Reference / Curriculum / 3B / Chapter 13

3B · Chapter 13 · Solutions

Complete worked answer key

Answers and explanations for every original lesson practice set and chapter assessment.

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Area

Prerequisite

  1. 60 — 6 groups of 10 make 6 × 10 = 60. Or find 5 × 10 and add 10.
  2. 24 — 6 groups of 4 make 6 × 4 = 24. Or find 5 × 4 and add 4.
  3. 24 — 6 groups of 4 make 6 × 4 = 24. Or find 5 × 4 and add 4.

Understanding checks

  1. 16 square units — There are 4 rows of 4 equal square units: 4 × 4 = 16 square units.
  2. 18 square units — There are 2 rows of 9 equal square units: 9 × 2 = 18 square units.
  3. 12 square units — There are 2 rows of 6 equal square units: 6 × 2 = 12 square units.

Guided

  1. 35 square units — There are 5 rows of 7 equal square units: 7 × 5 = 35 square units.
  2. 16 square units — There are 2 rows of 8 equal square units: 8 × 2 = 16 square units.
  3. 8 square units — There are 2 rows of 4 equal square units: 4 × 2 = 8 square units.
  4. 36 square units — There are 6 rows of 6 equal square units: 6 × 6 = 36 square units.
  5. 30 square units — There are 5 rows of 6 equal square units: 6 × 5 = 30 square units.

Independent

  1. 15 square units — There are 5 rows of 3 equal square units: 3 × 5 = 15 square units.
  2. 12 square units — There are 4 rows of 3 equal square units: 3 × 4 = 12 square units.
  3. 30 square units — There are 6 rows of 5 equal square units: 5 × 6 = 30 square units.
  4. 18 square units — There are 3 rows of 6 equal square units: 6 × 3 = 18 square units.
  5. 27 square units — There are 3 rows of 9 equal square units: 9 × 3 = 27 square units.
  6. 24 square units — There are 4 rows of 6 equal square units: 6 × 4 = 24 square units.
  7. 21 square units — There are 3 rows of 7 equal square units: 7 × 3 = 21 square units.
  8. 9 square units — There are 3 rows of 3 equal square units: 3 × 3 = 9 square units.

Challenge

  1. No — Both areas are equal, but different side dimensions usually give different boundary lengths.
  2. Perimeter 24 units — The area is 7 × 5 = 35 square units, but its perimeter is 2 × (7+5) = 24 units. Area and perimeter measure different things.

Exit

  1. 48 square units — There are 6 rows of 8 equal square units: 8 × 6 = 48 square units.
  2. 54 square units — There are 6 rows of 9 equal square units: 9 × 6 = 54 square units.
  3. 42 square units — There are 6 rows of 7 equal square units: 7 × 6 = 42 square units.

Units of Area

Prerequisite

  1. 16 square units — There are 4 rows of 4 equal square units: 4 × 4 = 16 square units.
  2. 8 square units — There are 2 rows of 4 equal square units: 4 × 2 = 8 square units.
  3. 18 square units — There are 6 rows of 3 equal square units: 3 × 6 = 18 square units.

Understanding checks

  1. 27 square units — There are 3 rows of 9 equal square units: 9 × 3 = 27 square units.
  2. 14 square units — There are 2 rows of 7 equal square units: 7 × 2 = 14 square units.
  3. 8 square units — There are 2 rows of 4 equal square units: 4 × 2 = 8 square units.

Guided

  1. 24 square units — There are 4 rows of 6 equal square units: 6 × 4 = 24 square units.
  2. 6 square units — There are 2 rows of 3 equal square units: 3 × 2 = 6 square units.
  3. 45 square units — There are 5 rows of 9 equal square units: 9 × 5 = 45 square units.
  4. 16 square units — There are 2 rows of 8 equal square units: 8 × 2 = 16 square units.
  5. 12 square units — There are 4 rows of 3 equal square units: 3 × 4 = 12 square units.

