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1.12: Transformations

Created by: CK-12

Translations

Pacing: This lesson should take one class period

Goal: The purpose of this lesson is to introduce the concept of translations (slides) in the coordinate plane.

Vocabulary! The word isometry refers to any type of transformation…

Did You Know? Translations are formed by a composite of reflections over parallel lines. To illustrate this in further detail, see the Reflections lesson.

Vectors Outside the Coordinate Plane! Vectors can be used to translate an object not on a coordinate plane. A vector in this case tells the length and direction to translate a figure. Use the diagram below.

The black ray represents the vector. Therefore, \triangle ABC should be translated NNE the length of the vector. The resulting translation is pictured below.

Additional Example:

Suppose A’B’C’D’ is the image of ABCD under a translation by vector c = (-8, 3). What are the vertices of the preimage is the image has vertices at the following locations: A’= (0,0), B’=(1, 5), C’=(-4,4), D’=(-7, -10)? A=-8,- 3), B=(9, 2), C=(4, 1), D=(-1, -13)

Matrix Addition

Pacing: This lesson should take one class period

Goal: Matrices are useful in geometry, as well as algebra and business. This lesson introduces students to the basics of matrices, namely, matrix addition.

Basics of Matrices! Matrices are referred to according to its dimensions, rows by columns. To get students thinking about which is which, use this phrase. “You row ACROSS a lake and columns hold UP houses.” By relating rowing across and columns up, students should correctly organize the information.

Matrices can only be added if the dimensions are equivalent. Because adding matrices requires adding the same cell, there must be equal numbers to combine.

Excel spreadsheets are excellent examples of matrices. If you have the ability to set up such spreadsheet matrices, students can see how businesses use these to organize and manipulative inventory.

Vocabulary! A 2 \times 1 matrix organizing a point is called a point matrix. The x-values should go into row 1 and the y-values should go into row 2. The columns represent the points of the figure in the coordinate plane.

Additional Example: Target is processing its baby items inventory. Arrange the following into a matrix. Shirts: 24 - 2T, 0 - 3T, 9 - 4T; shorts: 5 - 4T, 17 - 2T, 11 - 3T; pants: 8 - 3T, 0 - 4T, 3 - 2T.

Suppose another Target is shipping its excess inventory to this store. Write the sum of the two shipments into a single matrix.

2T 3T 4T
Shirts 25 6 7
Shorts 8 19 12
Pants 1 4 30

Reflections

Pacing: This lesson should take one class period

Goal: Reflections are an important concept in geometry. Many objects can be explained with reflections. This manipulation is also related to similarity and triangle congruence.

Matrix Multiplication! Matrix Multiplication can be quite difficult for students to compute by hand. Here is a way to use a graphing calculator to achieve the multiplication.

  1. Find the Matrix menu. If your students are using a Texas Instrument product, it is located by typing the 2^{\mathrm{nd}} and x^{-1} keys.
  2. Edit matrix A by moving to the right to the edit menu and choosing [A]. Input the dimensions and data.
  3. Choose 2^{\mathrm{nd}} & MODE to quit the menu
  4. Repeat steps 1 – 3 for the second matrix [B].
  5. Choose matrix [A] by repeating step 1 and touching enter under the Name Menu
  6. Choose the multiplication symbol
  7. Repeat step 7 but choose [B] instead.
  8. Your working menu should look like this: [A]*[B]
  9. Touch ENTER. The answer resulting is the product of the two matrices.

Extension In-Class Activity! Reflections can be performed without a coordinate plane, just as translations.

  1. Using patty paper (or tracing paper), have students draw a small scalene triangle (\triangle DEF) on the right side of the paper.
  2. Fold the paper so that \triangle DEF is covered.
  3. Trace \triangle DEF.
  4. Unfold the patty paper and label the vertices as D’, E’, and F’, the image points of the D, E, and F.
  5. Darken in the fold – this is the reflecting line. Label a point on this line Q.
  6. Use a ruler to draw \overline{FF'}. Mark the intersection of the reflecting line and \overline{FF'} point M.
  7. Measure the FM and F’M. What do you notice about these distances?
  8. Measure \angle FMQ and \angle F’MQ. What do you notice about these measurements?

Extension - Reflections and Translations! Use the diagram below. Reflect \triangle CAT over line m, obtaining \triangle C’A’T’. Now reflect \triangle C’A’T’ over line n, obtaining \triangle C”A”T”. The resulting image is a translation of the preimage, double the distance between the parallel lines.

Rotations

Pacing: This lesson should take one class period

Goal: Rotations are also an important concept in geometry. Tires rotate in 360\;\mathrm{degree} increments, as do the hands on a clock. This lesson presents the concept of rotations and how matrix multiplication is used to compute the image points.

