A word problem is a small story about quantities. Before choosing a calculation, explain what is known, what is happening and what you need to find. A few counters or a simple drawing can make the relationship easier to see.
These made-up classroom stories use whole numbers and one-step relationships. Work at your own pace. An adult can read a story aloud, explain unfamiliar everyday words or draw while you give directions. Being able to read every word independently is different from understanding the mathematics.
A routine for understanding the story
- Retell: say the story in your own words without changing its quantities.
- Find the question: name the unknown, including what it counts.
- Represent: move large paper counters, draw groups or sketch a labelled bar.
- Calculate: choose an operation that fits the relationship.
- Check: put the answer back into the story and write a sentence with its unit.
A question mark in a model stands for the amount you do not yet know. It might be the total, an amount left, the number in each equal group or the smaller of two compared quantities. Locate it before calculating.
Use drawn circles if you do not have counters. Avoid small objects if they create a swallowing or handling risk. A bar sketch does not need artistic detail: its job is to show how the quantities fit together. Label every known part and the unknown.
Worked case 1: join two parts
Story: a classroom display has 17 red flags and 12 blue flags. How many flags are there altogether?
Retell it as two separate collections being counted together. We know both parts, and the unknown is the total number of flags. Build one group of seventeen and another of twelve, then join them without counting any flag twice.
The matching equation is 17 + 12 = 29. One mental route is 17 + 10 = 27, then 27 + 2 = 29. Answer: There are 29 flags altogether.
Check by separating the total again: 29 − 12 = 17. The answer should exceed either part because both non-empty collections contribute to the total. An answer of five would describe a difference between the parts, which is not what this question asks.
Worked case 2: find what remains
Story: there are 28 picture cards in a folder. Nine cards are given away. How many cards remain?
Start with one collection of twenty-eight, then remove nine. Cross out nine circles or separate nine counters from the starting collection. The unknown is the number of cards still in the folder.
Calculate 28 − 9 = 19. You can remove ten to reach eighteen, then return one because only nine should have been removed. Answer: 19 cards remain.
Check by putting the two parts together: 19 remaining + 9 given away = 28 at first. The given-away cards still matter to the explanation, but they do not belong in the answer about cards remaining.
Worked case 3: find the total in equal groups
Story: four trays each hold six paper shapes. How many paper shapes are there in all?
Draw four tray outlines and put six circles inside every outline. The word “each” applies to every tray. Before calculating, check the groups: a tray with five or seven shapes would not match this story.
Add the equal groups: 6 + 6 + 6 + 6 = 24. Multiplication records the same grouping: 4 × 6 = 24, meaning four trays with six shapes per tray. Answer: There are 24 paper shapes.
The numbers four and six have different jobs: four counts trays; six counts shapes in each tray. Adding 4 + 6 would mix those jobs and would not count every shape in the picture.
Worked case 4: find an equal share
Story: 24 paper shapes are shared equally among four trays. How many shapes go on each tray?
This time the total is already known. Draw four empty trays and distribute the twenty-four shapes one at a time around them. Finish each round before starting another so that every tray receives the same number. Six full rounds use all the shapes.
Calculate 24 ÷ 4 = 6. Answer: Each tray receives six paper shapes. Check the whole: 4 × 6 = 24 shapes, with none left over.
Compare this with the previous story. Their finished pictures match, but their questions differ. In the multiplication story, the total is missing. In the sharing story, the amount per tray is missing. The unknown helps decide which calculation is useful.
Worked case 5: beware of a “more” shortcut
Story: A has eight more counters than B. A has 23 counters. How many counters does B have?
Say who has the larger amount: A does. Draw A’s bar and label its full amount 23. B’s bar must be shorter by eight. The missing amount is B’s smaller collection, not A’s larger collection.
The relationship is ? + 8 = 23. Find the unknown using 23 − 8 = 15. Answer: B has 15 counters. Check the comparison: 15 + 8 = 23, so A really does have eight more.
If you added 23 + 8, you would get 31. That would make B have more counters than A, contradicting the story. The word “more” is useful information about the comparison, but it does not choose the operation on its own.
Now change the known amount: “A has eight more counters than B. B has 23 counters. How many does A have?” The larger amount is now unknown, so 23 + 8 = 31 is appropriate. The same comparison words can accompany different calculations.
