NSW · Selective High School Placement Test · Maths

NSW selective test Mathematical Reasoning practice

The NSW Selective High School Placement Test has four sections, and Mathematical Reasoning is the one most families ask us about. It is 35 multiple-choice questions in 40 minutes, on a department computer, with no calculator. This page gives the format exactly as the NSW Department of Education publishes it, repeats plainly what the department says about coaching, points you to its free practice tests, and then teaches one tool that makes multi-step word problems manageable: the tape diagram. It contains no advice about school choice, placement or appeals, and no promise of any score.

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Format from the NSW Department of Education · Practice written by us · No placement advice

In short

Mathematical Reasoning is one of four sections in the NSW Selective High School Placement Test for Year 7 entry. Students answer 35 multiple-choice questions, each with five possible answers, in 40 minutes on a department computer. No calculator is allowed in any section. The department says questions draw on the NSW curriculum up to Year 6 and that coaching is not necessary.

Maths worth learning anyway

Three courses that build reasoning, not tricks

Useful for any capable upper-primary student, test or no test.

Problem Solving and Computational Thinking for Kids course thumbnail

NMR / 01

Problem solving

Multi-step thinking, drawn and explained before calculated.

Open the syllabus →
Elementary Maths for Grades 1-5 course thumbnail

NMR / 02

Primary maths in depth

Fractions, ratio and measurement from the upper-primary curriculum.

Open the syllabus →
Maths Olympiad & Competition Course course thumbnail

NMR / 03

Competition maths

For students who enjoy harder problems for their own sake.

Open the syllabus →

The section, as published

35 questions, 40 minutes, five options each

Every detail comes from the NSW Department of Education, read on 7 October 2026.

Selective High School Placement Test sections (department practice test page)
SectionQuestionsTimeType
Reading17 (3 questions have multiple parts)45 minutesMultiple choice
Mathematical Reasoning3540 minutesMultiple choice, five possible answers
Thinking Skills4040 minutesMultiple choice, four possible answers
Writing130 minutesOpen response

The test is computer-based and sat on computers the department provides, at test centres that are mostly local public high schools. Students can do working out on paper; the department's video for students says they "can use paper for working out" in Mathematical Reasoning.

Calculators are on the list of items not allowed, and the practice page states: "Students cannot use a calculator during this, or any other, test."

For Year 7 entry in 2027, the department lists the test on Friday 1 to Saturday 2 May 2026, with a make-up test on Friday 22 May 2026. Each student sits on one day only. From 2026 the placement tests are held only in New South Wales.

The department describes the section as testing "mathematical reasoning skills" in which a child "will use different concepts from maths to solve problems", and notes that the concepts are based on what the child already learns at school.

Sources: NSW Department of Education, Placement test; Selective high school practice tests. Checked on 7 October 2026. We are independent of the department.

What the department says

Coaching is not necessary, in the department's own words

We think families deserve to read this before paying anyone, including us.

From the NSW Department of Education

"It is important to know that coaching or tutoring for the placement test is not necessary. The test will not need your child to know any new or special knowledge. So, extra paid coaching outside of school will not give your child an advantage."

The department goes further: it "does not recommend or endorse coaching for the placement test", and it warns that too much coaching can harm a child's wellbeing. It provides free resources, including online computer-based practice tests and PDF practice tests, in its Student Resource Hub.

Our advice is the same. Start with the department's free practice tests. They show the real screens, the timer and the question styles, and they cost nothing.

So what is this page for? Some children genuinely love problem solving and want more of it, whether or not they sit the test. Others find upper-primary fractions and ratio shaky, which will matter in Year 7 regardless of school.

Our classes teach mathematics: fractions, ratio and multi-step reasoning from the curriculum the test already draws on. We do not run test-coaching courses, and we will tell you honestly if we think your child does not need us.

Source: Selective high school practice tests, "Helping your child prepare for the placement test". Checked on 7 October 2026.

Problems we wrote

Tape diagrams for multi-step problems

A tape diagram turns a word problem into equal units drawn as strips. Once the picture is right, the arithmetic is small. All problems below use only upper-primary content and need no calculator.

Draw equal units

If one amount is four times another, draw one strip of 1 unit and one strip of 4 units. Equal-sized boxes are the whole idea.

Label what you know

Write the total, or the difference, against the strips it belongs to. The question usually tells you one of these.

Find one unit

Divide to find the value of a single unit, then build every answer from it. Check by putting the numbers back into the story.

Original practice problems with the tape diagram reasoning
ProblemThe diagramOne unitAnswer
Three fifths of a class are girls. There are 12 more girls than boys. How many students are in the class?Girls 3 units, boys 2 units; the difference is 1 unit1260 students
Sam has 4 times as many cards as Lee. Together they have 85. How many does Sam have?Lee 1 unit, Sam 4 units: 5 units altogether1768 cards
Red and blue beads are in the ratio 2 to 3. There are 10 more blue beads than red. How many beads in total?Red 2 units, blue 3 units; the difference is 1 unit1050 beads
A tank is two fifths full. Adding 18 litres makes it seven tenths full. What does the tank hold?Tenths: 4 shaded, then 7 shaded; 3 tenths added6 litres per tenth60 litres
Ana uses a quarter of her stickers, then a third of what is left. She has 30 left. How many did she start with?Quarters: 1 used, 3 left; a third of those 3 used, 2 quarters left15 per quarter60 stickers
Three consecutive whole numbers add to 72. What is the largest?Three equal strips plus 0, 1 and 2 extra2325

Notice how the ratio and fraction problems collapse into the same move: find the gap or total in units, divide once, then rebuild. The sticker problem looks like two separate fraction steps, but on a diagram the second step simply halves what is left, and the whole question becomes "2 quarters are 30".

