PRIMARY 4 MATHEMATICS LEARNING GUIDE · BATCH 8 · GUIDE 31
Writing a mathematics question requires more than putting numbers into a story. A useful question has quantities with clear meanings, relationships that connect them, conditions that can be satisfied and a target the reader can identify. Its author should also know whether it has one answer, several possible answers or an intentionally impossible set of conditions.
Problem posing turns the learner from an answer finder into a question designer. Instead of only solving “What is three eighths of 56?”, the learner can ask which collection makes three eighths equal 21, how to change the whole while preserving a whole-number answer, or which wording would accidentally make the question ambiguous.
The MOE Primary Mathematics Syllabus connects posing questions with mathematical reasoning. This guide provides original classroom and home activities; its authoring checks are not an official assessment rubric.
Series route: return to the Primary 4 Mathematics Learning Hub. To explain a solution clearly, use Mathematical Communication and Complete Working. To test a small collection of possible answers, use Systematic Listing, Tables and Guess-and-Check.
Navigate: start with a relationship · choose valid numbers · control the conditions · revise and test · twenty authoring tasks · sample questions and answers · teaching routine.
1. Start with the mathematics before adding the story
Choose one relationship first: separate parts make a whole, an amount changes, equal groups form a total, one quantity is several times another, or a fraction selects part of a collection.
For example, four equal groups of 36 make 144. This relationship can become a question about boxes of pencils, rows of chairs or equal sets of counters. The objects change, but the equal-group structure remains.
A weak authoring route begins with a long story and then asks which operations can be made from the numbers. That can produce irrelevant information, missing conditions or an answer that is not what the story actually requests.
A stronger route writes the mathematical relationship in a private working line first. Then choose objects that suit it. Indivisible counters fit whole-number counts. Lengths and liquid amounts can be divided into fractional measurements when the taught arithmetic allows it.
The story should give the relationship a meaningful setting. It should not obscure the relationship with details that have no role in the learning task.
2. Choose which quantity the reader must find
From four groups of 36 totalling 144, three natural questions can be written.
Give four groups and 36 per group; ask for the total. Give 144 and four equal groups; ask for the amount per group. Give 144 and 36 per group; ask how many groups can be formed.
The relationship remains the same while the unknown changes. The required operation changes from multiplication to division, and the meaning of the division quotient changes between the last two questions.
This is a useful way to build a question family without simply enlarging the numbers. It tests whether the learner can move among the connected quantities.
Write the answer sentence before finalising the question. “There are four groups” and “There are 36 counters in each group” are different targets. The wording should make the intended target clear without giving away the operation.
Do not include the requested answer as an accidental fact earlier in the story. If the question asks for the total, stating “There are 144 counters altogether” may make the intended calculation unnecessary.
3. Build backwards from a desired answer
Suppose the desired answer is 36 counters in each group. Choose four groups, so the total after sharing must be 144. To make a two-step problem, choose a starting total of 170 and remove 26 before sharing.
A valid question is: “A teacher has 170 counters. She puts 26 aside. She shares all the remaining counters equally among four groups. How many counters does each group receive?”
Check forwards: 170 − 26 = 144; 144 ÷ 4 = 36. The story uses every chosen value for a specific purpose.
Designing from an answer is not cheating. It is a way to control whether the intended calculation is feasible. The learner still needs to understand the relationships to construct the inputs correctly.
However, the author’s intention does not prove that the written question has that answer. After writing the story, solve it again using only the words on the page. The wording may have accidentally changed a stage or omitted the equal-sharing condition.
4. Choose numbers that respect the objects
For three eighths of a set of counters, choose a total that divides into eight equal whole-number groups. A total of 56 works: one eighth is seven and three eighths is 21 counters.
If the total were 58, three eighths would not be a whole-number counter count. That is not an arithmetic contradiction in abstract numbers, but it may conflict with a story requiring an exact subset of indivisible counters.
Decide whether such a conflict is intentional. An ordinary practice question should not accidentally force the reader to split an indivisible object. A reasoning challenge can intentionally ask whether the conditions are possible, but it should say so.
