YOUNGSTOWN CITY SCHOOLS
MATH: GEOMETRY
UNIT 2A: SIMILARITY, AND PROOF (3 WEEKS) 2013-2014
Synopsis: Continuing with transformations, the concept of similarity will be presented as a dilation. Expanding on
similarity, theorems will be presented and problems involving similar figures will be calculated.
STANDARDS
G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor.
a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through
the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
G.SRT.2 Given two 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.
G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two
proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
MATH PRACTICES
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning
LITERACY STANDARDS
L.1
L.2
L.5
L.6
Learn to read mathematical text (including textbooks, articles, problems, problem explanations )
Communicate using correct mathematical terminology
Justify orally and in writing mathematical reasoning
Represent and interpret data with an without technology
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MOTIVATION
TEACHER NOTES
1. There are two excellent videos on similar polygons. Choose one of the two listed:
http://www.youtube.com/watch?v=YRD4gb0p5RM which is a video on Donald Duck in
mathmagic land .
http://app.discoveryeducation.com/player/view/assetGuid/A97CE027-77FB-4FBA-B2C7C97CC0EF5518 United streaming video with 12 segments of which you can choose as many as
you prefer.
After viewing the videos, introduce the concept of similarity. (G.SRT.2, MP.4, MP.7, MP.8, L.2)
2. Preview expectations for the end of the Unit
3. Have students set both personal and academic goals for this Unit or grading period.
TEACHING-LEARNING
Vocabulary
Similar
Rigid motion
Dilation
Scale factor
TEACHER NOTES
Center of dilation
Geometric mean
Pythagorean theorem
Ratio and proportion
1. Review the basic real-life definition of similar figures. Review transformations and then lead the
review into a discussion of dilations with a connection to similarity. (G.SRT.1, MP.1, MP.4, MP.8, L.2)
2. Discuss center of dilation, then have students work on the geometer’s sketchpad activity on “Similar
Polygons” which discusses moving the center of dilation. The activity is attached to the unit on
pages 7-9. Then refer to the textbook, section 9.5, pages 490-494 in the McGraw Hill Geometry text.
If more work is needed, refer students to the following web site which works on resizing polygons:
(G.SRT.1, G.SRT.2, MP.1, MP.2, MP.4, MP.5, MP.8, L.1, L.2)
http://www.mathsisfun.com/geometry/similar.html
3. Draw three pairs of similar triangles, acute, obtuse and right and discuss the relationship between the
angles and sides. Also include the proper procedure for naming the triangles, angles and sides.
(G.SRT.2, MP.4, MP.6, MP.8, L.2)
4. Have students create a similar triangle to one given with a given scale factor using rulers and
protractors. (teacher discretion as to the number of these examples presented) Included is a
worksheet on using similar polygons and working with scale factors on pages 10, 11 12, and 13.
(G.SRT.1.b, MP.1, MP.4, MP.6, MP.8, L.2)
5. At this time, students should be able to formally define similar polygons: A pair of polygons where one
is formed by a dilation of the other, the angles are congruent, and the sides are in proportion. Have
students examine two similar polygons and write the similarity statement for the two polygons, and the
correspondence relationships of the sides and angles. That is, if ∆ABC ∆RST (similarity
statement), then <A = <R, <B = <S, <C = <T and
=
=
(correspondence relationships).
Reinforce with additional examples. (G.SRT.2, MP.4, MP.7, MP.8, L.2 )
6. After being able to set up the similar correspondence relationships, the students will work with scale
factor. Discuss finding a scale factor given two similar polygons with the dimensions of at least one
pair of corresponding sides and also given a scale factor and the sides of one similar polygon, find the
sides of the other polygon. At this time, have students set up proportions for similar polygons and
solve them. (Included at the end of the unit is a review on solving proportions pages 14 and 15)
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TEACHING-LEARNING
TEACHER NOTES
which you can use if you feel it is appropriate.) Also given angles of one similar polygon, find the
corresponding angles of the other polygon. (G.SRT.1.b, G.SRT.5, MP.1, MP.2, MP.4, MP.7, MP.8,
L.2)
