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Math · Number Sense & Mathematical Reasoning
Two-Step Word Problems: Find the Unknown, Choose the Operation, Check
Grade 3–4 reasoning with equal groups, comparisons, two-step problems, misleading keywords, remainders and missing-information traps.
The idea
Before calculating, ask: What must I find? Which facts matter? What answer must I find first? Write a short sentence naming the unknown.
Words like 'more' and 'left' are clues, not automatic commands. If Mina has 9 more than Ava, you add or subtract depending on which girl's total you already know.
Show each step as a short equation or labelled diagram. Keep units such as apples, metres or cars, and check your answer with an inverse calculation.
Work from Foundation and Core toward Transfer. Challenge includes remainders, irrelevant numbers and questions that cannot be solved from the information given.
Worked examples
Example 1 · Groups and subtraction
Four bags contain 7 apples each. Six are eaten.
- 4 × 7 = 28 apples at first.
- 28 − 6 = 22 apples left.
- Check: 22 + 6 = 28.
22 apples.
Example 2 · More can mean subtract
Iris has 28 stickers, 9 more than Ben. Find Ben's number.
- Iris has the larger count.
- 28 − 9 = 19.
- Check: 19 + 9 = 28.
19 stickers.
Example 3 · People need a seat
Twenty-five children need tables with 4 seats each.
- Six tables seat 24 children.
- One child still needs a seat.
- We need another whole table.
7 tables.
Example 4 · Complete groups
Twenty-five beads make bracelets with 4 beads each.
- Six bracelets use 24 beads.
- One bead is left over.
- One bead cannot make a complete bracelet.
6 bracelets, 1 bead left.
Example 5 · Ignore a distractor
A 25 m pool has 6 lanes. A swimmer completes 4 lengths in one lane.
- The question asks for metres, not lanes.
- 4 × 25 = 100 m.
- Six lanes does not affect the distance.
100 m.
Common mistakes
These are tempting because they use familiar operations. Check what the question is actually measuring.
Why this fails: For '26 is 9 more than an unknown', the unknown is 26 − 9.
Why this fails: The second step needs to know whether the number means groups, items, metres or seats.
Why this fails: A partial group needs another whole vehicle or table.
Why this fails: The number of complete bracelets excludes the leftover beads.
Why this fails: Choose relevant facts using the question; colours and lane counts may not matter.
Foundation
Foundation practice
Find the unknown and choose an operation.
1. Five baskets each have 4 apples. How many apples?
2. Noah has 25 stickers; Leah has 9 fewer. How many does Leah have?
3. After buying 7 pencils the class has 24. How many did it have before?
4. Thirty counters are shared equally by 5 students. How many each?
5. Four packets each hold 6 cards. How many cards in all?
6. A swimmer completes 3 lengths in a 25 m pool. What distance?
7. A blue envelope holds 12 tickets. How many does it hold?
8. There are 36 crayons in groups of 6. How many groups?
Core
Core practice
Connect two steps and keep the units.
9. Four boxes hold 6 pencils each. The class uses 5. How many remain?
10. Thirty-six markers fill boxes of 6. Two boxes are given away. How many boxes remain?
11. Seven tables seat 4 children each, and 3 extra chairs are added. How many places?
12. Thirty-eight clips are in a tin. Twelve are used, then five more. How many remain?
13. Four lengths of a 25 m pool plus one more length: how far?
14. Three pieces of ribbon are 8 cm each. Cut 5 cm from the total. How much remains?
15. Four shelves hold 9 books each, with 12 more on a desk. How many books?
16. Twenty-four muffins fill boxes of 6. One empty box is added. How many boxes in all?
17. A shop has 42 balloons and gives away 12. Make bunches of 5. How many full bunches?
18. Six trays have 7 seeds each. Ten seeds fail to sprout. How many sprout?
Transfer
Transfer practice
Let the meaning—not a keyword—decide the operation.
