# 4.7: Similarity

## Ratios and Proportions

I. Section Objectives

- Write and simplify ratios.
- Formulate proportions.
- Use ratios and proportions in problem solving.

II. Multiple Intelligences

- Differentiate this lesson by providing opportunities to expand each example for the students. This will provide students with a way to practice the concepts as the information is presented.
- Ratio is a fraction that compares two things.
- Three ways to write a ratio. As a fraction, with a colon or using the word to.
- Expand Example 1: What is the ratio of everything bagels to sesame bagels?
- Answer:
- Expand the equation example with the dancers and the singers.
- “What if the ratio of dancers to singers was to and there were forty- five dancers? How many singers are there? How many dancers are there?
- Answer:
- Proportion- an equation that compares two equal ratios.
- Expand Barn Dimensions example.
- “What if the water line was actually instead of ten? What would the length be on a scale drawing?”
- Answer
- Intelligences- linguistic, logical- mathematical, visual- spatial

III. Special Needs/Modifications

- Write each definition and its examples on the board. Request students write down the information in their notebook.
- Explain the Cross Multiplication Theorem as something that the students already know from previous classes about how to solve a proportion. This is a formal way of writing it.

IV. Alternative Assessment

- Throughout the expansion of each exercise, allow students to contribute their answers to class discussion.

## Properties of Proportions

I. Section Objectives

- Prove theorems about proportions.
- Recognize true proportions.
- Use proportions theorem in problem solving.

II. Multiple Intelligences

- This is a brief lesson used to explain the corollary theorems associated with the Cross Multiplication Theorem.
- To make this lesson more interesting and interactive, use group work.
- Divide the students into five groups.
- After teaching the lesson and going through the material, assign each group one of the corollary theorems.
- Then put an example of two similar triangles (with measurements) on the board/overhead.
- Ask each group to use the example on the board to create an example that illustrates their corollary theorem.
- Allow time for the students to work and then have each group present their example to the class.
- When finished, encourage the other students to ask questions to see how well the students in the group can answer them.
- Answering questions is a great way to assess student understanding.
- Intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal, intrapersonal.

III. Special Needs/Modifications

- Write these notes on the board/overhead. Request that students copy them down.
- Cross Multiplication Theorem- defining property of proportions.
- Subtheorems are called corollary theorems.
- Review which terms in a proportion are the means and the extremes.
- Corollary 1- swap means.
- Corollary 2- swap extremes.
- Corollary 3- flip it upside down.
- Corollary
- Corollary

IV. Alternative Assessment

- Assessment can be done through student questions and answers following the exercise.

## Similar Polygons

I. Section Objectives

- Recognize similar polygons.
- Identify corresponding angles and sides of similar polygons from a statement of similarity.
- Calculate and apply scale factors.

II. Multiple Intelligences

- Activity to differentiate this lesson.
- Ask students to draw a polygon and label the lengths of the sides of their polygon.
- Then ask the students to exchange polygons with someone else. Students may exchange more than once just be sure that everyone has a different polygon than the one that they started with.
- Students need to complete the following with this new polygon.
- 1. Draw a similar polygon to the one that you have been given.
- 2. Write proportions to demonstrate that the side lengths are similar.
- 3. Determine the scale factor.
- 4. Determine the ratio of the perimeters.
- When finished, divide into small groups to share their findings.
- Use peers to correct any errors in the work of each individual.
- Intelligences- linguistic, logical- mathematical, visual- spatial, bodily- kinesthetic, interpersonal, intrapersonal

III. Special Needs/Modifications

- Define similar.
- Similar in the context of polygons.
- 1. Same number of sides
- 2. For each angle there is a corresponding angle in the other polygon that is congruent.
- 3. Lengths of all corresponding sides are proportional.
- Write all assignment directions on the board so that students can refer back to what is needed for each step.
- Use flexible grouping to assist students in understanding the activity.
- Ratios of similar perimeters- same as scale factor- be sure that students understand these two concepts.

IV. Alternative Assessment

- Walk around and listen in on group discussions.
- Interject important information, offer feedback or constructive criticism when needed.

## Similarity by AA

I. Section Objectives

- Determine whether triangles are similar.
- Understand AAA and AA rules for similar triangles.
- Solve problems about similar triangles.

II. Multiple Intelligences

- The technology integration is a nice way to differentiate this lesson by adding the interactive element of technology.
- In addition, after covering the material in the lesson, the shadow problems with the similar triangles are a fun way to help the students to gain a deeper understanding of the concepts in the lesson.
- You could have the students write their own problems and solve each other’s using diagrams and drawings.
- You could also take the students outside, use a tree or a flagpole, the height of a student, and their shadow to measure and actually create a real life problem.
- This can be a fun way to bring the outdoors, nature and real life into the math classroom.
- Intelligences- linguistic, logical- mathematical, visual- spatial, bodily- kinesthetic, interpersonal, intrapersonal

III. Special Needs/Modifications

- Walk students through the directions on how to use the technology.
- AAA Rule- if angles of a triangle are congruent to the corresponding angles of another triangle, then the triangles are similar.
- SAME AS: The AA Triangle Similarity Postulate except that it uses two angles and not three.
- Both are true and both work.
- Indirect measurement- provide students with a visual example of the two similar triangles and the proportions.

IV. Alternative Assessment

- Alternative Assessment in this lesson is done through observation and through interacting with the students during the activity.

## Similarity by SSS and SAS

I. Section Objectives

- Use SSS and SAS to determine whether triangles are similar.
- Apply SSS and SAS to solve problems about similar triangles.

