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Adult answer key

Picture Equation Balance Puzzles

This separate key keeps answers and worked steps out of the student printable while giving adults everything they need to check the fixed examples and generated practice set.

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Answer entries

One unknown — remove known units

Easy
2 circles + 3 unit cubes = 13 unit cubes
What is the value of 1 circle?
Expression: circle
Prompt answer
What is the value of 1 circle? 5
Shape values: circle = 5
Computed check: 5

Worked steps

  1. Take away 3 unit cubes from both sides. That leaves 2 circles = 10.
  2. Split 10 equally between the 2 matching circles. Each circle = 5.
  3. Check: 2 × 5 + 3 = 13.

Known substitution — two shapes

Medium
3 triangles + 2 squares = 18 unit cubes
1 square = 3 unit cubes
What is the value of 1 triangle?
Expression: triangle
Prompt answer
What is the value of 1 triangle? 4
Shape values: triangle = 4, square = 3
Computed check: 4

Worked steps

  1. Use the second balance first: 1 square = 3, so 2 squares = 6.
  2. Replace the 2 squares in the first balance with 6. Now 3 triangles + 6 = 18.
  3. Subtract 6 from 18. That leaves 3 triangles = 12.
  4. Split 12 into 3 equal groups. Each triangle = 4.
  5. Check: 3 × 4 + 2 × 3 = 18.

Two equations, two unknowns

Medium
2 stars + 1 moon = 13 unit cubes
1 star + 2 moons = 14 unit cubes
What is star + moon?
Expression: star + moon
Prompt answer
What is star + moon? 9
Shape values: star = 4, moon = 5
Computed check: 9

Worked steps

  1. Double the second equation: 2 stars + 4 moons = 28.
  2. Subtract the first equation, 2 stars + 1 moon = 13. That leaves 3 moons = 15, so 1 moon = 5.
  3. Put moon = 5 back into 2 stars + 1 moon = 13. Then 2 stars = 8, so 1 star = 4.
  4. Now add them: star + moon = 4 + 5 = 9.
  5. Check both balances: 2 × 4 + 5 = 13 and 4 + 2 × 5 = 14.

Equivalent groups / ratio challenge

Medium
2 diamonds = 3 circles
1 diamond + 1 circle = 20 unit cubes
What is diamond - circle?
Expression: diamond - circle
Prompt answer
What is diamond - circle? 4
Shape values: diamond = 12, circle = 8
Computed check: 4

Worked steps

  1. From 2 diamonds = 3 circles, think in equal groups: each diamond can be 3 equal groups, and each circle can be 2 equal groups. Then 2 diamonds = 6 groups and 3 circles = 6 groups.
  2. That means diamond + circle = 3 groups + 2 groups = 5 groups.
  3. The second balance says diamond + circle = 20, so 5 groups = 20. Each group = 4.
  4. A diamond is 3 groups, so diamond = 12. A circle is 2 groups, so circle = 8.
  5. Now subtract: diamond - circle = 12 - 8 = 4.
  6. Check: 2 × 12 = 24 and 3 × 8 = 24, and 12 + 8 = 20.

Three-picture challenge

Challenge
2 hexagons + 1 triangle = 16 unit cubes
1 triangle + 2 circles = 14 unit cubes
1 hexagon + 1 circle = 11 unit cubes
What is hexagon + triangle + circle?
Expression: hexagon + triangle + circle
Prompt answer
What is hexagon + triangle + circle? 15
Shape values: hexagon = 6, triangle = 4, circle = 5
Computed check: 15

Worked steps

  1. Add the first two equations. That gives 2 hexagons + 2 triangles + 2 circles = 30.
  2. Double the third equation: 2 hexagons + 2 circles = 22.
  3. Subtract to find 2 triangles = 8, so 1 triangle = 4.
  4. Put triangle = 4 into triangle + 2 circles = 14. Then 2 circles = 10, so 1 circle = 5.
  5. Use hexagon + circle = 11. Since circle = 5, hexagon = 6.
  6. Now add them: 6 + 4 + 5 = 15.
  7. Check all three balances: 2 × 6 + 4 = 16, 4 + 2 × 5 = 14, and 6 + 5 = 11.

