Balance Puzzles

Free puzzles to print

Each set: a Learn sheet with a worked example, 10 puzzles from easiest to hardest, and the answers.

• Level 1

  Ages 4–6 · Pre-K–K

  

  One scale (1 toy car = 3 buttons; how many buttons balance 2 toy cars?). Then two scales that each say what one object is worth, then an extra scale the question doesn't need. Answers stay within 10.

  On the sheet: “Both sides weigh the same. How many balance the last pan? Check it.”

  Print Level 1 (Letter)Other paper size: A4

• Level 2

  Ages 6–8 · Grades 1–2

  

  The easiest puzzles have two scales that each say what one shape is worth; the answer comes from reading them and adding (up to 10). One scale may not be needed.

  Then one swap. A shape may be worth shapes you already know, so swap each one for its worth and add. Or the same shape sits on both pans, so take it off. Answers go up to 12.

  The hardest puzzles have three scales and two or more such steps, sometimes with the same shape on both pans of one scale.

  On the sheet: “A level scale means both sides weigh the same. Find how many shapes balance the last pan.”

  Print Level 2 (Letter)Other paper size: A4

• Level 3

  Ages 8–10 · Grades 3–4

  

  Two shapes and number weights. First comes sharing (3 squares weigh 18, so 1 square weighs 6), with a second scale that uses it. Then the second scale holds more shapes. Then no shape is alone on any scale, so the child compares two scales and takes the difference. The hardest use three scales and totals up to 50.

  On the sheet: “Each scale is level. Find what each shape weighs.”

  Print Level 3 (Letter)Other paper size: A4

• Level 4

  Ages 10–12 · Grades 5–6

  

  Three shapes (square, circle, triangle) on three scales, up to 5 things on a pan. First a chain where each scale adds one new shape; then one take-away between scales. Next, two take-aways with no shape alone. The hardest also write each scale with letters (s + 2c + t = 21).

  On the sheet: “Each scale is level. Find what each shape weighs.”

  Print Level 4 (Letter)Other paper size: A4

The thinking it works on

• Equals means the same on both sides

  Reads a level scale as "both sides are worth the same".

  Rittle-Johnson and colleagues (2011) tested 175 children in grades 2 to 6: by grade 5, most saw the equal sign as 'both sides are worth the same'. Source

• Swapping equal things

  Replaces something with what it is worth.

• Sharing a total

  Divides a total weight equally among the same shapes.

  Squire and Bryant (2002) studied sharing problems with 5- to 8-year-olds. The children found them much easier when the items were already grouped, with one group for each person sharing. Source

• Comparing two scales

  Finds what one scale has that another doesn't.

  Robinson and Dubé (2009) studied children in grades 2 to 4. They varied widely in whether they noticed that adding and then taking away the same amount cancels out, with little change across the grades. Some used the shortcut and others calculated everything. Source

• Picking the useful facts

  Decides which scale to use first and which to leave out.

• Using letters for unknowns

  Reads a scale written with letters.

Where is this thinking used?

• Kitchen scales: A recipe weight can be made from different things that weigh the same.
• Seesaws: Two small children can balance one bigger one.
• Unit prices: If 3 packs cost $18, one pack costs $6, and that settles the rest of the bill.
• Fair trades: A swap is fair when both sides are worth the same.