The hardest PSLE math question: what makes it hard, and how to prepare calmly

Tabby: Every year, after the paper, parents ask us the same thing: "What was the hardest PSLE math question, and could my child have done it?"

It is a fair question. But searching for the hardest PSLE math question usually leads to a list of past puzzles to marvel at, which is interesting and not very useful. A child cannot prepare for one particular question. A child can prepare for the features that make a question hard.

Polly: So we are not collecting scary questions today?

Tabby: No, Polly. We are taking them apart to see how they work.

In this guide, we look at four features that hard PSLE math questions tend to share, work through three examples written by our teachers, and suggest a calm way to practise at home.

What makes the hardest PSLE math question hard?

When we look closely at difficult PSLE math questions, the arithmetic is seldom the problem. Most of the numbers are friendly. The difficulty sits in the thinking that comes before any calculating.

The topics themselves are set out in the MOE Primary Mathematics syllabus. A challenging question does not go outside those topics. It asks a child to use familiar ideas in a less familiar way.

Here are the four features we see most often.

Polly: None of those is "big numbers".

Tabby: That is the point. A child who is quick at sums can still feel stuck, and a child who is slower at sums can do very well here.

Example 1: something stays the same

Question. Mrs Lim is 38 years old and her son is 8 years old. In how many years will Mrs Lim be 3 times as old as her son?

Tabby: Many children try 38 ÷ 3 and get a number that does not help. The clue is that both of them grow older together, so the difference in their ages never changes.

• Step 1: The difference in their ages is 38 − 8 = 30 years. It will always be 30 years.
• Step 2: When Mrs Lim is 3 times as old, she is 3 units and her son is 1 unit. The difference is 2 units.
• Step 3: 2 units = 30 years, so 1 unit = 15 years. Her son will be 15.
• Answer: 15 − 8 = 7 years.

Sense check: In 7 years, Mrs Lim is 45 and her son is 15. 3 × 15 = 45. It matches.

What to say at home: "As the years go by, what changes and what stays the same?"

Example 2: the story runs backwards

Question. Jia had some beads. She used half of them and 4 more to make a bracelet. She then used half of the remaining beads and 3 more to make a keychain. She had 9 beads left. How many beads did Jia have at first?

Polly: We don't know the start. How can we begin?

Tabby: We begin at the end, because the end is the part we know. Each step is undone in reverse order.

• Step 1: Before using the extra 3 beads, Jia had 9 + 3 = 12 beads.
• Step 2: Those 12 beads were half of what she had after the bracelet. So she had 12 × 2 = 24 beads then.
• Step 3: Before using the extra 4 beads, she had 24 + 4 = 28 beads.
• Step 4: Those 28 beads were half of what she had at first.
• Answer: 28 × 2 = 56 beads.

Sense check: Start with 56. Half is 28, and 4 more makes 32 used, leaving 24. Half of 24 is 12, and 3 more makes 15 used, leaving 9. It matches.

What to say at home: "Which part of the story do we know for sure? Let's start there."

Example 3: two "what if" situations

Question. A teacher has some stickers for her class. If she gives each child 5 stickers, she will have 12 stickers left over. If she gives each child 7 stickers, she will be short of 18 stickers. How many children are in the class?

Tabby: This is one of the most challenging PSLE math question styles for children who have only practised by topic, because there is no fraction, ratio or percentage to hold on to. We compare the two situations instead.

• Step 1: Going from 5 stickers each to 7 stickers each, every child gets 2 more stickers.
• Step 2: To do that, the teacher uses up the 12 left over and still needs 18 more. That is 12 + 18 = 30 extra stickers.
• Step 3: 30 extra stickers, shared as 2 more for each child.
• Answer: 30 ÷ 2 = 15 children.

Sense check: 15 × 5 = 75, and 75 + 12 = 87 stickers. 15 × 7 = 105, and 105 − 18 = 87 stickers. Both give the same number of stickers.

What to say at home: "What is different between the first plan and the second plan?"

How to prepare for hard PSLE math questions without drilling

A child does not become steady on hard PSLE math questions by doing a great many of them quickly. In our lessons, we slow down and build four habits.

• Say it before solving it. Your child tells you, in one sentence, what the question is asking for. If that sentence is not clear, no method will help yet.
• Look for what stays the same. A total, a difference, or one person's amount. This single question unlocks a large share of the tricky ones.
• Draw before calculating. A quick model, a timeline or a table. It does not need to be neat.
• Sense check at the end. Put the answer back into the story, as we did above. Children who do this catch their own slips.

Choosing a method is a skill of its own. Our PSLE math heuristics checklist gives your child a short set of questions to ask when the first idea does not work.

A calm routine is also kinder than a long one. Two or three challenging questions in a sitting, talked through properly, is plenty. If you would like to fit this into a longer plan, our PSLE math roadmap shows how to set milestones and weekly reviews.

Common questions from parents

What is the hardest PSLE math question?

There is no single one. Each year, one or two questions get talked about, and they are usually different in topic. What they share are the features above: an unfamiliar story, something hidden that stays the same, or an ending given before the beginning.

Which topics do the difficult PSLE math questions come from?

All of them come from the primary syllabus. The longer problem sums often draw on fractions, ratio, percentage, speed and area, sometimes two of these in one question. The topic is rarely what makes it hard. The way the information is arranged is.

Should my child attempt the hardest questions in the paper?

Yes, calmly, and after the questions they are confident with are done and checked. A clear first step and tidy working are always worth writing down, even when the final answer is not reached.

My child gives up quickly on challenging questions. What can we do?

This is very common, and it is not a sign of ability. Start smaller. Ask only for the one-sentence summary and a drawing, and stop there. Once starting feels safe, finishing follows more easily.

One question at a time

The hardest PSLE math question is not something to fear or to predict. It is a familiar idea in an unfamiliar story, and that is something a child can learn to recognise.

You will find printable guides and checklists in our resources library. If your child would benefit from guided, bite-size practice in reading questions carefully and showing clear working, you can see how we teach in our Primary 5 and Primary 6 lessons.

Polly: So the hard ones are just ordinary ones wearing a costume?

Tabby: Quite often, Polly. We look under the costume first.