Independent

  1. 24 square units — There are 3 rows of 8 equal square units: 8 × 3 = 24 square units.
  2. 32 square units — There are 4 rows of 8 equal square units: 8 × 4 = 32 square units.
  3. 16 square units — There are 4 rows of 4 equal square units: 4 × 4 = 16 square units.
  4. 18 square units — There are 6 rows of 3 equal square units: 3 × 6 = 18 square units.
  5. 36 square units — There are 6 rows of 6 equal square units: 6 × 6 = 36 square units.
  6. 12 square units — There are 3 rows of 4 equal square units: 4 × 3 = 12 square units.
  7. 10 square units — There are 2 rows of 5 equal square units: 5 × 2 = 10 square units.
  8. 18 square units — There are 3 rows of 6 equal square units: 6 × 3 = 18 square units.

Challenge

  1. No — Both areas are equal, but different side dimensions usually give different boundary lengths.
  2. Perimeter 18 units — The area is 3 × 6 = 18 square units, but its perimeter is 2 × (3+6) = 18 units. Area and perimeter measure different things.

Exit

  1. 18 square units — There are 2 rows of 9 equal square units: 9 × 2 = 18 square units.
  2. 9 square units — There are 3 rows of 3 equal square units: 3 × 3 = 9 square units.
  3. 21 square units — There are 3 rows of 7 equal square units: 7 × 3 = 21 square units.

Area of Rectangles

Prerequisite

  1. 32 square units — There are 4 rows of 8 equal square units: 8 × 4 = 32 square units.
  2. 30 square units — There are 5 rows of 6 equal square units: 6 × 5 = 30 square units.
  3. 8 square units — There are 2 rows of 4 equal square units: 4 × 2 = 8 square units.

Understanding checks

  1. 12 square units — There are 2 rows of 6 equal square units: 6 × 2 = 12 square units.
  2. 30 square units — There are 5 rows of 6 equal square units: 6 × 5 = 30 square units.
  3. 18 square units — There are 3 rows of 6 equal square units: 6 × 3 = 18 square units.

Guided

  1. 6 square units — There are 2 rows of 3 equal square units: 3 × 2 = 6 square units.
  2. 15 square units — There are 3 rows of 5 equal square units: 5 × 3 = 15 square units.
  3. 27 square units — There are 3 rows of 9 equal square units: 9 × 3 = 27 square units.
  4. 18 square units — There are 2 rows of 9 equal square units: 9 × 2 = 18 square units.
  5. 54 square units — There are 6 rows of 9 equal square units: 9 × 6 = 54 square units.

Independent

  1. 15 square units — There are 5 rows of 3 equal square units: 3 × 5 = 15 square units.
  2. 18 square units — There are 6 rows of 3 equal square units: 3 × 6 = 18 square units.
  3. 12 square units — There are 3 rows of 4 equal square units: 4 × 3 = 12 square units.
  4. 40 square units — There are 5 rows of 8 equal square units: 8 × 5 = 40 square units.
  5. 10 square units — There are 2 rows of 5 equal square units: 5 × 2 = 10 square units.
  6. 48 square units — There are 6 rows of 8 equal square units: 8 × 6 = 48 square units.
  7. 12 square units — There are 4 rows of 3 equal square units: 3 × 4 = 12 square units.
  8. 14 square units — There are 2 rows of 7 equal square units: 7 × 2 = 14 square units.

Challenge

  1. No — Both areas are equal, but different side dimensions usually give different boundary lengths.
  2. Perimeter 20 units — The area is 4 × 6 = 24 square units, but its perimeter is 2 × (4+6) = 20 units. Area and perimeter measure different things.

Exit

  1. 32 square units — There are 4 rows of 8 equal square units: 8 × 4 = 32 square units.
  2. 9 square units — There are 3 rows of 3 equal square units: 3 × 3 = 9 square units.
  3. 16 square units — There are 2 rows of 8 equal square units: 8 × 2 = 16 square units.

