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Primary Mathematics: Exploring, Noticing and Finding Structure | Worked Learning Guide

Before a problem can be solved, something useful has to be noticed. Strong mathematical exploration is not random trial-and-error. It is a controlled way of changing, comparing, recording and asking what stays the same.

A learner may know multiplication, fractions, area, ratio and graphs yet still freeze when a question arrives in an unfamiliar form. Often the missing move is not another formula. It is the ability to enter the problem: make a small example, compare two cases, change one condition, record what happens and notice which relationship survives.

This guide develops that entry capability. It treats exploring and noticing as mathematical work in their own right. The aim is to move from “I tried some numbers” to “I changed one feature, observed a repeatable relationship, tested an edge case and now know what question to ask next.”

The approach is consistent with current rich-task practice in leading primary mathematics libraries, where exploring, noticing, working systematically, visualising, conjecturing, explaining and proving are treated as connected mathematical habits. The examples and practice here are independently written for the BTT Primary Mathematics route.

Observe before explaining · Compare cases · Change one thing · Find what stays fixed · Test boundaries · 24 questions · Worked solutions

1. Observation is not yet explanation

Consider the sequence 5, 8, 11, 14, 17. A learner can make several observations:

• Every term is three more than the previous term.
• The terms alternate odd, even, odd, even, odd.
• The fifth term is 17.

These are descriptions of what is visible in the listed terms. A different statement—“the 100th term will be 302”—is a prediction based on a rule. That prediction may be justified if the sequence is stated to continue by adding three each time. It is not justified merely because five displayed terms happen to fit that rule.

This distinction matters across Primary Mathematics. A graph that rises is an observation. “Temperature caused the rise” is an explanation. Two equal fractions in a diagram are an observation. “Multiplying numerator and denominator by the same non-zero number preserves value” is a relationship that explains why equivalent fractions can be generated.

Worked example: separate what you see from what you infer

A 3 × 4 rectangle has area 12 square units and perimeter 14 units. A 3 × 5 rectangle has area 15 square units and perimeter 16 units.

Observation: increasing the length from 4 to 5 increased area by 3 and perimeter by 2.

Structural explanation: the extra one-unit-wide strip contains three new unit squares, while only the two horizontal outer sides become one unit longer. The vertical sides do not change.

The observation points toward the relationship. The explanation identifies why it occurs.

2. Comparison turns examples into information

One example tells us what happened once. Two carefully chosen examples can reveal what changed. Three or four organised examples can expose a pattern worth testing.

Suppose we calculate:

Number	Double it	Add 1
2	4	5
5	10	11
8	16	17

A useful notice is that the final result is always odd in these three cases. Why? Doubling any whole number produces an even number, and adding one produces the next odd number. The examples helped us see the pattern; the parity relationship explains it generally.

Choose comparisons that differ in one important way

To study the effect of a rectangle’s length while keeping width three, compare 3 × 4, 3 × 5, 3 × 6 and so on. If both dimensions change at once—say 3 × 4 compared with 5 × 7—it becomes harder to know which change caused which numerical difference.

This is the mathematical analogue of a fair comparison. The aim is not always to keep everything fixed, but to control enough of the situation that the relationship being studied becomes visible.

Worked example: fractions with the same numerator

Compare 3/4, 3/5 and 3/8 using the same whole size. The number of selected parts remains three, but the unit fraction becomes smaller as the denominator increases. Therefore 3/4 > 3/5 > 3/8.

The useful comparison is not “8 is larger than 4”. It is “eighths are smaller pieces than quarters when the whole is the same.” Keeping the numerator fixed allows the denominator’s effect to be inspected.

3. Change one thing and ask what follows

A powerful exploration routine is change one thing. Change a number, a dimension, an operation, a boundary condition, a diagram orientation or a starting value while preserving the rest of the problem. Then ask:

• What changed?
• What did not change?
• Can I explain both?

Worked example: one extra term in an average

The average of four numbers is 12, so their total is 48. Add a fifth number equal to 12. The new total is 60 and the new average remains 12.

Now add 17 instead. The total becomes 65 and the average becomes 13. The changed value is five above the old average. Spread across five entries, the average rises by one.

Changing only the added number reveals how averages respond to new data.

