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3.7: 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. 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. 1.6 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 1” 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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