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Primary Mathematics: Working Systematically, Organised Search and Recording | Worked Learning Guide

Working systematically means choosing an order that makes missing cases and repeated cases difficult. The goal is not to make the page look neat. The goal is to make the reasoning inspectable.

A learner can understand the mathematics inside a problem and still lose control because the search is haphazard. Three-digit numbers are written in random order. Rectangle counts are circled by eye. Route possibilities are tried and forgotten. Ratio cases are generated without recording which condition changed. The final answer may be correct, but the learner has no convincing reason that every possibility has been considered.

This guide develops a broader capability than the existing Systematic Listing and Logical Counting article. Listing is one application. Working systematically also includes choosing an order of variation, keeping a state table, separating cases, naming what has already been checked, stopping when the search is complete, and selecting the smallest record that still preserves the information needed.

Choose an order · Record states · Split cases · Prove completeness · Make the search efficient · 24 questions · Worked solutions

1. Random trials create memory problems

Using digits 1, 2, 3 and 4 without repetition, list all two-digit numbers greater than 30.

A random search might produce 31, 42, 34, 41, 32, 43 and then pause. Are we finished? The difficulty is no longer arithmetic. It is bookkeeping.

A systematic search begins with the tens digit because the condition “greater than 30” acts there first.

• 3_: 31, 32, 34
• 4_: 41, 42, 43

Now completeness is visible. The tens digit must be 3 or 4, and for each, every remaining unit digit was used exactly once. Six numbers.

Worked example: fixed first feature

How many three-digit even numbers can be made from digits 1,2,3,4 without repetition?

Organise by final digit because evenness is controlled there.

• Ending 2: remaining first two positions use 1,3,4 → 6 arrangements.
• Ending 4: remaining first two positions use 1,2,3 → 6 arrangements.

Total = 12.

Choosing the controlling condition first reduces wasted search.

2. A record should remember what your head should not have to

Suppose a learner explores totals made from two coins chosen from values 2,5 and9, with repetition allowed. Writing outcomes in a table is more reliable than remembering previous trials.

First coin	Second coin	Total
2	2	4
2	5	7
2	9	11
5	5	10
5	9	14
9	9	18

If order does not matter, 2+5 and5+2 are the same selection and should not be listed twice. The table can be triangular rather than rectangular because the lower half would duplicate the upper half.

Record the smallest useful state

In a route problem, the full history may be unnecessary. If the only future choices depend on current position and which bridges remain unused, that pair is the state that matters. If two different histories lead to the same state, future possibilities are identical.

This idea is advanced in language but simple in practice: record only what affects the next legal move.

3. Split the search into non-overlapping cases

A strong case split has two properties:

1. Every valid solution belongs to at least one case.
2. No valid solution belongs to more than one case.

For whole-number factor pairs of 36, organise by the smaller factor:

1×36, 2×18, 3×12, 4×9, 6×6.

After 6×6, the next factor pair would reverse an earlier pair. Stopping there prevents duplication.

Worked example: rectangles by height

In a 2×3 grid, every rectangle has height 1 or2. Count all width possibilities within height1, then all within height2. Because no rectangle has another height, the two cases are complete.

The shape-counting guide develops this structure in detail.

4. Change one variable in a fixed order

Consider pairs of positive whole numbers with sum 12:

(1,11), (2,10), (3,9), (4,8), (5,7), (6,6).

We increase the first number by one and decrease the second by one. The sum remains fixed. If unordered pairs are sufficient, we stop at equality because later pairs reverse earlier ones.

This order is useful because the invariant and the stopping condition are visible simultaneously.

Worked example: fixed perimeter rectangles

Whole-number rectangles with perimeter 20 satisfy length + width = 10. List (1,9),(2,8),(3,7),(4,6),(5,5). Areas are 9,16,21,24,25. The systematic list shows the area increasing as dimensions become closer.

A random sample might miss the square and therefore miss the maximum.

5. Use a grid when two independent choices combine

If one choice is a row and another is a column, a grid can expose all combinations.