Worked example: move one unit between two addends

Compare 37 + 48 with 38 + 47. One unit moved from the second addend to the first. The total remains 85.

This is not coincidence. Adding one to one addend and subtracting one from the other preserves the sum. Mental compensation methods depend on this invariant. A learner who notices it can transform 37 + 48 into 35 + 50 or 40 + 45 without changing the answer.

4. Small cases reveal the machinery

Large questions often hide the relationship beneath arithmetic. Reduce the size while preserving the structure.

If a fence-post problem with 101 posts feels confusing, study three posts first:

Post ---- Post ---- Post

Three posts create two intervals. The small case reveals the relation: an open chain with both endpoints marked has one more mark than interval. Once understood, the same relation scales to 101 posts and 100 intervals.

Worked example: counting rectangles

A 6 × 8 grid contains too many rectangles to count comfortably by eye. Start with a 1 × 2 grid. There are two unit rectangles plus one whole rectangle: three. Then a 2 × 2 grid can be organised by size. The point of the small case is not to solve the large case directly; it is to identify what uniquely specifies one rectangle.

Once we notice that choosing two horizontal boundary lines and two vertical boundary lines determines one rectangle, the large count becomes structured. The Counting Rectangles, Squares and Triangles guide develops that method fully.

5. Organise observations so memory is not doing the counting

A table, labelled diagram or ordered list can expose structure that an unsorted collection hides. For a pattern of joined squares using matchsticks:

Squares	Matchsticks	Increase
1	4	—
2	7	+3
3	10	+3
4	13	+3

The table makes constant growth visible. It also helps separate term number from term value. The next question is no longer “what number comes next?” but “why does each new square add exactly three new matchsticks?”

That explanation comes from shared edges. Once the structure is understood, the general rule 3n + 1 has a meaning rather than being a guessed formula.

6. Notice what stays fixed

An invariant is something preserved while other features change. Primary Mathematics contains many:

• Moving one unit from one addend to another preserves their sum.
• Adding the same amount to two quantities preserves their difference.
• Multiplying both terms of a ratio by the same positive factor preserves the ratio.
• Multiplying numerator and denominator of a fraction by the same non-zero whole number preserves its value.
• Rearranging pieces can preserve area while changing perimeter.

Noticing an invariant often removes unnecessary calculation. The dedicated Invariants, Before-and-After and Unchanged Quantities guide provides a deeper worked route.

Worked example: equal additions

A is 17 greater than B. Add 50 to both quantities. The difference remains 17. We do not need to know either original value to know the new difference.

Why? (A + 50) − (B + 50) = A − B. The two added fifties cancel in the difference.

Worked example: scale a ratio

A:B = 3:5. Doubling both gives 6:10. Tripling both gives 9:15. The actual quantities grow, but the multiplicative comparison remains 3 to 5.

Changing only one side, such as 3:10, would change the ratio. The invariant depends on applying the same scale factor to both terms.

7. Notice when a representation creates the difficulty

The same quantity may be easier to see in another form:

• 0.75 = 75/100 = 3/4.
• 25% = 1/4.
• 18 × 5 = 9 × 10.
• 3:5 can be drawn as three equal units compared with five equal units.

If the arithmetic is correct but the learner cannot start, change the representation before adding more numbers. The next article in this batch, Visualising, Representing and Seeing the Same Mathematics, develops this capability directly.

8. Look for an anomaly, not only a pattern

Exploration becomes stronger when the learner also searches for what does not fit.

Suppose the first four values of a process are 6, 10, 14, 18. A learner predicts “add four forever”. If the rule generating the sequence was never stated, a fifth value of 23 would break that conjecture. The anomaly tells us that the first pattern was descriptive, not necessarily the underlying rule.

Worked example: average trap

Data Set A: 8, 8, 8. Data Set B: 2, 8, 14. Both averages are eight. If we notice only the average, the two sets look identical. If we notice spread, they are very different.

A useful exploration asks which features are hidden by a summary measure. This turns data reading into mathematical questioning rather than arithmetic alone.

9. Boundary cases test whether a rule really travels

A boundary case pushes a relationship to a smallest, largest, zero or extreme allowed value.

Worked example: zero groups

The rule “three sweets per bag gives 3n sweets for n bags” should still make sense when n = 0. Zero bags contain zero sweets, so 3 × 0 = 0. The boundary case supports the structure.

