Jeremy Bletterman, Kim Davis & Jennifer Mozet
ED 7204T – Advanced Pedagogy & Curriculum II
Fall 2011 – Module 1, Assignment 4
The National Council of Teachers of Mathematics (NCTM) is a global authority
in mathematics education, providing support for teachers through their core principles,
including curriculum, instruction and assessment, equity, advocacy, professional
development, and research, in order to ensure that all students are receiving the highest
quality mathematics teaching and learning. The organization’s overarching goal is to
enable a global environment wherein all teachers and students are enthusiastic about
mathematics, recognize its value, and are empowered by the possibilities mathematics
presents.
In pursuing this goal, NCTM has developed a set of standards for school
mathematics – divided into Process Standards and Content Standards – that provide
guidelines for teachers to use in planning, implementing, and assessing instructional
programs for all students from Pre-K through grade 12. These standards inform teachers
as to the mathematical understanding, knowledge, and skills that students should acquire
at each grade level.
The Process Standards, subdivided into 5 strands, including Problem Solving,
Reasoning and Proof, Communication, Connections, and Representation, apply to all
grade levels and outline the strategies all students should possess and be able to apply.
For example, instructional programs that address the Problem Solving strand should
enable all students from Pre-K through grade 12 to “build new mathematical knowledge
through problem solving; solve problems that arise in mathematics and in other contexts;
apply and adapt a variety of appropriate strategies to solve problems; and monitor and
reflect on the process of mathematical problem solving.” Similarly, instructional
programs that address the Reasoning and Proof strand should enable all students from
Pre-K through grade 12 to “recognize reasoning and proof as fundamental aspects of
mathematics; make and investigate mathematical conjectures; develop and evaluate
mathematical arguments and proofs; and select and use various types of reasoning and
methods of proof.” Comparably, instructional programs that address the Communication
strand should enable all students from Pre-K through grade 12 to “organize and
consolidate their mathematical thinking through communication; communicate their
mathematical thinking coherently and clearly to peers, teachers, and others; analyze and
evaluate the mathematical thinking and strategies of others; and use the language of
mathematics to express mathematical ideas precisely.” Instructional programs that
address the Connections strand should enable all students from Pre-K through grade 12 to
“recognize and use connections among mathematical ideas; understand how
mathematical ideas interconnect and build on one another to produce a coherent whole;
and recognize and apply mathematics in contexts outside of mathematics.” Finally,
instructional programs that address the Representation strand should enable all students
from Pre-K through grade 12 to “create and use representations to organize, record, and
communicate mathematical ideas; select, apply, and translate among mathematical
representations to solve problems; and use representations to model and interpret
physical, social, and mathematical phenomena.”
The Content Standards are also separated into five strands, including Number and
Operations, Algebra, Geometry, Measurement, and Data and Probability. Unlike the
Process Standards, which govern strategies, the Content Standards outline the specific
content area knowledge students should possess at each grade level from Pre-K through
grade 12. For example, within the Number and Operations strand, NCTM dictates that
instructional programs should enable all students from Pre-K through grade 12 to
“understand numbers, ways of representing numbers, relationships among numbers, and
number systems.” Within grades 3-5, specifically, students should “understand the placevalue structure of the base-ten number system and be able to represent and compare
whole numbers and decimals; recognize equivalent representations for the same number
and generate them by decomposing and composing numbers; develop understanding of
fractions as parts of unit wholes, as parts of a collection, as locations on number lines,
and as divisions of whole numbers; use models, benchmarks, and equivalent forms to
judge the size of fractions; recognize and generate equivalent forms of commonly used
fractions, decimals, and percents; explore numbers less than 0 by extending the number
line and through familiar applications; and describe classes of numbers according to
characteristics such as the nature of their factors.” The Number and Operations strand
goes on to state that instructional programs should enable all students from Pre-K through
grade 12 to “understand meanings of operations and how they relate to one another.”
