Student Exploration: Rotations, Reflections, and Translations

Prior Knowledge Questions (Do these BEFORE using the Gizmo.)

Little Johnnie is playing with a puzzle piece shaped like the letter “b.” He is surprised to discover he can make other letters with the same piece.

1. If he flips the “b” over to the left, what new letter is formed?

Draw a picture to the right.

2. If he flips the letter you drew above down, what new letter is formed?

Draw a picture to the right.

3. Little Johnnie puts the “b” back in its original position. Then he rotates it 90° clockwise. Draw the result to the right. Is this a letter? 

The manipulations of the letter “b” are examples of different transformations – rotations (turns), reflections (flips), and translations (slides). In the Rotations, Reflections, and Translations Gizmo™, you will rotate, reflect, and translate various figures on a coordinate plane. To begin, select Segment from the Figure type menu and Rotate around Origin from the Operation menu.

1. Drag the Rotation (in degrees) slider. What happens to ?

In all of the transformations in this Gizmo, is the image and is the preimage.

2. Select Reflect over x-axis. Drag points A and B so they are both above the x-axis. What do you notice about ?

3. Select Translate. Drag the x translation and y translation sliders. What happens to ? 

Activity A: Translations

Get the Gizmo ready: · Under Figure type, select Point. · Under Operation, select Translate.

1. Recall that point A is the preimage, and point E is the translated image of point A.

A. With the y translation slider set to 0, drag the x translation slider. How does this affect point E?

B. Now set the x translation slider to 0 and drag the y translation slider. How does this affect point E?

C. Set both translation sliders to a positive value. Drag point A around. How does this affect point E?

2. Turn on Show table. Set x translation to –5 and y translation to 6. Drag point A to (–2, 3).

A. What are the coordinates of point E? ( , )

B. How can you calculate the coordinates of point E?

C. Suppose a point has coordinates (x, y). What are the coordinates of the image if the x translation is a and the y translation is b? ( , )

3. The endpoints of are at A(–5, 6) and B(4, 0). Predict the endpoints of the image for the translations listed in the table below. Then sketch and each image on the grid. Click on Show table to check your answers.

x and y translation	Point E Image of A(–5, 6)	Point F Image of B(4, 0)
x translation: 3 y translation: 0
x translation: –1 y translation: –5
x translation: 1 y translation: –6

Activity B: Reflections

Get the Gizmo ready: · Under Figure type, select Point. · Under Operation, select Reflect over x-axis. · Turn off Show table.

1. Recall that point E (the image) is the reflection of point A (the preimage).

A. Drag point A up, down, left, and right. Fill in the table to describe how point E moves when you do this.

Point A	Up	Down	Left	Right
Point E

B. Turn on Show table. Watch the coordinates in the table as you drag point A around. How do the coordinates of point E compare to the coordinates of point A?

C. A point has coordinates (x, y). What are the coordinates of the image if (x, y) is reflected over the x-axis? ( , )

2. Turn off Show table. Select Reflect over y-axis.

A. What do you think will happen to point E when point A is moved to the right?

B. Turn on Show table. How do the coordinates of point E compare to those of point A?

C. A point has coordinates (x, y). What are the coordinates of the image if (x, y) is reflected over the y-axis? ( , )

3. Under Operation, select None. Under Figure type, select Triangle. Drag the vertices of ΔABC to A(7, 5), B(–10, 2), and C(2, –8). Predict the coordinates of the vertices of the image ΔEFG for the reflections listed below. Then check your answers in the Gizmo.

Over the x-axis: E( , ) F( , ) G( , )

Over the y-axis: E( , ) F( , ) G( , )

Activity C: Rotations

Get the Gizmo ready: · Under Figure type, select Point. · Under Operation, select Rotate around Origin. · Turn off Show table.

1. Drag point A, the preimage, to (10, 5).

A. Drag the Rotation (in degrees) slider. What shape does point E, the image of A, trace as you drag the slider to the right? 

Angle of rotation (θ)	cos θ	sin θ
30°
45°
60°
90°

 

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