What are heuristics in Singapore maths?

Heuristics are general problem-solving strategies, such as drawing a bar model, working backwards or making a supposition, that help children tackle non-routine word problems where the method isn't obvious.

How many heuristics are there in primary maths?

Singapore's primary maths syllabus lists about a dozen, grouped into four families: give a representation, make a calculated guess, walk through the process, and change the problem.

What is the difference between guess and check and supposition?

Guess and check tries values and adjusts until every condition fits. Supposition assumes everything is one kind, measures the gap from the real total, and fixes it in one calculation, so it is usually faster for 'two kinds of item' problems.

Does my child have to name the heuristic in the PSLE?

No. Markers look for clear, correct working that leads to the answer, whatever the method. Naming the heuristic just helps your child decide what to try.

Which heuristic should my child learn first?

The bar model. It underpins comparison, part-whole, fraction and ratio problems, and several other heuristics, such as before and after, are usually drawn with bars.

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Singapore Maths Heuristics: All 12 Strategies, With Worked Examples (P3–P6)

Heuristics are the problem-solving strategies Singapore primary maths teaches for word problems you can't solve with a single sum. The syllabus groups them into four families: give a representation (draw a diagram or model, make a list, use an equation), make a calculated guess (guess and check, look for patterns, make a supposition), walk through the process (act it out, work backwards, before and after), and change the problem (restate it, simplify it, solve part of it). Below, each one has the clue words that signal it and a fully worked example.

Most PSLE Maths Paper 2 questions are multi-step word problems. Children who know the heuristics recognise the shape of a problem quickly and pick a method; children who don't often freeze or guess. The good news is that the same dozen strategies cover almost every question, so learning to spot them pays off across P3 to P6.

Your child does not have to name the heuristic in an exam. What matters is clear working that leads to the answer. Knowing the names simply makes it easier to talk about which method to try next.

Give a representation

Turn the words into a picture, list or equation you can work with.

Draw a diagram or model (bar model)

Draw bars, a number line or a sketch to show how the quantities relate. The bar model is the signature Singapore method for part-whole, comparison and ratio problems.

Clue words: times as many · more than / less than · altogether · fraction of · ratio

Example: Ali has 3 times as many stickers as Ben. Together they have 48 stickers. How many stickers does Ali have?

Draw Ben's bar as 1 unit and Ali's bar as 3 units.
Together that is 4 units, so 4 units = 48.
1 unit = 48 ÷ 4 = 12.
Ali has 3 units = 3 × 12 = 36 stickers.

Answer: 36 stickers

Make a systematic list

Write out the possibilities in an orderly way so none is missed or counted twice.

Clue words: how many ways · possible combinations · different arrangements

Example: How many different ways can you pay exactly 30¢ using only 10¢ and 5¢ coins?

Start with the most 10¢ coins and work down.
3 × 10¢.
2 × 10¢ + 2 × 5¢.
1 × 10¢ + 4 × 5¢.
0 × 10¢ + 6 × 5¢.

Answer: 4 ways

Use an equation

Let a letter stand for the unknown and write the relationship as an equation. This becomes common in P6 once algebra is taught.

Clue words: a number · think of a number · find the value

Example: When a number is multiplied by 4 and then 7 is added, the result is 43. What is the number?

Let the number be n, so 4n + 7 = 43.
Subtract 7: 4n = 36.
Divide by 4: n = 9.

Answer: 9

Make a calculated guess

Try a sensible value, see how far off it is, and adjust.

Guess and check

Guess an answer, test it against every condition, and use the result to make a better guess. A small table keeps the guesses organised.

Clue words: heads and legs · two kinds of item with a total count and a total value

Example: A farm has chickens and goats. There are 10 heads and 28 legs. How many goats are there?

Guess 5 goats and 5 chickens: 5 × 4 + 5 × 2 = 30 legs. Too many.
Fewer goats: 4 goats and 6 chickens: 4 × 4 + 6 × 2 = 16 + 12 = 28 legs. Correct.

Answer: 4 goats

Look for patterns

Work out the first few cases, spot the rule, then use the rule instead of drawing everything.

Clue words: figure 1, figure 2… · what comes next · the 50th term

Example: Figure 1 is one square made of 4 matchsticks. Each new figure adds one square in a row, using 3 more matchsticks. How many matchsticks are in Figure 10?

Figure 1: 4. Figure 2: 7. Figure 3: 10.
The rule: start with 4 and add 3 for each figure after the first.
Figure 10: 4 + 3 × 9 = 31.