Area of Composite Figures

Prerequisite

  1. 30 square units — There are 5 rows of 6 equal square units: 6 × 5 = 30 square units.
  2. 27 square units — There are 3 rows of 9 equal square units: 9 × 3 = 27 square units.
  3. 25 square units — There are 5 rows of 5 equal square units: 5 × 5 = 25 square units.

Understanding checks

  1. 16 square units — Whole rectangle: 6 × 3 = 18. Missing corner: 2 × 1 = 2. Remaining: 18 − 2 = 16.
  2. 10 square units — Whole rectangle: 6 × 2 = 12. Missing corner: 2 × 1 = 2. Remaining: 12 − 2 = 10.
  3. 12 square units — Whole rectangle: 6 × 4 = 24. Missing corner: 4 × 3 = 12. Remaining: 24 − 12 = 12.

Guided

  1. 25 square units — Whole rectangle: 9 × 3 = 27. Missing corner: 2 × 1 = 2. Remaining: 27 − 2 = 25.
  2. 23 square units — Whole rectangle: 9 × 3 = 27. Missing corner: 4 × 1 = 4. Remaining: 27 − 4 = 23.
  3. 21 square units — Whole rectangle: 5 × 5 = 25. Missing corner: 1 × 4 = 4. Remaining: 25 − 4 = 21.
  4. 33 square units — Whole rectangle: 9 × 5 = 45. Missing corner: 3 × 4 = 12. Remaining: 45 − 12 = 33.
  5. 12 square units — Whole rectangle: 3 × 6 = 18. Missing corner: 2 × 3 = 6. Remaining: 18 − 6 = 12.

Independent

  1. 26 square units — Whole rectangle: 9 × 4 = 36. Missing corner: 5 × 2 = 10. Remaining: 36 − 10 = 26.
  2. 40 square units — Whole rectangle: 9 × 6 = 54. Missing corner: 7 × 2 = 14. Remaining: 54 − 14 = 40.
  3. 15 square units — Whole rectangle: 3 × 6 = 18. Missing corner: 1 × 3 = 3. Remaining: 18 − 3 = 15.
  4. 41 square units — Whole rectangle: 7 × 6 = 42. Missing corner: 1 × 1 = 1. Remaining: 42 − 1 = 41.
  5. 19 square units — Whole rectangle: 6 × 4 = 24. Missing corner: 5 × 1 = 5. Remaining: 24 − 5 = 19.
  6. 11 square units — Whole rectangle: 4 × 5 = 20. Missing corner: 3 × 3 = 9. Remaining: 20 − 9 = 11.
  7. 13 square units — Whole rectangle: 5 × 3 = 15. Missing corner: 1 × 2 = 2. Remaining: 15 − 2 = 13.
  8. 13 square units — Whole rectangle: 4 × 4 = 16. Missing corner: 1 × 3 = 3. Remaining: 16 − 3 = 13.

Challenge

  1. 68 square units — Subtract the missing four squares from the bounding rectangle, or split the remaining shape into two nonoverlapping rectangles and add.
  2. It is outside the figure — Counting the full bounding rectangle would count squares not inside the figure.

Exit

  1. 8 square units — Whole rectangle: 6 × 2 = 12. Missing corner: 4 × 1 = 4. Remaining: 12 − 4 = 8.
  2. 30 square units — Whole rectangle: 9 × 4 = 36. Missing corner: 6 × 1 = 6. Remaining: 36 − 6 = 30.
  3. 8 square units — Whole rectangle: 6 × 3 = 18. Missing corner: 5 × 2 = 10. Remaining: 18 − 10 = 8.

Perimeter

Prerequisite

  1. 6 square units — Whole rectangle: 5 × 2 = 10. Missing corner: 4 × 1 = 4. Remaining: 10 − 4 = 6.
  2. 17 square units — Whole rectangle: 7 × 3 = 21. Missing corner: 2 × 2 = 4. Remaining: 21 − 4 = 17.
  3. 26 square units — Whole rectangle: 8 × 4 = 32. Missing corner: 2 × 3 = 6. Remaining: 32 − 6 = 26.

