## Ratios and Proportions

I. Section Objectives

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

II. Cross- curricular-Greek Architecture

- Provide students with an image of the Parthenon from Wikipedia.
- This is Figure 08.01.01
- www.en.wikipedia.org/wiki/File:Parthenon-2008.jpg
- Then provide students with an image of the Acropolis from Wikipedia.
- www.en.wikipedia.org/wiki/File:AthensAcropolisDawnAdj06028.jpg
- Now use the images as a discussion about the golden ratio of approx. and how this is shown in the dimensions of each building.

III. Technology Integration

- Students can look at this website using The Golden Ratio and talking about how beauty has to do with ratios. Check it out first.
- www.intmath.com/Numbers/mathOfBeauty.php
- Students can also use this website which looks at ratios in nature.
- www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibInArt.html
- This website has a ton of different links for students to explore when looking at how ratios play into different topics.

IV. Notes on Assessment

- The content of this lesson is assessed through the discussion.
- You want students to understand that when you compare different facets, the ratios impact the design.
- With the golden ratio, the ratio is the same.
- Then you want the students to begin to make the connections on their own.
- Students can see the real life examples of ratios, especially the golden ratio.

## Properties of Proportions

I. Section Objectives

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

II. Cross- curricular- Astronomy

- Use the following map of the constellations in this activity.
- This is Figure 08.02.01
- www.nightskyinfo.com/sky_highlights/july_nights/july_sky_map.png
- Use the image of Ursa Major and Ursa Minor to explore the concepts of proportions.
- Are the two images in proportion?
- How can we tell?
- Complete an in class discussion on what makes two images or two ratios a proportion.
- What kinds of measurements would we need to prove that the two constellations were proportional?
- Encourage students to work with the concepts of proportions and apply it to the constellation map.

III. Technology Integration

- Students can use this youtube video to study the planets in proportion.
- www.youtube.com/watch?v=PZNrQGCEXzs
- Students can follow this up by researching and comparing two planets.
- Have them choose two to compare and write ratios and proportions to compare them both.
- Allow time for students to share their work when finished.

IV. Notes on Assessment

- Assess student work through the discussion and through student notes.
- Were the students able to decide how to write proportions and ratios on the planets and constellations?
- Then provide students with feedback on their work.

## 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. Cross- curricular-Model Design

- This is a great opportunity to include scale and design into the mathematics classroom.
- You can work with this lesson in two different ways.
- The first way is to have the students choose a polygon and to build a model of two polygons that are similar using a scale model.
- This way, the students can actually have a hands- on experience of figuring out the dimensions of a scale model and then put these measurements to work building the model.
- The second way is to choose a mountain or a building for the students to use to create a scale design or model of.
- For example, if you chose the Empire State Building, the students would figure out the actual measurements, and then build a model or draw a design using a scale.
- You could do per foot, etc.
- Allow time for students to share their work when finished.

III. Technology Integration

- Students can go to the following website to explore similar polygons.
- www.saskschools.ca/curr_content/byersjmath/geometry/students/polygon/intmovie.html
- When the students go to this website, they need to go to the section on similar polygons.
- From there, they can watch the animation which explains all how to determine similar polygons and how to create similar polygons.

IV. Notes on Assessment

- Check student work for accuracy.
- Is the scale accurate?
- Does the model or design match the scale?
- Do the students have a good understanding of similar polygons?
- Provide students with correction/feedback on their work.

## 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. Cross- curricular-Pyramids

- This lesson will work best with the technology integration.
- Have students complete the study of Thales first and then move to a hands- on activity.
- Once students have selected a pyramid, they are going to work on this activity.
- Students are going to use the researched dimensions of the pyramid to build a model to scale.
- Students can build this model out of sugar cubes and glue.
- Sugar cubes tend to work well.
- After completing the model, use a darkened room and a high powered flashlight to demonstrate the shadow of the pyramid.
- Is it accurate according to Thales?
- See if the students can develop a way to test out this theory.
- Allow time for students to share their work when finished.

III. Technology Integration

- Have students complete some research on Thales and on indirect measurement.
- Students can read about Thales at the following website.
- www.phoenicia.org/thales.html
- Conduct a discussion on Thales and on how he discovered and figured out the height of the pyramids using indirect measurement.
- Once students have a good understanding of this, move on to the next part of this lesson.
- Then have the students do a search and choose a pyramid.
- Students are going to use the dimensions of this pyramid to build a model.

IV. Notes on Assessment

- Assess student work through discussion and observation.
- Do the students understand who Thales was and the significance of his discovery?
- Is the student model to scale?
- Were the students able to come up with a way to test Thales’ findings?
- What are students sharing about this assignment?
- Is higher level thinking involved?
- Provide students with feedback as needed.

