Working Backward with Operations (Grade 5)
Working backward is a strategy for stories and puzzles where you know how things ended and want how they started: "Mia had some stickers, gave 4 away, then doubled what was left and ended with 10."
Understanding working backward with operations
Start at the end and undo each step, last step first. Here, undo the doubling (10 ÷ 2 = 5), then undo the giving away (5 + 4 = 9), so Mia started with 9. Finish by running the story forward from your answer to check it lands on the ending: 9 − 4 = 5, then 5 × 2 = 10.
Seeing it in action
Worked example
A number is multiplied by 2, then 3 is added, giving 11. Find it.
Undo in reverse: 11 − 3 = 8, then 8 ÷ 2 = 4.
Undo the operations in reverse order.
A common mistake, and the fix
Undoing the steps in the order they happened.
For “times 2, then plus 3, gives 11”, halving first gives 5.5, and 5.5 − 3 = 2.5, which is wrong. Undo the last step first: 11 − 3 = 8, then 8 ÷ 2 = 4, and check: 4 × 2 + 3 = 11.
Try a few
×3 then −1 gives 14. Start?
undo −1: 15, then undo ×3: 15 ÷ 3.
+5 then ×2 gives 20. Start?
undo ×2: 10, then undo +5: 10 − 5.
÷2 then +4 gives 9. Start?
undo +4: 5, then undo ÷2: 5 × 2.
Make the Target
A calm order-of-operations game: choose the signs, watch the steps happen in order, and add parentheses.
Try it in Numeris.
A short lesson, then practice with feedback on every answer.
Put the reasoning on paper.
Try six fraction, angle, perimeter and area questions with explained answers, for grades 4–6.
Free printable · 3 pagesText version (screen-reader friendly): Draw it. Explain it. Math sample
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