Walk through the first problem slowly. The class is split into fifths, so draw five equal boxes: three for girls, two for boys. The girls have one box more than the boys, and the story says that difference is 12 students, so one box is 12. Five boxes make 60. To check, three fifths of 60 is 36 girls and two fifths is 24 boys, and 36 is indeed 12 more than 24. Every problem in the table can be read back into its story the same way.

With five options per question, a tape diagram also helps eliminate answers. If the class must be a multiple of 5 units, any option that is not a multiple of 5 can be crossed out before any calculation.

Pacing

About 68 seconds per question

Our arithmetic: 40 minutes divided by 35 questions is roughly 68.6 seconds each.

Habits that help

Read the last sentence of a question first, so you know what is being asked before reading the story.

Draw small and fast. A tape diagram on scrap paper takes five seconds and saves a muddled calculation.

Use the five options as a check, not a shortcut. If your answer is not among them, the diagram needs another look.

Habits to avoid

Spending three minutes on one problem. A question left for later is better than five questions never reached.

Guessing from the look of the numbers. Options are often built from common mistakes.

Practising under time pressure every day. Calm, accurate reasoning first; speed follows.

Building multi-step reasoning

Four stages across Years 4 to 6

Most capable students move through these over two or three years of ordinary good teaching.

Reasoning with units
StageWhat the student doesSecure when
1. Part and wholeDraws a total split into partsSolves "together they have 85" problems
2. ComparisonDraws two strips and marks the differenceSolves "12 more girls than boys" problems
3. Fractions of a wholeDivides a strip into fifths or tenthsSolves the tank problem without converting
4. Two-step changesRedraws the strip after each changeSolves the sticker problem in one diagram

Keeping it healthy

The department warns that too much coaching can harm wellbeing, and we agree. An hour a week of enjoyable maths is plenty for most children.

If your child is stressed, stop practising tests and go back to puzzles they choose themselves.

The other sections

Thinking Skills is a separate 40-minute section; our Thinking Skills page covers it in the same honest way.

For the Year 4 Opportunity Class test, see OC test maths practice.

Courses for upper primary

Maths courses for Years 4 to 6

Grouped by focus. Each card links to a full syllabus.

I

Reasoning

Problem solving and logic

NMR / R / 01

Problem solving and computational thinking

Breaking problems into clear steps.

Open the syllabus

NMR / R / 02

Competition maths

Harder problems for keen students.

Open the syllabus

NMR / R / 03

Maths through coding

Patterns and logic explored in code.

Open the syllabus
II

Core maths

Fractions, ratio and measurement

NMR / C / 01

Primary maths

The upper-primary curriculum, taught deeply.

Open the syllabus

NMR / C / 02

Mental maths

Fast, careful calculation without a device.

Open the syllabus

NMR / C / 03

Statistics and probability

Data and chance for upper primary.

Open the syllabus
III

Looking ahead

For students ready for secondary maths

NMR / A / 02

Algebra foundations

Units in tape diagrams become letters.

Open the syllabus

NMR / A / 03

Abacus mental maths

Visual calculation for younger learners.

Open the syllabus

Our lessons

Mathematics first, for children who enjoy it

Teachers work from India, which is behind Sydney. During daylight saving a 4.30 pm Sydney class starts at 11 am India time, so after-school slots are simple to keep.

After school

The usual slot for Years 4 to 6.

Saturday

For families with weekday commitments.

Holidays

Short blocks on fractions or ratio.

Diagrams before numbers

Students draw the problem before they calculate anything.

Explain every answer

A correct answer without a reason counts as half done.

Groups of fifteen to twenty

Children at a similar level learn together.

One-to-one available

For a child who wants a particular topic strengthened.

No test coaching

We teach maths from the curriculum, as the department advises.

No placement promises

Selection is the department's decision; we make no claims about it.

Student work

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Made by learners who wanted to build something. More in the student labs.

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Questions about Mathematical Reasoning

What NSW families ask us

How many questions are in the selective test Mathematical Reasoning section?

35 multiple-choice questions, each with five possible answers, in 40 minutes, according to the NSW Department of Education.

Can students use a calculator?

No. The department says students cannot use a calculator during Mathematical Reasoning or any other section, though they can work out on paper.

Is the selective test on a computer?

Yes. It is computer-based and sat on computers provided by the department at test centres, mostly local public high schools.

What maths does it cover?

The department says questions draw on concepts from the NSW curriculum up to Year 6, and that children do not need to study extra concepts.

Does my child need coaching?

The department says coaching or tutoring for the placement test is not necessary and that it does not recommend or endorse coaching. Its free practice tests are the place to start.

What is a tape diagram?

A drawing of a word problem as equal-sized strips or boxes, so that totals, differences and fractions can be seen as units before any calculation.

When is the test?

For Year 7 entry in 2027 the department listed Friday 1 to Saturday 2 May 2026, with a make-up test on Friday 22 May 2026. Check the department's site for later years.

Can you advise on which school to choose?

No. We do not give school-choice, placement or appeals advice. Those questions belong with the department and the schools.

Will lessons improve my child's chances?

We make no promise about scores or placement. We teach mathematics that is worth knowing regardless of the test.

When would lessons run?

We agree one weekly time after the free trial, usually after school. During daylight saving Sydney is five and a half hours ahead of India time.

Related pages

More for NSW families

Other test sections, NAPLAN and coding.

Selective test Thinking Skills

The 40-question reasoning section.

OC test maths

The Year 4 Opportunity Class test.

Coding and AI in NSW

Our state page for New South Wales.

NAPLAN Year 5 numeracy

Upper-primary maths in NAPLAN.

Maths tutoring in Australia

Live classes for every year.

Competitions calendar

Contests for students who love problems.

Start here

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