Understanding checks

  1. 16 cm — Walk along every outer edge once: 4 + 4 + 4 + 4 = 16 cm.
  2. 24 cm — Walk along every outer edge once: 8 + 4 + 8 + 4 = 24 cm.
  3. 24 cm — Walk along every outer edge once: 9 + 3 + 9 + 3 = 24 cm.

Guided

  1. 26 cm — Walk along every outer edge once: 9 + 4 + 9 + 4 = 26 cm.
  2. 28 cm — Walk along every outer edge once: 9 + 5 + 9 + 5 = 28 cm.
  3. 12 cm — Walk along every outer edge once: 3 + 3 + 3 + 3 = 12 cm.
  4. 18 cm — Walk along every outer edge once: 3 + 6 + 3 + 6 = 18 cm.
  5. 22 cm — Walk along every outer edge once: 8 + 3 + 8 + 3 = 22 cm.

Independent

  1. 14 cm — Walk along every outer edge once: 5 + 2 + 5 + 2 = 14 cm.
  2. 20 cm — Walk along every outer edge once: 4 + 6 + 4 + 6 = 20 cm.
  3. 18 cm — Walk along every outer edge once: 7 + 2 + 7 + 2 = 18 cm.
  4. 20 cm — Walk along every outer edge once: 7 + 3 + 7 + 3 = 20 cm.
  5. 22 cm — Walk along every outer edge once: 7 + 4 + 7 + 4 = 22 cm.
  6. 12 cm — Walk along every outer edge once: 4 + 2 + 4 + 2 = 12 cm.
  7. 22 cm — Walk along every outer edge once: 9 + 2 + 9 + 2 = 22 cm.
  8. 28 cm — Walk along every outer edge once: 8 + 6 + 8 + 6 = 28 cm.

Challenge

  1. 18 cm — Add four edges: 6+3+6+3=18 cm. Boundary distance is one-dimensional.
  2. It is internal — When the pieces are combined the shared edge lies inside the new 12-by-1 figure. Only the exposed boundary is counted.

Exit

  1. 14 cm — Walk along every outer edge once: 3 + 4 + 3 + 4 = 14 cm.
  2. 24 cm — Walk along every outer edge once: 7 + 5 + 7 + 5 = 24 cm.
  3. 18 cm — Walk along every outer edge once: 6 + 3 + 6 + 3 = 18 cm.

Perimeter of Rectangles

Prerequisite

  1. 18 cm — Walk along every outer edge once: 7 + 2 + 7 + 2 = 18 cm.
  2. 24 cm — Walk along every outer edge once: 8 + 4 + 8 + 4 = 24 cm.
  3. 20 cm — Walk along every outer edge once: 6 + 4 + 6 + 4 = 20 cm.

Understanding checks

  1. 22 cm — Walk along every outer edge once: 5 + 6 + 5 + 6 = 22 cm.
  2. 24 cm — Walk along every outer edge once: 8 + 4 + 8 + 4 = 24 cm.
  3. 22 cm — Walk along every outer edge once: 6 + 5 + 6 + 5 = 22 cm.

Guided

  1. 18 cm — Walk along every outer edge once: 6 + 3 + 6 + 3 = 18 cm.
  2. 20 cm — Walk along every outer edge once: 7 + 3 + 7 + 3 = 20 cm.
  3. 16 cm — Walk along every outer edge once: 4 + 4 + 4 + 4 = 16 cm.
  4. 28 cm — Walk along every outer edge once: 9 + 5 + 9 + 5 = 28 cm.
  5. 16 cm — Walk along every outer edge once: 3 + 5 + 3 + 5 = 16 cm.

Independent

  1. 20 cm — Walk along every outer edge once: 5 + 5 + 5 + 5 = 20 cm.
  2. 18 cm — Walk along every outer edge once: 5 + 4 + 5 + 4 = 18 cm.
  3. 26 cm — Walk along every outer edge once: 7 + 6 + 7 + 6 = 26 cm.
  4. 20 cm — Walk along every outer edge once: 8 + 2 + 8 + 2 = 20 cm.
  5. 28 cm — Walk along every outer edge once: 8 + 6 + 8 + 6 = 28 cm.
  6. 20 cm — Walk along every outer edge once: 4 + 6 + 4 + 6 = 20 cm.
  7. 26 cm — Walk along every outer edge once: 8 + 5 + 8 + 5 = 26 cm.
  8. 24 cm — Walk along every outer edge once: 9 + 3 + 9 + 3 = 24 cm.