Suppose shirts are red, blue or green and shorts are black or white. A 3×2 table has six cells, one for each outfit. No outfit can occupy two cells because its colour pair identifies exactly one location.

The same structure appears in multiplication facts, coordinate pairs, two-stage choices and input-output tables.

Worked example: sums from two dice-style spinners

Spinner A: 1,2,3. Spinner B: 4,5,6. A 3×3 grid records nine equally likely ordered outcomes if each spinner is fair. The grid can then be coloured by total or parity.

6. Use a tree when choices happen in sequence

A tree diagram preserves order. If choosing A then B is different from choosing B then A, branches should keep that chronology.

Example: a two-step code uses first one letter from X,Y and then one digit from 1,2,3. Branch X to three digits and Y to three digits. Six codes appear.

Do not use a tree to create duplicate branches when order does not matter.

7. Record before-and-after states

Many multi-step problems become systematic when each state is written after one operation.

Stage	Amount A	Amount B	Total
Start	30	50	80
Move 12 from B to A	42	38	80
Move 5 from A to B	37	43	80

The table shows both the changing values and the invariant total. It also reduces the risk of applying the second move to the original amounts by mistake.

8. Systematic trial is different from blind trial

Guess-and-check can be rigorous when each guess is chosen from information gained by the previous one.

Question: two types of tickets cost $4 and $7. Ten tickets cost $55. Start by assuming all ten are $4: total $40. We need $15 more. Replacing one $4 ticket by a $7 ticket increases the total by $3. Therefore five replacements are needed. Five tickets of each type.

This is not random guessing. It uses a fixed baseline and a known effect per replacement.

9. A complete search needs a stopping reason

“I cannot think of more” is not a stopping rule.

Good stopping reasons include:

• all permitted first digits have been used,
• factor pairs have reached or crossed equality,
• every row and column of the choice table is complete,
• the changing quantity reached its allowed maximum,
• the next case would duplicate an earlier case,
• all branches of the decision tree terminate.

Worked example: integer solutions

Find positive whole-number pairs (x,y) satisfying x+y=9 and x≤y.

Start x=1 and increase:

(1,8),(2,7),(3,6),(4,5).

At x=5, y=4, violating x≤y and reversing the previous pair. The search is complete.

10. Duplicate control is part of the method

Suppose we choose two different books from A,B,C,D. The pair AB is the same selection as BA if reading order is irrelevant.

A systematic list can enforce alphabetical order inside each pair:

AB, AC, AD, BC, BD, CD.

By refusing to write BA after AB, the method prevents duplicates by design.

11. Working systematically across topics

Topic	Systematic organisation
Factors	increasing factor pairs
Fractions	fix numerator or denominator while varying the other
Geometry	classify by size, angle, side or position
Data	table stages, common units, ordered categories
Word problems	stage table or before-after record
Probability enrichment	outcome grid or tree
Routes	record current state and unused connections
Optimisation	ordered candidates plus lower/upper bound

12. Systematic work can reveal patterns that random work hides

List square numbers and their differences:

n	n²	Increase
1	1	—
2	4	3
3	9	5
4	16	7
5	25	9

Ordered recording exposes consecutive odd differences. The pattern can then be explained structurally by adding an L-shaped border to grow one square into the next.

13. Efficiency means preserving enough information, not writing everything

A full table may be wasteful when symmetry halves the search. A long list may be unnecessary when a multiplication structure proves completeness. A diagram may replace several lines of words.

Systematic does not mean slow. It means organised enough that you can justify what has and has not been considered.

Worked example: six-person handshakes

If every pair of six people shakes hands once, one systematic route is to let Person A shake with five later people, B with four later people, then 3,2,1. Total 5+4+3+2+1=15.

Listing every ordered pair would produce 30 entries and then require dividing by two. The triangular list avoids duplicates from the start.

14. When systematic work is not enough

An organised search can prove a finite result if every case is included. It does not prove an infinite universal claim by checking many examples. If the statement says “for every whole number”, systematic examples may suggest the rule, but a general explanation is still needed.