Continuous measurements offer different possibilities. Sharing 26 litres equally among four containers gives 6.5 litres per container, which is a meaningful decimal amount. Sharing 26 indivisible counters equally among four children with none left over does not have the same physical interpretation.
The author must match the numerical relationship to the type of quantity. Changing only the noun can sometimes change whether a problem is valid.
5. Keep the reference whole visible in fraction questions
“Three eighths are red” is incomplete until the reader knows three eighths of what. The fraction could refer to all the counters, the counters in one box or a remainder after some were removed.
A clear question might say: “A box contains 56 counters. Three eighths of the counters are red, and all the others are blue. How many are blue?” Red counters total 21, so blue counters total 35.
The phrase “all the others are blue” closes the category structure. Without it, non-red counters could include other colours and a question specifically asking for blue might be underdetermined.
Changing “of the counters” to “of the remaining counters” introduces a new reference whole. The author must specify how that remainder was produced.
When designing a sequence, write a label beside each fraction before drafting the prose: fraction of original set, fraction of remaining set or fraction of one group. The labels help prevent an unintended shift of reference.
6. Make decimal questions exact about notation and purpose
A decimal addition problem can use 4.75 and 2.60 to give 7.35. A money story might ask for a combined cost, while a length story might ask for a combined length. Each needs an appropriate unit.
A rounding question needs a requested place: nearest whole number, one decimal place or two decimal places. “Round 6.284” does not identify one intended answer.
A comparison question should not accidentally depend on a visual formatting convention. Ask the learner to compare the values of 0.6 and 0.56, not to decide which printed string is longer.
When designing within current learning, keep the operations inside the taught scope. Larger numbers or extra decimal places are not automatically a better challenge. A changed unknown, a different representation or a request to explain can deepen the task without adding untaught calculation.
Write whether the answer is exact or approximate. A fictional price of $4.75 used as an exact teaching input should not be rounded silently when calculating exact change.
7. Construct geometry from dimensions, then derive the givens
Choose a rectangle measuring twelve centimetres by seven centimetres. Its area is 84 cm² and its perimeter is 38 cm. These values form a consistent family.
A direct question can give the two sides and ask for area. A reverse question can give area 84 cm² and width seven centimetres, then ask for the length. A two-stage question can use those same givens and ask for perimeter.
The reverse route finds twelve centimetres first, then calculates 12 + 7 + 12 + 7 = 38 cm. The intermediate side has a necessary role.
Do not choose area, perimeter and sides independently without checking that they fit one shape. A square with side eight centimetres cannot also have perimeter 28 cm. An inconsistency may be useful for a deliberate error-detection task but should not occur accidentally.
For composite figures described in words, specify the position and dimensions of each removed or attached rectangle. “Remove a small piece” is not enough to determine an exact area or boundary.
8. Decide whether the task is closed, open or deliberately impossible
A closed question is intended to produce one determinate answer under its conditions. For example, two equal amounts total eighteen dollars, so each is nine dollars.
An open question may allow several valid responses. “Choose two non-negative whole-dollar prices that total eighteen dollars” has many possible pairs. The instruction should invite such possibilities rather than pretend one pair is uniquely required.
A deliberately impossible question asks the learner to identify a conflict. For example, two equal whole-number groups cannot together contain an odd total of indivisible counters.
All three forms can be useful. The problem is not that a question has multiple solutions; the problem is failing to tell the reader what kind of answer is wanted or failing to recognise the multiplicity in the answer key.
Write the intended outcome type on the author’s copy. Then test whether the words actually produce it. Do not repair a multiple-solution question by quietly rejecting valid alternatives just because they differ from the author’s preferred answer.
9. One added condition can change an entire answer set
Start with “Two positive whole numbers total twelve.” Ignoring order, the possible pairs are one and eleven, two and ten, three and nine, four and eight, five and seven, and six and six.
Add “Both numbers are even.” Only two and ten, four and eight, and six and six remain. Add “They are different.” The equal pair disappears. Add “Their difference is four.” Only four and eight remain.
This sequence shows how conditions narrow the possible answers. The final condition should not be chosen at random; it needs to leave the intended answer while excluding alternatives.