A question–model–equation–sentence grid
Use one row for a story. The equation should describe the quantities in the model, and the sentence should answer the original question. This completed grid brings the first four cases together.
| Question: find… | Model | Equation | Answer sentence |
|---|---|---|---|
| Total flags | Join parts of 17 and 12 | 17 + 12 = 29 | There are 29 flags. |
| Cards remaining | Remove 9 from a whole of 28 | 28 − 9 = 19 | 19 cards remain. |
| Total shapes | 4 equal groups of 6 | 4 × 6 = 24 | There are 24 shapes. |
| Shapes on each tray | Share 24 equally among 4 groups | 24 ÷ 4 = 6 | Each tray has 6 shapes. |
Complete two partly finished examples
Example A: the equation is missing
A board has 13 triangle cards and 15 circle cards. How many cards are on the board? The model has two joined parts labelled 13 and 15, with a question mark over the whole. Complete: equation ___; answer sentence ___.
Completion: 13 + 15 = 28. There are 28 cards on the board. Both part amounts are known, so join them to find the missing whole. Check: 28 − 13 = 15.
Example B: the unknown is missing
Twenty paper flowers are shared equally among five display panels. Complete the model labels: whole = 20 flowers; number of equal groups = 5 panels; unknown = ___. Then finish 20 ÷ 5 = ___.
Completion: the unknown is the number of flowers on each panel. The quotient is four. Each panel receives four paper flowers. Check: five groups of four give twenty flowers. An answer of “four panels” would use the wrong unit.
Four new practice stories with explanations
Practice 1: a missing part
A mural uses 35 paper leaves. Twenty-two are green and the rest are yellow. How many leaves are yellow?
Solution: the whole is 35 and the known part is 22. Draw one whole bar with a 22 part and an unknown part. Calculate 35 − 22 = 13 yellow leaves. Check: 22 + 13 = 35. We subtract because a part is missing, even though nothing is being given away.
Practice 2: repeated equal groups
Five envelopes each contain seven puzzle pieces. How many puzzle pieces are in the envelopes?
Solution: draw five groups, each labelled seven. The total is unknown, so 5 × 7 = 35 puzzle pieces. Check by repeated addition: 7 + 7 + 7 + 7 + 7 = 35.
Practice 3: equal sharing
Thirty-six bookmarks are placed equally into six folders. How many bookmarks go into each folder?
Solution: the total is 36, and six is the number of equal groups. Calculate 36 ÷ 6 = 6 bookmarks per folder. Six folders containing six bookmarks each account for all thirty-six. No bookmarks are left unassigned.
Practice 4: a smaller unknown
A long display row has 32 paper stars. It has nine more stars than a short row. How many stars are in the short row?
Solution: the short row plus nine equals 32. Draw the longer bar as the short bar and an extra nine. Calculate 32 − 9 = 23 stars in the short row. Check: 23 + 9 = 32, and 23 is smaller than 32 as required.
Questions about word problems
Should I underline words such as “more”?
You may mark useful information, but read the whole relationship. “More” identifies a comparison; the unknown determines whether addition or subtraction will help.
What if I can calculate but cannot read the story?
Ask someone to read it aloud without naming the operation. Then retell it, point to the quantities and choose a model yourself.
Do I have to draw every object?
No. Small groups can be drawn as circles; larger amounts can use labelled bars or group outlines. Keep each quantity and its role clear.
Can two operations describe the same story?
Yes. The relationship ? + 8 = 23 can be solved using 23 − 8. One equation shows how the parts fit; the other directly finds the unknown.
Why must I include a unit in my answer?
The unit says what the number counts. Six shapes per tray answers a sharing question; six trays would describe a different quantity.
Does subtraction always mean taking objects away?
No. It can also find a missing part or a difference between quantities. The model should show which meaning fits the story.
How do I check a word-problem answer?
Use it in the original relationship, check the unit and ask whether its size makes sense. Rebuilding the whole or reversing the operation can help.
For an adult supporting the activity
Ask “What are we finding?” before “Which operation?” Deliberate instruction on word problems and carefully chosen representations are recommended in the Institute of Education Sciences mathematics practice guide. Let the learner explain a model before expecting a written calculation.
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