Worked example: perimeter growth

A row of n unit squares has perimeter 2n + 2. Check n = 1: the formula gives four, the perimeter of one square. If a proposed rule gave 2n + 4, it would already fail at the smallest case.

Worked example: fraction denominator

The statement “for the same positive numerator, a larger denominator gives a smaller fraction” requires the same positive whole and positive denominators. It does not make sense with denominator zero, because division by zero is undefined. Boundary testing includes checking whether the rule’s conditions still hold.

10. Reverse a relationship

After finding a forward relation, ask the reverse question.

If a row of joined squares uses 3n + 1 matchsticks, forward use asks how many matchsticks Figure 12 needs. Reverse use asks which figure uses 37 matchsticks. Solving 3n + 1 = 37 gives n = 12.

Reverse questions are valuable because they expose whether the learner understands the quantities inside the relationship. A child who can continue a pattern may still confuse figure number with number of matchsticks when the direction is reversed.

11. Ask “what if?” with discipline

A useful what-if question changes a stated condition while leaving enough of the original structure intact to compare.

• What if the path becomes a loop?
• What if one endpoint is not counted?
• What if the fraction refers to a different whole?
• What if the graph axis scale changes?
• What if the same total is shared among one more group?

The goal is not to generate endless variations. It is to identify which condition controlled the original method.

Worked example: path becomes loop

Eight equal intervals in an open row require nine endpoint-inclusive posts. Join the final post back to the first and treat the route as a closed loop. The same eight intervals now need eight distinct posts because the first and last endpoint are the same physical location.

The changed answer tells us exactly which assumption the “intervals + 1” rule depended on: an open chain with two different endpoints.

12. Exploration should eventually narrow

Exploration is productive when it reduces uncertainty. After several trials, stop and write a provisional relationship:

1. What have I noticed?
2. What do I think is causing it?
3. Which new case could test that idea?
4. What result would make me change my mind?

This turns wandering into inquiry. It also prepares the next two Working Mathematically guides: Conjecturing and Generalising and Explaining, Convincing and Proving.

13. Common exploration errors

• Trying random numbers without recording what changed.
• Changing several conditions at once and then guessing which mattered.
• Treating a visible pattern as a proven rule.
• Looking only for supporting examples and never for a failure.
• Using a large example when a small case would reveal the structure.
• Recording answers but not the relationship between them.
• Stopping after finding one successful case when the question asks what always happens.
• Continuing to explore after the decisive structure is already clear.

14. A six-move exploration protocol

1. Shrink: make a smaller case.
2. Label: name the quantities and conditions.
3. Vary: change one feature.
4. Record: use a table, diagram or ordered list.
5. Notice: state what changed and what stayed fixed.
6. Test: try a boundary or contrasting case.

Only after these moves should the learner decide whether a conjecture, formula, proof, model or calculation is the appropriate next step.

15. Practice: 24 original questions

Questions 1–8: Notice and compare

1. Sequence: 3, 6, 9, 12, 15. Write two observations that are definitely true of the displayed terms and one prediction that needs a continuation rule to be justified.
2. Compare 3 × 4 and 3 × 5 rectangles. What changes in area and perimeter when only the length increases by one?
3. Compare 3/4, 3/5 and 3/8 using the same whole. What is held fixed and what changes?
4. Compare 37 + 48 with 38 + 47. What stays invariant?
5. Data sets A = 8,8,8 and B = 2,8,14 have the same average. State one other feature that differs.
6. Matchstick counts are 4,7,10,13 for Figures 1–4. Record the first differences and state a structural question you should ask before accepting 3n+1.
7. A learner notices 2+4=6, 4+6=10 and 8+10=18 and says “two even numbers always add to an even number.” What new type of example would be most useful to test the word always?
8. A graph rises from 20 to 50 while another rises from 200 to 260. What should be checked before deciding which graph shows the larger proportional increase?