Removing a condition reverses that process. It may turn a unique problem into an open one.
A table or systematic list is a useful authoring tool here. See Systematic Listing for ways to show that all candidates have been considered.
10. Use precise comparison language
“A has three times as many counters as B” states a multiplicative comparison. “A has three more counters than B” states an additive difference. They are not interchangeable.
Avoid “three times more” when the intended interpretation could be disputed. Write the relationship explicitly in standard school language.
Similarly, “at most twenty dollars” permits exactly twenty, whereas “less than twenty dollars” does not. “At least one of each” excludes zero counts. “Different digits” prohibits repetition; “use each digit once” also controls which symbols appear.
A word can be short and still carry a major condition. Removing it may change the answer set or make the question impossible to determine.
Ask a second reader to restate the problem before solving it. If their restatement differs from the intended relationship, revise the wording. Do not assume the reader made an error when the author’s sentence allowed a reasonable alternative interpretation.
11. Irrelevant information should have a deliberate teaching purpose
A question about pencils does not become mathematically richer merely because it mentions the colour of the classroom walls, the weather and the names of several shops. Those details may increase reading without changing the relationship.
Sometimes selecting relevant information is the intended skill. In that case include a small amount of clearly irrelevant information and explain in the answer key why it is not used.
For example, a box contains 48 pencils and weighs 300 g. The pencils are shared equally among six pupils. The mass is not needed to find eight pencils per pupil. The task can ask which information is unnecessary.
Be careful with details that look irrelevant but actually affect the answer. A statement that two pencils are broken matters if only usable pencils are shared. A price reduction matters if the final cost is requested.
The author should know the role of every supplied fact: essential, contextual, intentionally irrelevant or potentially confusing. Remove accidental clutter that has no learning purpose.
12. Derive an answer key independently from the finished wording
After writing the question, hide the private planning notes and solve the wording as a reader would. This catches details that were intended but never actually stated.
Suppose the plan required equal group sizes but the final sentence says only “put the counters into four boxes”. The reader cannot infer that each box has 36. The answer key must not depend on an unstated equality.
Check the final answer against every independent condition. For a total-and-difference question, test both. For a digit construction, inspect length, permitted digits, repetition and ordering. For a geometry question, test the shape definition as well as the numbers.
If several answers are valid, say so and give an acceptance rule. A sample answer is not automatically the only acceptable answer.
This independent solve is especially important when changing a previously valid problem. One altered number can break divisibility, create a negative remainder or make two conditions inconsistent.
13. Change one feature so you can see its effect
Begin with a reliable base question: “There are 56 counters. Three eighths are red. How many are red?” The answer is 21.
Change only the whole to eighty, giving thirty red counters. Change only the requested part to the non-red counters, giving 35 for the original whole. Change only the unknown to the original whole when three eighths are 21; the equal-unit route recovers 56.
Each variation has a clear purpose. Changing the whole tests the fraction relationship with new values. Changing the target tests selection of the requested quantity. Changing the unknown tests reverse reasoning.
If the author changes the numbers, story, representation, unknown and number of steps all at once, it becomes harder to tell which change caused a difficulty.
This is a design choice rather than a demand that every lesson use only tiny changes. Start with controlled variations when the aim is to inspect a particular relationship. Larger integrated changes can follow when that relationship is secure.
14. Design an error-detection question fairly
An error-detection task presents an incorrect claim and asks the learner to examine it. Label the claim as a proposed solution, not as an instruction that the reader should copy.
For example: “A pupil says a square of side eight centimetres has perimeter 28 cm. Is the claim correct? Explain.” The answer is no: four sides of eight make 32 cm.
A stronger version asks the learner to repair only one given value. Changing the side to seven centimetres makes perimeter 28 cm valid. Changing the perimeter to 32 cm preserves the original side.
Both repairs are legitimate because the task permits changing one value. An answer key should not reject the second just because the author imagined the first.
Avoid trick questions whose only difficulty is unclear language. A fair challenge makes the mathematical issue available for inspection: a missing condition, an inconsistent number, a false operation or a claim that extends beyond the evidence.