Challenge

  1. 20 cm — Add four edges: 7+3+7+3=20 cm. Boundary distance is one-dimensional.
  2. It is internal — When the pieces are combined the shared edge lies inside the new 14-by-1 figure. Only the exposed boundary is counted.

Exit

  1. 14 cm — Walk along every outer edge once: 3 + 4 + 3 + 4 = 14 cm.
  2. 14 cm — Walk along every outer edge once: 4 + 3 + 4 + 3 = 14 cm.
  3. 12 cm — Walk along every outer edge once: 4 + 2 + 4 + 2 = 12 cm.

Area and Perimeter

Prerequisite

  1. 12 cm — Walk along every outer edge once: 4 + 2 + 4 + 2 = 12 cm.
  2. 10 cm — Walk along every outer edge once: 3 + 2 + 3 + 2 = 10 cm.
  3. 18 cm — Walk along every outer edge once: 7 + 2 + 7 + 2 = 18 cm.

Understanding checks

  1. 12 square units; 14 units — Area counts square units: 3 × 4 = 12. Perimeter counts the boundary: 2(3+4) = 14. They have different units.
  2. 32 square units; 24 units — Area counts square units: 8 × 4 = 32. Perimeter counts the boundary: 2(8+4) = 24. They have different units.
  3. 36 square units; 24 units — Area counts square units: 6 × 6 = 36. Perimeter counts the boundary: 2(6+6) = 24. They have different units.

Guided

  1. 25 square units; 20 units — Area counts square units: 5 × 5 = 25. Perimeter counts the boundary: 2(5+5) = 20. They have different units.
  2. 30 square units; 22 units — Area counts square units: 5 × 6 = 30. Perimeter counts the boundary: 2(5+6) = 22. They have different units.
  3. 36 square units; 26 units — Area counts square units: 9 × 4 = 36. Perimeter counts the boundary: 2(9+4) = 26. They have different units.
  4. 15 square units; 16 units — Area counts square units: 5 × 3 = 15. Perimeter counts the boundary: 2(5+3) = 16. They have different units.
  5. 24 square units; 20 units — Area counts square units: 4 × 6 = 24. Perimeter counts the boundary: 2(4+6) = 20. They have different units.

Independent

  1. 27 square units; 24 units — Area counts square units: 9 × 3 = 27. Perimeter counts the boundary: 2(9+3) = 24. They have different units.
  2. 20 square units; 18 units — Area counts square units: 5 × 4 = 20. Perimeter counts the boundary: 2(5+4) = 18. They have different units.
  3. 24 square units; 20 units — Area counts square units: 6 × 4 = 24. Perimeter counts the boundary: 2(6+4) = 20. They have different units.
  4. 18 square units; 18 units — Area counts square units: 6 × 3 = 18. Perimeter counts the boundary: 2(6+3) = 18. They have different units.
  5. 6 square units; 10 units — Area counts square units: 3 × 2 = 6. Perimeter counts the boundary: 2(3+2) = 10. They have different units.
  6. 45 square units; 28 units — Area counts square units: 9 × 5 = 45. Perimeter counts the boundary: 2(9+5) = 28. They have different units.
  7. 8 square units; 12 units — Area counts square units: 4 × 2 = 8. Perimeter counts the boundary: 2(4+2) = 12. They have different units.
  8. 54 square units; 30 units — Area counts square units: 9 × 6 = 54. Perimeter counts the boundary: 2(9+6) = 30. They have different units.

Challenge

  1. No — Both areas are equal, but different side dimensions usually give different boundary lengths.
  2. Perimeter 12 units — The area is 3 × 3 = 9 square units, but its perimeter is 2 × (3+3) = 12 units. Area and perimeter measure different things.

