Lesson Plan
“One of These Things IS Like the Other”
Instructor(s): GWAEA Math Consultants
Objective(s)
Understand the definition of similarity in terms of similarity transformations.
Grade Level
Understand the relationship between scale factor and corresponding areas for
similar geometric figures.
8, 9-12
Est. Time
90 minutes
Pre-requisite
Knowledge:
Rotation, reflection, translation, Pythagorean theorem
Vocabulary:
Materials
Needed:
Dilation, similarity, scale factor, betweeness of points
Student handouts
Straightedges
Blank Paper
Set of tangrams
GeoGebra or equivalent access
Geometry
8.G
Understand congruence and similarity using physical models,
transparencies, or geometry software.
3. Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates.
4. Understand that a 2 dimensional figure is similar to another if the second can
be obtained from the first by a sequence of rotations, reflections, translations,
dilations; given 2 similar two dimensional figures, describe a sequence that
exhibits the similarity between them.
Iowa Core
Geometry
HS
Similarity, Right Triangles, and Trigonometry
G-SRT
2. Given 2 figures, use the definition of similarity in terms of similarity
transformations to decide if they are similar; explain using similarity
transformations the meaning of similarity for triangles as the equality of all
corresponding pairs of angles and the proportionality of all corresponding
pairs of sides.
van Hiele
Framework
Iowa Core
Mathematics
Standards of
Mathematical
Practice
Visual
Analysis
Informal
Deduction
Deduction
Rigor
Problem
Solving
Reasoning
Viable
Arguments
Model
Tools
Precision
Structure
Regularity
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Launch
Examine the seven pieces of a tangram puzzle. What relationships exist among the
pieces of the puzzle? (Given a set of paper tangrams)
Explore
Divide the group into pairs. Each student has his/her own worksheet and set of tangrams.
Have them complete exercises 2 – 19 on the One of These Things IS Like the Other
worksheet. Teacher check points after exercises 7, 12, 19.
Regroup the students so they have computer access. Make the student to computer ratio
as small as possible. Access GeoGebra or another program that performs dilation.
Help the students create a rigid triangle and a point. Complete exercises 20 – 26 on the
One of These Things IS Like the Other worksheet.
Key Ideas
Key ideas/important points
Teacher strategies/actions
Ratio of the areas of similar figures is related by
the square of the scale factor.
Each time figures are compared ask questions so
that students compare lengths and areas.
Dilation is determined by a point and a scale factor.
Constructions done with all three tangram triangles
and the parallelogram should clarify the definition of
dilation.
Similarity is defined by dilation, rotation, and/or
reflection.
Ask if any of our activities dealt with non-aligned
polygons. (Question in the summary.)
Similarity of 2-D geometric figures implies
congruent corresponding angles and proportional
corresponding sides.
Connect with the properties of a dilation and the
scale factor of a dilation.
.
Guiding Questions
Good questions to ask
What does the scale factor
relate to?
Possible student responses
or actions
It tells how much bigger the new
figure is from the original.
Possible teacher
responses
What do you mean by bigger?
Does that refer to sides? To the
areas?
Is the new figure always larger?
How do the areas of the image
and pre-image compare?
7027 a One Like Other_Lesson Plan
The area changes.
Is there any certain amount? If
you knew the area of the original
figure, can you determine the
area of the new figure in any way
other than measuring?
The area changes with the scale
factor.
The same way that the lengths of
the sides change?
2/4
I don’t know.
What if the scale factor is 2? Can
you draw the new triangle and
show me how many of the original
triangles will fit inside?
The new figure is the scale factor
squared in area.
So, if the scale factor is 3, the
area of the new triangle will be
how many times the area of the
original? What if the scale factor
is .5? What if the scale factor is
-1?
Misconceptions, Errors, Trouble Spots
Possible errors or trouble spots
Teacher questions/actions to resolve them
Students have trouble determining the √2
relationship in the tangram. (Small to medium
triangle and medium to large triangle.)
The students will need to explore the sides of triangles
using the Pythagorean theorem. Another method would
be to take 4 small triangles together to form a square
(the right angles would all meet and the hypotenuses
would be sides of the square. If the small triangle is an
area of ½, then 4 triangles would have an area of 2. A
square’s area is side x side = 2, so the side of the new
figure is √2
Students conclude that area is double the scale
factor.
Have students use a scale factor of 2 to create a new
figure. Using at least 2 copies of the original figure,
students cover the new figure to see that it will take 4 to
do so. If students do not see that the area of the new is
scale factor squared, try using scale factor of 3.
Students may not understand that comparison
of the triangles relates to area.
Have students cover the large triangle with the small
triangles.
Students are unfamiliar with notation of preimage and image (P to P’).
Show them that common math notation is to take figure
ABC, transform it, and label the transformation A’B’C’,
read A prime B prime C prime instead of using different
letters so that the transformed figure clearly has
originated from the original because letters are the
same.
Exercise 9 may result in misunderstanding for
students struggling with visualization. They may
not easily locate points X, Y, and Z with the
math language given.
Be prepared to have student sketch out what is asked
or to give alternate directions such as “Measure the
distance QA. Multiply it by 1.5. On the ray QA, place
point X the new distance from Q.”
Summarize and Clarify
How are the scale factor and the ratio of areas related in similar triangles?
How many dilations will map two similar polygons to each other?
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How is the definition Two figures are similar if you can rotate and/or reflect one of them so
that you can dilate it onto the other. an improvement over the definition given in exercise
14?
Accommodation(s)
Provide solid plastic or wood tangrams for students that have major difficulty with the paper
ones.
Provide extended free play with tangrams to determine relationships among the pieces.
Extension(s)
Provide two similar polygons and have students locate the center of dilation and the scale
factor.
Given two similar polygons in different orientations find the rotation and/or relfection and then
the dilation needed to map one to the other.
Given two polygons determine if they are similar by definitions.
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