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# 4.12: 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. Multiple Intelligences

• To differentiate this lesson, make it very interactive.
• Be sure that students have graph paper, colored pencils and rulers at their seats. Work through this lesson on the overhead projector with graph paper yourself so that the students can model the examples and work them out themselves at their seats.
• Begin by reviewing some information about translation.
• Have students draw a translation. Use the example from the text or create one of your own.
• Use the distance formula as was done in the text to review finding the coordinates to graph. Go the extra step and graph an example with the students.
• Define isometry- explain how the distance between the two points of an image is the same as the distance between the two images.
• When you look at the example that the students have just drawn, illustrate this.
• Then name it with the Translation Isometry Theorem.
• Move on to vectors. Begin by having students graph two line segments on a coordinate grid. You can use the same line segments as in Example 2, or you can create your own.
• The key is that you want the students to understand that the vector is the horizontal and vertical direction connected with each graphed line segment.
• Drawing in vectors is a great opportunity for students to use color to differentiate the vectors.
• Then ask students to name the horizontal component and the vertical component of the graphed line segments.
• Expand Example 3. Read through it with the students and then give them time to graph the two triangles and to explore what happens to the triangles.
• Allow time for student sharing.
• Intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal

III. Special Needs/Modifications

• Review translations and what makes a translation.
• Review using the distance formula with a translation.
• Define isometry.
• Define the Translation Isometry Theorem.
• Define Vector.

IV. Alternative Assessment

• Assess student understanding through discussion.
• Allow time for student feedback.
• Assess understanding again with the work done to expand Example 3. How did the students do with this? Do they understand vectors?
• What conclusions did they draw from the example?

## Matrices

I. Section Objectives

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

II. Multiple Intelligences

• Begin by teaching the material in the lesson. Then move on to the activity.
• Students will need graph paper, rulers and colored pencils.
• Students can work with a partner for this activity.
• Ask students to draw a triangle, a square or a rectangle on the coordinate grid.
• Then have them create a matrix of the coordinates of their polygon.
• Next, students are going to create a translation of the polygon that is two units down and three units to the right.
• Note if this doesn’t work with the student’s image change it to two units up and three units to the left.
• Then have the students design a matrix to represent this translation.
• Finally students will add the two matrices together.
• Ask them to exchange papers with a peer for a check of their work.
• After their peer review, make any necessary changes.
• Allow time for student’s to share their work.
• Intelligences- linguistic, logical- mathematical, visual- spatial, bodily- kinesthetic, interpersonal.

III. Special Needs/Modifications

• Notes on Matrices
• 1. A multidimensional way to show data.
• 2. They have their own arithmetic.
• 3. In brackets, a matrix is an array of numbers.
• 4. Numbers are arranged in rows and columns.
• You add the elements of a matrix by adding the value in each place in one matrix with the matching value in the same place in the other matrix.
• Matrices can represent real- life data.
• Matrices can represent the vertices of a polygon.
• Operation with matrices and translations = ADDITION

IV. Alternative Assessment

• Collect student work and use it as a way to check student understanding.

## 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. Multiple Intelligences

• To differentiate this lesson, teach the material in the lesson first, and then use this activity to give the students a hands- on way of practicing multiplying matrix reflections.
• Have the students work in pairs.
• They will need graph paper, rulers and colored pencils.
• Students may choose a polygon and draw it on the coordinate grid.
• Then have them show that it is reflected by the line $y = x$.
• Students take the vertices of their polygon to create a matrix for it.
• Then multiply the matrix of the polygon by the matrix represented by $y = x$.
• Finally, students show the product in a new matrix.
• Allow time for student sharing in a whole class discussion or in small groups.
• Intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal.

III. Special Needs/Modifications

• Operation with reflections = MULTIPLICATION
• Matrix multiplication-
• Multiply firsts by firsts and seconds by seconds- then add the products
• You can’t multiply a smaller matrix by a larger one.
• You can multiply a larger matrix by a smaller one.

IV. Alternative Assessment

• Observe students as they work through the assignment.
• Offer assistance when necessary.
• Check for student understanding when discussing the activity in the whole class discussion.
• If students are discussing in small groups, walk around and check in with them.

## 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. Multiple Intelligences

• To differentiate this lesson, teach the material in the lesson first, and then let students practice working with rotations.
• The students are going to play a game called “Pass the Image”
• To prepare this activity, you will need to prepare some small square images. You can design a square where it is divided on the diagonal and one- half of it is blue and the other half is red, etc.
• Students are going to be given an image. They need to draw the image according to the rotation specified in the instruction.
• For example, a student is given an image card, they start by drawing the image as it is.
• Then they are told to draw it at $180^\circ$ rotation.
• Then they draw it at a $90 \;\mathrm{degree}$ rotation.
• A $45 \;\mathrm{degree}$ rotation.
• A $270 \;\mathrm{degree}$ rotation.
• Then when finished, they pass the image and are given a new one.
• You can do this several times and the students can then compare their work with other students who had the same images.
• Break up students in pairs to have them compare and discuss their work.
• Next, you can move to working with an image on the coordinate plane.
• You can have students draw their own or work with the exercises in the text.
• If you do this activity first, the students will have an excellent understanding of a rotation before moving to the coordinate grid.
• Intelligences- linguistic, logical- mathematical, interpersonal, intrapersonal, visual- spatial, bodily- kinesthetic.

III. Special Needs/Modifications

• Review the basics of matrices.
• Review how to multiply a matrix.
• Review how to draw a matrix using the vertices of a polygon.
• Define rotation
• 1. Center at the origin with an angle of rotation of $n^\circ$.
• 2. Point moves counterclockwise along an arc of a circle.

