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How to Solve Picture Equation Balance Puzzles

In a picture equation, matching shapes have the same value within that puzzle. A balanced scale says the total on the left equals the total on the right. Work with equal groups and known unit cubes before guessing a shape.

This is fourth-grade enrichment and early pre-algebra, not a claim that solving two equations with two unknowns is a required fourth-grade standard. Try the questions together; save the worked answer until your child has had a turn.

Two connected balances

Each unit cube is worth 1. The same circle appears in both balances, so its value does not change between them.

Balance A

2 circles + 4 unit cubes = 18 unit cubes

Balance B

1 circle + 2 triangles = 17 unit cubes

Ask before revealing the steps

  1. On Balance A, what can you remove from both sides without tipping it?
  2. How many equal groups remain? What would one circle be worth?
  3. Where can that circle value replace a picture on Balance B?
  4. How could you check your answers on both balances?

If a child is stuck, place two drawn circles beside 14 marks and divide the marks evenly before writing an equation.

Work it through together

  1. Make equal groups on Balance A. Remove 4 unit cubes from each side. Two circles now equal 14 unit cubes. Split 14 into two equal groups: one circle is worth 7.
  2. Substitute into Balance B. Replace its circle with 7: 7 + 2 triangles = 17. Remove 7 from each side; two triangles equal 10. Split 10 into two equal groups: one triangle is worth 5.
  3. Check both balances. A: 2 × 7 + 4 = 18. B: 7 + 2 × 5 = 17. Both sides match in each balance, so circle = 7 and triangle = 5 solve this example.

These values belong to this two-balance example. A circle or triangle in another puzzle can have a different value.

Try a new balance puzzle

Explore the picture-equation activity and generator, then open the answer-free student sheet. Keep the separate adult answer key for checking the sheet's puzzles; it does not repeat this guide's example. You can also browse more fourth-grade math activities.