CHAPTER 7
5
Sorting Polygons
STUDENT BOOK PAGES 222–223
Guided Activity
Goal Sort polygons by line symmetry.
Prerequisite Skills/Concepts
Expectations
• Know what lines of symmetry are.
• Know how to find lines of
symmetry on polygons using
a transparent mirror.
• Know how to sort using a
Venn diagram.
• classify and construct polygons and angles
• sort and classify quadrilaterals by geometric properties related to symmetry, angles,
and sides, through investigation using a variety of tools (e.g., geoboard, dynamic
geometry software) and strategies (e.g., using charts, using Venn diagrams)
• sort polygons according to the number of lines of symmetry [and the order of
rotational symmetry], through investigation using a variety of tools (e.g., tracing
paper, dynamic geometry software, Mira)
Assessment for Feedback
What You Will See Students Doing…
Students will
When Students Understand
If Students Misunderstand
• know how to identify the lines of symmetry for
each polygon shape
• Students will use transparent mirrors or folding
to identify and check the lines of symmetry in
each polygon.
• Demonstrate how to use a mirror and folding
procedures in order to find and check lines of
symmetry. Ask students to number the lines to
be sure that they count each line only once.
• make reasonable predictions regarding the number
of lines of symmetry in regular polygons and in
those which are not regular
• Students will notice that in regular polygons, the
number of lines of symmetry is equal to the number
of sides, and that this same pattern does not hold
true for polygons that are not regular.
• Ask students to create a table to identify regular
polygons as those having equal sides and equal
angles, and to discover that the number of lines
of symmetry in regular polygons is the same as
the number of sides.
• use a Venn diagram to sort polygons according to
given criteria
• Students will use a Venn diagram to sort the
polygons according to line symmetry.
• Ask students to use a ruler to find those polygons
that have at least four equal sides and list them.
Then ask them to use the transparent mirror to
find the polygons that have two or more lines of
symmetry, and list them. Have them place the
polygons in each list in their respective circles in
the Venn diagram, with any on both lists going
into the intersection part of the Venn diagram.
Preparation and Planning
Pacing
5–10 min Introduction
20–25 min Teaching and Learning
15–25 min Consolidation
Materials
•ruler (1/student)
•scissors (1/student)
•polygon shapes sheets (1/student)
•transparent mirror (1/student)
Masters
•Venn Diagram 1, Masters Booklet
p. 55
•Polygon Shapes, p. 67
•Optional: Chapter 7 Mental Math
p. 59
Workbook
p. 67
Key
Assessment
of Learning
Question
Question 5, Application of Learning
Mathematical Reasoning and Proving
Processes
Copyright © 2006 by Thomson Nelson
Meeting Individual Needs
Extra Challenge
• Have students predict the number of lines of symmetry for regular polygons
that have so many sides (for example, 50 or more) that it is impossible to
make and test them in a single class period. Ask students to justify their
predictions based on what they have already learned about lines of
symmetry and simpler regular polygons.
Extra Support
• Review the concept of lines of symmetry. Demonstrate how to find and
check them for each polygon shape by folding that shape in order to make
congruent halves and by using a transparent mirror.
• Have students trace and cut out a square and an equilateral triangle. Ask
them to find where their lines of symmetry are by using a transparent
mirror or by folding. Then ask them to repeat the procedure with a
regular pentagon and hexagon. When successfully accomplished, have
them trace and cut out the kite shapes in the lesson and follow the same
procedure to find their lines of symmetry.
Lesson 5: Sorting Polygons
33
1.
Introduction (Whole Class)
➧ 5–10 min
Introduce the lesson by placing two pattern blocks, such as
a hexagon and a trapezoid, on the overhead projector. Ask
students to look at the two shapes and consider their various
properties. Remind them that polygon properties include
angle measures, side lengths, number of sides, and lines of
symmetry. Give students a moment or two to think about
the properties of the two shapes, and then engage the class
in a discussion along the following lines.
Sample Discourse
“What do these shapes have in common?”
• They both have obtuse angles. The trapezoid has two and
the hexagon has six.
“How are these shapes different?”
• The trapezoid has four sides and the hexagon has six.
• The trapezoid does not have equal sides and the hexagon does.
• The hexagon has all equal angles and trapezoid does not.
Now replace the hexagon pattern block on the overhead
with a square pattern block and continue the discourse.
“How are these two shapes alike and different?
• They both have four sides, but the trapezoid has unequal
sides and unequal angles and the square has equal sides
and equal angles.
“Do they both have the same number of lines of symmetry?
How could you tell?”
• I could trace the shapes, fold them, and count the lines
and compare.
• I could use a mirror to find the lines and then compare.
Tell students that, in this lesson, they will be sorting
polygons according to line symmetry.
2.
Teaching and Learning (Whole Class/Small Groups) ➧ 20–25 min
Ask students to turn to Student Book page 222. As a class,
read Kurt’s statement about his kite collection and the central
question. Discuss the various shapes of Kurt’s kites, drawing
particular attention to those that have equal sides and those
that do not. Ask students to name, to the best of their
ability, the polygon shapes represented by the kites, and
write the names on the chalkboard. Review with students the
definitions of regular and irregular polygons, and have
them read Ayan’s plans for sorting the polygons.
