Representing Transformations in a Plane

Representing Transformations in a Plane

The Lesson Activities will help you meet these educational goals:

Content Knowledge—You willrepresent transformations in a plane and compare transformations that preserve distance and angle to those that do not.
Mathematical Practices—You will make sense of problems and solve them.
STEM—You will apply mathematical and technology tools and knowledge to analyze real-world situations.
21stCentury Skills—You will use critical-thinking and problem-solving skills.

Directions

You will evaluate some of these activities yourself, and your teacher may evaluate others. Please save this document before beginning the lesson and keep the document open for reference during the lesson. Type your answers directly in this document for all activities.

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Self-CheckedActivities

Read the instructions for the following activities and type in your responses. At the end of the lesson, click the link to open the Student Answer Sheet. Use the answers or sample responses to evaluate your own work.

Translations

In this activity, you will use GeoGebra to explore the properties of translations as you complete the steps below.If you need help, follow these instructions for using GeoGebra.

Move so it lies on top of and How many units do you have to move from its original location to coincide with each of the other triangles? Specify how moves up, down, left, or right.

Type your response here:

Image / Movement of ∆ A′B′C′

What are the coordinates of the vertices of ∆A′B′C′ after each movement?

Type your response here:

Image / Coordinates of Vertices for ∆A′B′C′

Find the distance from the vertices of to the corresponding vertices of the other three triangles, and enter them in the table. For you'll need to use the distance formula Verify your calculations using the tools available in GeoGebra.

Type your response here:

Original Vertex / New Vertex / Distance Calculated / Distance Using GeoGebra
A / D
B / E
C / F
A / G
B / H
C / I
A / J
B / K
C / L

What do you observe about the distances between the three pairs of corresponding vertices for each movement? Based on your observations, how would you define a translation? How can you extend what you know about the vertices of a shape to all the points on a shape during a translation?

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How did you do? Check a box below.

Nailed It!—I included all of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

Reflections

Use GeoGebra to explore the properties of reflections,and complete each step below.

To produce a reflection, or mirror image, you need to specify a line of reflection. When mirroring objects on a plane, a line of reflection is represented by a straight line. Move polygon ABCD around the coordinate plane, and describe the three lines of reflection that you see.

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Place the vertices of polygon ABCDwherever you wish, and record the coordinates of its vertices. What are the coordinates of the vertices of the reflected polygons when the x-axis and y-axis are used as the lines of reflection? Record the coordinates in the table.

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Vertex / Coordinate / Vertex / Coordinate / Vertex / Coordinate
A / /
B / /
C / /
D / /

Review your answers to part b. What happens to the coordinates of a shape when the shape reflects about the x-axis? What happens to the coordinates when the shape reflects about the y-axis? Explain.

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Move point F to the origin, (0, 0), so the equation of is y = x. Now record the coordinates of the vertices of the polygon ABCD. What are the coordinates of the vertices of the reflected polygon where is the line of reflection? Enter the coordinates in the table.

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Vertex / Coordinate / Vertex / Coordinate
A /
B /
C /
D /

Review your answers to part d. What happens to the coordinates of a shape when the shape reflects about the line y = x? Explain.

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Draw a line segment between each vertex of ABCD and the corresponding vertex of There are four line segments in all. Then move polygon ABCD around the coordinate plane. What do you observe about the angle formed between the line segments you drew and the line of reflection, Use the tools available in GeoGebra to confirm your hypothesis.

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Move polygon ABCD to a location of your choice on the coordinate plane, and change the orientation of Be sure that the preimage ABCD does not overlap the image when you're through. Next, find the perpendicular distance between each vertex of ABCD and Enter the values in the table. Do the same for the perpendicular distance between each vertex of the reflected shape and Record all of your answers to one decimal place.

Type your response here:

Vertex / Distance to / Vertex / Distance to
A /
B /
C /
D /

What conclusions can you draw from the distances that you recorded in part g?

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How did you do? Check a box below.

Nailed It!—I included all of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

Rotations

Use GeoGebra to explore the properties of rotations, and complete each step below.

Use the slider to rotate about point P through different angles of rotation (α). As you move the slider, you'll see the image, move on the screen. At what angles of rotation does coincide with each of the other triangle images?

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Image / α°

Find the distance from each vertex on the preimage, to the center of rotation, P. Enter the values in the table. Do the same for each vertex on the other images.

Type your response here:

Vertex / Distance to P / Vertex / Distance to P
A / G
B / H
C / I
D / J
E / K
F / L

What can you conclude from your observations in part b?