Within grades 3-5, specifically, students should “understand various meanings of
multiplication and division; understand the effects of multiplying and dividing whole
numbers; identify and use relationships between operations, such as division as the
inverse of multiplication, to solve problems; and understand and use properties of
operations, such as the distributivity of multiplication over addition.” The Number and
Operation strand concludes by asserting that instructional programs should enable all
students from Pre-K through grade 12 to “compute fluently and make reasonable
estimates.” Within grades 3-5, specifically, students should “develop fluency with basic
number combinations for multiplication and division and use these combinations to
mentally compute related problems; develop fluency in adding, subtracting, multiplying,
and dividing whole numbers; develop and use strategies to estimate the results of wholenumber computations and judge the reasonableness of such results; develop and use
strategies to estimate computations involving fractions and decimals in situations relevant
to students' experience; use visual models, benchmarks, and equivalent forms to add and
subtract commonly used fractions and decimals; and select appropriate methods and tools
for computing with whole numbers from among mental computation, estimation,
calculators, and paper and pencil according to the context and nature of the computation
and use the selected method or tools.”
With regard to the Algebra strand, NCTM dictates that instructional programs
should enable all students from Pre-K through grade 12 to “understand patterns, relations,
and functions.” Within grades 3-5, specifically, students should “describe, extend, and
make generalizations about geometric and numeric patterns; and represent and analyze
patterns and functions, using words, tables, and graphs.” The strand goes on to state that
instructional programs should enable all students from Pre-K through grade 12 to
“represent and analyze mathematical situations and structures using algebraic symbols.”
Within grades 3-5, specifically, students should “identify such properties as
commutativity, associativity, and distributivity and use them to compute with whole
numbers; represent the idea of a variable as an unknown quantity using a letter or a
symbol; and express mathematical relationships using equations.” Progressing, the strand
states that instructional programs should enable all students from Pre-K through grade 12
to “use mathematical models to represent and understand quantitative relationships.”
Within grades 3-5, specifically, students should “model problem situations with objects
and use representations such as graphs, tables, and equations to draw conclusions.”
Finally, the strand states that instructional programs should enable all students from PreK through grade 12 to “analyze change in various contexts.” Specifically, within grades
3-5, students should “investigate how a change in one variable relates to a change in a
second variable; and identify and describe situations with constant or varying rates of
change and compare them.”
Within the Geometry strand, NCTM dictates that instructional programs should
enable all students from Pre-K through grade 12 to “analyze characteristics and properties
of two- and three-dimensional geometric shapes and develop mathematical arguments
about geometric relationships.” Specifically, within grades 3-5, students should
“identify, compare, and analyze attributes of two- and three-dimensional shapes and
develop vocabulary to describe the attributes; classify two- and three-dimensional shapes
according to their properties and develop definitions of classes of shapes such as triangles
and pyramids; investigate, describe, and reason about the results of subdividing,
combining, and transforming shapes; explore congruence and similarity; and make and
test conjectures about geometric properties and relationships and develop logical
arguments to justify conclusions.” Moving on, the strand states that instructional
programs should enable all students from Pre-K through grade 12 to “specify locations
and describe spatial relationships using coordinate geometry and other representational
systems.” Specifically, within grades 3-5, students should “describe location and
movement using common language and geometric vocabulary; make and use coordinate
systems to specify locations and to describe paths; and find the distance between points
along horizontal and vertical lines of a coordinate system.” Next, the strand states that
instructional programs should enable all students from Pre-K through grade 12 to “apply
transformations and use symmetry to analyze mathematical situations.” Specifically,
within grades 3-5, students should “predict and describe the results of sliding, flipping,
and turning two-dimensional shapes; describe a motion or a series of motions that will
show that two shapes are congruent; and identify and describe line and rotational
symmetry in two- and three-dimensional shapes and designs.” Finally, the strand dictates
that instructional programs should enable all students from Pre-K through grade 12 to
“use visualization, spatial reasoning, and geometric modeling to solve problems.”