II. Multiple Intelligences

- There are two great ways to differentiate this lesson included in the text. One is the technology integration and one is the hands- on activity.
- Here is how we can take the hands- on activity and expand it a bit to include an interactive part.
- Complete the hands- on activity as it has been written in the text.
- When finished, ask students to write a few observations that they have made into their notebooks.
- Then ask students to contribute their observations to a class discussion.
- Write all of the student observations on the board. Point out the SSS for Similar Triangles.
- SSS for Similar Triangles- if the lengths of the sides of the two triangles are proportional, the triangles are similar.
- SAS for Similar Triangles- use Cheryl’s examples to create your own example to illustrate the SAS for Similar Triangles.
- Intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal

III. Special Needs/Modifications

- Write the intention of the lesson on the board.
- Intention is to explore relationships between proportional side lengths and congruent angles of similar triangles.
- Review the directions for using the technology.
- Write the directions to the hands- on activity on the board.

IV. Alternative Assessment

- Observe student work through the technology integration activity.
- Observe student work through the hands- on activity.

## Proportionality Relationships

I. Section Objectives

- Identify proportional segments when two sides of a triangle are cut by a segment parallel to the third side.
- Divide a segment into any given number of congruent parts.

II. Multiple Intelligences

- Complete the technology integration as one way to differentiate this lesson.
- After working through the technology integration ask students to write their observations down in their notebooks.
- Conduct a sharing session and write student observations on the board.
- Midsegment of a Triangle- midsegment that divides the sides of a triangle proportionately.
- Activity- have the student use construction paper to design a triangle.
- Then using a ruler and a pencil, draw in the midsegment of the triangle.
- Ask students to make notes about the measurements of the triangle and how they have been altered with the midsegment.
- Then, use scissors to actually divide the triangle into two proportionate sections.
- Request that students write their observations down in their notebooks.
- The Lined Notebook Paper Corollary- demonstrate this as a whole class. Ask students to use notebook paper and a ruler.
- Write student observations on the board.
- Intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal.

III. Special Needs/Modifications

- Review ways to figure out if two triangles are similar using side lengths and angles.
- Define midsegment of a triangle on the board.
- Write out the Triangle Proportionality Theorem.
- Go through each step of each proof and explain each “Reason” as it is presented.
- Do not assume that the students remember previously learned information.

IV. Alternative Assessment

- Create an observation checklist to use during both activities.
- For the technology piece, make a list of things that you would like students to gain from the activity.
- For the hands- on piece, make a list of things that you would like students to gain.
- Notice during the discussion sessions whether or not these goals have been met.
- If not, make these points to the students and explain how they were discovered.

## Similarity Transformations

I. Section Objectives

- Draw a dilation of a given figure.
- Plot the image of a point when given the center of dilation and scale factor.
- Recognize the significance of the scale factor of a dilation.

II. Multiple Intelligences

- Teach the information in this lesson.
- Then use the information on dilations to complete the following activity.
- Divide the students into groups and have them measure the classroom.
- Each group is working on the same assignment. Having each group work on the same assignment will produce different diagrams to explain the same information.
- Students are going to complete a drawing that shows the measurement of the classroom. They can create a scale to represent the measurement of the room.
- Next, students create a dilation of the classroom where the scale factor is .
- Students need to draw a diagram of this and label it with the correct measurements.
- Then, students create a dilation of the classroom with a scale factor of .
- Next, students create a dilation of the classroom with a scale factor of .
- Then have students complete the area and the perimeter of the room.
- Find a perimeter with a scale factor of .
- Find an area with a scale factor of .
- Allow time for the students to share their work and any observations.
- Intelligences- linguistic, logical- mathematical, visual- spatial, bodily- kinesthetic, interpersonal, intrapersonal.

III. Special Needs/Modifications

- Review congruence transformations that preserve length- translations, rotations, and reflections.
- Define a dilation.
- Define scale factor.
- Review finding the perimeter of a figure.
- Review finding the area of a figure.

IV. Alternative Assessment

- Walk around and observe students as they work.
- Collect diagrams and use for a classwork or a quiz grade.

## Self- Similarity (Fractals)

I. Section Objectives

- Appreciate the concept of self- similarity.
- Extend the pattern in a self- similar figure.

II. Multiple Intelligences

- Differentiate this lesson by completing a construction of each of the fractals. Ue rulers and measure to ensure accuracy.
- Here are the steps.
- 1. Begin with a segment.
- 2. Divide the segment into three congruent parts.
- 3. Remove the middle part leaving two parts.
- 4. Divide each segment into three congruent parts.
- 5. Remove the middle part of each segment.
- Allow students to work in groups.
- You can extend this lesson by having students work with both horizontal lines and vertical lines.
- With the pattern for the Sierpinski Triangle, have students see which other polygons can be used to create a similar pattern.
- Allow students to work in pairs or groups on this assignment.
- Display student work.

III. Special Needs/Modifications

- Write the steps for each fractal on the board/overhead.
- Pattern for Sierpinski Triangle.
- 1. Draw a triangle.
- 2. Connect the midpoints of the sides of the triangle. Shade in the center triangle.
- 3. Repeat this process with each triangle.

### Image Attributions

## Description

## Authors:

## Tags:

## Categories:

## Date Created:

Feb 22, 2012## Last Modified:

Feb 23, 2012**You can only attach files to None which belong to you**

If you would like to associate files with this None, please make a copy first.