, and verbal representations and analyze and understand patterns, relations, and
functions
Patterns and Algebra, Concept 2
K-12 Enduring Understanding: Situations and structures can be represented, modeled, and analyzed using algebraic
symbols.
Grade Band Expectations:
In grade band K-2:
Illustrate and model general principles and properties of operations
In grade band 3-5:
Investigate, describe, and represent various rates of change
In grade band 6-8:
Solve problems and understand that relationships among quantities can often be expressed symbolically and represented in
more than one way
In grade band 9-12:
Fluently manipulate algebraic expressions by combining them and re-expressing them in alternative forms, including identification
and selection of relevant features in applications of real-world situations
Mathematics Vertical Team, May, 2007
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Patterns and Algebra, Concept 3
K-12 Enduring Understanding: Change, in various contexts, both quantitative and qualitative, can be identified and
analyzed.
Grade Band Expectations:
In grade band K-2:
Describe change both qualitatively and quantitatively
In grade band 3-5:
Propose and justify conclusions and predictions that are based on data and design studies to further investigate the conclusions
or predictions
In grade band 6-8:
Use graphs to analyze the nature of changes in quantities in linear relationships
In grade band 9-12:
Approximate and interpret rates of change from graphical and numerical data
Mathematics Vertical Team, May, 2007
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Russell, S. J.; Clements, D.; & Sarama, J. (1998). Investigations in Number, Data, and Space, Quilt squares and block towns.
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