That is the handoff to Conjecturing and Generalising and Explaining, Convincing and Proving.

15. Common systematic-working errors

• Starting with random cases and sorting them only after duplicates appear.
• Using an order that does not match the strongest condition.
• Changing two variables at once without recording both.
• Writing results but not the inputs that generated them.
• Stopping because the list feels long enough.
• Counting symmetric duplicates as different.
• Using a tree when order is irrelevant or a triangular list when order matters.
• Keeping so much information that the record becomes harder to read than the problem.

16. A five-step working-systematically protocol

1. Define: what counts as one valid case?
2. Order: which variable or condition should be fixed first?
3. Record: choose list, table, grid, tree or labelled state.
4. Stop: state the rule proving the search is complete.
5. Check: verify no case appears twice and every final case satisfies the original conditions.

17. Practice: 24 original questions

Questions 1–8: Ordered cases

1. Using digits 1,2,3,4 without repetition, list all two-digit numbers greater than 30.
2. Using digits 1,2,3,4 without repetition, count all three-digit even numbers.
3. List positive whole-number factor pairs of 48 without duplication.
4. List unordered positive whole-number pairs with sum 14.
5. List ordered pairs (x,y) of positive whole numbers satisfying x+y=6.
6. Choose two different letters from A,B,C,D without order. List all pairs.
7. Choose one letter from X,Y and one digit from 1,2,3. List all ordered two-part codes.
8. Find all positive whole-number pairs x≤y satisfying x+y=11.

Questions 9–16: Tables and states

9. A:B has total 48 and ratio 3:5. Create an equal-unit table and find A and B.
10. Two boxes contain 80 counters. Start 30 and50. Move12 from B to A, then5 from A to B. Record every state.
11. Ten tickets cost $55; each ticket costs either $4 or $7. Use a systematic replacement method to find the counts.
12. Whole-number rectangles have perimeter24. List all dimension pairs and their areas.
13. For n=1 to6, record n² and consecutive differences.
14. Spinner A has 1,2,3 and Spinner B has4,5. Create an outcome grid and count all ordered outcomes.
15. A row has 7 posts. How many neighbouring intervals? Extend the table to 8,9,10 posts and notice the relationship.
16. Record the first six triangular numbers and the amount added at each stage.

Questions 17–24: Completeness and efficiency

17. Explain why factor-pair listing for 36 can stop at6×6.
18. Explain why AB and BA should not both appear when choosing an unordered pair.
19. Six people each shake hands with every other person once. Use a no-duplicate systematic count.
20. A 2×3 grid has rectangles of height1 or2. Explain why splitting by height is complete.
21. Find all positive whole-number pairs x≤y with xy=40.
22. A learner tries random whole numbers to solve x+y=20. Propose a more systematic search.
23. Give one example where a tree diagram is more appropriate than an unordered list.
24. Create a finite Primary Mathematics search problem and state a stopping rule that proves your list is complete.

18. Worked solutions

Solutions 1–8

1. 31,32,34,41,42,43. Organise by tens digit 3 then4.

2. 12. Final digit2 gives6 arrangements of the other three digits taken two at a time; final digit4 gives another6.

3. (1,48),(2,24),(3,16),(4,12),(6,8). Stop after the smaller factor would exceed the larger.

4. (1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(7,7).

5. (1,5),(2,4),(3,3),(4,2),(5,1). Order matters, so reverse pairs remain distinct.

6. AB,AC,AD,BC,BD,CD.

7. X1,X2,X3,Y1,Y2,Y3.

8. (1,10),(2,9),(3,8),(4,7),(5,6).

Solutions 9–16

9. Eight ratio units represent48, so one unit=6. A=18 andB=30.

10. Start(30,50), after first move(42,38), after second(37,43). Total80 at every stage.

11. All-$4 baseline=40. Need15 more. Each replacement adds3, so5 replacements. Five $7 tickets and five $4 tickets.

12. length+width=12. Pairs(1,11),(2,10),(3,9),(4,8),(5,7),(6,6). Areas11,20,27,32,35,36.

