Mathematical Reasoning: Topics & Strategies

Mathematical Reasoning for the Selective Test: Topic Breakdown & Strategies

What Is the Mathematical Reasoning Section?

The Mathematical Reasoning section is one of four equally weighted components of the NSW Selective High Schools Placement Test (SHSPT), contributing 25% of your overall placement score. It contains 35 multiple-choice questions (each with 5 options) to be completed in 40 minutes.

This section assesses your ability to apply mathematical understanding to solve problems, not just recall memorised procedures. Calculators are not allowed — students may use scratch paper for working out, but all answers are entered digitally.

The questions draw from Year 5–6 mathematics, but the key difference from school maths is that every question requires reasoning, not just computation. A student who can perform the four operations perfectly but can't apply them to unfamiliar word problems will struggle.

The Seven Topic Areas

1. Number and Arithmetic Word Problems

The most fundamental topic area. These questions test operations with whole numbers applied to real-world contexts.

What you'll encounter:

• Multi-step word problems involving addition, subtraction, multiplication and division
• Problems involving money, time, measurement or everyday situations
• Questions requiring estimation and reasonableness checks
• Problems where you need to choose the most efficient operation or sequence of operations

Key strategies:

• Read the entire question before starting calculations — many students lose marks by solving the wrong problem
• Identify what the question is actually asking (the final answer, not an intermediate step)
• Use scratch paper to write out your working — mental arithmetic errors are the #1 mark killer
• For estimation questions, round numbers to make calculation easier, then check which option is closest

2. Fractions, Decimals and Percentages

A heavily tested topic that connects to many other question types.

What you'll encounter:

• Comparing and ordering fractions and decimals
• Converting between fractions, decimals and percentages
• Percentage of a quantity, percentage increase and decrease
• Fraction-of-amount problems in real contexts (sharing, discounts, recipes)
• Problems combining fractions with other operations

Key strategies:

• Know the common fraction-decimal-percentage equivalents by heart: 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 1/5 = 0.2 = 20%, 3/4 = 0.75 = 75%, 1/3 ≈ 0.333 = 33.3%, 1/8 = 0.125 = 12.5%
• For comparing fractions, convert to a common denominator or convert both to decimals
• For percentage problems, remember: "of" means multiply. 30% of 240 = 0.3 × 240 = 72
• Watch for percentage increase vs. new total — "20% more than 50" is 60, not 70 (a common trap)

3. Patterns and Sequences

These questions test your ability to identify rules and predict what comes next.

What you'll encounter:

• Number patterns with additive rules (e.g., +3, +5, +7…)
• Number patterns with multiplicative rules (e.g., ×2, ×3…)
• Mixed or two-rule patterns (e.g., alternating +2, ×3)
• Completing tables or sequences with a missing element
• Patterns in shapes or grids
• Finding the nth term or applying a rule to find a specific value

Key strategies:

• Write out the differences between consecutive terms — this reveals additive rules
• If differences aren't constant, check ratios between consecutive terms — this reveals multiplicative rules
• If neither works, check if the pattern alternates between two different rules
• For grid or shape patterns, look at rows, columns and diagonals separately
• Always verify your rule works for all given terms before selecting an answer

4. Algebraic and Relational Reasoning (Pre-Algebra)

These questions use unknowns, equations and relational thinking without requiring formal algebra skills.

What you'll encounter:

• Simple equations with unknowns: "If 3 × ☐ + 5 = 23, what is ☐?"
• Balance equations: "If A + B = 15 and A = 9, what is B?"
• Word problems that translate to equations
• Inequalities and comparisons: "Which is greater: 3 × 8 or 5 × 5?"
• Problems where changing one variable affects another

Key strategies:

• For "find the unknown" problems, work backwards from the answer: 23 - 5 = 18, 18 ÷ 3 = 6
• For balance equations, whatever you do to one side, you must do to the other
• Translate word problems into number sentences: "three times a number, increased by five, equals twenty-three" → 3 × ☐ + 5 = 23
• If stuck, try substituting each answer option back into the equation — one of the five options will work

5. Measurement and Geometry

These questions test spatial understanding and the application of measurement formulas.