Answer: 31 matchsticks

Make a supposition (assumption)

Suppose everything is of one kind, see how far the total is from the real total, and fix the difference one item at a time. Faster than guess and check for 'two kinds of item' problems.

Clue words: all … suppose · two kinds of item with a total count and a total value

Example: A car park has 20 cars and motorcycles with 64 wheels altogether. How many cars are there?

Suppose all 20 are motorcycles: 20 × 2 = 40 wheels.
That is 64 − 40 = 24 wheels short.
Swapping a motorcycle for a car adds 2 wheels, so 24 ÷ 2 = 12 swaps.
Check: 12 × 4 + 8 × 2 = 48 + 16 = 64.

Answer: 12 cars

Walk through the process

Follow the story step by step, forwards or backwards.

Act it out

Physically or mentally play out what happens, often with objects or people standing in for the items.

Clue words: each person … once · takes turns · moves

Example: 5 friends each shake hands once with every other friend. How many handshakes are there?

The first friend shakes 4 new hands, the second 3 new hands, the third 2, the fourth 1.
4 + 3 + 2 + 1 = 10.

Answer: 10 handshakes

Work backwards

Start from the final amount and undo each step in reverse order.

Clue words: had … left · in the end · how much at first

Example: Mei spent half her money on a book, then $6 on lunch. She had $9 left. How much money did she have at first?

Undo the lunch: $9 + $6 = $15 before lunch.
Undo 'spent half': $15 is the half that was left, so $15 × 2 = $30.

Answer: $30

Before and after

Draw the situation before and after a change (giving, spending, transferring) and use what stays the same.

Clue words: at first … in the end · gave … to · after

Example: Ravi had 3 times as many marbles as Sam. After Ravi gave Sam 10 marbles, they had the same number. How many marbles did Ravi have at first?

Before: Ravi = 3 units, Sam = 1 unit.
After: Ravi has 3 units − 10, Sam has 1 unit + 10, and these are equal.
So 2 units = 20 and 1 unit = 10.
Ravi had 3 × 10 = 30 at first. Check: 30 − 10 = 20 and 10 + 10 = 20.

Answer: 30 marbles

Change the problem

Make the problem easier to see before you solve it.

Restate the problem

Say the problem another way to reveal what is really being asked, often something that stays constant.

Clue words: years' time · the difference between · still

Example: John is 5 years older than Kim. How much older will John be than Kim in 8 years' time?

Restate: does an age difference change over time?
Both get 8 years older, so the difference stays the same.

Answer: 5 years

Simplify the problem

Try a smaller or neater version of the problem, find the method, then scale it up.

Clue words: 1 + 2 + 3 + … · large numbers · many repeated steps

Example: Find 1 + 2 + 3 + … + 100.

Try a small case: 1 + 2 + 3 + 4 = (1 + 4) + (2 + 3) = 2 pairs of 5.
For 1 to 100, pair the ends: 1 + 100, 2 + 99, … each pair makes 101.
There are 50 pairs, so 50 × 101 = 5050.

Answer: 5050

Solve part of the problem

Break a long question into smaller questions and answer them in order.

Clue words: multi-step · then · how much change

Example: Pens cost $2 for 3. Lisa buys 12 pens and pays with a $10 note. How much change does she get?

Part 1: 12 pens is 12 ÷ 3 = 4 groups of 3.
Part 2: 4 groups × $2 = $8.
Part 3: change = $10 − $8 = $2.

Answer: $2

Frequently asked questions

What are heuristics in Singapore maths?: Heuristics are general problem-solving strategies, such as drawing a bar model, working backwards or making a supposition, that help children tackle non-routine word problems where the method isn't obvious.
How many heuristics are there in primary maths?: Singapore's primary maths syllabus lists about a dozen, grouped into four families: give a representation, make a calculated guess, walk through the process, and change the problem.
What is the difference between guess and check and supposition?: Guess and check tries values and adjusts until every condition fits. Supposition assumes everything is one kind, measures the gap from the real total, and fixes it in one calculation, so it is usually faster for 'two kinds of item' problems.
Does my child have to name the heuristic in the PSLE?: No. Markers look for clear, correct working that leads to the answer, whatever the method. Naming the heuristic just helps your child decide what to try.
Which heuristic should my child learn first?: The bar model. It underpins comparison, part-whole, fraction and ratio problems, and several other heuristics, such as before and after, are usually drawn with bars.

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