Questions 9–16: Change one thing

9. The average of four numbers is 12. Add a fifth number equal to 12. Find the new average and state what stayed fixed.
10. The same four-number set has total 48. Add 17. Find the new average and compare it with the old average.
11. A:B = 3:5. Multiply both terms by 4. What changes and what remains the same?
12. A is 17 greater than B. Add 50 to both. What is the new difference?
13. An open path has 8 equal intervals and both endpoints marked. How many marks? Now close the path into a loop using the same 8 intervals. How many distinct marks?
14. A row of 5 unit squares has perimeter 12. Add one square to the end. Predict the new perimeter and explain the change.
15. Five bags contain 3 counters each. Write the total. Change the number of bags to zero and check whether the rule still works.
16. For the same positive whole, compare 2/3 and 2/9. Explain why changing only the denominator changes the fraction size.

Questions 17–24: Test structure and limits

17. A sequence begins 6,10,14,18. Give one possible continuation rule and one reason the first four terms do not prove it is the intended rule.
18. A learner claims every rectangle with larger area also has larger perimeter. Find a counterexample using whole-number side lengths.
19. Eleven posts stand in an open row. How many intervals are between the first and last? Explain using a three-post small case.
20. A rule for a row of joined squares is P=2n+2. Check the boundary case n=1.
21. Figure n uses 3n+1 matchsticks. Which figure uses 37 matchsticks?
22. A quantity is doubled and then 1 is added. Explain why the result is always odd for whole-number inputs.
23. A 3×4 grid problem feels too large to count by eye. Describe one smaller case and one representation that could help discover a general rectangle-counting structure.
24. Write one mathematically useful “what if?” question for a ratio, fraction, geometry or data problem and state what you would hold fixed while testing it.

16. Worked solutions

Solutions 1–8

1. Examples of definite observations: all displayed terms are multiples of 3; consecutive displayed terms differ by 3. A statement such as “the 100th term is 300” needs the rule “continue adding 3” or another equivalent generating rule. The displayed terms alone do not force a unique continuation.

2. Area increases from 12 to 15, a change of 3 square units. Perimeter increases from 14 to 16, a change of 2 units. The extra one-unit strip contains three new unit squares, while two horizontal sides each lengthen by one.

3. The numerator and the whole are held fixed. The denominator increases, creating smaller equal parts. Thus 3/4 > 3/5 > 3/8.

4. The sum remains 85. One addend rises by one while the other falls by one, so the changes cancel.

5. Their spread differs. A has no variation; B ranges from 2 to 14. Equal averages do not make the datasets identical.

6. The differences are +3,+3,+3. Ask why adding one new square creates exactly three new matchsticks. The shared-edge construction is the structural justification for 3n+1.

7. Try arbitrary even numbers, including zero or very large even numbers, but more importantly explain the parity structure: each even number is made of pairs, so combining two collections of pairs still gives pairs. Searching for a counterexample is appropriate because the claim says always.

8. Check starting values and use a common basis. The first rises by 30 on a base of 20, or 150%; the second rises by 60 on a base of 200, or 30%. Larger absolute rise need not mean larger proportional rise.

Solutions 9–16

9. Original total is 48. New total 60 across five numbers gives average 12. Adding a value equal to the old average preserves the average.

10. New total = 48+17=65. Average = 13. The average rises by one.

11. The quantities become 12:20. Their common scale is four times larger, but the ratio relationship remains equivalent to 3:5.

12. The difference remains 17 because the same amount is added to both quantities.

13. Open path: 9 marks. Closed loop: 8 distinct marks. The open route has two different endpoints; the loop identifies the final endpoint with the first.

14. New perimeter = 14. A new square contributes four sides but one shared side removes two boundary copies, so net change is +2.

15. Five bags contain 15 counters. Rule T=3n gives T=0 when n=0, consistent with zero bags containing zero counters.

16. Thirds are larger unit fractions than ninths when the whole is the same. Selecting two thirds therefore covers more of the whole than selecting two ninths.

Solutions 17–24

17. One possible rule is add four each time. But finitely many displayed terms can fit more than one continuation rule unless the generating process is stated or structurally justified.

18. A 1×12 rectangle has area 12 and perimeter 26. A 3×5 rectangle has larger area 15 but smaller perimeter 16. Therefore larger area does not always imply larger perimeter.

19. Ten intervals. A three-post row has two gaps; each additional post adds one additional interval at the end. Thus eleven posts have ten intervals.

