<meta http-equiv="refresh" content="1; url=/nojavascript/"> Similarity | CK-12 Foundation

# 4.7: Similarity

Created by: CK-12

## 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: $\frac{50}{25}= \frac{2}{1}$
• Expand the equation example with the dancers and the singers.
• “What if the ratio of dancers to singers was $5$ to $4$ and there were forty- five dancers? How many singers are there? How many dancers are there?
• Answer: $5n = \mathrm{dancers}$
• $4n = \mathrm{singers}$
• $5n+4n=45$
• $9n=45$
• $n=5$
• $5(5)=25 \;\mathrm{dancers}$
• $4(5)=20 \;\mathrm{singers}$
• Proportion- an equation that compares two equal ratios.
• Expand Barn Dimensions example.
• “What if the water line was actually $20 \;\mathrm{ft}$ instead of ten? What would the length be on a scale drawing?”
• Answer$x=5 \;\mathrm{inches}$
• 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

• 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 $4- \frac{a+b}{b} = \frac{c+d}{d}$
• Corollary $5- \frac{a-b}{b} = \frac{c-d}{d}$

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 $\frac{1}{2}$.
• 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 $\frac{1}{4}$.
• Next, students create a dilation of the classroom with a scale factor of $3$.
• Then have students complete the area and the perimeter of the room.
• Find a perimeter with a scale factor of $\frac{1}{3}$.
• Find an area with a scale factor of $\frac{1}{3}$.
• 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.

## Date Created:

Feb 22, 2012

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.