Specifically, within grades 3-5, students should “build and draw geometric objects; create
and describe mental images of objects, patterns, and paths; identify and build a threedimensional object from two-dimensional representations of that object; identify and
draw a two-dimensional representation of a three-dimensional object; use geometric
models to solve problems in other areas of mathematics, such as number and
measurement; and recognize geometric ideas and relationships and apply them to other
disciplines and to problems that arise in the classroom or in everyday life.”
The fourth strand, Measurement, dictates that instructional programs should
enable all students from Pre-K through grade 12 to “understand measurable attributes of
objects and the units, systems, and processes of measurement.” Specifically, within
grades 3-5, students should “understand such attributes as length, area, weight, volume,
and size of angle and select the appropriate type of unit for measuring each attribute;
understand the need for measuring with standard units and become familiar with standard
units in the customary and metric systems; carry out simple unit conversions, such as
from centimeters to meters, within a system of measurement; understand that
measurements are approximations and how differences in units affect precision; and
explore what happens to measurements of a two-dimensional shape such as its perimeter
and area when the shape is changed in some way.” The strand goes on to dictate that
instructional programs should enable all students from Pre-K through grade 12 to “apply
appropriate techniques, tools, and formulas to determine measurements.” Specifically,
within grades 3-5, students should “develop strategies for estimating the perimeters,
areas, and volumes of irregular shapes; select and apply appropriate standard units and
tools to measure length, area, volume, weight, time, temperature, and the size of angles;
select and use benchmarks to estimate measurements; develop, understand, and use
formulas to find the area of rectangles and related triangles and parallelograms; and
develop strategies to determine the surface areas and volumes of rectangular solids.”
The fifth and final strand, Data and Probability, states that instructional programs
should enable all students from Pre-K through grade 12 to “formulate questions that can
be addressed with data and collect, organize, and display relevant data to answer them.”
Specifically, within grades 3-5, students should “design investigations to address a
question and consider how data-collection methods affect the nature of the data set;
collect data using observations, surveys, and experiments; represent data using tables and
graphs such as line plots, bar graphs, and line graphs; and recognize the differences in
representing categorical and numerical data.” Moving on, the strand states that
instructional programs should enable all students from Pre-K through grade 12 to “select
and use appropriate statistical methods to analyze data.” Specifically, within grades 3-5,
students should “describe the shape and important features of a set of data and compare
related data sets, with an emphasis on how the data are distributed; use measures of
center, focusing on the median, and understand what each does and does not indicate
about the data set; and compare different representations of the same data and evaluate
how well each representation shows important aspects of the data.” Next, the strand
professes that instructional programs should enable all students from Pre-K through grade
12 to “develop and evaluate inferences and predictions that are based on data.”
Specifically, within grades 3-5, students should “propose and justify conclusions and
predictions that are based on data and design studies to further investigate the conclusions
or predictions.” And finally, the strand states that instructional programs should enable
all students from Pre-K through grade 12 to “understand and apply basic concepts of
probability.” Specifically, within grades 3-5, students should “describe events as likely
or unlikely and discuss the degree of likelihood using such words as certain, equally
likely, and impossible; predict the probability of outcomes of simple experiments and test
the predictions; and understand that the measure of the likelihood of an event can be
represented by a number from 0 to 1.”
NCTM believes that in order to effectively teach mathematics, educators must
acknowledge that every student deserves to be challenged to achieve the skills necessary
to become productive citizens. To accomplish this, NCTM expects that teachers be
highly qualified, possess sound knowledge of mathematics, and understand how children
with diverse needs and learning styles learn mathematics. Furthermore, teachers must
strive to maintain the highest level of expectations for themselves and their students.
Beyond the individual educator, school districts must develop a complete and
coherent mathematics curriculum that focuses, at every grade level, on the Process and
Content Standards set forth by NCTM. It is then the teacher’s responsibility to
understand how the content they teach fits within the development of these standards.
The teacher guides the learning process and they should employ a variety of instructional
approaches that directly reflect the mathematics content being taught as well as the
diverse needs of the individual students in their class. NCTM believes that in order to
maximize mathematics learning, educators should focus on mathematical thinking and
reasoning. This is clearly enhanced when connections are made to students’ everyday
experiences as well as to other content areas.
