One of the strongest ways to test whether a Primary 3 student understands mathematics is to ask the learner to create a problem, not only solve one. Problem posing requires the child to understand which quantities belong together, what information is necessary, where the unknown can be placed, what operation the structure supports and whether the problem has enough information to be solved.
This is Guide 31 in the Primary 3 Mathematics Learning Hub. It develops mathematical creativity through question generation, changing conditions, moving the unknown, creating examples and non-examples, testing solvability and building multiple questions from the same mathematical situation.
To create a valid problem, the learner must understand the structure well enough to control it.
What Problem Posing Means
Problem posing can begin very simply. Give a number sentence, picture, table, graph or real-world situation and ask the student to create a matching mathematical question.
| Starting point | Problem-posing task |
|---|---|
| 7 × 8 = 56 | Create an equal-groups story. |
| 56 ÷ 7 = 8 | Create a sharing story. |
| 3/5 | Create a part-whole situation. |
| $20 − $12.95 | Create a change problem. |
| bar graph | Create one direct and one multi-step question. |
From Number Sentence to Story
Take 6 × 7 = 42. A valid story might be:
There are 6 boxes with 7 pencils in each box. How many pencils are there altogether?
The learner must preserve equal groups. A story with 6 pencils in one box and 7 pencils in another would not match the multiplication structure.
One Fact Family, Several Problems
From 7 × 8 = 56, create three different problems:
- 7 groups of 8 → find total.
- 56 shared among 7 groups → find amount in each group.
- 56 packed 8 per group → find number of groups.
This teaches that the same numbers can support different questions depending on the unknown.
Move the Unknown
Moving the unknown is one of the most powerful Primary 3 problem-posing moves.
Start with:
Hana has 79 more stamps than Mei. Mei has 145 stamps. How many does Hana have?
Then move the unknown:
Hana has 224 stamps. She has 79 more stamps than Mei. How many does Mei have?
The relationship remains the same, but the operation changes.
Create a Reverse Problem
Start with a forward problem:
Mei had 213 cards and gave away 68. How many remain?
Reverse it:
After giving away 68 cards, Mei has 145 left. How many did she have at first?
Problem posing makes inverse relationships visible because the student actively relocates the unknown.
Create a Fraction Problem
Given the fraction 3/8, ask the learner to create a valid situation. For example:
A cake is divided into 8 equal pieces. Three pieces are eaten. What fraction of the cake is eaten?
The word “equal” is essential. Without equal parts, the fraction model is invalid.
Create Equivalent-Fraction Questions
Ask students to create two different pictures that both represent 1/2, such as one whole split into 2 equal parts with 1 shaded and another split into 4 equal parts with 2 shaded.
The creation task reveals whether the learner understands equivalence rather than merely recognising it.
Create Money Problems
Give prices $8.75 and $2.60 and payment of $20. The learner can generate several questions:
- What is the total cost?
- How much more does the first item cost?
- How much change is received?
- If the second item is removed, how does the change change?
One set of quantities can support multiple mathematical relationships.
Create Measurement Problems
Give a ribbon length of 4 m and two used lengths of 175 cm and 125 cm. Ask the learner to create:
- a total-used question;
- a remaining-length question;
- a unit-conversion question;
- a comparison question.
The student must maintain compatible units while changing the mathematical job.
Create Time Questions From One Schedule
If an activity starts at 13:25 and ends at 15:05, ask the learner to create questions about duration, start time, finish time or an inserted break.
Changing the unknown teaches that the same schedule can produce different operations.
Create Area and Perimeter Questions From One Rectangle
Use a rectangle measuring 8 cm by 5 cm. Ask the student to create:
- a perimeter question;
- an area question;
- a fencing story;
- a covering story;
- a question with one side length missing but perimeter known.
The same diagram supports several relationships. Problem posing reveals whether the student can distinguish them.
Create Geometry Questions
Show a rectangle or a simple set of intersecting lines. Ask the learner to create one question about right angles, one about parallel lines and one about perpendicular lines.
The student must use properties rather than visual appearance when constructing valid questions.
Create Bar-Graph Questions
Given a graph, require several question types:
- one direct reading question;
- one greatest/least question;
- one difference question;
- one total question;
- one multi-step question using a later operation.
This teaches that data representation can feed several forms of reasoning.
Change One Condition
After solving a problem, ask “What if?”
- What if there were 8 boxes instead of 7?
- What if the payment were $50 instead of $20?
- What if the graph scale changed from 5 to 10?
- What if the rectangle were rotated?
- What if the unknown moved to the starting amount?
The learner compares which parts of the method stay the same and which must change.
Create a Problem With Irrelevant Information
Ask the learner to add one fact that sounds plausible but is not needed. Then ask another student to identify it.
This strengthens relevance filtering and exposes the belief that every printed number must be used.