19. Mina has 26 beads, which is 9 more than Ava. How many does Ava have?
20. Raj has 14 marbles, which is 6 fewer than Mei. How many does Mei have?
21. Six packets have 4 crackers each; three children share them equally. How many each?
22. I had some stickers. I gave away 8, found 5, and now have 17. How many did I start with?
23. Twenty-six balls go into bags of 4. How many full bags and leftovers?
24. Sara has twice as many shells as Tom. Together they have 18. How many does Tom have?
25. Five players score 8 points each. Jerseys are green. A 6-point penalty follows. Score?
26. A 25 m pool has six lanes. A swimmer completes three lengths in one lane. How far?
27. Ben has 32 markers and buys 9 more. He subtracts because he sees the word 'more'. Correct total?
Challenge
Challenge practice
Explain remainder rules and identify insufficient information.
28. A team wins 5 matches and practises 40 minutes. How many goals does it score?
29. Twenty-nine students need cars with 4 passenger seats. How many cars are needed?
30. Twenty-nine beads make bracelets of 4 beads each. How many complete bracelets?
31. Seven cartons contain 6 eggs each. Nine break. How many unbroken eggs?
32. Ava reads 12 pages Monday, 8 more Tuesday than Monday, and 6 fewer Wednesday than Tuesday. Wednesday?
33. Three boxes hold 8 bottles each. Two break and 5 are used. How many remain?
34. Three rows have 8 seats each; two rows have 6 each. How many seats?
35. Ella has 27 cards, 9 more than Jack. Jack buys 6 more. How many does Jack now have?
36. Blue scores 4 groups of 7 points. Red scores 6 fewer than blue. How many more does blue have?
Answer key
- 1. 20 apples
5 × 4 = 20. - 2. 16 stickers
25 − 9 = 16. - 3. 17 pencils
24 − 7 = 17. - 4. 6 counters
30 ÷ 5 = 6. - 5. 24 cards
4 × 6 = 24. - 6. 75 m
3 × 25 = 75 m. - 7. 12 tickets
Colour is irrelevant; 12 is already given. - 8. 6 groups
36 ÷ 6 = 6. - 9. 19 pencils
4 × 6 = 24; 24 − 5 = 19. - 10. 4 boxes
36 ÷ 6 = 6; 6 − 2 = 4. - 11. 31 places
7 × 4 + 3 = 31. - 12. 21 clips
38 − 12 − 5 = 21. - 13. 125 m
4 × 25 + 25 = 125 m. - 14. 19 cm
3 × 8 − 5 = 19 cm. - 15. 48 books
4 × 9 + 12 = 48. - 16. 5 boxes
24 ÷ 6 + 1 = 5. - 17. 6 bunches
(42 − 12) ÷ 5 = 6. - 18. 32 seeds
6 × 7 − 10 = 32. - 19. 17 beads
26 − 9 = 17; Mina has the larger count. - 20. 20 marbles
14 + 6 = 20. - 21. 8 crackers
6 × 4 = 24; 24 ÷ 3 = 8. - 22. 20 stickers
Work backward: 17 − 5 + 8 = 20. - 23. 6 full bags, 2 left
6 × 4 = 24, leaving 2. - 24. 6 shells
One share plus two shares = three shares: 18 ÷ 3 = 6. - 25. 34 points
5 × 8 − 6 = 34; colour is irrelevant. - 26. 75 m
3 × 25 = 75 m; lane count does not matter. - 27. 41 markers
Buying increases the quantity: 32 + 9 = 41. - 28. Cannot be determined
Neither fact gives the goals scored. - 29. 8 cars
Seven cars seat 28; one more student needs another car. - 30. 7 bracelets
Seven use 28 beads; one is left over. - 31. 33 eggs
7 × 6 − 9 = 33. - 32. 14 pages
Tuesday 20; Wednesday 20 − 6 = 14. - 33. 17 bottles
3 × 8 − 2 − 5 = 17. - 34. 36 seats
3 × 8 + 2 × 6 = 36. - 35. 24 cards
27 − 9 + 6 = 24. - 36. 6 points
Blue 28, red 22; difference 6.