Recycling Drive: Line Motion Tests

Test how moving an entire layout affects parallel guide lines.

Translate the Can-Sorting Guides

The recycling drive uses pairs of parallel guide lines for cans, bottles, and paper. Lines extend forever; labeled points mark only part of each line. Use a ruler and tracing paper. A prime mark (′) identifies an image point.

Draw the can-sorting guide lines through A and B and through C and D. Trace both lines on one sheet. Slide the sheet 2 units right and 1 unit down without turning it. Mark all four image points on the grid. Use this trial for Questions 1–4.

Compare the direction from A′ to B′ with the direction from C′ to D′.

From A′ to B′, move right units and up units. From C′ to D′, move right units and up units.

Which observation from the tracing experiment is evidence that the image lines are parallel?

A. Both image lines lie entirely above the x-axis.B. Both image lines pass through the same grid point.C. Both distinct image lines have the same rise for the same run.D. Both image lines have endpoints on the edge of the grid.

The can team needs to move both guides 3 more units left to clear a collection cart. Which procedure tests the move without changing the parallel relationship?

A. Slide one line 3 units left and turn the other line 90°.B. Hold one point fixed and turn just one of the two lines.C. Slide the tracing of both lines 3 units left without turning it.D. Move each marked point 3 units toward the origin.

Repeat the experiment with a different translation of the entire pair. Select every statement that must remain true.

Reflect the Bottle-Sorting Guides

Start with the original bottle layout for each trial. Reflect every point of both lines across the same mirror line.

Draw lines EF and GH. Trace both lines and the y-axis. Turn the tracing paper over, aligning the traced y-axis with the grid's y-axis. Mark the reflected lines. Use this trial for Questions 5–7.

Which table correctly records the reflection across the y-axis?

A. E′(4, −3), F′(1, 0), G′(2, 2), H′(−1, 5)B. E′(−4, 3), F′(−1, 0), G′(−2, −2), H′(1, −5)C. E′(4, 3), F′(1, 0), G′(2, −2), H′(−1, −5)D. E′(−3, −4), F′(0, −1), G′(2, −2), H′(5, 1)

Read each reflected line from left to right. Use a horizontal run of 3 units on each line. Record each signed vertical change; a downward change is negative.

Vertical change from F′ to E′ (left to right): units. Vertical change from H′ to G′ (left to right): units.

A volunteer compares each reflected line with its own original line. The volunteer finds that their directions differ. What comparison actually checks whether the two reflected guides are parallel?

A. Compare the x-coordinate of E with the y-coordinate of H′.B. Compare the number of labeled points in each quadrant.C. Compare the length of EF with the length of GH.D. Compare the directions of E′F′ and G′H′ with each other.

For a second bottle area, the guides are the vertical lines x = −5 and x = 2. Reflect both across the y-axis. Sketch the result on scratch paper. Which description matches the image lines?

A. They are x = 5 and x = −2, two parallel vertical lines.B. They are x = −5 and x = −2, two parallel vertical lines.C. They are x = 5 and y = −2, two perpendicular lines.D. They are y = 5 and y = −2, two parallel horizontal lines.

Rotate the Paper-Sorting Guides

Rotate the entire pair of paper guides together. Use a tracing-paper turn to check your coordinate work.

The paper guides pass through J(−3, −4), K(1, −2) and through L(−4, 0), M(0, 2). Draw both original lines on scratch paper. Rotate their tracing 90° clockwise about the origin. On the grid below, plot and label J′, K′, L′, and M′. Draw the two image lines through their corresponding point pairs, extending each to the grid edges.

10.

Use your rotated paper layout from Question 9. Compare the signed vertical changes from J′ to K′ and from L′ to M′.

Each pair moves 2 units right. From J′ to K′, the signed vertical change is units. From L′ to M′, it is units.

11.

A new paper station has two distinct horizontal guides. Volunteers rotate the entire layout 35° counterclockwise about one center. Which result should their tracing experiment show?

A. The two image guides are perpendicular to each other.B. The two image guides have the same new direction.C. The two image guides cross at the rotation center.D. The two image guides remain horizontal on the page.

12.

The drive coordinator tests three complete layouts. Select every trial whose final pair of guides must be parallel.

13.

Sam tests the parallel guides y = 1 and y = 4. Sam rotates only y = 1 by 90° clockwise about the origin and leaves y = 4 fixed. The final lines cross. Sam concludes, “Rotations do not preserve parallel lines.” Explain why this is not a valid test of the claim. State the image equations when BOTH original lines receive that rotation, and explain what they show.