Primary 3 Mathematics Learning Guide | Heuristics II: Act It Out, Draw a Diagram, Simplify & Look for a Pattern

Some Primary 3 Mathematics problems resist a direct calculation because the structure is hidden. When that happens, the student needs a way to change the problem without changing its meaning. Four particularly useful heuristics are to act it out, draw a diagram, simplify the problem and look for a pattern.

This is Guide 54 in the Primary 3 Mathematics Learning Hub. It complements Guide 21 by extending the heuristic toolbox rather than repeating work backwards, guess-and-check or make-a-table.

A heuristic is not a guaranteed trick. It is a disciplined way to make a difficult problem more visible.

Heuristic 1 | Act It Out

Acting out converts a verbal description into a sequence of visible actions. It is especially useful when the problem contains movement, exchanging, distributing, rearranging or several changing states.

The goal is not play for its own sake. Each action must correspond to a mathematical state in the problem.

Worked Example | Sharing With Remainder

38 counters are placed into groups of 6. How many complete groups can be formed, and how many counters remain?

• Act out equal groups of 6.
• Six complete groups use 36 counters.
• 2 counters remain.
• Therefore 38 ÷ 6 = 6 remainder 2.

The physical grouping makes the meaning of quotient and remainder visible.

Acting Out a Change Problem

A learner begins with an unknown number of counters, gives away 7 and has 15 left. Acting out the reverse route can show that the starting amount must be 22. The action helps the child see why addition is used even though the story says “gave away”.

When to Stop Acting It Out

Once the learner can represent the same relationship with a diagram or number sentence, move away from physical action. Acting out is a bridge toward abstraction, not a permanent requirement.

Heuristic 2 | Draw a Diagram

A diagram is useful when relationships are easier to see spatially than verbally. It can show routes, groups, lengths, areas, comparisons, time sequences and arrangements.

The diagram should be labelled. An unlabeled sketch may look helpful while still hiding the mathematical roles.

Worked Example | Route Length

A child walks 250 m east, then 180 m north, then 250 m west. How far has the child walked altogether?

• Draw the three connected route segments.
• Label them 250 m, 180 m and 250 m.
• Total distance = 250 + 180 + 250.
• Answer = 680 m.

The diagram prevents the student from confusing final position with total distance travelled.

Worked Example | Rectilinear Figure

An L-shaped figure can be divided into two rectangles. Drawing the split line and labelling dimensions makes the area problem manageable. The split is a tool for calculation; it is not part of the outside perimeter.

Diagram Versus Bar Model

A bar model is a specialised diagram for quantity relationships. A route sketch, shape decomposition or clock timeline serves different jobs. Students should choose the representation that exposes the hidden structure.

Heuristic 3 | Simplify the Problem

Simplifying does not mean solving an easier but unrelated question. It means replacing difficult numbers or a large case with a smaller version that preserves the important structure.

Change the size of the problem, not the relationship you are trying to understand.

Worked Example | Same Perimeter, Different Area

If a problem asks what happens to area among rectangles with a fixed perimeter, first test a small perimeter such as 12 cm.

• 1 × 5 gives area 5.
• 2 × 4 gives area 8.
• 3 × 3 gives area 9.

The small case reveals a pattern worth testing in the larger problem: shapes closer to a square may produce larger area for the same perimeter.

Simplify Multiplication

If 8 × 47 feels difficult, first think about 8 × 40 and 8 × 7. The problem is simplified by decomposing 47 while preserving the total product: 320 + 56 = 376.

Simplify Word-Problem Numbers

If the relationship is confusing, temporarily replace 224 and 79 with 20 and 5. Once the learner identifies larger, smaller and difference, restore the original numbers. This separates structural understanding from calculation load.

Heuristic 4 | Look for a Pattern

Patterns can appear in number sequences, multiplication facts, fraction equivalence, shape growth, repeated measurement cases and organised searches.

The learner should identify what changes, what stays fixed and whether the rule continues.