15. Let students exchange questions without using personal data
A small classroom exchange can have three roles: author, solver and reviewer. The author supplies the question and a separate answer key. The solver attempts the question without seeing the intended answer. The reviewer checks the wording, solution and conditions.
Use fictional names and invented quantities. A mathematics activity does not need classmates’ real marks, family income or private household details.
When the solver finds another valid answer, treat that as information about the question. The author may choose to make the task explicitly open or add a condition that selects one intended result.
The reviewer should distinguish mathematical disagreement from stylistic difference. A table and a bar model can both be valid. A short explanation can be sufficient. A different sample story can meet every requested condition.
Keep the exchange focused: one relationship, one clear target and one review question. The aim is to inspect mathematical design, not to turn a short activity into an elaborate publishing project.
16. Twenty question-design tasks
Write a question and solve it. Several tasks intentionally allow multiple valid constructions. The next section gives sample responses and explains what a different valid answer must preserve.
- Create an equal-sharing question whose answer is 36 objects per group.
- Use the same relationship to ask for the number of groups instead.
- Create a two-step removal-then-sharing question whose answer is 36 objects per group.
- Write a question about three eighths of 56 counters. Ask for the selected amount.
- Change Question 4 to ask for the unselected amount without changing the whole or fraction.
- Create a reverse question in which three eighths of an unknown collection are 21.
- Repair “26 indivisible counters are shared equally among four children with none left” by changing only the total.
- Keep the number 26 and change the quantity type so four equal shares can be expressed meaningfully as decimals.
- Write a decimal addition story using 4.75 and 2.60, with appropriate units.
- Repair the instruction “Round 6.284” so that it has a clear requested precision, then answer it.
- Write a question asking for all positive whole-number rectangles with area 24, counting rotations once. Supply the full answer list.
- Create a reverse rectangle question using area 84 cm² and one side seven centimetres. Ask for perimeter.
- Repair the claim that a square has side eight centimetres and perimeter 28 cm by changing exactly one value.
- Write an eighteen-dollar total-price question with a unique equal-price answer.
- Remove the equality condition from Question 14 and explain what changes about the answer.
- Starting from two positive whole numbers that total twelve, add conditions that select four and eight uniquely when order is ignored.
- Create an all-answers digit question using zero, two and four once each in a three-digit number. State whether a leading zero is allowed.
- Add one intentionally irrelevant measurement to a question about sharing 48 pencils equally among six pupils. Identify the irrelevant fact in the answer key.
- Write a comparison question using “three times as many” and give a contrasting question using “three more”. Use the same reference amount.
- Create one problem with more than one valid answer and write an answer-key rule that accepts all valid responses rather than only your favourite example.
17. Sample constructions and answer-key checks
1. “144 counters are shared equally among four groups. How many does each group receive?” The answer is 36. Other totals and group counts are acceptable when the division is exact and gives 36.
2. “144 counters are arranged with 36 in each group. How many groups are formed?” The answer is four groups. The unknown is now group count rather than group size.
3. “From 170 counters, 26 are put aside. All the rest are shared equally among four groups. Find each share.” The remainder is 144, giving 36 per group. Another construction must preserve the exact post-removal total needed for its chosen divisor.
4. “A box contains 56 counters. Three eighths are red. How many are red?” One eighth is seven, so 21 counters are red. The fraction refers to all 56.
5. “How many counters are not red?” gives 56 − 21 = 35. Asking for a particular other colour would require a condition saying all non-red counters have that colour.
6. “Three eighths of a collection are 21 counters. How many are in the whole collection?” Three units total 21, so one is seven and eight units make 56. This is an optional reverse-reasoning challenge.
7. Change 26 to 28. Four equal shares then contain seven counters each. Other non-negative totals divisible by four can work, provided the intended object count is meaningful.
8. “Twenty-six litres of water are shared equally among four containers. How much goes in each?” The answer is 6.5 L. Liquid volume permits equal decimal shares in this model.
9. “Two ribbons measure 4.75 m and 2.60 m. Find their combined length.” The answer is 7.35 m. A money or another compatible measurement context is also valid if both values describe addable parts.
