## 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.

### Image Attributions

## Description

## Authors:

## Tags:

## Categories:

## Date Created:

Feb 22, 2012## Last Modified:

Apr 08, 2013**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.