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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

  • 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 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.

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