Some Primary 3 Mathematics problems are difficult because there is no single obvious calculation to perform first. The learner may need to search through possibilities, organise cases, eliminate impossible choices and decide when the search is complete. This is different from guessing randomly. A systematic search has an order.
This is Guide 51 in the Primary 3 Mathematics Learning Hub. It develops organised lists, tables, constraints, case-making, elimination, duplicate control and age-appropriate completeness reasoning.
Start Here | The Search Route
- Define: state the conditions.
- Organise: choose a list, table or ordered sequence.
- Generate: produce possibilities in a fixed order.
- Eliminate: remove cases that break a condition.
- Check completeness: explain why no valid case was missed.
Guessing becomes mathematical when the guesses follow a system and the system tells you when to stop.
What Counts as a Constraint?
A constraint is a condition that every valid answer must satisfy. A problem may say that a number is even, between 30 and 40, and a multiple of 6. Each condition narrows the search.
| Condition | Effect on candidates |
|---|---|
| greater than 30 | removes 30 and below |
| less than 40 | removes 40 and above |
| even | keeps 32, 34, 36, 38 |
| multiple of 6 | leaves 36 |
Use an Organised List
Suppose a rectangle has perimeter 24 cm and whole-number side lengths. Instead of drawing rectangles randomly, list possible lengths in order.
- 1 cm and 11 cm
- 2 cm and 10 cm
- 3 cm and 9 cm
- 4 cm and 8 cm
- 5 cm and 7 cm
- 6 cm and 6 cm
After 6 and 6, the next case 7 and 5 would duplicate an earlier rectangle with the dimensions reversed. The order itself reveals when to stop.
Why Duplicate Control Matters
In some searches, 3 and 9 represents the same case as 9 and 3. In others, order matters. Students should decide whether reversing the order creates a genuinely new mathematical object.
For rectangle side lengths, 3 by 9 and 9 by 3 describe the same rectangle dimensions. For a two-step route where the first and second actions are different, reversing order may create a different case.
Use a Table When Two Quantities Change Together
The perimeter-24 search becomes especially useful when area is added.
| Length | Width | Perimeter | Area |
|---|---|---|---|
| 1 | 11 | 24 | 11 |
| 2 | 10 | 24 | 20 |
| 3 | 9 | 24 | 27 |
| 4 | 8 | 24 | 32 |
| 5 | 7 | 24 | 35 |
| 6 | 6 | 24 | 36 |
The table exposes a second pattern: among these whole-number rectangles, the square has the greatest area.
Search by Holding One Quantity Fixed
If a total must be 20, keep one part fixed and vary the other systematically:
- 1 + 19
- 2 + 18
- 3 + 17
- …
- 10 + 10
This produces every unordered pair of positive whole numbers that sums to 20 without random guessing.
Search With Multiple Conditions
“Find a two-digit number that is even, greater than 50, less than 70 and has digits that add to 9.”
Even candidates are 52, 54, 56, 58, 60, 62, 64, 66 and 68. Check digit sums. Only 54 has digits that add to 9. The search is small because each constraint removes candidates.
Elimination Can Be Faster Than Construction
Sometimes it is easier to start with a complete candidate set and remove impossible cases than to build the answer from nothing. This is useful when the allowed range is small and clear.
Systematic Search in Money
Suppose a student must spend between $12 and $15 using two items from a short price list. Create a table of item pairs, calculate each total once and eliminate totals outside the range.
The method is stronger than choosing combinations randomly because every candidate can be audited.
Systematic Search in Fractions
Find fractions with denominator 8 that are greater than 1/2 and less than 1. Since 4/8 = 1/2 and 8/8 = 1, the valid fractions are 5/8, 6/8 and 7/8.
Benchmarks turn the open question into a bounded interval search.
Systematic Search in Multiplication Facts
Find all factor pairs within the known multiplication facts that make 36:
- 1 × 36
- 2 × 18
- 3 × 12
- 4 × 9
- 6 × 6
For Primary 3, the emphasis is on known factor relationships and organised search rather than formal factor theory.
Systematic Search in Number Patterns
If a sequence follows a repeated rule, list terms in order and mark where the condition is first satisfied. Search should follow the pattern’s structure rather than skip unpredictably between guesses.
Systematic Search in Geometry
Ask students to draw all different rectangles with a fixed perimeter using whole-number side lengths. Then classify them by area. This combines geometry, multiplication, organisation and completeness.
Search and Word Problems
Some word problems can be solved by guess-and-check, but the guesses should be organised. If a total number of objects and groups is known, construct a table of candidate group sizes and test each against the total.
Work Backwards to Shrink the Search
If the final state is known, reverse the last condition first. This can remove many impossible starting values before any forward testing begins.
This connects systematic search with the work-backwards heuristic in Guide 21.
How to Know the Search Is Complete
“I found several answers” is not the same as “I found all answers.” A complete search needs a stopping rule.
- The allowed range has been exhausted.
- The next case would duplicate an earlier case.
- Every value in one controlled variable has been tested.
- Every remaining candidate violates at least one constraint.
Completeness comes from the structure of the search, not from feeling that enough examples were found.
Student Route | Four Columns
For difficult open problems, divide rough work into four mental columns:
- conditions;
- candidate;
- pass/fail reason;
- valid result.
This keeps failed cases useful because the reason for rejection remains visible.
Parent Route | Ask “How Do You Know There Are No More?”
When a child finds several answers, ask how the search was ordered and why it ended. This encourages mathematical completeness without requiring advanced terminology.
Teacher Route | Design Low-Floor, High-Ceiling Searches
Choose a task where a student can find one valid case quickly, then extend the job to all cases, an optimisation question or an explanation of the stopping rule. The same problem can therefore support several depths of reasoning.
Diagnostic Map
| Observed behaviour | Likely break | Repair |
|---|---|---|
| random repeated guesses | no search order | fix one variable and change it systematically |
| same case counted twice | duplicate control weak | define whether order matters |
| stops after first answer | reader job misunderstood | highlight “all possible” |
| cannot explain completeness | no stopping rule | use range or duplicate boundary |
| too many cases | constraints not used early | apply strongest filters first |
Common Search Mistakes
- guessing without order;
- changing two variables unpredictably;
- counting duplicates;
- ignoring one constraint;
- stopping after one valid answer when all are required;
- claiming completeness without a stopping rule.
Exam Craft | Organise Before the Search Expands
If a question asks for possibilities, spend a moment choosing the structure of the search. A two-column table or ordered list can prevent repeated work and make the final answer easier to verify.
Next Route
Continue with Guide 21: Problem-Solving Heuristics, Guide 35: Open-Ended Enrichment, and Guide 31: Problem Posing.
Return to the Primary 3 Mathematics Learning Hub.