What you'll encounter:

• Perimeter and area of rectangles, triangles and composite figures
• Angle properties: straight lines (180°), right angles (90°), full turn (360°), angles in a triangle (180°)
• Units and conversions: length (mm, cm, m, km), mass (g, kg), capacity (mL, L), time (sec, min, hr)
• Reading scales on rulers, thermometers, speedometers and measuring jugs
• Volume of rectangular prisms (length × width × height)
• Coordinate grids and positions

Key strategies:

• For composite area problems, break the shape into rectangles and triangles, calculate each area, then add (or subtract)
• Know your conversions: 1 km = 1000 m, 1 m = 100 cm, 1 cm = 10 mm, 1 kg = 1000 g, 1 L = 1000 mL
• For angle problems, identify what you know and use angle facts: angles on a straight line sum to 180°, angles in a triangle sum to 180°
• When reading scales, first determine what each small division represents before reading the value
• Draw diagrams on scratch paper for geometry problems — visualising the shape makes calculation much easier

6. Data and Chance (Statistics and Probability)

These questions require reading and interpreting information presented in tables, graphs and charts.

What you'll encounter:

• Reading and interpreting bar graphs, line graphs, pie charts and tables
• Calculating totals, differences and simple averages from data displays
• Drawing conclusions from data (e.g., "Which month had the highest rainfall?")
• Basic probability: equally likely outcomes (dice, spinners, card selection)
• Understanding probability language: "certain," "likely," "unlikely," "impossible"

Key strategies:

• Read the title, axis labels and legend before trying to answer — many errors come from misreading what the graph represents
• For averages, remember: mean = total ÷ number of items
• For probability: P(event) = number of favourable outcomes ÷ total number of outcomes
• For pie charts, remember the total is 100% (or 360°) — use this to calculate unknown sections
• Watch for graphs with non-zero starting points or irregular scales designed to mislead

7. Multi-Step Problem Solving and Puzzle-Style Questions

The hardest questions combine multiple topic areas or present mathematical puzzles that require logical reasoning.

What you'll encounter:

• Problems combining geometry with fractions (e.g., "What fraction of the shape is shaded?")
• Problems combining time with arithmetic (e.g., journey problems with stops and different speeds)
• Logic-based number puzzles requiring deduction from constraints
• Questions where you must work backwards from the answer
• Problems requiring systematic listing of possibilities

Key strategies:

• Break complex problems into smaller steps — solve one part at a time
• Use scratch paper to organise your working — these problems have multiple steps and it's easy to lose track
• If the question seems impossible, try working backwards from the answer options
• For systematic counting problems, organise your list so you don't miss or double-count possibilities
• These questions often appear near the end of the section — if you're running low on time, make an educated guess and move on

Time Management: 40 Minutes for 35 Questions

You have just over 1 minute per question on average. Some questions take 30 seconds; harder ones may take 2 minutes. The key is not to get stuck.

The 3-Pass Strategy

Pass 1 (0–25 minutes): Work through all 35 questions

• Answer each question you can solve within 60–90 seconds
• If a question looks like it will take longer, make your best guess, flag it and move on
• Don't leave any question blank — there's no penalty for wrong answers

Pass 2 (25–35 minutes): Return to flagged questions

• These are the questions worth fighting for
• Spend up to 2 minutes on each flagged question
• Use scratch paper — many flagged questions are solvable with more careful working

Pass 3 (35–40 minutes): Review

• Check for careless errors: Did you answer what was asked? Did you circle the right option?
• Double-check any calculations you did mentally
• Don't change answers unless you're confident

Which Questions to Skip

Not all questions are equal. If you encounter any of these, flag and move on quickly:

• Complex multi-step puzzles with many constraints
• Questions that require drawing or visualising elaborate diagrams
• Questions where you can't see a clear starting point after 30 seconds

Come back to these in Pass 2 with fresh eyes.