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Point P is presently located at the origin. Move P, and observe how the rotated images change. Be sure that neither the images nor the preimage overlap. Record the distances of the vertices of the preimage from P. Do the same for the other images. Round all of your answers to the hundredths place.

Type your response here:

Vertex / Distance from P / Vertex / Distance from P / Vertex / Distance from P / Vertex / Distance from P
A / D / G / J
B / E / H / K
C / F / I / L

What can you conclude about your observations in part d? What happened as you moved P, the center of rotation, about the coordinate plane? How can you extend your discoveries about the vertices to all points that lie on a figure or shape?

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How did you do? Check a box below.

Nailed It!—I included all of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

Dilations

Use GeoGebra to explore the properties of dilations, and complete each step below.

Move the slider to change the value of n. Watch what happens to the preimage and the image of the triangle. Set n to any value that you wish, and enter it in the table. Then enter the coordinates of the vertices for Do this for at least three different values of n.

Type your response here:

n / A / A′ / B / B′ / C / C′
(1, 1) / (3, 2) / (4, 1)
(1, 1) / (3, 2) / (4, 1)
(1, 1) / (3, 2) / (4, 1)

What is the relationship between the coordinates of the vertices for ΔABC and the corresponding vertices for? Explain the relationship in terms of n.

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Change the value of n some more, and observe how the lengths of the sides of the image change. Once you've settled on a value for n, record it in the table. Measure the lengths of the sides on Do this for at least three different values of n. What is the relationship between the lengths of the sides of and? Explain the relationship in terms of n.

Type your response here:

n / a / a′ / b / b′ / c / c′
1.41 units / 3 units / 2.24 units
1.41 units / 3 units / 2.24 units
1.41 units / 3 units / 2.24 units

Based on your observations in parts a through c, what is the significance of a dilation when 0 < n < 1? What is the significance when n = 1? What about when n > 1? Explain in terms of the size of the image and the preimage.

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Now take a look at the angle measurements of and Move the slider, and observe the values of the angle measurements of the image and the preimage. Measure and record at least two sets of angle measurements in the table. Choose one set of measurements wheren > 1and one set wheren < 1. Round the values to the nearest degree.

Type your response here:

n / / / / / /

What can you conclude about dilations from your observations in part e?

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Throughout this activity, the scale factor, n, has been a positive value. However, a scale factor for dilations can also be negative. When a set of coordinates on a preimage is multiplied by -n instead of n, the coordinates of the image rotate about the center of dilation by 180°. Consider all the potential intervals of n in the table. Enter information in the table to indicate whether the scale factor produces an enlargement, a reduction, or no size change. Also indicate whether the image rotates 180° about the center of dilation. The first one has been done for you.

Type your response here:

Interval / Enlargement, reduction, orno size change? / Does the image rotate 180° about the center of dilation?
0 < n < 1 / reduction / no
n = 1
n > 1
-1 < n < 0
n = -1
n < -1

How did you do? Check a box below.

Nailed It!—I included all of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

Horizontal and Vertical Stretches

Use GeoGebra to explore the properties of horizontal andvertical stretches,and complete each step below.

Move the horizontal and vertical sliders to stretch the image of the triangle. When you've settled on a pair of transformed images, record the coordinates of the images in the table. Be sure to list coordinates in both the horizontal (x) and vertical (y) directions. Also record the scale factors, nx and ny. Do this atleast three times to fill the table.

Type your response here:

nx / A / A′ / B / B′ / C / C′
(2, 1) / (4, 2) / (5, 1)
(2, 1) / (4, 2) / (5, 1)
(2, 1) / (4, 2) / (5, 1)
ny / A / A″ / B / B″ / C / C″
(2, 1) / (4, 2) / (5, 1)
(2, 1) / (4, 2) / (5, 1)
(2, 1) / (4, 2) / (5, 1)

In part a, what do you notice about the y-coordinates before and after a horizontal stretch? What do you notice about the x-coordinates before and after a vertical stretch?

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In part a, what do you notice about the x-coordinates before and after a horizontal stretch? What do you notice about the y-coordinates before and after a vertical stretch? Express your answer in terms of nxand ny.

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Move the horizontal and vertical sliders some more to stretch the image of the triangle. As you move the sliders, observe what happens to the interior angles of each triangle. Based on your observations, can you say that stretches preserve angle measurements?

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How did you do? Check a box below.

Nailed It!—I included all of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.