13. Squares1,4,9,16,25,36. Differences3,5,7,9,11.

14. 3×2=6 ordered outcomes: (1,4),(1,5),(2,4),(2,5),(3,4),(3,5).

15. Posts7,8,9,10 correspond to intervals6,7,8,9. Open endpoint-inclusive rows have one fewer interval than posts.

16. 1,3,6,10,15,21. Added amounts2,3,4,5,6.

Solutions 17–24

17. Factor pairs reverse after equality. Once the smaller factor passes6, any later pair is a reversed copy already listed.

18. They contain the same two selected letters. Enforce alphabetical order to give each pair one canonical representation.

19. A shakes with5 later people, B with4, then3,2,1: total15.

20. Every rectangle in a2-row grid must have height exactly1 or2. These cases cannot overlap and cover all possible heights.

21. (1,40),(2,20),(4,10),(5,8).

22. Let x increase from1 to19 and set y=20−x, or if unordered with x≤y, stop at(10,10). This guarantees completeness.

23. Example: a two-stage outfit or code where the first choice and second choice have different roles. The branches preserve sequence.

24. Answers vary. A valid stopping rule must be tied to the allowed cases—for example, “I have used every possible hundreds digit and every permitted tens digit under each.”

19. Transfer laboratory: one problem, three records

Find all whole-number rectangles with area 60 and identify the one with minimum perimeter.

Ordered factor-pair record: (1,60),(2,30),(3,20),(4,15),(5,12),(6,10). Stop before the factors reverse.

Perimeter column: 122,64,46,38,34,32. Minimum is32 for6×10.

Why complete: every positive whole-number rectangle of area60 corresponds to one factor pair. Listing smaller factors in increasing order until they exceed their partners includes each unordered pair exactly once.

The table does more than calculate. It proves that the 6×10 candidate is being compared with every permitted competitor.

20. Parent and tutor guide

When a learner misses cases, do not begin by saying “be more careful”. Ask what variable could be fixed first. When duplicates appear, ask whether order matters and whether a canonical naming rule could prevent repetition.

If the learner produces an enormous table, ask what information actually affects the next step. Efficient systematic work removes unnecessary state rather than displaying maximal effort.

When a search seems complete, require a stopping sentence. “The next factor pair would reverse one already counted” or “every allowed final digit has been used” is stronger than “I think that’s all”.

Delayed return

Three days later, give one factor-pair problem, one digit-arrangement problem, one before-after problem and one small optimisation. Ask the learner to choose the record type before calculating.

21. Mastery receipt

• I choose an order that matches the strongest condition.
• I use lists, tables, grids or trees deliberately.
• I distinguish ordered from unordered cases.
• I prevent duplicates by design.
• I record changing states when later moves depend on them.
• I know why my search can stop.
• I can explain why every valid finite case is represented.
• I reduce the record when a simpler structure preserves all necessary information.

Sources and scope

University of Cambridge NRICH — Working Systematically, Primary Students describes systematic work as a methodical, organised and logical approach rather than haphazard trial. NRICH — Thinking Mathematically, Primary Students places it alongside exploring, conjecturing, visualising and proving.

NSW Department of Education — Mathematics K–6 Effective Teaching Approaches similarly describes Working Mathematically as the thinking and doing of mathematics through interconnected processes.

For Singapore subject scope, use the MOE Primary Mathematics Syllabus P1–P6, updated October 2025 alongside the learner’s current school programme.

Continue the next capability batch

• Mathematical Communication, Precision and Multiple Forms
• Mathematical Connections, Transfer and the Same Structure
• Choosing Strategies, Tools and Efficient Methods
• BTT Primary Mathematics Learning Hub

The Quiet Return

Systematic work turns “I found an answer” into “I know where every possible answer would have had to appear.” That difference is small on the page and enormous in the reasoning.