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2.12: Transformations

Created by: CK-12

Translations

Translation or Transformation – The words translation and transformation look and sound quite similar to students at first. Emphasis their relationship. A translation is just one of the many transformations the students will be learning about in this chapter.

Point or Vector – There are two mathematical objects being use in this lesson that have extremely similar looking notation. An ordered pair is use to represent a location on the coordinate plane, and it also is used to represent movement in the form of a vector. Some texts represent the translation vector as a mapping. The vector v = (-3, 7) would be written (x, y) \rightarrow (x - 3, y + 7). This makes distinguishing between the two easier, but does not introduce the student to the important concept of a vector. It can be used though if the students are having a really hard time with notation.

The Power of Good Notation – There is a lot going on in these exercises. There are the points that make the preimage, the corresponding points of the image, and the vector that describes the translation. Good notation is the key to keeping all of this straight. The points of the image should be labeled with capital letters, and the prime marks should be used on the points of the image. In this way it is easy to see where each point has gone. This will be even more important when working with more complex transformations in later sections. Start good habits now. A vector should be named with a bold, lower case letter, usually from the end of the alphabet. Just writing (9, 6) is a bit ambiguous, but labeling the vector u = (9, 6), will make the meaning clear. In time students will be able to understand the meaning from the context, but when they are first learning good notation can avoid confusion and frustration.

Use Graph Paper and a Ruler – When making graphs of these translations by had, insist that the students use graph paper and a ruler. If students try to graph on binder paper, the result is frequently messy and inaccurate. It is beneficial for students to see that the preimage and image are congruent to reinforce the knowledge that a translation is an isometry. It is also important that students take pride in producing quality work. They will learn so much more when they take the time to do an assignment well, instead of just rushing through the work.

Translations of Sketchpad – Geometers’ Sketchpad uses vectors to translate figures. The program will display the preimage, vector, and image at the same time. Students can type in the vector and can also drag points on the screen to see how the image moves when the vector is changed. It is a quick and engaging way to explore the relationships. If the students have access to Sketchpad and there is a little class time available, it is a worthwhile activity.

Matrices

Rows Then Columns – The dimensions of a matrix are stated by first stating the number of rows and then the number of columns. It may take some time for the students to remember this convention. Give them many opportunities to practice. In this lesson it is important to state the dimensions correctly because only matrices with the same dimensions can be added. The next lesson requires students to decide if two matrices can be multiplied. The order of the dimensions is critical in making that determination.

Additional Exercises:

1. What are the dimensions of any matrix that translates the vertices of a heptagon on the coordinate plane? Explain the significance of the numbers you used.

Answer: 7 \times 2 The number of rows is seven because each of the seven vertices of the heptagon must be moved. The second number is two because each vertex has an x-coordinate and a y-coordinate that must be changed.

2. The additive identity for real numbers is the number zero because:

a + 0 = a, for all real numbers a.

What is the additive identity for 3 \times 2 matrices?

Answer: \begin{bmatrix}  0 &  0  \\  0 & 0 \\   0 & 0\end{bmatrix}

3. If the matrix \begin{bmatrix}  2 &  -3  \\  2 & -4 \\   2 & -3\end{bmatrix} were added to matrix a 3 \times 2 matrix containing the vertices of a triangle would the resulting transformation be an isometry? Why or why not?

Answer: No, in an isometry each point must be moved the same distance in order to preserve size and shape. This matrix moves the second vertex farther down than it moves the other two vertices.

Reflections

Matrix Multiplication – It will take some time and practice for students to become proficient with matrix multiplication. On first inspection of the formula it is sometimes hard to see where all the numbers are coming from and where they are going. For many students a spatial representation is more useful. Here are some guidelines that will help students master matrix multiplication.

  1. Use the rows of the first matrix and the columns of the second matrix.
  2. Move across and down using each number only once.
  3. The resulting sum of the products goes in the slot determined by the row of the first matrix and the column of the second matrix.

Think Don’t Memorize – Many students will try to learn the reflection matrices using rote memory. This is difficult to begin with since the matrices are fairly similar, but it is also not a good method of learning the material because the knowledge will not last. As soon as the students stop regularly using the matrices, they will be forgotten. Instead have the students think about why the matrices produce their intended effect. When the students really understand what is happening, they will not need to memorize patterns of ones, negative ones, and zeros. The knowledge will be long lasting, and they will be able to develop new matrices that represent other types of transformation and other operations.

