Name: Steven PavlakisDate: 11/11/13Grade Level: 8Course: Pre-AlgebraTime Allotted: 70 minutesNumber of Students: 24

4-5 SimilarityI. Goal(s)

1. Students will investigate the properties of similar polygons.II. Objectives

1. Students will be able to identify corresponding parts of similar polygons.

2. Students will be able to define similarity in their own words.3. Students will be able to determine whether two polygons are similar

by measuring angles and sides.4. Students will be able to calculate the scale factor of two similar

polygons.III. Materials and Resources

Notebooks (30) Three pairs of similar polygons, one pair of dissimilar polygons

printed on colored paper, with each pair of the same color. (15) Rulers (30) Protractors (30)

IV. Motivation (5 minutes)1. Find an open space on the wall. Tell the class, “You know, I know you

guys miss me on the days I’m not here working with you guys. So I have the perfect solution. I am going to get a giant picture of my face and hang it in a frame and hang it right here. I have a frame that is 36”x60”, but the only picture I have of myself is a 2”x4” wallet picture. I want to make sure that the picture will fit into the frame if I blow it up, without any white space. Will it fit?”

2. Ask what they think, have a short discussion about it.3. TRANSITION: Say, “Well, if it’s going to fit, the frame and the picture

have to be similar polygons.”V. Lesson Procedure

1. (All slides should take ~30 minutes to get through) Definitions—have the students copy these down into their journals (10 minutes to cover definitions)

i. Similar polygons:=polygons that1. Have corresponding angles with the same measure

(congruent).2. Have proportional sides, that is, the corresponding sides

have the same scale factor.ii. corresponding parts:= the parts of similar figures that “match”

iii. scale factor:=the ratio of the lengths of two corresponding sides of two similar polygons.

2. (10 minutes) Show a slide (1) with a pair of similar, regular pentagons (appended). Show that all of the angles are the same measurement, and set up the proportion on side lengths, show that the scale factor is the same for any two corresponding sides.

i. Link back to their previous knowledge of proportions.1. “Over the last few days, we’ve been setting up

proportions between quantities. Today, we’re setting up these proportions between the corresponding parts of these similar polygons, and we can use the same method that we’ve been using to do that.”

ii. Be prepared for students to think that scale factors and proportionality are contained within the sides of just one polygon. Stress the importance that the scale factor is comparative ACROSS figures, not within one. Relate this to a scale model that they may have built. While the proportions of the sides of similar polygons are the same, the scale factor is comparative in nature, and describes the dilation between two similar polygons.

3. (5 minutes) Next, show a slide (2) with a pair of squares. Ask students: Are they similar? Why? Are squares always similar? (Yes) Have a short discussion. I would expect students to not believe me at first, but as we go through the definition of similarity as applied to squares, I believe that they will become convinced. Is this true for all rectangles? (No). I anticipate that students would try to apply what we just covered about squares to the general case of rectangles. I must demonstrate the difference between squares and rectangles, which has to do with side length. Since this is standard on a square, we don’t have to worry about sides not being corresponding or not having the same scale factor, both issues we have with rectangles. Show on the next slide. (3)

4. (5 minutes) Last slide (4)! A pair of scalene triangles that are similar. Have a student identify the corresponding parts, and have another come up with the scale factor.

i. Here, we must stress the importance of labeling the corresponding parts correctly. They may label them based on their orientation, but in this slide, I rotated the one triangle so that they are not in the same orientation.

5. (20 minutes) Tell students to sit with their partners (they have assigned partners for the week in the class). Distribute an envelope containing all four pairs of shapes for students to determine if they are similar to each pair. Instruct students to get a ruler and a protractor for the pair. Remind students what they need to find to determine whether two polygons are similar: same angle measure and proportional sides. Have them copy the following table down into

their notebooks to fill out during their investigation. (Answers inserted) Have students repeat the instructions to you to make sure they understand the

procedure. During this time, go around to different groups, making sure that different groups set up their proportions differently, and that students are measuring correctly. Correct conceptual errors by walking through some shapes with pairs.

6. (10

minutes) After 15 minutes of letting them try the shapes, bring the attention back to the board. Have the following table on the SMARTboard. Take a vote on each pair for similarity. Use random name generator to pick students for scale factor values. Make sure that students agree.

ii. I intentionally did not mark which scale factor I wanted, small to big, or big to small. This is a great opportunity to orient different students’ answers, saying as long as you are consistent, it doesn’t matter which order you put your proportion.

VI. Closure (5 min)1. “So now we’re masters of similar polygons. Let’s take a look at that

picture I want to hang again. I said the only way I won’t get any white space on the sides of the picture is if the frame and the picture are similar polygons. What do you think? Pull out a your journals, and write me a few sentences saying whether or not the picture will work, and why. Show your work!”

2. Assign homework: 4-5: # 2, 4, 5, 9, 10, 13, 22VII. Extension

1. Post a slide with images of the Mona Lisa, the Parthenon, and the Taj Mahal, each with a golden rectangle around parts of it. Since they are eighth graders, I don’t think I should go into the golden ratio at all, but I can preface the extension activity by saying, “The Ancient Greeks found the golden rectangle to be the most beautiful rectangle, and others, such as Leonardo Da Vinci agreed and used it, so it is common in art and architecture. All golden rectangles are similar. I marked the height and width of the Parthenon, and one dimension of the other two. Find the missing sides!” Note that the Mona Lisa’s rectangle is rotated, to emphasize matching the corresponding parts. Slide attached.

VIII. Assessment Summary While walking around and monitoring groups, the teacher will be

making sure that students are actually applying the ideas correctly. The journal reflection, done at the end of class, will be reviewed by the

teacher before the next class to make sure students have absorbed the material.

Pair Red Blue Green OrangeSimilar? No Yes Yes YesScale factor? N/A 2:1 or 1:2 7:2 or 2:7 1:1

The homework assigned will be discussed at the beginning of the next class, and then collected for a grade.

IX. Standards1. CC.8.G.1 Understand congruence and similarity using physical models,

transparencies, or geometry software. Verify experimentally the properties of rotations, reflections, and translations: -- a. Lines are taken to lines, and line segments to line segments of the

same length.-- b. Angles are taken to angles of the same measure.-- c. Parallel lines are taken to parallel lines

I accomplish this standard by the activity in step v of section V of this lesson plan. They use a physical model to explore these characteristics to gain understanding of similarity.

2. CCSS.Math.Practice.MP6 Attend to precision. Students must be precise in their measurements of

angles and sides. Otherwise, the polygons may not turn out to be similar, or they may measure an inaccurate scale factor.

3. CCSS.Math.Practice.MP5 Use appropriate tools strategically. Students will be allowed to use measurement tools to

calculate side length and angle measure.

AttachmentsSlide (1)

(2)

120°

120°

120°120°

120°

2cm

2cm

2cm

2cm

2cm

1cm

7 in

7 in 7 in

7 in

90°

90°90°

90°

8 in

8 in

8 in

90°

90°

(3)

(4)

Pairs :Red

Blue

7 in

7 in 7 in

7 in

90°

90°90°

90°

9 in

9 in

9 in

90°

90°

1.5 ft

3 ft

4 ft

3.75 ft

10 ft

7.5 ft

Green

Orange

Extension Slide: