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# 3.12: Transformations

Created by: CK-12

## Translations

I. Section Objectives

• Graph a translation in a coordinate plane.
• Recognize that a translation is an isometry.
• Use vectors to represent a translation.

II. Cross- curricular-Sculpture

• Use this image of Rinus Roelof’s tetrahedron sculpture.
• This is Figure 12.01.01
• www.mathpaint.blogspot.com/2007/04/structures-by-rinus-roelofs.html
• Use this to show the students the vectors that can be drawn from one tetrahedron to the next tetrahedron.
• This shows length and direction.
• Discuss the various components of the sculpture.
• Ask students to identify all of the different elements of the sculpture.
• Students can then draw a design of their own using different three- dimensional solids or one solid as Roelof did.
• Have students identify any and all solids as well as the vectors in the design.
• Allow time for the students to share their designs.

III. Technology Integration

• This is a great website to explore translations.
• www.cut-the-knot.org/Curriculum/Geometry/Translation.shtml
• Then there is an interactive section where the students can manipulate the figure in the box.
• Students can use this to demonstrate their understanding.
• Have students work in pairs on this task.

IV. Notes on Assessment

• Assess student understanding through discussion.
• Ask questions to be sure that the students understand the key elements of this lesson.
• They should have an understanding of isometry, vectors and translations.

## Matrices

I. Section Objectives

• Use the language of matrices.
• Apply matrices to translations.

II. Cross- curricular-Progressive Matrices

• This website has an image of a progressive matrix.
• www.en.wikipedia.org/wiki/File:RavenMatrix.gif
• Have the students discuss the elements of how this image is representative of a matrix.
• Then have them use this as a model to create their own matrix pattern.
• Students should complete at least three steps of the progressive matrix.
• Students can use black and white or color.
• Students could expand this idea into a design with dimensions.
• Students can also use paint chips from a hardware store, or small mosaic tiles.
• Students could use elements of nature such as rocks or small leaves.
• This could be a very creative assignment.
• Allow time for the students to share their work when finished.

III. Technology Integration

• Complete a research assignment on how the banking industry uses matrices.
• Students will need to visit several different websites to do this.
• Have them write a short essay and include examples on how the banking industry relies on matrices to support their work.

IV. Notes on Assessment

• Assess each design.
• Is it modeled off of the example on Wikipedia?
• Does it show a progression?
• What would be the next step in the progression?
• Provide students with feedback on their work.

## Reflections

I. Section Objectives

• Find the reflection of a point in a line on a coordinate plane.
• Multiple matrices.
• Apply matrix multiplication to reflections.
• Verify that a reflection is an isometry.

II. Cross- curricular-Art

• Students are going to create a reflection.
• They can choose a picture, a symbol, a shape or an image.
• The key thing is that they can reproduce it as a reflection.
• The students are going to show how this image is reflected in a horizontal or vertical plane.
• They can work in small groups on this.
• The task will involve spatial thinking and organization to be sure that the students can “see” the correct positioning of the image.
• Then they need to reproduce this.
• Students can choose to use as simple or as complicated an image as they choose.
• The key is that they need to be able to explain their work and have it be accurate.
• Allow time for students to share their work when finished.

III. Technology Integration

• Have students use the following website to work on reflections.
• www.mathwarehouse.com/transformations/compositions/reflections-in-math.php
• The website provides a tutorial on how to create a reflection.
• It also provides students with an interactive way to work on reflections.
• Students can practice designing reflections.
• Provide an opportunity for students to ask questions as they work.

IV. Notes on Assessment

• Check student work on reflections.
• Is the reflection accurate?
• Is there anything missing in its representation?
• Is the image too complicated?
• Provide students with feedback on their work.

## Rotations

I. Section Objectives

• Find the image of a point in a rotation in a coordinate plane.
• Recognize that a rotation is an isometry.
• Apply matrix multiplication to rotations.

II. Cross- curricular- Sports

• Provide students with three or four copies of this image of a skateboarder.
• www.en.wikipedia.org/wiki/File:Skateboarder1.jpg
• This is Figure 12.04.01.
• Tell the students that they are to use these images to create a scene showing the rotations of a skateboarder.
• Students can create this any way that they choose.
• Pair these students up with students who don’t consider themselves knowledgeable.
• Then have the students work together to create the scenes.
• Students can show as many different rotations as they would like.
• Be sure to give students an opportunity to share their work.
• Some students may want to extend this scene to include other skateboarding images- that is fine as long as the concept of rotations is included.

III. Technology Integration

• A great website to explore rotations.
• www.cut-the-knot.org/Curriculum/Geometry/Rotation.shtml
• Students can review information on rotations here.
• Then they can work to manipulate and create different rotations.
• There are directions on the screen which help them in accomplishing this task.
• You can use this as extra practice or for a student who needs remedial work in this area.

IV. Notes on Assessment

• Assess the student rotation scenes.
• Did the students accomplish the task of showing the skateboarder in different rotations?
• If not, what would have worked better?
• Did the students expand on the assignment?
• Were the students able to explain the use of rotations in their scene?
• Provide students with feedback on their work.