7. Short mini quiz.
8. Review the sum of the angles of a triangle is 1800. Then have students work similar problems such
as: A triangle has angles measured 1050, 250, and 500. Another triangle has two angles measured
250 and 500. Are the two triangles similar? Explain. A second problem is: A triangle has angles
measured 900, 450, and 450. Another triangle has two angles measured 900 and 450. Are the two
triangles similar? Explain. After discussing these examples, use them to have students lead into the
angle-angle theorem for similar triangles. Ask the question “Do you need all three pairs of angles
congruent or are two angles sufficient? Then add the SAS and SSS similarity theorems. At this time
students should be proficient in proving triangles similar by AA, SAS, and SSS. Included is a
worksheet on similar triangles on pages 16, 17, and 18. (G.SRT.3, MP.1, MP.2, MP.3, MP.4,
MP.7, MP.8, L.2, L.5)
9. Relate the AA Theorem to dilation: Teacher uses all three pairs of angles separately and/or combined
to place similar triangles on top of each other with rigid motions. Give students the vertices of two
similar shapes on the coordinate system and have them explain the relationships between the sides
and the angles. Then have them slide one triangle on top of the other with one pair of corresponding
vertices on top of each other and one side parallel to the other. Discuss with students the fact that
there are corresponding angles that are congruent and sides that are in proportion. Have this lead
into the theorem: If a line is parallel to a side of a triangle and intersects the other two sides, it forms
two similar triangles. Work on practice problems in section 6-4 of the textbook.
Next look at right triangles that use this theorem, such as the following example:
BC is parallel to DC, <ACB and <D are right angles.
Find AC, AB and BC (note: need to use Pythagorean
Theorem to find AC and then similarity to find AB.)
A
4
C
B
2
D
C
8
Add few more examples and use this to lead into the Pythagorean proof. At this time students can
check out Plato content, Plato course geometry semester A, Unit 2, proportionality, make sure popups are enabled. (G.SRT.2. G.SRT.3, G.SRT.4, MP.2, MP.4, MP.7, MP.8, L.2 )
10.
C
A
D
B
Mark <ACD as B’ and <BCD as A’. Put the above triangles on cardstock (separated and together)
and laminate so students can write on them. Review definition of altitude. List all the angles that are
congruent in the drawing. Have students find the triangles that are similar. ( ∆ ADC
∆ ACB by AA
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TEACHING-LEARNING
TEACHER NOTES
and ∆ACB
∆CDB by AA, so by transitive, property, ∆ ADC ∆ CDB.) To simplify you can
assign b to side AC, a to side BC and c to side AB. To prove the Pythagorean theorem, set up the
proportions from the similar triangles, cross multiply, add two equations together and the proof is
complete:
From ∆ACD
∆ABC, we get the proportion
=
, cross multiplying gives
AC2 = AD*AB. From ∆CBD ∆ABC, we get the proportion
=
, cross multiplying gives CB2
2
2
= DB * AB. Adding those two equations gives AC + CB = AD*AB + DB*AB. Using the distributive
property AC2 + CB2 = AB(AD+DB). We know AD+DB=AB so AC2 + CB2 = AB2.
Have students complete the Similar Triangle Proof which is attached to the unit on pages 8 and 9. It
shows the detail in proving the Pythagorean theorem. This activity can be done before showing the
proof in class or after.
After the Pythagorean theorem is proven, have students examine Pythagorean triples: 3-4-5, 5-12-13,
7-24-25, 8-15-17 and look at multiples of each triple. Also work on problems such as: given a triangle
with sides 3-4-5 and the smallest side of a similar triangle is 27, find the other two sides of the similar
triangle. Included is a worksheet on the Pythagorean theorem and its converse on pages 19 and
20. (G.SRT.2, G.SRT.3,G.SRT.4, G.SRT.5, MP.1, MP.2, MP.3, MP.4, MP.7, MP.8, L.1, L.2, L.5)
11. Short quiz
12. Solve the following story problems dealing with similar triangles (problems attached on pages 2122):
a. The roof of a house has a slope of . What is the width of the house if the height of the roof is
8 ft.?
b. Jake looked at the plans for a new house he was building. The plans were drawn to scale of ¼
in = 1ft.. He measured the size of a room on the plans and found it to be 3.45 in by 4.75 in.