Exit

  1. 9 square units; 12 units — Area counts square units: 3 × 3 = 9. Perimeter counts the boundary: 2(3+3) = 12. They have different units.
  2. 48 square units; 28 units — Area counts square units: 8 × 6 = 48. Perimeter counts the boundary: 2(8+6) = 28. They have different units.
  3. 16 square units; 20 units — Area counts square units: 8 × 2 = 16. Perimeter counts the boundary: 2(8+2) = 20. They have different units.

Practice A

  1. 10 square units — There are 2 rows of 5 equal square units: 5 × 2 = 10 square units.
  2. 24 square units — There are 4 rows of 6 equal square units: 6 × 4 = 24 square units.
  3. 6 square units — There are 2 rows of 3 equal square units: 3 × 2 = 6 square units.
  4. 23 square units — Whole rectangle: 7 × 5 = 35. Missing corner: 4 × 3 = 12. Remaining: 35 − 12 = 23.
  5. 30 square units — There are 5 rows of 6 equal square units: 6 × 5 = 30 square units.
  6. 36 square units — There are 6 rows of 6 equal square units: 6 × 6 = 36 square units.
  7. 36 square units — There are 6 rows of 6 equal square units: 6 × 6 = 36 square units.
  8. 13 square units — Whole rectangle: 3 × 6 = 18. Missing corner: 1 × 5 = 5. Remaining: 18 − 5 = 13.
  9. 18 square units — There are 6 rows of 3 equal square units: 3 × 6 = 18 square units.
  10. 15 square units — There are 3 rows of 5 equal square units: 5 × 3 = 15 square units.
  11. 45 square units — There are 5 rows of 9 equal square units: 9 × 5 = 45 square units.
  12. 20 square units — Whole rectangle: 6 × 4 = 24. Missing corner: 4 × 1 = 4. Remaining: 24 − 4 = 20.
  13. 48 square units — There are 6 rows of 8 equal square units: 8 × 6 = 48 square units.
  14. 32 square units — There are 4 rows of 8 equal square units: 8 × 4 = 32 square units.
  15. 25 square units — There are 5 rows of 5 equal square units: 5 × 5 = 25 square units.
  16. 4 square units — Whole rectangle: 3 × 2 = 6. Missing corner: 2 × 1 = 2. Remaining: 6 − 2 = 4.
  17. Perimeter 18 units — The area is 5 × 4 = 20 square units, but its perimeter is 2 × (5+4) = 18 units. Area and perimeter measure different things.
  18. No — Both areas are equal, but different side dimensions usually give different boundary lengths.

Practice B

  1. 30 square units — There are 6 rows of 5 equal square units: 5 × 6 = 30 square units.
  2. 27 square units — There are 3 rows of 9 equal square units: 9 × 3 = 27 square units.
  3. 10 square units — There are 2 rows of 5 equal square units: 5 × 2 = 10 square units.
  4. 10 square units — Whole rectangle: 4 × 4 = 16. Missing corner: 2 × 3 = 6. Remaining: 16 − 6 = 10.
  5. 18 cm — Walk along every outer edge once: 4 + 5 + 4 + 5 = 18 cm.
  6. 30 cm — Walk along every outer edge once: 9 + 6 + 9 + 6 = 30 cm.
  7. 12 square units; 14 units — Area counts square units: 4 × 3 = 12. Perimeter counts the boundary: 2(4+3) = 14. They have different units.
  8. 28 square units — There are 4 rows of 7 equal square units: 7 × 4 = 28 square units.
  9. 35 square units — There are 5 rows of 7 equal square units: 7 × 5 = 35 square units.
  10. 8 square units — There are 2 rows of 4 equal square units: 4 × 2 = 8 square units.
  11. 23 square units — Whole rectangle: 9 × 3 = 27. Missing corner: 2 × 2 = 4. Remaining: 27 − 4 = 23.
  12. 20 cm — Walk along every outer edge once: 4 + 6 + 4 + 6 = 20 cm.
  13. 18 cm — Walk along every outer edge once: 6 + 3 + 6 + 3 = 18 cm.
  14. 25 square units; 20 units — Area counts square units: 5 × 5 = 25. Perimeter counts the boundary: 2(5+5) = 20. They have different units.
  15. 27 square units — There are 3 rows of 9 equal square units: 9 × 3 = 27 square units.
  16. 6 square units — There are 2 rows of 3 equal square units: 3 × 2 = 6 square units.
  17. Perimeter 24 units — The area is 6 × 6 = 36 square units, but its perimeter is 2 × (6+6) = 24 units. Area and perimeter measure different things.
  18. No — Both areas are equal, but different side dimensions usually give different boundary lengths.