10. “Round 6.284 to one decimal place” gives 6.3. Requesting two decimal places instead gives 6.28. Either is a clear repaired instruction; the key must match the chosen precision.
11. “Find all rectangles with positive whole-number sides and area 24 square units, counting rotations once.” The complete pairs are 1 and 24; 2 and 12; 3 and 8; 4 and 6. Reversed pairs do not create new answers under the stated rule.
12. “A rectangle has area 84 cm² and width 7 cm. Find its perimeter.” The length is twelve centimetres, and the perimeter is 38 cm. Both stages have a defined role.
13. Keep perimeter 28 cm and change the side to 7 cm, or keep side eight centimetres and change the perimeter to 32 cm. Both change exactly one value and restore consistency.
14. “Two items have equal prices and cost eighteen dollars altogether. What is each price?” Each costs nine dollars. The equality condition is what justifies two equal shares.
15. Without equality, there are multiple possible price pairs. Five and thirteen dollars or seven and eleven dollars both work. The total alone no longer determines each price.
16. Add “both are even and their difference is four”. The unordered pair is uniquely four and eight. A complete check shows the sum is twelve, both values are even and their difference is four.
17. “Use zero, two and four once each to make ordinary three-digit numbers; a leading zero is not allowed. List all.” The answers are 204, 240, 402, 420.
18. “A box containing 48 pencils has mass 300 g. The pencils are shared equally among six pupils. How many pencils does each receive?” Each receives eight. The box mass is not required for this count question.
19. With B having eighteen counters, “A has three times as many as B” gives A 54. “A has three more than B” gives A 21. The multiplier and fixed excess are different relationships.
20. One example is “Give two positive whole numbers that total ten.” An appropriate key accepts any pair satisfying positivity, whole-number values and total ten. It should not reject two and eight because the sample response was four and six.
18. A short author–solver–reviewer lesson
Begin with a solved relationship rather than a blank page. Invite the learner to change the unknown while keeping the numbers consistent. Solve the new wording and compare it with the original.
Next change one condition. Ask whether the answer remains unique, becomes multiple or becomes impossible. A small list can make this visible.
Then exchange the question with a reader who has not seen the planning notes. The reader should identify the target and restate the conditions before solving. This reveals ambiguity early.
The reviewer checks three things: whether the question is clear, whether its conditions can be satisfied and whether the answer key actually follows from the text. Do not judge primarily by story length or decorative vocabulary.
A suggested independent return is to ask the learner to construct one valid question with a chosen answer and one deliberately invalid question that differs in only one detail. The learner must explain the detail that changes validity. This tests control over the relationship rather than ability to imitate a sentence frame.
19. Keep creativity attached to mathematical responsibility
A creative question can use an unusual setting, a new target or an unexpected constraint. It still owes the reader a fair statement of the problem.
When a construction fails, locate the authoring issue. An indivisible-object problem may need a divisible total. A reverse problem may need an additional condition. A geometry problem may need consistent dimensions. A rounding problem may need its precision specified.
Do not erase every interesting difficulty. An open problem can remain open when the instructions and answer key recognise that form. A contradiction can remain when the task deliberately asks the learner to find it.
Question quality comes from matching intention, wording, mathematical structure and evaluation. A sound author can explain not only the answer but why the question supports that answer and what a nearby changed question would do.
This is the handover from copying mathematics to taking responsibility for a mathematical statement.
20. Continue into independent home learning
Problem posing can be a small part of home practice: solve a question, change one condition and test the new version. It should not become an additional heavy assignment every time homework is completed.
Continue to Home Learning, Homework Independence and Parent Support. For strengthening a particular mathematical concept before designing new questions, return to its owner guide in the Primary 4 Mathematics Learning Hub.
Final checkpoint: can the learner write a clear target, choose consistent quantities, state necessary conditions, solve the finished wording and accept every answer that genuinely satisfies it?
Editorial approach: Wintour House V1.0 · CivDJ · eduKate Publishing. Original authoring activities with optional extensions; no official endorsement, compulsory chapter status or guaranteed outcome is implied.