Additional Exercises:

1. Let matrix A = \begin{bmatrix}   1 &  2  \\  2 & 1   \end{bmatrix} and matrix B = \begin{bmatrix}   4 &  3  \\  3 & 4   \end{bmatrix}. Calculate AB and BA. What do you notice about the products? Will this happen with all 2 \times 2 matrices? What is special about matrix A and matrix B that allowed this special result?

Answer: AB = BA = \begin{bmatrix}   10 &  11  \\  11 & 10   \end{bmatrix} . Matrix A and B commute. This property does not hold for most matrices. These matrices commute because they are symmetric matrices. A symmetric matrix is one in which the rows and columns of the matrix are the same.

Rotation

The Trigonometry – The general rotation matrix uses the trigonometric functions, sine and cosine of the angle of rotation. Students are generally introduced to right triangle trigonometry in the third quarter of geometry. This means they understand the meaning of the trigonometric ratios sine, cosine, and tangent for acute angles in right triangles. In later courses students are introduced to the unit circle which enables them to expand the domain of the trigonometric functions to all real numbers. This is a good point to preview the upcoming material. Tell students that they will shortly learn a method for finding \mathrm{sin}\ 90^\circ and \mathrm{cos}\ 120^\circ; let them know that it is a brilliant method used to expand these extremely useful definitions. For now though, they will have to trust the number given to them by their calculator when using the general rotation matrix. It is not practical to give them the full explanation now, but letting them know that there is an explanation, and that they will learn it soon, will avoid confusion.

Degrees or Radians – If students have followed standard high school mathematical curriculum, they have no idea what a radian is, but somehow calculators frequently end up in radian mode. Students are not familiar enough with trigonometric functions to realize that they are not getting the correct ratio, and continue with their calculations resulting in incorrect answers. Show the students how to check if their calculators are in radian or degree mode and how to change the mode. This is also a good opportunity to start students thinking about different ways to measure angles. Let them know that radians are another unit used to measure rotation. Similar to how both inches and centimeters both measure length. Making students aware of radians now, will help them avoid errors and ready their minds to learn about radians in the future.

Additional Exercises:

1. Graph \triangle {ABC} with A(2, 1), B(5, 3), and C(4, 4).

2. Put the vertices of the triangle in a 3 \times 2 matrix and use matrix multiplication to rotate the triangle 45 \;\mathrm{degrees}. Graph the image of \triangle {ABC} on the same set of axis using prime notation.

3. Use the matrix multiplication to rotate \triangle A’B’C’\ 60 \;\mathrm{degrees}. To do this, take the matrix produced in #2 and multiply it by the rotation matrix for 60 \;\mathrm{degrees}. Graph the resulting triangle on the same set of axis as \triangle A’’B’’C’’.

4. What single matrix could have rotated \triangle ABC to \triangle A’’B’’C’’ in one step? How does this matrix compare to the 45 \;\mathrm{degree} and 60 \;\mathrm{degree} rotation matrix?

Answers:

1.

2. \begin{bmatrix}  2 &  1  \\  5 & 3 \\   4 & 4\end{bmatrix} \begin{bmatrix}  .707 &  .707  \\  -.707 & .707 \\   \end{bmatrix} = \begin{bmatrix}  .707 &  2.12  \\  1.41 & 5.66 \\   0 & 5.66\end{bmatrix}

3. \begin{bmatrix}  .707 &  2.12  \\  1.41 & 5.66 \\   0 & 5.66\end{bmatrix} \begin{bmatrix}  .5 &  .866  \\  -.866 & .5 \\   \end{bmatrix} = \begin{bmatrix}  -1.48 &  1.67  \\  -4.20 & 4.05 \\   -4.90 & 2.83\end{bmatrix}

4. \begin{bmatrix}  -0.26 &  1.00  \\  -1.00 & -0.26 \\  \end{bmatrix}= \begin{bmatrix}   .707 &  .707  \\  -.707 & .707 \\   \end{bmatrix}  \begin{bmatrix}  .5 &  .866  \\  -.866 & .5 \\   \end{bmatrix}

Composition

Use the Image – When first working with compositions student often try to apply both operations to the original figure. Emphasis that a composition is a two-step process 83 \mathrm{s}. The second step of which is performed on the result of the first step. Remind them of the composition of two functions if they have already learned about this topic.

Associative Property of Matrices – Students have heard about the commutative and associate properties many times during their education in mathematics. These are important properties in the study of matrices and become more meaningful for the students when applied to this new set. The fact that the commutative property does not hold for matrix multiplication is surprising at first, and is a concept that needs to be revisited. Although it has been discussed in recent lessons, it would be beneficial to go over it again here before discussing the property that is really of interest in this section, the associate property of matrix multiplication. So far the students have seen that the image of points can be found under a rotation or reflection by multiplying a matrix made up of the coordinates of the points by a matrix specific to the chosen transformation. In this section they should discover that because matrix multiplication is associative, they can multiply two or more transformation matrices together to get a matrix for the composition.

Key Exercises:

Consider the triangle with matrix representation A = \begin{bmatrix}2 &  1  \\5 & 3 \\ 4 & 4 \end{bmatrix}.

Matrix Multiplication is NOT Commutative

1. Use matrix multiplication to rotate the triangle 90 \;\mathrm{degrees}, then take the image and reflect it in the x-axis.

2. Use matrix multiplication to reflect the original triangle in the x-axis, then take the image and rotate it 90 \;\mathrm{degrees}.

3. Are the results of #1 and #2 the same?

Matrix Multiplication is Associate

4. Multiply the matrix used to rotate the triangle 90 \;\mathrm{degrees} by the matrix used to reflect the triangle in the line y = x.

5. Use the matrix found in #4 to transform the triangle. Is the result the same as that in #1?

Answers:

1. \begin{bmatrix}  2 &  1  \\5 & 3 \\ 4 & 4  \end{bmatrix} \begin{bmatrix}   0 &  1  \\  -1 & 0 \\   \end{bmatrix} = \begin{bmatrix}  -1 &  2  \\  -3 & 5 \\ -4 & 4  \end{bmatrix} \begin{bmatrix}  1 &  0  \\  0 & -1 \\   \end{bmatrix} =  \begin{bmatrix}  -1 &  -2  \\  -3 & -5 \\ -4 & -4  \end{bmatrix}

2. \begin{bmatrix}  2 &  1  \\5 & 3 \\ 4 & 4  \end{bmatrix} \begin{bmatrix}   1 &  0  \\  0 & -1 \\   \end{bmatrix} = \begin{bmatrix}  2 &  -1  \\  5 & -3 \\ 4 & -4  \end{bmatrix} \begin{bmatrix}  0 &  1  \\  -1 & 0 \\   \end{bmatrix} =  \begin{bmatrix}  1 &  2  \\  3 & 5 \\ 4 & 4  \end{bmatrix}

3. no

4. \begin{bmatrix}  0 &  1  \\-1 & 0 \\  \end{bmatrix} \begin{bmatrix}   1 &  0  \\  0 & -1 \\   \end{bmatrix} = \begin{bmatrix}  0 &  -1  \\  -1 & 0 \\   \end{bmatrix}

5. \begin{bmatrix}  2 &  1  \\5 & 3 \\ 4 & 4 \end{bmatrix} \begin{bmatrix}   0 &  -1  \\  -1 & 0 \\   \end{bmatrix} = \begin{bmatrix}  -1 &  -2  \\  -3 & -5 \\-4 & -4   \end{bmatrix} , yes

Tessellations

Review Interior Angles Measures for Polygons – Earlier in the course students learned how to calculate the sum of the measures of interior angles of a convex polygon, and how to divide by the number of angles to find the measures of the interior angle of regular polygons. Now would be a good time to review this lesson. The students will need this knowledge to see which regular polygons will tessellate and the final is fast approaching.

Move Them Around – When learning about regular and semi-regular tessellation it is helpful for students to have a set of regular triangles, squares, pentagons, hexagons, and octagons that they can slide around and fit together. These shapes can be bought from a mathematics education supply company or made with paper. Exploring the relationships in this way gives the students a fuller understanding of the concepts.

Use On-Line Resources – A quick search on tessellations will produce many beautiful, artistic examples like the work of M. C. Escher and cultural examples like Moorish tiling. This bit of research will inspire students and show them how applicable this knowledge is to many areas.

Tessellation Project – A good long-term project is to have the students create their own tessellations. This is an artistic endeavor that will appeal to students that typically struggle with mathematics, and the tessellations make nice decorations for the classroom. Here are some guidelines for the assignment.

  1. Fill an 8 \frac{1}{2} by 11\;\mathrm{inch} piece of solid colored paper with a tessellation of your own creation.
  2. Make a stencil from cardboard and trace it to make the figures congruent.
  3. Be creative. Make your tessellation look like something.
  4. Color your design to enhance the tessellation.
  5. Your tessellation will be graded on complexity, creativity, and presentation.
  6. Write a paragraph explaining how you made your tessellation, and why your design is a tessellation. Use vocabulary from this section.