Chapter Review

  1. 16 square units — There are 4 rows of 4 equal square units: 4 × 4 = 16 square units.
  2. 54 square units — There are 6 rows of 9 equal square units: 9 × 6 = 54 square units.
  3. 9 square units — There are 3 rows of 3 equal square units: 3 × 3 = 9 square units.
  4. 16 square units — Whole rectangle: 7 × 4 = 28. Missing corner: 6 × 2 = 12. Remaining: 28 − 12 = 16.
  5. 14 cm — Walk along every outer edge once: 5 + 2 + 5 + 2 = 14 cm.
  6. 20 cm — Walk along every outer edge once: 8 + 2 + 8 + 2 = 20 cm.
  7. 14 square units; 18 units — Area counts square units: 7 × 2 = 14. Perimeter counts the boundary: 2(7+2) = 18. They have different units.
  8. 12 square units — There are 2 rows of 6 equal square units: 6 × 2 = 12 square units.
  9. 40 square units — There are 5 rows of 8 equal square units: 8 × 5 = 40 square units.
  10. 12 square units — There are 4 rows of 3 equal square units: 3 × 4 = 12 square units.
  11. 16 square units — Whole rectangle: 5 × 4 = 20. Missing corner: 2 × 2 = 4. Remaining: 20 − 4 = 16.
  12. 16 cm — Walk along every outer edge once: 3 + 5 + 3 + 5 = 16 cm.
  13. 24 cm — Walk along every outer edge once: 6 + 6 + 6 + 6 = 24 cm.
  14. 12 square units; 16 units — Area counts square units: 6 × 2 = 12. Perimeter counts the boundary: 2(6+2) = 16. They have different units.
  15. 12 square units — There are 3 rows of 4 equal square units: 4 × 3 = 12 square units.
  16. 28 square units — There are 4 rows of 7 equal square units: 7 × 4 = 28 square units.
  17. 24 square units — There are 6 rows of 4 equal square units: 4 × 6 = 24 square units.
  18. 10 square units — Whole rectangle: 6 × 5 = 30. Missing corner: 5 × 4 = 20. Remaining: 30 − 20 = 10.
  19. 12 cm — Walk along every outer edge once: 3 + 3 + 3 + 3 = 12 cm.
  20. 22 cm — Walk along every outer edge once: 9 + 2 + 9 + 2 = 22 cm.
  21. 15 square units; 16 units — Area counts square units: 3 × 5 = 15. Perimeter counts the boundary: 2(3+5) = 16. They have different units.
  22. 15 square units — There are 5 rows of 3 equal square units: 3 × 5 = 15 square units.
  23. 9 square units — There are 3 rows of 3 equal square units: 3 × 3 = 9 square units.
  24. 35 square units — There are 5 rows of 7 equal square units: 7 × 5 = 35 square units.
  25. Perimeter 16 units — The area is 3 × 5 = 15 square units, but its perimeter is 2 × (3+5) = 16 units. Area and perimeter measure different things.
  26. No — Both areas are equal, but different side dimensions usually give different boundary lengths.

