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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
  • Students can read all about vectors and isometry.
  • 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.
  • Add 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.
  • Ask for students who are knowledgeable about skateboarding.
  • 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.
  • Draw your own tessellation.

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.

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