This project could also be done on a piece of legal sized paper. The tessellation can fill the top portion and the paragraph written on the lower part.

Symmetry

360 \;\mathrm{Degrees} Doesn’t Count – When looking for rotational symmetries students will often list 360 \;\mathrm{degree} rotational symmetry. When a figure is rotated 360 \;\mathrm{degrees} the result is not congruent to the original figure, it is the original figure itself. This does not fit the definition of rotational symmetry. This misconception can cause error when counting the numbers of symmetries a figure has or deciding if a figure has symmetry or not.

Review Quadrilateral Classifications – Earlier in the course students learned to classify quadrilaterals. Now would be a good time to break out that Venn diagram. Students will have trouble understanding that some parallelograms have line symmetry if they do not remember that squares and rectangles are types of parallelograms. As the course draws to an end, reviewing helps students retain what they have learned past the final. It is possible to redefine the classes of quadrilaterals based on symmetry. This pursuit will make the student use and combine knowledge in different ways making what they have learned more flexible and useful.

Applications – Symmetry has numerous applications both in and outside of mathematics. Knowing some of the uses for symmetry will motivate student, especially those who are not inspired by pure mathematics, to spend their time and energy learning this material.

Biology – Most higher level animals have bilateral symmetry, starfish and flowers often have 72 \;\mathrm{degree} rotational symmetry. Naturally formed nonliving structures like honeycomb and crystals have 60 \;\mathrm{degree} rotational symmetry. These patterns are fascinating and can be used for classification and study.

Trigonometry – Many identities of trigonometry are based on the symmetry of a circle. In the next few years of mathematics the students will see how to simplify extremely complex expressions using these identities.

Advertising – Many company logos make use of symmetry. Ask the students to bring in examples of logos with particular types of symmetry and create a class collection. Analyze the trends. Are certain products more appropriately represented by logos that contain a specific type of symmetry? Does the symmetry make the logo more pleasing to the eye or more easily remembered?

Functions – A function can be classified as even or odd based on the symmetry of its graph. Even functions have symmetry around the y-axis, and odd functions have 180 \;\mathrm{degree} rotational symmetry about the origin. Once a function is classified as even or odd, properties and theorems can be applied to it.

Draw – Have students be creative and create their own logos or designs with specific types of symmetry. Using these concepts in many ways will build a deeper understanding and the ability to apply the new knowledge in different situations.

Dilations

Naming Conventions – Mathematics is a language, an extremely precise method of communication. While matrices are named with uppercase letters, scalars are represented by lower case letters. Many times students do not know to look for these types of patterns. Point out these conventions when an appropriate example arises and tell the students to look for the subtle differences that have major significance when communicating with mathematics.

The Scale Factor and Area – Students frequently forget to square the scale factor when comparing the area of the original figure to the image under a dilation. This relationship has been covered several times before when studying area, similar figures, and when dilation was first introduced. Make a point of it again. This omission is quite common and the concept is often used on the SAT and other standardized tests.

Additional Exercises:

1. Does the multiplication of a scalar and a 2 \times 2 matrix commute? If so, write a proof. If not give a counterexample.

Answer: Yes, multiplication of a scalar with a 2 \times 2 matrix does commute.

Let k be a scalar and A = \begin{bmatrix}a &  b  \\c & d \\  \end{bmatrix}.

Then kA = k\begin{bmatrix}  a &  b  \\c & d  \end{bmatrix}= \begin{bmatrix}   ka &  kb  \\  kc & kd    \end{bmatrix} = \begin{bmatrix}  ak &  bk  \\  ck & dk   \end{bmatrix} = \begin{bmatrix}  a &  b  \\  c & d   \end{bmatrix} k = Ak

Here a 2 \times 2 matrix was used, but this same proof can be done with a matrix of any dimensions.

2. What scalar could be multiplied by a matrix containing the vertices of a polygon to produce an image with half the area as the original figure?

Answer: \frac{1}{\sqrt{2}} since \left (\frac{1}{\sqrt{2}} \right )^2 = \frac{1}{2}

3. Will a dilation followed by a reflection produce the same image if the order of the transformations is reversed? Why or why not?

Answer: The image will be the same regardless of the order in which the transformations are applied. This can be justified with the commutative property of scalar multiplication.

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Date Created:

Feb 22, 2012

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Apr 29, 2014
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CK.MAT.ENG.TE.1.Geometry.2.12

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