Transformations and Similarity unit pack (Grade 8 · CCSS 8.G.A.1, 8.G.A.3, 8.G.A.4): every worksheet and exit ticket for this unit, ready to print in one go. The answer keys print at the end, each on its own page, so you can leave them off. To make a PDF, click Print and choose "Save as PDF." Games: Let's Get to the Point · Similarity Builder
Transformations on the Coordinate Plane
Grade 8 · CCSS 8.G.A.1 · Pairs with the Let's Get to the Point game
Name
Date
Translation: add to the coordinates, (x, y) → (x + h, y + k).
Reflection over the x-axis: (x, y) → (x, −y); over the y-axis: (x, y) → (−x, y). Rotation about the origin: 90° counterclockwise (x, y) → (−y, x); 180° (x, y) → (−x, −y); 90° clockwise (x, y) → (y, −x).
Key vocabulary
transformation: a rule that moves or changes a figure on the coordinate plane
pre-image / image: the figure before a transformation / the figure after it
translation: a slide: every point moves the same distance in the same direction, (x, y) → (x + 3, y − 2)
reflection: a flip over a line; over the y-axis, (x, y) → (−x, y); over the x-axis, (x, y) → (x, −y)
rotation: a turn of 90°, 180° or 270° around a fixed point
rigid motion: a transformation that keeps size and shape the same: translations, reflections and rotations
congruent: exactly the same size and shape
Part A: Translations
1. Move (2, −1) by the rule (x − 4, y + 3).
Image: ( )
2. Move (−3, 5) 6 units right and 2 units down.
Image: ( )
3. Move (0, −4) by the rule (x + 5, y + 6).
Image: ( )
4. What rule moves (1, 1) to (−3, 4)?
(x , y )
Part B: Reflections
5. Reflect (5, 3) over the x-axis.
Image: ( )
6. Reflect (−2, 4) over the y-axis.
Image: ( )
7. Reflect (−6, −1) over the x-axis.
Image: ( )
8. Reflect (3, −7) over the y-axis.
Image: ( )
Part C: Rotations about the origin
9. Rotate (4, 1) 90° counterclockwise.
Image: ( )
10. Rotate (2, 5) 180°.
Image: ( )
11. Rotate (−3, 2) 90° clockwise.
Image: ( )
12. Rotate (6, −1) 270° counterclockwise.
Image: ( )
Part D: Put it on the grid
13. Plot A(−3, 2). Reflect it over the y-axis and label the image A′. Then rotate A′ 180° about the origin and label that image A″.
A′ ( ) A″ ( )
14.Find the mistake. Alex reflected (4, −2) over the x-axis and got (−4, −2). What did Alex do wrong? What's the correct image?
Correct: ( )
15. After any translation, reflection or rotation, is the new figure congruent to the original? Explain.
2. Rotate (2, −3) 90° counterclockwise about the origin.
( )
3. Reflect (−4, 5) over the x-axis. Which coordinate changed, and why?
( )
Similar Figures and Scale Factor
Grades 7–8 · CCSS 7.G.A.1, 8.G.A.4 · Pairs with the Similarity Builder game
Name
Date
Similar figures have equal angles and proportional sides. Scale factor k = image side ÷ original side.
k > 1 is an enlargement; k < 1 is a reduction. Image side = k × original side. Always match sides that are in the same position.
Key vocabulary
similar figures: figures with the same shape: equal corresponding angles and proportional corresponding sides
dilation: a transformation that enlarges or reduces a figure by a scale factor
scale factor (k): image side length ÷ pre-image side length (the new figure compared with the original)
corresponding sides: sides in matching positions on the two figures
corresponding angles: angles in matching positions; equal in similar figures
proportional: having the same ratio: every pair of corresponding sides has the same scale factor
Part A: Find the scale factor (from the first figure to the second)
1. Two similar rectangles.
k =
2. A triangle with sides 3, 4 and 5 is enlarged to sides 9, 12 and 15.
k =
3. A triangle with sides 8, 10 and 12 is shrunk to sides 4, 5 and 6.
k =
4. A square with side 3 is dilated to a square with side 12.
k =
Part B: Use the scale factor
5. k = 4. The original side is 6. Find the image side.