Mental Maths Skills to Master

Since calculators are not allowed, these mental maths techniques will save you significant time:

Times Tables (Essential)

Know your times tables to 12 × 12 instantly. This isn't optional — slow multiplication cascades into time pressure across the entire section.

Doubling and Halving

• To multiply by 5: multiply by 10 then halve (e.g., 36 × 5 = 360 ÷ 2 = 180)
• To multiply by 25: multiply by 100 then divide by 4
• To multiply by 50: multiply by 100 then halve

Working with Percentages

• 10% = divide by 10
• 5% = half of 10%
• 1% = divide by 100
• Build other percentages from these: 15% = 10% + 5%, 30% = 3 × 10%

Checking Your Answer

• Estimation: Does your answer make sense? If you're calculating the area of a small room and get 5,000 m², something went wrong
• Odd/even check: odd × odd = odd, odd × even = even, even × even = even
• Last digit check: multiply the last digits to verify the last digit of your answer

Practice Schedule

8+ Weeks Before the Test

4 Weeks Before the Test

• One full timed Mathematical Reasoning section per week
• Daily mental maths practice (5–10 minutes)
• Targeted work on remaining weak topics
• Practice the 3-pass strategy under timed conditions

Key Resources

• Official NSW DoE practice tests on the Janison platform
• Past selective school test papers (2020 onwards for closest alignment)
• Year 5–6 maths problem-solving workbooks (focus on word problems, not just computation)
• Mental maths apps and exercises

Common Mistakes and How to Avoid Them

Not reading the question carefully

The most common error. Students calculate correctly but answer the wrong thing. For example, the question asks "How many more?" and the student gives the total instead of the difference.

Careless arithmetic

Without a calculator, arithmetic errors are the biggest mark killer. Always write your working on scratch paper, especially for multi-step problems. Double-check any calculation you do mentally.

Getting stuck on hard questions

Spending 3–4 minutes on one question means 2–3 easier questions go unanswered. Flag and move on — it's better to answer 32 questions confidently than 20 questions perfectly.

Not using all five options

With 5 answer options, you have a 20% chance of guessing correctly. Never leave a question blank. If you're stuck, eliminate options that are clearly too large, too small or don't make sense, then guess from the remaining options.

Forgetting units and conversions

A question might give measurements in centimetres but ask for the answer in metres. Always check that your answer matches the units the question asks for.

Frequently Asked Questions

Is this the same as school maths?

The content is based on Year 5–6 curriculum topics, but the questions are more challenging. School maths often tests procedures ("Calculate 3/4 + 1/2"), while the selective test presents unfamiliar situations that require you to figure out which procedure to use and when.

Can my child use a calculator?

No. Calculators are not permitted. Students can use scratch paper for written working out, but all calculations must be done by hand or mentally. This is why strong mental maths skills are so important.

Are there any tricks or shortcuts?

The best "trick" is understanding the maths deeply enough to see efficient solutions. That said, techniques like estimation, working backwards from answer options and substitution are genuinely useful strategies. The mental maths shortcuts listed above (doubling/halving, percentage building) also save real time.

My child is good at maths but runs out of time. What should we do?

This is usually a speed issue with basic operations. Practise times tables until they're instant (under 2 seconds per fact), build mental maths habits (5 minutes daily), and use the 3-pass strategy to avoid getting stuck on hard questions. Many students lose time not on the maths itself but on re-reading questions they didn't understand the first time — practise reading word problems efficiently.

How important is geometry compared to number work?

All topic areas appear in the test, but number and arithmetic, fractions/decimals/percentages, and patterns typically have the most questions. Geometry and data questions appear regularly but usually in smaller numbers. However, every mark counts — don't skip geometry preparation entirely.

Last Updated: March 2026

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