Practice Exam

  1. 48 square units; 28 units — The suggested answer is not correct. Area counts square units: 8 × 6 = 48. Perimeter counts the boundary: 2(8+6) = 28. They have different units. (3 points: 1 point for identifying the given answer as incorrect; 1 for explaining the error; 1 for giving the corrected solution and appropriate units.)
  2. 32 square units; 24 units — Area counts square units: 8 × 4 = 32. Perimeter counts the boundary: 2(8+4) = 24. They have different units. (3 points: 1 point for area, 1 for perimeter, 1 for both appropriate units.)
  3. 15 square units — There are 5 rows of 3 equal square units: 3 × 5 = 15 square units. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  4. 16 cm — Walk along every outer edge once: 4 + 4 + 4 + 4 = 16 cm. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  5. 15 square units — The suggested answer is not correct. There are 3 rows of 5 equal square units: 5 × 3 = 15 square units. (3 points: 1 point for identifying the given answer as incorrect; 1 for explaining the error; 1 for giving the corrected solution and appropriate units.)
  6. 20 cm — Walk along every outer edge once: 7 + 3 + 7 + 3 = 20 cm. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  7. 20 square units — There are 4 rows of 5 equal square units: 5 × 4 = 20 square units. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  8. 42 square units — There are 6 rows of 7 equal square units: 7 × 6 = 42 square units. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  9. 24 square units — The suggested answer is not correct. There are 3 rows of 8 equal square units: 8 × 3 = 24 square units. (3 points: 1 point for identifying the given answer as incorrect; 1 for explaining the error; 1 for giving the corrected solution and appropriate units.)
  10. 26 cm — Walk along every outer edge once: 8 + 5 + 8 + 5 = 26 cm. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  11. 21 square units — There are 3 rows of 7 equal square units: 7 × 3 = 21 square units. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  12. 22 cm — Walk along every outer edge once: 6 + 5 + 6 + 5 = 22 cm. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  13. 24 square units — The suggested answer is not correct. There are 4 rows of 6 equal square units: 6 × 4 = 24 square units. (3 points: 1 point for identifying the given answer as incorrect; 1 for explaining the error; 1 for giving the corrected solution and appropriate units.)
  14. 23 square units — Whole rectangle: 5 × 5 = 25. Missing corner: 1 × 2 = 2. Remaining: 25 − 2 = 23. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  15. 24 square units — There are 6 rows of 4 equal square units: 4 × 6 = 24 square units. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  16. 24 square units — There are 6 rows of 4 equal square units: 4 × 6 = 24 square units. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  17. 16 cm — The suggested answer is not correct. Walk along every outer edge once: 5 + 3 + 5 + 3 = 16 cm. (3 points: 1 point for identifying the given answer as incorrect; 1 for explaining the error; 1 for giving the corrected solution and appropriate units.)
  18. 20 square units — There are 4 rows of 5 equal square units: 5 × 4 = 20 square units. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  19. 12 square units — There are 2 rows of 6 equal square units: 6 × 2 = 12 square units. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  20. 15 square units; 16 units — Area counts square units: 5 × 3 = 15. Perimeter counts the boundary: 2(5+3) = 16. They have different units. (3 points: 1 point for area, 1 for perimeter, 1 for both appropriate units.)
  21. 16 cm — The suggested answer is not correct. Walk along every outer edge once: 6 + 2 + 6 + 2 = 16 cm. (3 points: 1 point for identifying the given answer as incorrect; 1 for explaining the error; 1 for giving the corrected solution and appropriate units.)
  22. 7 square units — Whole rectangle: 3 × 3 = 9. Missing corner: 2 × 1 = 2. Remaining: 9 − 2 = 7. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  23. Perimeter 16 units — The area is 4 × 4 = 16 square units, but its perimeter is 2 × (4+4) = 16 units. Area and perimeter measure different things. (3 points: 1 point for the correct claim; 1 for a mathematical justification; 1 for a clear diagram, example, or relevant calculation when applicable.)
  24. No — Both areas are equal, but different side dimensions usually give different boundary lengths. (3 points: 1 point for the correct claim; 1 for a mathematical justification; 1 for a clear diagram, example, or relevant calculation when applicable.)