Create an Impossible Problem
Creating an intentionally unsolvable or inconsistent problem can sharpen understanding.
For example: “A rectangle has perimeter 20 cm. What is its area?” Without more information, many rectangles can have perimeter 20 cm but different areas. The problem is underdetermined.
The learner can then add a condition, such as length = 6 cm, to make the problem solvable.
Enough Information Versus Too Little Information
| Problem state | Meaning |
|---|---|
| Sufficient | enough information exists to determine the answer |
| Insufficient | more information is needed |
| Irrelevant extra | some information is unnecessary |
| Inconsistent | the conditions cannot all be true together |
Problem posing makes these distinctions concrete because the student controls the information set.
Create a Problem With More Than One Answer
Example: “A number is greater than 30, less than 50 and even.” Many answers are possible. The student can then add another condition such as “multiple of 6” to reduce the possibilities.
This develops logical elimination and awareness of constraints.
Problem Posing and Number Patterns
Give the sequence 100, 125, 150, 175, … and ask the student to:
- state the rule;
- create a missing-term version;
- create a “find the tenth term” version;
- create a story that could produce the pattern.
Problem Posing and Heuristics
Ask students to create a problem that would benefit from:
- a table;
- work backwards;
- a bar model;
- guess and check;
- logical elimination.
To create such a problem, the learner must understand what kind of structure makes the heuristic useful.
Problem Posing as a Diagnostic Tool
Errors in created problems reveal misconceptions:
- unequal groups in a multiplication story;
- non-equal parts in a fraction story;
- wrong units in measurement;
- area language paired with perimeter calculation;
- bar-graph questions that ignore scale;
- insufficient information for the requested unknown.
The created problem is therefore evidence of conceptual structure.
Use Constraints to Control Creativity
Open-ended creation can be difficult. Provide bounded prompts:
- Create a problem using 7, 8 and 56.
- Create a money problem where change is the unknown.
- Create a fraction comparison with the same numerator.
- Create a time problem where the start time is unknown.
- Create a graph question requiring two operations.
Constraints give the learner a structure within which to be creative.
Ask Another Student to Solve the Created Problem
A created problem should be tested. If another learner cannot tell what is being asked, the wording may be ambiguous. If several answers are possible unexpectedly, information may be missing.
A good mathematical question survives being handed to someone else.
Revise the Question
Problem posing includes editing. Ask:
- Is the final unknown clear?
- Are the units consistent?
- Is there enough information?
- Is any information accidentally contradictory?
- Does the story match the intended operation?
- Is the answer sensible?
Common Problem-Posing Mistakes
- Numbers without roles. The quantities are not connected.
- Wrong unknown. The question asks for a value already given.
- Insufficient information. Several answers are possible unintentionally.
- Unit mismatch. The calculation combines incompatible quantities.
- Story-operation mismatch. The narrative does not represent the equation.
- Overcomplication. Too many conditions obscure the core relationship.
- Copying surface wording. The learner changes nouns but not mathematical structure.
Diagnostic Questions
- Can the learner create a story for 7 × 8 = 56?
- Can the student create both sharing and grouping division problems?
- Can the learner move the unknown in a comparison problem?
- Can the student create a valid fraction problem with equal parts?
- Can the learner create multiple questions from one graph?
- Can the student detect insufficient information?
- Can the learner create and then repair an ambiguous problem?
- Can the student explain how changing one condition changes the solution?
A Weekly Problem-Posing Cycle
- one equation-to-story task;
- one move-the-unknown task;
- one fraction creation task;
- one graph question-generation task;
- one “what if?” condition change;
- one insufficient-information repair;
- one problem created for a named heuristic;
- one peer-solve-and-revise task.
Exam Craft | Understand Questions by Learning to Build Them
Students who practise constructing questions become more sensitive to how unknowns, conditions and units are encoded. That can make unfamiliar assessment questions easier to unpack because the learner has experience seeing problems from the inside.
Checkpoint | Can the Student Control the Structure?
- Can the learner generate a valid question from a number sentence?
- Can the student move the unknown while preserving the relationship?
- Can the learner create solvable and intentionally unsolvable problems?
- Can the student use conditions to control possible answers?
- Can the learner create questions from diagrams and graphs?
- Can the student test another learner’s interpretation?
- Can the learner revise a weak question?
- Can the student explain what mathematical structure the created problem contains?
How This Connects to the Primary 3 Mathematics System
This guide extends transfer in Guide 16, heuristics in Guide 21, mathematical modelling in Guide 28, and justification in Guide 30.
Final Thought
A student who can create, test and revise a mathematical problem is doing more than practising arithmetic. The learner is controlling quantities, conditions, relationships and unknowns. That is mathematical creativity grounded in structure.
Solve problems to understand mathematics. Create problems to test whether you truly control it.
Return to the Primary 3 Mathematics Learning Hub.