With respect to the ever-increasing impact of technology on our world, NCTM
feels that students should be able to use calculators and computers to investigate concepts
and increase their mathematical understanding. Teachers should also recognize that
students use differing strategies to solve problems. These strategies should be used to
help students develop a deeper understanding of mathematics content.
Finally, assessment of mathematics understanding must be congruent with the
content being taught and should consist of various sources of feedback, including but not
limited to testing, informal observations, performance tasks, and mathematical
investigations. In order to improve as a teacher of mathematics, educators should engage
in critical self-reflection, as well as stay abreast of ongoing research, in order to evaluate
the effectiveness of mathematics curriculum and their own mathematics instruction.
SAMPLE LESSON
Process Standards:

Problem Solving
o Build new mathematical knowledge through problem solving.
o Solve problems that arise in mathematics and in other contexts.
o Apply and adapt a variety of appropriate strategies to solve problems.
o Monitor and reflect on the process of mathematical problem solving.

Reasoning and Proof
o Recognize reasoning and proof as fundamental aspects of mathematics.
o Select and use various types of reasoning and methods of proof.

Communication
o Organize and consolidate their mathematical thinking through
communication.
o Communicate their mathematical thinking coherently and clearly to peers,
teachers, and others.
o Analyze and evaluate the mathematical thinking and strategies of others.
o Use the language of mathematics to express mathematical ideas precisely.

Connections
o Recognize and use connections among mathematical ideas.
o Understand how mathematical ideas interconnect and build on one another to
produce a coherent whole.

Representation
o Create and use representations to organize, record, and communicate
mathematical ideas.
Content Standards:
Geometry:
Analyze characteristics and properties of two- and three-dimensional geometric shapes
and develop mathematical arguments about geometric relationships.
Specifically, within grades 3-5:
 Identify, compare, and analyze attributes of two- and three-dimensional shapes and
develop vocabulary to describe the attributes.
 Classify two- and three-dimensional shapes according to their properties and develop
definitions of classes of shapes.
+++
Summary: Students will create, classify and sort quadrilaterals.
Grade: 4th
Main Curriculum Tie: Identify and describe attributes of two-dimensional geometric
shapes (grade 4)
Materials:
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Geoboards and geobands
Geodot Paper
Various quadrilateral shapes
Quadrilateral Family Tree
Quadrilateral Pieces
Quadrilateral Venn Diagram
Yarn or string
Background For Teachers:
A common activity involving geometry is for students to recognize and name various
polygons. Their experiences with four-sided polygons may lack depth or may have some
misconceptions. For example, students are often taught to categorize rectangles and squares
separately. Typically, a polygon with four equal sides and four equal angles is referred to as
a square; whereas, a polygon with four equal angles but one pair of long sides and one pair
of short sides is referred to as a rectangle. We hear students refer to rectangles as being
“long” or “tall.” Their system for differentiating between squares and rectangles is based
on narrow experiences with a few specific examples.
These constructions may cause confusion later as students learn that squares also fit the
description of rectangles. This new information does not fit logically to what they have
already learned, and it does not allow for growth in understanding that a square is a more
specific classification of a rectangle; just as a rectangle is a more specific classification of a
parallelogram; and that a parallelogram is a specific classification of a quadrilateral. These
shapes all fit in the quadrilateral “family.”
To aid understanding, teach quadrilaterals as a whole. Define quadrilaterals as a four-sided
figure and give students the opportunity to create a variety of quadrilaterals. They look for
similarities and differences and sort them into several different categories according to their
attributes. The sorting activity offers insight into the mathematical hierarchy used in
classifying quadrilaterals. It will become clear that every quadrilateral falls into three
categories:
 Those with two pairs of parallel sides,
 Those with only one pair of parallel sides, and
 Those with no parallel sides.
This activity will set the stage for students to understand that many types of quadrilaterals
exist and that these shapes have some elements in common.