What are the dimensions of the room?
c. A tree is casting a 25 ft. shadow. A nearby flagpole was casting a 15 ft. shadow. If the flagpole
is 20 ft. high, how high is the tree?
d. Tina is standing on the bank of a river at a place where the river narrows to a point. There is a
bridge down the river that is 50 ft. long and Tina knows she is 150 ft. from the bridge. On the
bank between her and the bridge 75 ft. down the river is a spot that might be a good place for
another bridge. How wide the river at this point? (G.SRT.5, MP.1, MP.4, MP.5, MP.6, MP.8,
L.1, L.2, L.6)
new bridge
Tina
bridge
13. This section deals mainly with proofs. Before discussing proofs, review vertical angles are
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TEACHER NOTES
congruent, parallel lines with alternate interior angles, alternate exterior angles, corresponding
angles, and same side interior supplementary. The proofs can be done formally or the teacher can
give the students the given and the prove statement and have the steps and reasons on separate
pieces of paper for the students to match up.
a. Prove: A line parallel to one side of a triangle divides the other two sides proportionally. Also
prove the converse and use it for the next proof.
b. Prove: The quadrilateral formed by joining the midpoints of the sides of a quadrilateral
consecutively forms a parallelogram.
c. Prove: The altitude drawn to the hypotenuse of a right triangle divides the triangle into two
triangles that are similar to the original triangle. (This is review from step #11.)
d. Discuss the definition of geometric mean and work a few examples. After students feel
comfortable with the new term, prove the theorem: The altitude to hypotenuse of a right
triangle is the geometric mean of the two segments into which its foot divides the
hypotenuse.
e. Prove: ∆ABC
∆DEC, Given: AB is parallel to DE
A
E
C
B
f. Prove: ∆ABE
D
∆ACD, given: AB = 2, AC = 4, AE = 3, AD = 6
g. Work on examples in textbook pages 345 and 346. Also included is a worksheet on similar
right triangles, many of which deal with section d.
(G.SRT.4, G.SRT.5, MP.1, MP.2, MP.3, MP.4, MP.7, MP.8, L.2, L.5)
TEACHER CLASSROOM ASSESSMENT
1. Quizzes
2. In-class participation and practice problems for each concept
3. 2- and 4-Point Questions
TEACHER NOTES
TRADITIONAL ASSESSMENT
1. Paper-pencil test with M-C questions with 2- and 4-Point Questions
TEACHER NOTES
AUTHENTIC ASSESSMENT
1. Students evaluate goals set at the beginning of the unit or on a weekly basis.
TEACHER NOTES
2. Students are to draw three pairs of similar figures, each with a different scale factor. The
figures are to be chosen from triangles, rectangles, trapezoids, parallelograms and squares;
students then state the rule that applies. They find the perimeters of the original figure and
its dilation and find the relationship between perimeters and the sides of the similar figures.
They find the areas of the original figure and its dilation and find the relationship between the
areas and the sides of the similar figures, then state the rule that applies. Give an
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explanation and state the formulas. Materials needed for this assessment are rulers, graph
paper, and protractors. (G.SRT.1, G.SRT.2, G.SRT.3, G.SRT.5, MP.1, MP.2, MP.3, MP.4,
MP.8, L.2, L.5)
RUBRIC
Elements of the Project
Drew 3 pairs of similar
figures with 3 different
scale factors
Found perimeters
Stated rule for perimeters
Found areas
Stated rules for areas
Explained perimeters and
areas
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0
Did not
attempt
Did not
attempt
Did not
attempt
Did not
attempt
Did not
attempt
Did not
attempt
1
Drew 3 pairs of
similar figures with
one scale factor
Found perimeters of
2 figures
Stated rule, but there
were errors
Found areas of 2
figures
Stated rule, but there
were errors
Gave explanation for
perimeters OR areas
2
Drew 3 pairs of similar
figures with two different
scale factors
Found perimeters of 4
figures
NA
3
Drew 3 pairs of similar
figures with 3 different
scale factors
Found perimeters of 6
figures
State rule with no errors
Found areas of 4
figures
NA
Found areas of 6
figures
Stated rule with no
errors
Gave explanation for
both perimeters and
areas
NA
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