Mathematics Unit Group Project: Picturing Polygons
Erika Cannady, Kari Holliman-McGregor and Cindy Sanders
C I 402
Wichita State University
Unit Theme: Picturing Polygons
Grade Level: 5th
Standards: NCTM Geometry Standards:
 Analyze characteristics and properties of two- and three-dimensional
geometric shapes and develop mathematical arguments about
geometric relationships
o Identify, compare, and analyze attributes of two- and threedimensional shapes and develop vocabulary to describe the
attributes
o Explore congruence and similarity
o Make and test conjectures about geometric properties and
relationships and develop logical arguments to justify
conclusions
 Use visualization, spatial reasoning, and geometric modeling to solve
problems
o Build and draw geometric shapes
o Recognize geometric ideas and relationships and apply them to
other disciplines and to problems that arise in the classroom or
in everyday life.
NCTM Communication Standard
 Communicate their mathematical thinking coherently and clearly to
peers, teachers, and others
 Analyze and evaluate the mathematical thinking and strategies of
others
NCTM Problem Solving Standards
 Build new mathematical knowledge through problem solving
 Apply and adapt a variety of appropriate strategies to solve problems
 Monitor and reflect on the process of mathematical problem solving
NCTM Reasoning and Proof Standards
 Make and investigate mathematical conjectures
 Develop and evaluate mathematical arguments and proofs
NCTM Representations Standards
 Create and use representations to organize, record, and communicate
mathematical ideas
 Use representations to model and interpret physical, social and
mathematical phenomena
Vocabulary:
Polygon, Tangram, Triangle, Square, Rectangle, Hexagon, Similar
Integration Ideas:
 Art: “Picasso’s Polygons” handout
 Math: fractions with different polygons


Art: Use a triangle or quadrilateral and make tessellations, you can
discuss the properties of each as you change, flip, and slide the shapes
Language Arts: Make a children’s story about a triangle or
quadrilateral. The students will have to include the properties that
define each in a fun way with illustrations
Explanation of How Each Lesson Fits in the Unit:
What Can We Do With Polygons? Lesson:
This lesson acts as an introduction to the unit Picturing Polygons. Students will
review the basics of a polygon, classify polygons from non-polygons, and use
polygons as tangrams. This lesson will assess the students’ prior knowledge, as
well as, increase their knowledge on polygons, which will prepare them for our
next lesson in the unit, Sorting Polygons.
What Makes a Polygon a Polygon? Lesson:
This lesson allows student’s to use previous knowledge from “What Can We Do
With Polygons?” to determine the characteristics that define a triangle and
quadrilateral. They have to reason and communicate with their peers to proof that
the definition of their triangle and quadrilateral is correct. They are also using
problem solving and manipulatives to decide what type of polygon they created
on the geoboard. This will prepare them for comparing similar shapes with
polygons.
Building Similar Shapes Lesson:
This lesson is the last lesson in the Picturing Polygons Math Unit. It allows
students to build on their prior knowledge about polygons to learn the
mathematical meaning of similarity as they look as similar polygons. The students
will build larger and larger polygons using similar shapes of the starting shape.
They will also make generalizations about what happens to the area of a shape
when the lengths of all its sides are doubled and tripled.
Unit Reflection:
This unit allows students to explore polygons using different process standards
and manipulatives. Students are able to create polygons and discuss the
properties that define each shape. This will help students to identify and connect
the patterns and relationships between the sides and area of polygons, as they
make similar shapes. The students will have to reason and communicate with
their peers to determine what characteristics define a polygon and use this
information to evaluate similar shapes.
WHAT CAN WE DO WITH POLYGONS? – LESSON PLAN
Name: Erika Cannady
Date of Lesson: September 16, 2004
Subject Area/Topic: Mathematics/Polygons, Tangrams
Grade Level: 5th Grade
Standards:

NCTM Geometry Standard
o Analyze characteristics and properties of two and three dimensional
geometric shapes and develop mathematical arguments about
geometric relationships

Identify, compare, and analyze attributes of two and three=
dimensional shapes and develop vocabulary to describe the
attributes.
o Use visualization, spatial reasoning, and geometric modeling to solve
problems

Use geometric models to solve problems in other areas of
mathematics

NCTM Communication Standard – Communicate their mathematical
thinking coherently and clearly to peers, teachers, and others

NCTM Problem Solving Standard
o Build new mathematical knowledge through problem solving
o Apply and adapt a variety of appropriate strategies to solve problems
o Monitor and reflect on the process of mathematical problem solving
Outcomes/Objectives:

The main idea that students will understand is that polygons can be used to
create different shapes.

The student will distinguish between polygons and shapes that are not
polygons.