## Composition

I. Section Objectives

• Understand the meaning of composition.
• Plot the image of a point in a composite transformation.
• Describe the effect of a composition on a point or polygon.
• Supply a single transformation that is equivalent to a composite of two transformations.

II. Cross- curricular-Movie Posters

• Have students watch the video first.
• Discuss the elements of a great movie poster.
• What works and what doesn’t work.
• Tell the students that their job is to create a movie poster for a new movie.
• You can use one that is popular with the students at this time or use an old favorite like “Star Wars” that probably all of the students have seen.
• Tell the students that they are going to create a poster for this movie using the elements of transformations.
• There needs to be a use of rotation, translation and reflection in their posters.
• Students can work in pairs on this task.
• Have students share their work when finished.

III. Technology Integration

• If possible, have the students watch this short video first.
• www.thefutureschannel.com/dockets/hands-on_math/movie_posters/
• You want the students to be looking for elements or transformations in the posters.
• Students are going to use this information in the activity.

IV. Notes on Assessment

• Assess student work based on the student’s use of transformations.
• Are there rotations in the poster?
• Are there translations in the poster?
• Are there reflections in the poster?
• Did the student focus on one or all of the elements?
• How successful were they?
• Provide students with feedback on their work.

## Tessellations

I. Section Objectives

• Understand the meaning of tessellation.
• Determine whether or not a given shape will tessellate.
• Identify the regular polygons that will tessellate.

II. Cross- curricular-Honeycombs

• Show students the following images of honey combs.
• This is Figure 12.06.01.
• www.en.wikipedia.org/wiki/File:Cubic_honeycomb.png
• This is Figure 12.06.02
• www.en.wikipedia.org/wiki/File:Honey_comb.jpg
• This is Figure 12.07.03
• www.en.wikipedia.org/wiki/File:Apis_florea_nest_closeup2.jpg
• Tell students that their task is to create a honeycomb piece of art.
• They can use any shape that will tessellate as they saw with the cubic honeycomb.
• The key is that the honeycomb, according to a Wikipedia definition, is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps.
• Students can use any size that they choose and can incorporate color too.
• They will need rulers, pencils, colored pencils or markers, paper and scissors.

III. Technology Integration

• Have students study the work of M.C. Escher who was famous for his tessellations.
• www.en.wikipedia.org/wiki/M._C._Escher
• They can begin with the Wikipedia site, but there are so many other sites to work with as well.
• Have the students select one piece of his work as a favorite piece and share in small groups the elements that tessellate and how they tessellate.
• Conduct a small group discussion on the power of tessellations.

IV. Notes on Assessment

• Assess student honeycombs.
• Were they successful in their tessellations?
• Provide students with feedback on their work.

## Symmetry

I. Section Objectives

• Understand the meaning of symmetry.
• Determine all the symmetries for a given plane figure.
• Draw or complete a figure with a given symmetry.
• Identify planes of symmetry for three- dimensional figures.

II. Cross- curricular-Puzzle Creation

• Review the basics of symmetry with the students.
• Have them define symmetry and describe what makes something symmetrical.
• Review symmetry in nature or in other objects or buildings.
• Then assign students the task of taking a symmetrical image and making it into a puzzle.
• Students can use an image from a magazine, a computer image, or a hand drawn image.
• They are going to use cardboard to create a puzzle.
• They can make it as simple or complex as they wish.
• Have students create their puzzle and then exchange puzzles with a peer and work to assemble the other person’s puzzle.
• Allow time for students to share their work when finished.

III. Technology Integration

• Explore the sculptures of Quark Park on the following website.
• www.symmetrymagazine.org/cms/?pid=1000396
• Have the students work to discuss each different sculpture in small groups.
• Students should make notes on the symmetrical elements of each sculpture and be prepared to share them with the class.

IV. Notes on Assessment

• Assess student work through observation.
• Observe students as they create their puzzles.
• Inquire into how symmetry can assist someone in creating or putting together a puzzle.
• Then listen in as students discuss the symmetrical elements of the sculpture in Quark Park.

## Dilations

I. Section Objectives

• Use the language of dilations.
• Calculate and apply scalar products.
• Use scalar products to represent dilations.

II. Cross- curricular-Dilations in context

• Ask the students to think about the concept of dilations and to come up with one career where people would use dilation in their work.
• Students need to write a hypothesis on how they think that this profession would use dilations.
• Have them write down their hypothesis.
• Ask students to make a list of questions that they are going to explore.
• If you have use of technology, complete this with the use of the computer.
• If not, visit the school library so that students can research there.

III. Technology Integration

• Have students research their chosen profession.
• They need to prove that their hypothesis is true or not.
• Each student should have reasons and explanations on how dilations are used in the chosen profession.
• Students need to write a short essay and provide one diagram or image to support their findings.
• Ask students to keep track of websites that they visit for documentation purposes.
• Allow time for students to present their work when finished.

IV. Notes on Assessment

• Assess student work.
• Was the student able to prove their hypothesis?
• What corrections did he/she make?
• Is the essay well written?
• Does it explain how this profession uses dilations?
• Does the diagram support student research?
• Provide students with feedback on their work.

## Date Created:

Feb 22, 2012

Apr 08, 2013
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