Intended Learning Outcomes:
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Demonstrate a positive learning attitude toward mathematics
Reason mathematically
Communicate mathematically
Make mathematical connections
Instructional Procedures:
Invitation to Learn
Provide each student with a geoboard and geoband. Ask them to create several four-sided
polygons then choose their most unique quadrilateral to share with their group.
Instructional Procedures
1. Ask the students to compare their quadrilateral with those made by other members in
their group. Are all quadrilaterals different? If not, agree on how to make them look
different. Record quadrilateral on Geodot Paper and cut shape out for display.
2. Invite each group to post their quadrilaterals in one of three columns:
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Those with one pair of parallel sides,
Those with two pairs of parallel sides, and
Those with no parallel sides.
Give students time to determine if all the quadrilaterals are in their appropriate
columns. Discuss congruent and similar shapes and remove any duplicates.
3. Identify the columns with the appropriate headings: trapezoids (one pair of parallel
sides), parallelograms (two pair of parallel sides), and trapeziums (no parallel sides).
4. Use the Quadrilateral Family Tree handout to discuss the properties, attributes, and
characteristics, as well as the interconnective and hierarchical commonalities and
differences, between and among quadrilateral shapes.
i. Have the students look at the relationship between squares and rectangles. What
are the characteristics of each? Is a square a rectangle? (Yes, it has four equal
angles.) Are all rectangles squares? (No, many rectangles do not have four equal
angles and four equal sides.)
ii. Have the students look at the relationship between squares and rhombuses. What
are the characteristics of each? Is a square a rhombus? (Yes, it has four equal
sides.) Are all rhombuses squares? (No, many rhombuses do not have four equal
sides and four equal angles.)
iii. A Venn diagram is a good visual aid to illustrate that a square is both a rectangle
and a rhombus.
5. Further explore the relationships between quadrilaterals by having the students work
with roping quadrilaterals. Provide each pair of students a set Quadrilateral Pieces and
two or three pieces of string to make a Quadrilateral Venn Diagram. Ask them to place
the appropriate quadrilateral pieces in each ring according to the following labels:
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Ring 1 (Left side): At least one pair of parallel sides
Ring 2 (Right side) No sides parallel
Ask students to justify their placement of different pieces. What do all the shapes in one
ring have in common? How might the shapes in one ring be different? (Some shapes in
Ring 1 are trapezoids, and some are parallelograms.) What different label would
eliminate one or more of the shapes from a ring? (Only one pair of parallel sides.) If we
drew a giant circle around everything, including any shapes that are outside the rings,
what might the label for this new ring be? (Quadrilaterals) Try further explorations
using the following labels: 
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Ring 1 (Inner ring): All sides of equal length
Ring 2 (Outer ring): At least one pair of parallel sides
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Ring 1 (Left side): At least one right angle
Ring 2 (Right side): No right angles
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Ring 1 (Left side): All sides the same length
Ring 2 (Right side): At least one acute angle
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Ring 1 (Left side): At least one set of parallel sides
Ring 2 (Right side): At least one obtuse angle
Extensions:
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Have students make their own labels and then challenge a partner to use them to create
quadrilateral rings.
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Have students make “mystery rings” for their partner to solve. Simply sort
quadrilaterals into the Venn Diagram rings according to some characteristic and have a
partner try to decide how the quadrilateral pieces have been sorted.
Family Connections:
Have students take home the quadrilateral pieces to share with their family. Show them
how to sort the pieces in each ring according to the labels given. They may need to overlap
some rings to form intersections. Make “mystery rings” for family members to solve.
Assessment Plan:
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Have students justify the placement of quadrilaterals in the Venn diagram. Journal
reflections explaining the placement of quadrilaterals are useful for checking students’
understanding.
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Have students explain the relationship among the rectangle, rhombus, and square.
REFERENCES
The National Council of Teachers of Mathematics (2011). Math Standards and
Expectations. Retrieved from
http://www.nctm.org/standards/content.aspx?id=4294967312
Utah Lesson Plans (2004). Quadrilaterals. Retrieved from
http://www.uen.org/Lessonplan/preview.cgi?LPid=11235