## 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. Cross- curricular-Literature/Poetry

- In this activity, students need to create a poem, song or story that explains the three ways to figure out if two triangles are similar.
- The first is AA- angle angle
- The second is side- side- side.
- The third is side- angle- side.
- You can begin this lesson by reviewing the definitions of each and how to use them to figure out if two triangles are similar.
- Then divide students into groups of three.
- Have the groups work on their expression of figuring out if two triangles are similar.
- When finished, allow time for the students to share their work.

III. Technology Integration

- Students can go to the following class zone website and see the animation on similar triangles.
- www.classzone.com/cz/books/geometry_2007_na/get_chapter_group.htm?cin=2&rg=animated_math&at=animations&var=animations
- This is a fun interactive way to see the work done.
- Because class zone is affiliated with another textbook, the students can have a difficult time navigating the site.
- Use the link above for it.
- This will bring the students to the animation.
- If you don’t wish to use class zone, students can also go to futureschannel.com and see a short movie on triangles and architecture.

IV. Notes on Assessment

- Assess each group’s poem or story.
- Does it explain how to figure out if triangles are similar?
- Is each theorem well explained?
- Provide students with feedback as needed.

## 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. Cross- curricular-Proportional Divisions

- Have students participate in a hands- on activity to explore the section objectives.
- Students are going to work with several different triangles.
- The triangles should all be the same size.
- You can either prepare the triangles ahead of time or have the students cut them out themselves.
- Then have students work in small groups.
- In each group, the students are going to explore the proportional segments that are created when two sides of a triangle are cut by a segment parallel to the third side.
- They should try this will three different line segments each parallel to a different side.
- This means that the activity will get repeated with three different triangles.
- The students need to measure each side and write proportions to represent the different sections of the triangle.
- For example, when the triangle is cut, there are two polygons- how do the side lengths compare? Are they in proportion?
- Students need to make notes on these comparisons and share them with the other students.

III. Technology Integration

- Use Wikipedia to explore the concept of proportionality.
- www.en.wikipedia.org/wiki/Proportionality
- Students can look at proportionality in mathematics, but also in human design and architecture.
- There are several different links to explore.

IV. Notes on Assessment

- Assess student understanding by observing their work in small groups.
- Were the students able to successfully cut the triangles into proportions?
- Were they able to write proportions that demonstrate that the two polygons are similar?
- Provide feedback as needed.

## 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. Cross- curricular- Art

- The name of this activity is “Honey I Shrunk the Polygon!”
- Students are going to take any polygon that they would like to and create an art piece that shows the dilations of the polygon.
- The polygon that is the beginning polygon should be in red.
- That way you can tell which polygon is being transformed.
- Students should create dilations which are smaller and larger.
- The scale factor can be decided by the student.
- The scale should be the same whether the polygon is being dilated smaller or larger.
- Allow students time to work.
- Display student work when finished.

III. Technology Integration

- To look at different dilations, students can do some research on Christmas Tree Farms.
- Because farms often use the same kind of tree, there will be small versions of the tree and large versions of the tree.
- This is a real life look at dilations.
- Students can do some work drawing different trees.
- Have them choose one to begin with and then dilated two or three times.
- This will show a “growth progression” of the tree.

IV. Notes on Assessment

- Ask the students to share their dilated polygons.
- What works about the polygon and what doesn’t work?
- Is there an accurate scale factor?
- Are both images correctly dilated?
- Provide students with feedback.

## Self- Similarity (Fractals)

I. Section Objectives

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

II. Cross- curricular- T-shirt Design

- Review the concept of fractals and what makes a fractal image.
- Then show students the image on this website.
- This is Figure 07.08.01
- www.redbubble.com/people/archimedesart/art/3390955-4-bright-lights
- Then show students this second fractal.
- This is Figure 07.08.02
- www.zazzle.com/right_angles_tshirt-235230222951842274
- Discuss these fractals with the students.
- Notice the quadrilaterals in the image.
- This is a T- shirt design.
- Have students design their own fractal t- shirt.
- This can be as complicated or simple as you wish.
- Students can use fabric paint and fabric markers to actually draw their fractal on their shirt.
- They could also create a pattern with a piece of cardboard and then use fabric paint to paint over the image and have it displayed on the shirt.

III. Technology Integration

- Have students research vegetable fractals.
- There are so many interesting images of fractals.
- Ask the students to select a few and write about why they chose the one that they did.
- Also, ask the students to explain, to the best of their ability, how the image is a fractal.
- What characteristics/qualities make it a fractal?
- Allow time for students to share their thinking when finished.

IV. Notes on Assessment

- When looking at student t-shirt designs, you are looking for a representation of a fractal.
- This can be assessed by looking at each t- shirt.
- Provide students with feedback when finished.

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