IV. Alternative Assessment

• Create a “key” of what each image looks like after it is rotated.
• Then use this key to check student 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. Multiple Intelligences

• Begin by introducing the concept of a composition.
• A composition is when transformations are “put together”. In this lesson, we will be putting together translations, reflections and rotations.
• Glide Reflection- a composition of a reflection and a translation. The translation is in a direction parallel to the line of reflection.
• Expand Example 1- Before moving to the matrix, have the students draw out this glide reflection. This will give them a hands- on way to see the two images without first moving to the matrices. This will keep it in a visual way, before moving to an arithmetic way.
• Once students have practices drawing in the glide reflection, move to using the matrix to figure out the same information. At this point, you can refer back to the text.
• The technology integration in this chapter is also a great way to provide students with a visual and hands- on way of working with the material.
• Provide time for feedback, discussion and questions after completing the work with technology.
• Intelligences- linguistic, logical- mathematical, bodily- kinesthetic, visual spatial, interpersonal.

III. Special Needs/Modifications

• Review translations, reflections and rotations.
• Review matrices.
• Review the matrix for a $180^\circ$ rotation $- \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \end{bmatrix}$
• Review the matrix for a $90^\circ$ rotation $- \begin{bmatrix} 0 & 1 \\ -1 & 0 \\ \end{bmatrix}$

IV. Alternative Assessment

• Allow plenty of time for the students to ask questions during this lesson.
• Spend time on reviewing previously learned skills (ie. How to multiply a matrix) if necessary.

## 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. Multiple Intelligences

• This is a fun lesson and students usually love working with tessellations.
• Provide students with some information on tessellations and then have them work on two activities.
• The first activity, you will need to prepare.
• Students will work in groups, so you will need a few polygons/shapes for each group. If you can provide different polygons/shapes for each group- great.
• Give each group their polygons and tell them that they will need to prove whether each one tessellates or not.
• Students need to demonstrate that it has no gaps, no overlapping shapes, that the entire plane is covered in all directions.
• Also have students demonstrate that it surrounds a point.
• Allow students time for this exploration and then have students share their work.
• The second activity is to have students create their own tessellation.
• Encourage students to be creative and design a colorful tessellation of their own creation.
• Intelligences- linguistic, logical- mathematical, visual- spatial, bodily- kinesthetic, interpersonal, intrapersonal.

III. Special Needs/Modifications

• Write the notes of how to tell if a shape tessellates on the board.
• Provide students with a few visual examples of some tessellations.
• Have them use actual pattern blocks to explore.

IV. Alternative Assessment

• You will be able to tell if the students executed the concept well by looking at their tessellations.
• Be sure to include design and color in your evaluation.
• Also check the edges of the plane- did the students successfully fill- in partial images?

## 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. Multiple Intelligences

• To differentiate this lesson, divide the students into groups.
• Ask each group to come up with an example to explain line symmetry, rotational symmetry, point symmetry and planes of symmetry.
• When finished, have each group share their images.
• Then move on to the next part of the activity.
• Ask each group to draw half of an image that has line symmetry, rotational symmetry, point symmetry. Students can use objects in the room to help them brainstorm which image to draw for each. Students may use examples from biology as well.
• Then they are going to pass their papers to a group near them.
• The next group must finish the drawings according to each description.
• When finished, allow time for sharing.
• Intelligences- linguistic, logical-mathematical, visual- spatial, bodily- kinesthetic, interpersonal, intrapersonal.

III. Special Needs/Modifications

• Two- dimensional
• 1. Line symmetry- left- right symmetry. Divides the figure into two congruent halves. When flipped over the line of symmetry, it is exactly the same.
• 2. Rotational symmetry- rotated image looks exactly like it did before the rotation.
• 3. Point symmetry- looks the same right side up and upside down. It looks the same from the left and from the right.
• Three dimensional
• Planes of symmetry- divide a $3D$ figure into two parts that are reflections of each other. Think of a cylinder or cube.

IV. Alternative Assessment

• There are several ways to assess student understanding in this lesson.
• The first way is with the images to represent each type of symmetry.
• The second is with the partial images and completions.
• Collect student work and check to see that student work is complete and accurate.
• Provide students with feedback or corrections.

## Dilations

I. Section Objectives

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

II. Multiple Intelligences

• To differentiate this lesson, begin by teaching the concepts in the lesson to the students.
• Then, students are going to create their own dilations using scalar multiplication.
• Students will need graph paper, rulers and colored pencils.
• Ask the students to show all of their work.
• Here are the steps to the activity.
• 1. Draw a polygon of choice on the coordinate grid.
• 2. Use the vertices of the polygon to create a matrix.
• 3. Select or use a given scale factor.
• 4. Multiply the scale factor with the matrix.
• 5. The product is a new matrix- the new matrix is the vertices of the dilated matrix.
• 6. Draw in the figure on the coordinate grid.
• Allow time for the students to share their work when finished.
• Intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal, intrapersonal

III. Special Needs/Modifications

• Review that a dilation is an image “blown up” or decreased in size.
• Transformations are also dilations.
• Dilations can be written as a matrix.
• Review scale factor.
• Scalar Multiplication- Take the real number and multiply it with each element in a matrix. The product is a new matrix.
• To create a dilation on the coordinate grid
• 1. Design a matrix based on the vertices of a polygon drawn on the coordinate grid.
• 2. Decide on a scale factor for the dilation.
• 3. Multiply the scale factor with the matrix.
• 4. The product is a new matrix that is the vertices of the dilated figure.
• 5. Draw in the new figure on the coordinate grid.

IV. Alternative Assessment

• Collect student work.
• Check to be sure that the scalar multiplication is accurate.
• Be sure that the images match the elements of each matrix.
• Assign students a classwork grade based on their work.

## Date Created:

Feb 22, 2012

Feb 23, 2012
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