Shape Shifter Worksheet
Math 8 • Geometry Scale Factors
Shape Shifter Worksheet
Dilations on the Coordinate Plane • Center at the Origin (0, 0)
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The Coordinate Rule for Dilation (x, y) → (k • x, k • y)
To dilate a point from the origin (0, 0), multiply both the x-coordinate and the y-coordinate by the scale factor (\(k\)).
• Enlargement (\(k > 1\)): Multiply by numbers greater than 1 (e.g., \(k = 2, 3\)) → Shape gets bigger
Grows
• Reduction (\(k < 1\)): Multiply by fractions between 0 and 1 (e.g., \(k = \frac{1}{2}, \frac{1}{3}\)) → Shape gets smaller
Shrinks
WORKED EXAMPLE Enlarge Triangle ABC by Scale Factor \(k = 2\)
Multiply each coordinate by 2
| Original Vertex | Work: Multiply by \(k = 2\) | New Vertex (Image) |
|---|
| A (1, 1) | | |
| 1 • 2 = 2 | | |
1 • 2 = 2
| A' ( 2 , 2 ) |
| B (3, 1) |
3 • 2 = 6
1 • 2 = 2
| B' ( 6 , 2 ) |
| C (1, 3) |
1 • 2 = 2
3 • 2 = 6
| C' ( 2 , 6 ) |
Step 1: List original coordinates \(A(1,1)\), \(B(3,1)\), \(C(1,3)\).
Step 2: Multiply both \(x\) and \(y\) by 2 to find \(A', B', C'\).
Step 3: Plot new points and connect them. Notice the shape is twice as big!
x y 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 A B C A' B' C'
Original Dilated Image
PROBLEM 1 Enlarge Triangle DEF by Scale Factor \(k = 3\)
Multiply by 3
Directions: Multiply both coordinates by 3 in the work column. Then plot and connect D', E', F' on the grid!
| Original | Work: Multiply by 3 | New Image Coordinates |
|---|
| D (1, 1) | | |
| 1 • 3 = 3 | | |
1 • 3 = 3
| D' ( 3 , 3 ) |
| E (3, 1) | | E' ( , ) |
| F (1, 2) | | F' ( , ) |
Tip: Base \(DE\) is 2 units wide. With \(k = 3\), base \(D'E'\) will be \(2 \times 3 = 6\) units wide!
x y 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 D E F ↑ Plot D', E', F' and connect them!
CHECKPOINT Big or Small? Circle the correct word for each scale factor (\(k\)):
k = 3
Enlargement Reduction
k = 1/2
Enlargement Reduction
k = 1/3
Enlargement Reduction
Shape Shifter Worksheet • Grade 8 Dilation Practice Page 1 of 2
Multiplying by Fractions: Halves, Thirds & Four Quadrants
Fractional Dilations on the Coordinate Plane • Center at the Origin (0, 0)
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Multiplying by Fractions: Multiply by \(\frac{1}{2}\) or \(\frac{1}{3}\)!
E.g., \(6 \cdot \frac{1}{2} = 3\) and \(6 \cdot \frac{1}{3} = 2\).
Quadrant II Tip: In Quadrant II, \(x\) is negative and \(y\) is positive:
E.g., \(-3 \cdot \frac{1}{3} = -1\)!
PROBLEM 2 Dilate Rectangle ABCD by Scale Factor \(k = \frac{1}{2}\)
Multiply each coordinate by \(\frac{1}{2}\)
| Original | Work: Multiply by \(\frac{1}{2}\) | New Image Coordinates |
|---|
| A (2, 6) | | |
| 2 • ½ = 1 | | |
6 • ½ = 3
| A' ( 1 , 3 ) |
| B (8, 6) | | B' ( , ) |
| C (8, 2) | | C' ( , ) |
| D (2, 2) | | D' ( , ) |
Plot and connect A', B', C', D' on the grid to create the shrunken rectangle.
x y 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 A B C D
PROBLEM 3 Dilate Triangle JKL in Quadrant II by Scale Factor \(k = \frac{1}{3}\)
All 4 Quadrants
| Original (Quad II) | Work: Multiply by \(\frac{1}{3}\) | New Image Coordinates |
|---|
| J (-3, 6) | | |
| -3 • ⅓ = -1 | | |
6 • ⅓ = 2
| J' ( -1 , 2 ) |
| K (-6, 3) | | K' ( , ) |
| L (-3, 3) | | L' ( , ) |
Plot and connect J', K', L' on the 4-quadrant grid in Quadrant II.
II (-,+) I (+,+) III (-,-) IV (+,-) x y 0 -6 -4 -2 2 4 6 -6 -4 -2 2 4 6 J K L
CHALLENGE Scale Factor Detective
1. Find the Scale Factor \(k\):
Original point P(-2, 3) in Quadrant II is multiplied by \(k\) to get P'(-6, 9).
What was \(k\)? \(k\) =
2. Quadrant II Multiplication:
Point Q(-6, 4) is dilated by multiplying both coordinates by \(k = \frac{1}{2}\).
New Point: Q' ( , )
Shape Shifter Worksheet • Grade 8 Dilation Practice Page 2 of 2
Shape Shifter Answer Key
TEACHER ANSWER KEY Math 8 • Geometry
Shape Shifter Answer Key
Complete Solutions, Step-by-Step Work & Coordinate Plots • Origin (0,0)
For Teacher Grading & Review
Unit: Dilations on Coordinate Plane
Essential Rule & Misconception Check (x, y) → (k • x, k • y)
• Enlargement (\(k > 1\)): Multiply by \(k > 1\) → Shape grows
• Reduction (\(k < 1\)): Multiply by fractions → Shape shrinks
WORKED EXAMPLE REVIEW Triangle ABC Enlarged by \(k = 2\)
Modeled Reference
| Original Vertex | Work: Multiply by 2 | Image Coordinates |
|---|
| A (1, 1) | | |
| 1 • 2 = 2 | | |
1 • 2 = 2
| A' ( 2 , 2 ) |
| B (3, 1) |
3 • 2 = 6
1 • 2 = 2
| B' ( 6 , 2 ) |
| C (1, 3) |
1 • 2 = 2
3 • 2 = 6
| C' ( 2 , 6 ) |
Key Teaching Point: Remind students that side lengths double: base \(AB = 2\) becomes \(A'B' = 4\), and height \(AC = 2\) becomes \(A'C' = 4\).
x y 0 1 3 6 A' B' C'
PROBLEM 1 SOLUTION Enlarge Triangle DEF by Scale Factor \(k = 3\)
Rule: (x, y) → (3x, 3y)
| Original | Completed Work (Multiply by 3) | Solution Coordinates |
|---|
| D (1, 1) | | |
| 1 • 3 = 3 | | |
1 • 3 = 3
| D' ( 3 , 3 ) |
| E (3, 1) |
3 • 3 = 9
1 • 3 = 3
| E' ( 9 , 3 ) |
| F (1, 2) |
1 • 3 = 3
2 • 3 = 6
| F' ( 3 , 6 ) |
Common Error: Students may add 3 instead of multiplying (e.g., getting (4, 4) instead of (3, 3)). Remind them that dilation requires multiplication.
x y 0 1 3 9 D E F D' E' F' • D'(3,3), E'(9,3), F'(3,6) correctly plotted
CHECKPOINT KEY Correctly circled responses:
k = 3
Enlargement ✓ Reduction
k = 1/2
Enlargement Reduction ✓
k = 1/3
Enlargement Reduction ✓
Shape Shifter Answer Key • Teacher Grading Guide Page 1 of 2
TEACHER ANSWER KEY Fractional Reductions & Quadrant II
