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

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.

Image Attributions

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.

Reviews

Please wait...
You need to be signed in to perform this action. Please sign-in and try again.
Please wait...
Image Detail
Sizes: Medium | Original
 
CK.MAT.ENG.TE.1.Geometry.3.7

Original text