6. k = 3. The image side is 21. Find the original side.
7. k = ½. The original side is 18. Find the image side.
8. k = 1.5. The original side is 8. Find the image side.
Part C: Find k first, then the missing side
9. The rectangles are similar. Find the missing side.
? =
10. A small triangle has sides 4, 6 and 7. In a similar big triangle, the side that matches 4 is 10. Find the other two sides of the big triangle.
k = sides ,
Part D: Similar triangles in the real world
11. A line parallel to the base of a triangle cuts off a small triangle with base 4. The big triangle's base is 10. A side of the small triangle is 6. Find the matching side of the big triangle.
12. A 5 ft student casts a 3 ft shadow. At the same time, a tree casts an 18 ft shadow. How tall is the tree?
ft
Part E: Similar or not?
13. Are a 2 × 3 rectangle and a 4 × 6 rectangle similar? Are a 2 × 3 rectangle and a 4 × 5 rectangle similar? Explain both.
14.Find the mistake. Casey said: "3 grew to 9, which is 6 more, so the side of 4 grows to 10." What went wrong? What's the right answer?
2. k = 3 and the original side is 7. Find the image side.
3. Are all squares similar to each other? Explain.
Answer Key: Transformations
For the teacher
(2 − 4, −1 + 3) → (−2, 2)
(−3 + 6, 5 − 2) → (3, 3)
(0 + 5, −4 + 6) → (5, 2)
x changes by −4, y by +3 → (x − 4, y + 3)
(5, −3)
(2, 4)
(−6, 1)
(−3, −7)
(−y, x) → (−1, 4)
(−x, −y) → (−2, −5)
(y, −x) → (2, 3)
270° counterclockwise is the same as 90° clockwise: (y, −x) → (−1, −6)
A′ = (3, 2); A″ = (−3, −2)
Alex changed the sign of x, which is a reflection over the y-axis. Over the x-axis only y changes: (4, 2)
Yes. Translations, reflections and rotations are rigid motions: they keep every length and angle, so the image is congruent.
The game names these same mistakes after a miss: wrong direction, wrong axis, swapped coordinates, and moving only one coordinate. Real tracing paper makes #9–12 much easier for students who are stuck.
(−4, −5). Only y changes: the point flips to the other side of the x-axis but stays the same distance left of the y-axis.
Quick read: (−3, −2) on #2 is a clockwise rotation (wrong direction); (4, 5) on #3 is a reflection over the wrong axis. Both are named mistakes in the game's Practice mode.
Answer Key: Similar Figures and Scale Factor
For the teacher
10 / 4 = 15 / 6 → k = 2.5
9 / 3 → k = 3
4 / 8 → k = ½ (a reduction)
12 / 3 → k = 4
6 × 4 = 24
21 ÷ 3 = 7
18 × ½ = 9
8 × 1.5 = 12
k = 15 / 5 = 3; 8 × 3 = 24
k = 10 / 4 = 2.5; 6 × 2.5 = 15 and 7 × 2.5 = 17.5
k = 10 / 4 = 2.5; 6 × 2.5 = 15
Height : shadow is the same for both. k = 18 / 3 = 6; 5 × 6 = 30 ft
2 × 3 and 4 × 6: similar (both sides × 2). 2 × 3 and 4 × 5: not similar (4/2 = 2 but 5/3 ≠ 2).
Casey added 6 instead of multiplying. The scale factor is 9 / 3 = 3, so 4 × 3 = 12
In the game, the answer is the size the bridge is built at, so an additive answer like Casey's builds a bridge that's the wrong size and fails. Matching sides share a color in the game, which helps with turned figures.
Yes. Every square has four 90° angles, and all four sides grow by the same factor, so one square is always a dilation of another.
Quick read: k = 0.4 on #1 means the student divided original by image; 10 on #2 means they added 3. Practice mode in the game lets students pick the "use the scale factor" questions.