Worked Example | Growing Pattern

A pattern has 4 tiles in Stage 1, 7 tiles in Stage 2, 10 tiles in Stage 3 and 13 tiles in Stage 4. How many tiles are in Stage 7?

• The pattern increases by 3 each stage.
• Stage 5 = 16.
• Stage 6 = 19.
• Stage 7 = 22.

The rule is repeated addition of 3.

Worked Example | Multiplication Pattern

• 6 × 5 = 30
• 6 × 6 = 36
• 6 × 7 = 42
• 6 × 8 = 48

Each next product is 6 greater because one more group of 6 has been added. This pattern can help recover a forgotten fact.

Worked Example | Equivalent-Fraction Pattern

• 1/2 = 2/4
• 2/4 = 3/6?

Rather than guessing from the visual pattern of numerators and denominators alone, test the value. Both 1/2, 2/4 and 3/6 represent the same point halfway between 0 and 1. A genuine mathematical pattern must preserve the relationship, not just the appearance of the symbols.

Pattern Is a Hypothesis, Not Proof

Seeing three cases behave the same way suggests a rule. It does not automatically guarantee every case will follow. At Primary 3 level, students can learn the habit: notice, predict, test another case, explain why it makes sense.

Combining Heuristics

Difficult problems may need more than one heuristic. A student might simplify a problem, draw a diagram of the smaller case, notice a pattern, then return to the original numbers.

Heuristics are not separate boxes. They are tools that can be sequenced.

Worked Example | Combining Simplify + Pattern

How many handshakes occur if 5 children each shake hands once with every other child?

• Start smaller: 2 children → 1 handshake.
• 3 children → 3 handshakes.
• 4 children → 6 handshakes.
• When the 5th child joins, that child adds 4 new handshakes.
• Total = 6 + 4 = 10.

The smaller cases reveal how each new person changes the total. The student should draw or act out if duplicate handshakes are hard to control.

Student Route | What Is Blocking Me?

• If I cannot imagine the action, act it out.
• If I cannot see the relationship, draw it.
• If the numbers are hiding the idea, simplify them.
• If several cases are connected, look for what changes predictably.

Parent Route | Ask for the Reason the Heuristic Helps

After the child chooses a heuristic, ask what became clearer. “I drew a diagram because I could not track the route” is stronger than “Teacher says always draw.” The learner should know the purpose of the tool.

Teacher Route | Teach Selection, Not Ritual

Present two problems where the same heuristic helps, then one where it adds unnecessary work. Ask students to compare. This prevents a useful strategy from becoming an automatic ritual.

Diagnostic Map

Observed behaviour	Likely issue	Next move
cannot imagine changing states	story remains abstract	act it out
mixes positions or lengths	spatial relation hidden	draw and label
understands small numbers but not large	calculation load masks structure	simplify then restore
repeats cases without prediction	pattern not abstracted	state change rule
uses heuristic on every problem	strategy selection weak	compare useful versus unnecessary use

Common Heuristic Mistakes

• drawing an unlabelled picture that does not encode quantities;
• acting out without recording the mathematical state;
• simplifying until the original structure disappears;
• assuming a visual pattern must always continue;
• using the first familiar heuristic regardless of the obstacle;
• failing to return from the smaller case to the original problem.

Practice Progression

• one problem where acting out is clearly useful;
• one labelled-diagram problem;
• one large-number problem simplified to a small case;
• one growing pattern;
• one problem where two heuristics work;
• one problem where a tempting heuristic is unnecessary;
• one mixed problem requiring strategy selection.

Exam Craft | Choose the Smallest Useful Tool

Under assessment conditions, a heuristic should reduce uncertainty faster than it adds working. Draw enough to expose the structure, simplify enough to see the rule, and stop acting out once the relationship is understood.

Next Route

Continue with Guide 21: Work Backwards, Guess & Check, Make a Table, Guide 51: Systematic Search, and Guide 16: Mixed Problems and Strategy Choice.

Return to the Primary 3 Mathematics Learning Hub.