Now distribute the rulers, scissors, polygon shapes sheets,
and transparent mirrors, and have students work through
Prompts A to C in small groups. After students find the lines
of symmetry of each polygon with a mirror, have them fold
each polygon along those lines in order to check their results.
34
Chapter 7: 2-D Geomtery
Finally, have them record their results from Prompt C.
From their results, have them identify the shapes that
have four lines of symmetry.
Reflecting
Use these questions to draw students’ attention to the
relationship between the number of equal sides and the
number of lines of symmetry in each of Kurt’s kite shapes, and
in particular to the number of equal sides and the number of
lines of symmetry in those shapes which are regular polygons.
Copyright © 2006 by Thomson Nelson
1. For example, I started looking at shapes with four or
more sides, because when I recorded the number of lines
of symmetry for each polygon, I found that the number
was the same or less than the polygon’s number of sides.
2. Yes. The kite, isosceles triangle, octagon, cross, and rhombus
had more sides than lines of symmetry. There were not
any polygons with more lines of symmetry than sides.
3. The number of lines of symmetry in a regular polygon is
the same as its number of sides.
(Lesson 5 Answers continued on p. 80)
3.
Consolidation ➧ 15–25 min
Checking (Whole Class/Pairs)
For intervention strategies, refer to Meeting Individual
Needs or the Assessment for Feedback chart.
4. Have students identify and list those polygons that have
at least four equal sides, and then do the same for those
polygons that have more than one line of symmetry.
Have them compare their lists and record the names
or draw pictures of any polygons that are on both lists
in the intersection, and any polygons not on either list
outside the two circles. Then have them record the names
or draw pictures of the remaining polygons that have
at least four equal sides in the left circle, and those that
have more than one line of symmetry in the right circle.
Key Assessment of Learning Question. (See chart on next page.)
Practising (Individual)
5. Have students use the same strategy given above
for Question 4.
6. Ask students to remember and apply their answer to
Question 3 to this question. Have them sketch a decagon,
and then draw its lines of symmetry.
Answers
A. & B.
1
3
2
1
2
3 4 5
3
2
6
4
1
2
3
4
1
Related Questions to Ask
1
2
1
1
C.
3
4
2
5
3
1
Name of polygon
Lines of symmetry
Equilateral triangle
3
Kite
Regular hexagon
1
6
Square
4
Octagon
4
Isosceles triangle
1
Regular pentagon
5
Cross
4
Rhombus
2
Copyright © 2006 by Thomson Nelson
4
1
2
Ask
Possible Responses
About Question 6
• What do we know about regular
polygons that will help us to
answer this question?
• The number of equal sides is
equal to the number of lines of
symmetry in a regular polygon.
Closing (Whole Class)
Have students summarize what they have learned by
responding to this question: “How does sorting polygons in
different ways help you to understand their properties better?”
• Sorting lets me identify which polygons have equal sides
and which ones have unequal sides, and which ones have
certain numbers of lines of symmetry.
• Sorting lets me relate the number of equal sides to the
number of lines of symmetry.
Lesson 5: Sorting Polygons
35
Assessment of Learning—What to Look for in Student Work…
Assessment Strategy: Written Answer
Application of Learning
Key Assessment Question 5
• a) Identify the lines of symmetry of each of these polygons.
b) Sort the polygons. Use the Venn diagram from Question 4.
1
2
• demonstrates limited ability to
apply mathematical knowledge and
skills in familiar contexts (e.g., has
difficulty using lines of symmetry to
sort polygons)
• demonstrates some ability to apply
mathematical knowledge and skills
in familiar contexts (e.g.,
demonstrates some ability to use
lines of symmetry to sort polygons)
3
4
• demonstrates considerable ability
to apply mathematical knowledge
and skills in familiar contexts
(e.g., uses lines of symmetry to
sort polygons)
• demonstrates sophisticated ability
to apply mathematical knowledge
and skills in familiar contexts
(e.g., demonstrates sophisticated
ability to use lines of symmetry
to sort polygons)
Extra Practice and Extension
At Home
• You might assign any of the questions related to this lesson,
which are cross-referenced in the chart below.
• Have students look for regular and other polygon shapes at
home and identify the lines of symmetry in them. Then
have them test to see if the relationship between the
number of sides and the number of lines of symmetry
holds true for the regular polygons they have found.
Skills Bank
Student Book p. 226, Questions 6 & 7
Problem Bank
Student Book p. 228, Question 3
Chapter Review
Student Book p. 231, Questions 7 & 8
Workbook
p. 67, all questions
Nelson Web Site
Visit www.mathK8.nelson.com and follow
the links to Nelson Mathematics 6, Chapter 7.
Venn Diagram 1,
Masters Booklet p. 55
Polygon Shapes, p. 67
Math Background
A line of symmetry is a line that divides a shape
perpendicularly into congruent halves. These halves must
fit exactly onto each other when the shape is folded and
they are superimposed. In other words, if a transparent
mirror is put on the fold line, the halves must be mirror
images of one another. Thus, the diagonals across a rectangle
are not lines of symmetry even though they divide the
rectangle into congruent halves. These halves are orientated
in a direction that does not allow them to fit exactly when
superimposed during folding. Their reflected images using
a transparent mirror are also not mirror images.
Therefore, some lines that create congruent halves in
polygons are not lines of symmetry.
36
Chapter 7: 2-D Geomtery
Optional: Chapter 7
Mental Math p. 59
Copyright © 2006 by Thomson Nelson