Extension – Reflections and Rotations! Just as translations are a composite of reflections over parallel lines, rotations are a composite of reflections. The only difference is that rotations occur when the reflecting lines intersect. Have your students complete the following:

Using the diagram below, reflect ABCD over line f, resulting in A’B’C’D. Reflect A’B’C’D’ over line e, resulting in A”B”C”D”. The final image represents a rotation of the preimage double the acute angle formed by the intersecting reflecting lines.

Composition

Pacing: This lesson should take one class period

Goal: This lesson introduces students to the concept of composition. Composition is the process of applying two (or more) operations to an object. This concept is also in Advanced Algebra.

Look Out! Students can get easily confused when applying compositions. They may attempt to perform the composition from left to right, as in reading a sentence. Point out to the students they must begin with the object and, according to the order of operations, should perform the operation occurring within the parentheses first.

Notation! Composition notation can take two forms. To write the reflection of ABCD over line m following a reflection over line n, you could:

A. Write r_n \ ^\circ \ r_m (ABCD)

B. Write r_n(r_m(ABCD))

The lowercase “r” stands for reflection and the subscript refers to the reflecting line.

Watch Your Feet! Your feet are a prime example of glide reflections. When you walk, one foot is translated above the other and are reflected about your body’s center line.

Tessellations

Pacing: This lesson should take one class period

Goal: This lesson introduces students to how tessellations are formed and which type of polygons will tessellate the plane.

Research! Have your students research M.C. Escher, an artist who has designed numerous pieces of artwork using tessellations. Have each student choose a piece of artwork, outline its preimage figures and give a short presentation.

Create Your Own Escher Print! An activity many students love to do is designed an unique piece of art. Complete the following steps:

  1. Cut out a 2” square from a sheet of copy paper.
  2. Draw a curve between two consecutive vertices. Be careful to not cut off a vertex!
  3. Cut out the curve and slide the cutout to the opposite side of the square and tape it in place.
  4. Repeat this process with the remaining two sides of the square.
  5. This is your template, or preimage. Begin with an 11” \times 14” piece of copy paper. Trace your preiamge and continue the pattern by rotating and translating until you cover the entire sheet.
  6. Color and post for a bulletin board.

Did You Know? The first person to discover how to tessellate with a pentagon was….

Symmetry

Pacing: This lesson should take one class period

Goal: The purpose of this lesson is to introduce the concept of symmetry. Students have experienced symmetry in previous lessons: isosceles triangles and various quadrilaterals. This lesson incorporates two-dimensional and three-dimensional figures and their lines of symmetry.

Maximum Points! Have your students write the alphabet in uppercase letters. Using one colored pencil, show which letters possess horizontal symmetry by drawing in the symmetry line. For example, B has a line of symmetry, as does E and K. Using a second color, draw in the vertical lines of symmetry. Have a contest to determine who can write the longest word possessing one type of symmetry. For example, MAXIMUM is a word where all the letters have vertical symmetry. KICKBOXED is another.

Project! Using a digital camera, have students (or groups thereof) take photographs of objects possessing symmetry, either rotational or reflective. Give points for the most original, the most nature made, etc. Create a slide show presentation in PowerPoint or Microsoft Movie Maker.

Vocabulary! When discussing rotational symmetry, some textbooks may refer to it as n-fold rotational symmetry. This simply means that the n is the number of times the figure can rotate onto itself. For example, a regular pentagon has 5-fold rotation symmetry, because it can be rotated 5\;\mathrm{times} of 108\;\mathrm{degrees} before returning to its original position.

The Return of the Breadknife! The notion of cutting through a 3-dimensional object with a breadknife was used in an earlier lesson to demonstrate cross sections. This concept can be used to illustrate planes of symmetry. The plane of symmetry essentially “cuts” through the 3-dimensional solid so that each piece is identical.

Dilations

Pacing: This lesson should take one class period

Goal: The purpose of this lesson is to illustrate how scalar multiplication yield dilations. Figures under dilation are similar figures; all properties of the similarity chapter apply to these objects.

Vocabulary! Scale factors of the same value, such as S_2, are also called size changes. All properties of similar figures hold for size changes.

Extension! Scalar multiplication can be extended to multiplying the x-values and y-values by different values, yielding a non-similar figure. For example, you could multiply the points (0, 2), (1, 7), (4, 5), (6, 2) by S_{2, \ 3}. The first value in the subscript is the multiplier for the x-values and the second value in the subscript is the multiplier for the y-values. The resulting ordered pairs are (0, 6), (2, 21), (8, 15), and (12, 6).

Technology! To find the image points of a size change, input a 2 \times 2 matrix in matrix [A], such as \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}. For a scale change, a matrix could look like this: \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}. Multiply the two matrices using the process found in Reflections lesson.

Additional Example:

  1. Suppose the image JAR has the following matrix: \begin{bmatrix}  2 & -4 & 8 \\  6 & -3 & 5 \end{bmatrix}, occurring under a size change S_{\frac{1}{2}}. What are the coordinates of the preimage TIP?

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CK.MAT.ENG.TE.1.Geometry.1.12

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