20. P=2(1)+2=4, matching the perimeter of one unit square. The smallest case is consistent with the rule.

21. Solve 3n+1=37. Then 3n=36 and n=12. Figure 12 uses 37 matchsticks.

22. Doubling any whole number gives an even number. Every even number plus one is odd. Therefore 2n+1 is odd for any whole-number n.

23. Use a 1×2 or 2×2 grid. Label horizontal and vertical boundary lines, then count rectangles by choosing top/bottom and left/right boundaries. The small case helps reveal what uniquely determines one rectangle.

24. Example: “What if both terms of a 3:5 ratio are doubled?” Hold the relationship type fixed and change both quantities by the same scale factor. Compare what happens to the actual values and to the ratio.

17. Transfer laboratory: one object, four ways to explore

Take the expression n(n+1), using whole-number values n=1,2,3,4,5.

1. Record the values in a table.
2. Notice whether each value is odd or even.
3. Draw n rows of n+1 counters and inspect the rectangle.
4. Ask what happens when the rectangle is split into two equal staircase shapes.

The values are 2,6,12,20,30. They are all even because two consecutive whole numbers cannot both be odd; one of them is always even. The rectangular representation also makes an even split possible because the total contains an even factor. If the rectangle is divided along an appropriate staircase boundary, each half contains n(n+1)/2 counters, connecting the exploration to triangular numbers.

The important outcome is not the triangular-number formula alone. It is the sequence of mathematical actions: calculate, notice, represent, explain, then generalise. A different route could begin from the staircase picture and arrive at the same structure.

18. Parent and tutor guide

When a learner says “I don’t know how to start,” do not immediately reveal the operation. Ask for a smaller case, a labelled sketch or two examples that differ in only one feature. The purpose is to create information from the problem before selecting a method.

If the learner produces many examples, ask for organisation. “Put them in order” or “show me what changed from one row to the next” often converts trial-and-error into a table or invariant. If the learner finds a pattern quickly, ask for a case that might break it.

Do not reward only the final discovery. Useful notices include recognising a constant quantity, finding an impossible case, noticing that a diagram is misleading, or realising that a question does not contain enough information. These are mathematical achievements because they reduce uncertainty.

The first weak link

If exploration repeatedly fails, diagnose where: does the learner struggle to generate examples, label quantities, organise data, compare cases, distinguish observation from explanation, or test a boundary? These are different problems and need different repairs.

For a learner who can explore but cannot communicate why a pattern should continue, continue to Explaining, Convincing and Proving. For one who does not see the same structure after the diagram changes, use Visualising and Representing.

19. Delayed return

Three days later, give the learner four unfamiliar prompts: a short number pattern, a fraction comparison, a grid-counting task and an open/closed route problem. Before any full solution, require six notes only: a smaller case, the quantities, one controlled variation, one observation, one invariant or changing feature, and one test case.

If those six notes can be produced independently, the learner is building a reusable entry routine rather than remembering the surface of today’s examples.

20. Mastery receipt

• I distinguish observation from explanation and prediction.
• I compare cases that differ in one useful feature.
• I reduce large problems to smaller structurally similar cases.
• I record examples in tables, diagrams or ordered lists.
• I notice invariants as well as changing quantities.
• I test anomalies, boundaries and contrasting cases.
• I use “what if?” questions to expose assumptions.
• I stop exploring when the next mathematical question becomes clear.

Sources and scope

University of Cambridge NRICH — Thinking Mathematically, Primary Students organises rich primary tasks around exploring and noticing, working systematically, conjecturing and generalising, visualising and representing, and explaining, convincing and proving. NRICH — Developing Mathematical Thinking, Primary Teachers provides the corresponding teacher-facing route.

For official Singapore subject documents, use the MOE Primary curriculum and syllabus page and the learner’s current school programme. This guide is an eduKate/BTT worked-learning resource, not an official examination specification.

Continue the Working Mathematically collection

• Visualising, Representing and Seeing the Same Mathematics
• Conjecturing, Generalising and Testing a Rule
• Explaining, Convincing and Proving
• BTT Primary Mathematics Learning Hub

The Quiet Return

Exploration is not the stage before mathematics begins. It is mathematics learning to see itself. Change one thing, record what happens, find what survives, and the problem starts telling you what kind of solution it needs.