The student will develop an understanding that polygons can be put
together to create different shapes.
Vocabulary: polygon, tangram
Materials/Technology: Scissors, tangram handout, expand tangram handout
Procedure:
Engage:


Explore:



Review what a polygon is by reading over facts on the chart and
completing “Is it a Polygon?” handout as a class.
Ask “What can we do with polygons?”
Hand out materials.
Students work to make the large shape using the polygon pieces.
Possible teacher questions:
o What else could you try?
o Can you find another arrangement that works?
Explain:


Expand:


Students will show what arrangements they found that worked.
Possible teacher questions:
o Can you think of any hints you could give to someone who
isn’t figuring it out.
Students will be given a sheet of new polygons. They will design a
shape of their own using polygons.
Students will share what they come up with.
Evaluate:

Each student will make a sketch of how they accomplished two of
the tangram pictures.
Expand 2:

Students will be given the handout Picasso’s Polygons.
Adaptations/Modifications: Students can work in groups to figure out the tangrams.
References:

Akers, J., Clements, D., Murray, M., Tierney, C., & Samara, J.
(1998). 2-D geometry: picturing polygons. White
Plains, NY: Dale Seymour Publications, 2-8, 162-163,
167.

“Tangrams.” Available: http://www.tangrams.ca [September
19, 2004].

“Tangram Puzzles.” Available: http://www.funorama.com
September 19, 2004].

The National Council of Teachers of Mathematics. (2000).
Principles and standards for school mathematics.
Virginia: The National Council of Teachers of
Mathematics, Inc., 164, 402
Reflection: This lesson is appropriate for 5th graders because the material is familiar to
them. They have already learned what a polygon is, so they are now increasing their
knowledge by doing a hands-on activity.
Ideas for Integration:

Art- “Picasso’s Polygons” handout

Math- fractions with different polygons.
Student Handouts/ Worksheets Will Go Here
WHAT MAKES A POLYGON A POLYGON?
Kari Holliman
Math, Polygons
Grade Level: 5
Estimated Length: 1-1 ½ Hours
September 17, 2004
Standards:
NCTM StandardsGeometry
 Make and test conjectures about geometric properties and relationships
and develop logical arguments to justify conclusions
 Recognize geometric ideas and relationships and apply them to other
disciplines and to problems that arise in the classroom or in everyday
life.
 Explore congruence and similarity
Reasoning and Proof
 Make and investigate mathematical conjectures
 Develop and evaluate mathematical arguments and proofs
Communication
 Communicate their mathematical thinking coherently and clearly to
peers, teachers, and others
 Analyze and evaluate the mathematical thinking and strategies of others
Outcomes/Objectives:
1. Students will understand what the characteristics of a triangle and
quadrilateral are. They will know what is required for a shape to be a
rectangle and square.
2. Students will
o Explore attributes of triangles and quadrilaterals
o Classify triangles and quadrilaterals
Vocabulary:
Polygon, Triangle, Square, Rectangle
Materials:
 Geoboard
 Rubber bands
 Pictures of 3 sided Polygons (given to each student)
 Pictures of 4 sided Polygons (given to each student)
 “Polygons on the Geoboard” handout
 “Is Every Three-Sided Polygon a Triangle” handout
 “Is Every Square a Rectangle, Is Every Rectangle a Square” handout
Procedure
1. ENGAGE
a. Every student will be given a Geoboard and the handout Polygons on the
Geoboard, they will work together will a partner to complete this
activity (CONCRETE)
b. The students will make different polygons according to the
characteristics given from the handout
c. After everyone has completed, review the different types of polygons
 “What are some ways we can use to identify the type of polygon?”
2. EXPLORE
a. Give each student about 3-4 pictures of three sided polygons and have
them examine them (PICTORIAL)
b. “What do we call three sided polygons?”
c. “Are all of these three-sided figures triangles? Why or why not?”
d. The students will work in small groups to discover if all three sided
figures are triangles-(use handout to record their data)
 They will also discuss in their groups the different categories of
triangles
3. EXPLAIN
a. The students will report to the class their finding of three sided figures
and triangles
b. They will get back into their groups and figure out the different
characteristics of All Triangles and Some Triangles-(they will record their
results on a chart)
c. When the students have ran out of ideas, post their lists, the students
will explain their results
d. Tell them that you will continue to add to the list throughout the unit as
they come up with more ideas
4. EXPAND
a. Give the students pictures of four-sided shapes
b. “What are these shapes?”
c. “Do all quadrilaterals have four sides? What else do all quadrilaterals
have to have?
d. Record the data of All Quadrilaterals and Some Quadrilaterals
e. Possible Teacher Questions
 “What are some of the differences among quadrilaterals?”
 “What is true of the sides of some quadrilaterals?”
 “What kinds of angles do some quadrilaterals have?”
5. EVALUATE
a. Now that you have discussed different characteristics of triangles and
quadrilaterals the students will complete the handout “Is Every Square a
Rectangle, Is Every Rectangle a Square”
b. Have the students communicate their results and discuss when every one
is completed
6. EXPAND (2)
 Students will complete “Can You Make These Triangles” handout
References
National Council of Teachers of Mathematics (2000). Principles and standards
for school mathematics. Portland, OR: Graphic Arts Center.
Akers, J., Clements, D., Murray, M., Tiernay, C. & Samara, J. (1998). 2-D
geometry: Picturing polygons. White Plains, NY: Dale Seymour
Publications, 31-45, 169-170, 172
Reflection
This lesson is appropriate for fifth graders because they have to use prior
knowledge of polygons to determine the characteristics that define a triangle
and quadrilateral. They have to reason and communicate with their peers to
proof that the definition of their triangle and quadrilateral is correct. They
also are using problem solving and manipulatives to decide what type of
polygon they created on a geoboard.
Ideas for Integration
 Art: Take a triangle or quadrilateral and make a tessellation
 Language Arts: Make a children’s story about a triangle or quadrilateral.
The students will have to include the properties that define each in a
fun way with illustrations.
Student Handouts/ Worksheets Will Go Here
Building Similar Shapes
Name: Cindy Sanders
Date: September 28, 2004
Subjects:
 Mathematics
Grade Level: 5th
Standards: National Mathematics Standards:
 Analyze characteristics and properties of two- and three-dimensional
geometric shapes and develop mathematical arguments about
geometric relationships
o Identify, compare, and analyze attributes of two- and threedimensional shapes and develop vocabulary to describe the
attributes
o Explore congruence and similarity
 Use visualization, spatial reasoning, and geometric modeling to solve
problems
o Build and draw geometric shapes
 Create and use representations to organize, record, and
communicate mathematical ideas
 Use representations to model and interpret physical, social and
mathematical phenomena
Objectives:
1) Students will be able to explain the mathematical meaning of similarity.
2) Students will be able to create geometric patterns that grow in regular
ways.
Vocabulary:
Materials:







Polygon
Hexagon
Similar
Overhead Projector
Power Polygons
Student Worksheets “Building Similar Shapes” and “Length of Sides
Versus Area” (1 for each student)
Transparency of Student Worksheet “Building Similar Shapes”
Procedures:
Engage:
Have several pictures that have different shapes in them. Ask students to pick out
similar shapes that appear in the pictures.
Explore:
Show three polygons of different sizes from the set of Power Polygons on the
overhead. Ask students what is the same and what is different about the shapes
(Students may notice that the shapes share the same size angles, the same number
of sides, etc. Students may notice a general difference in size.). Introduce the term
similar as a word to describe the shapes. Place a rectangle on the overhead. Tell
the students that we want to make larger and larger that are similar to this one,
using only rectangle pieces this size.
Explain:
Have a student come up and build one using rectangle pieces on the overhead. If
the student builds a rectangle that is 2x2, ask if it is really similar to the first (It
is.). Build a rectangle that is 2x3; ask if the third rectangle is similar to both the
first and second rectangles (It’s not). Ask the students to explain their reasoning.
Expand/Extend/Apply:
Hand out the Building Similar Shapes worksheet. Explain that to the students that
they will be building larger and larger similar shapers, just as they did with the
rectangle in the explain phase. Put a transparency of the worksheet on the
overhead projector and show them how to record the data needed using the
rectangle from the explain phase. For each shape on the worksheet, students will
consider that the smallest similar figure takes one piece (the shape itself). Each
time the next larger shape is built they record how many pieces it took. Students
may work together, but should each independently build the shapes. Warn
students that the hexagons are more difficult and will require a different strategy
than the rest of the figures. Give the students some time to work then call them
together to compare notes. Ask them what happened when they built larger and
larger shapes. Many will observe that the numbers were the same; some will
notice that the numbers were square numbers. Have the students explain what
happened when they tried to make a similar hexagon. Did their first attempt
work? What shapes did they need to use?
Evaluate:
Informally observe the students during the building time and class discussion.
Analyze the responses on students’ worksheet. Have students write a paragraph
about what they learned and include examples from their task.
Expand/Extend/Apply 2:
Hand out the worksheet “Length of Sides Versus Area” for homework. Students’
work on this sheet will serve as a good checkpoint of their understanding of
similarity and the relationship of area to perimeter in similar figures.
Resources:
National Council of Teachers of Mathematics (2000). Principles and standards for
school mathematics. Reston, VA: Authors, 164 and 402.
Akers, J., Clements, D., Murray, M., Tierney, C., & Samara, J. (1998). 2-D
geometry: picturing polygons. White Plains, NY: Dale Seymour
Publications, 93-99 & 189-190.
Reflection:
This lesson is a good way to explain similarity of shape for students because it gives the
students visual as well as kinesthetic ways to learn the topic. It also allows for small
group work therefore fostering a community of learners in the classroom.
Ideas for Integration:

Art: Have students make their own art piece using similar
polygons like Picasso.
Student Handouts/ Worksheets Will Go Here