Easy generated balance puzzle

Seed: student-sheet-easy-1
Easy
4 circles + 5 unit cubes = 33 unit cubes
What is the value of 1 circle?
Expression: circle
Prompt answer
What is the value of 1 circle? 7
Shape values: circle = 7
Computed check: 7

Worked steps

  1. Take away 5 unit cubes from both sides. That leaves 4 circles = 28.
  2. Split 28 into 4 equal groups. Each circle = 7.
  3. Check: 4 × 7 + 5 = 33.

Easy generated balance puzzle

Seed: student-sheet-easy-2
Easy
4 circles + 3 unit cubes = 31 unit cubes
What is the value of 1 circle?
Expression: circle
Prompt answer
What is the value of 1 circle? 7
Shape values: circle = 7
Computed check: 7

Worked steps

  1. Take away 3 unit cubes from both sides. That leaves 4 circles = 28.
  2. Split 28 into 4 equal groups. Each circle = 7.
  3. Check: 4 × 7 + 3 = 31.

Medium generated balance puzzle

Seed: student-sheet-medium-1
Medium
2 stars + 3 moons = 31 unit cubes
1 moon = 7 unit cubes
What is the value of 1 star?
Expression: star
Prompt answer
What is the value of 1 star? 5
Shape values: star = 5, moon = 7
Computed check: 5

Worked steps

  1. Replace 3 moons with 21 unit cubes because 1 moon = 7.
  2. Now subtract 21 from 31. That leaves 2 stars = 10.
  3. Split 10 into 2 equal groups. Each star = 5.
  4. Check: 2 × 5 + 3 × 7 = 31.

Medium generated balance puzzle

Seed: student-sheet-medium-2
Medium
4 triangles + 2 moons = 42 unit cubes
1 moon = 3 unit cubes
What is the value of 1 triangle?
Expression: triangle
Prompt answer
What is the value of 1 triangle? 9
Shape values: triangle = 9, moon = 3
Computed check: 9

Worked steps

  1. Replace 2 moons with 6 unit cubes because 1 moon = 3.
  2. Now subtract 6 from 42. That leaves 4 triangles = 36.
  3. Split 36 into 4 equal groups. Each triangle = 9.
  4. Check: 4 × 9 + 2 × 3 = 42.

Challenge generated balance puzzle

Seed: student-sheet-challenge-1
Challenge
2 squares + 1 diamond = 25 unit cubes
1 diamond + 2 stars = 25 unit cubes
1 square + 1 star = 16 unit cubes
What is square + diamond + star?
Expression: square + diamond + star
Prompt answer
What is square + diamond + star? 25
Shape values: square = 8, diamond = 9, star = 8
Computed check: 25

Worked steps

  1. Add the first two balances. That gives 2 squares + 2 diamonds + 2 stars = 50.
  2. Double the third balance: 2 squares + 2 stars = 32. Subtract that from the big total, and 2 diamonds = 18. So 1 diamond = 9.
  3. Put diamond = 9 into the second balance. Then 2 stars = 16, so 1 star = 8.
  4. Use the third balance. square + 8 = 16, so 1 square = 8.
  5. Now add all three values: 8 + 9 + 8 = 25.
  6. Check all three balances to make sure they still match.

Challenge generated balance puzzle

Seed: student-sheet-challenge-2
Challenge
2 moons + 1 diamond = 25 unit cubes
1 diamond + 2 hexagons = 13 unit cubes
1 moon + 1 hexagon = 10 unit cubes
What is moon + diamond + hexagon?
Expression: moon + diamond + hexagon
Prompt answer
What is moon + diamond + hexagon? 19
Shape values: moon = 8, diamond = 9, hexagon = 2
Computed check: 19

Worked steps

  1. Add the first two balances. That gives 2 moons + 2 diamonds + 2 hexagons = 38.
  2. Double the third balance: 2 moons + 2 hexagons = 20. Subtract that from the big total, and 2 diamonds = 18. So 1 diamond = 9.
  3. Put diamond = 9 into the second balance. Then 2 hexagons = 4, so 1 hexagon = 2.
  4. Use the third balance. moon + 2 = 10, so 1 moon = 8.
  5. Now add all three values: 8 + 9 + 2 = 19.
  6. Check all three balances to make sure they still match.