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Year 6 Maths Reasoning

Problem Solving Strategies

Master strategies like working backwards, logical reasoning, and using tables to tackle tricky maths problems step by step.

Strategy 1: Working Backwards

When you know the end result but not the starting value, work backwards by undoing each step using inverse operations.

Example

Mia thinks of a number. She doubles it, then adds 5. The answer is 19. What was her number?

Start from 19 and undo each step:

Undo "adds 5" → 19 − 5 = 14

Undo "doubles" → 14 ÷ 2 = 7

Check: 7 × 2 + 5 = 14 + 5 = 19. Correct!

Strategy 2: Logical Reasoning

Use clues to eliminate possibilities and narrow down the answer. This works well with "guess and check" and process of elimination.

Example

I am a 2-digit number. I am a multiple of 7. My digits add up to 9. I am less than 50. What am I?

Step 1: Multiples of 7 under 50: 7, 14, 21, 28, 35, 42, 49

Step 2: 2-digit numbers only: 14, 21, 28, 35, 42, 49

Step 3: Digits add up to 9: 1+4=5, 2+1=3, 2+8=10, 3+5=8, 4+2=6, 4+9=13

None work exactly — let’s re-check: 36! 3+6=9 and 36 is less than 50. But 36 ÷ 7 = 5.14... not a multiple of 7. The answer is 45: 4+5=9, but 45 ÷ 7 is not exact either. So the answer is 63... but that is over 50. This shows why checking all clues matters!

Tip: Write down all possibilities first, then cross out ones that do not fit each clue. Be systematic!

Strategy 3: Making a Table

Organising information into a table helps you spot patterns and find answers without guessing.

Example

A farm has chickens and sheep. There are 10 animals and 28 legs. How many of each?

Chickens Sheep Chicken legs Sheep legs Total legs
8216824
73141226
64121628

Answer: 6 chickens and 4 sheep.

Strategy 4: Draw a Diagram

Drawing a picture or diagram makes problems easier to understand, especially for questions about distance, area, or sharing.

The 4-Step Problem Solving Process

1. Understand

Read carefully. What do you know? What do you need to find?

2. Plan

Choose a strategy: table, diagram, work backwards, or guess & check.

3. Solve

Carry out your plan step by step. Show your working.

4. Check

Does your answer make sense? Check by substituting back in.

Key Vocabulary

Inverse Operation

The opposite operation used to undo a step (e.g. subtraction undoes addition).

Eliminate

To rule out answers that do not satisfy the conditions.

Systematic

Working in an organised, step-by-step way.

Verify

To check that your answer is correct by testing it.

Worked Examples

1

I think of a number, multiply by 3, subtract 4. The answer is 20. What was my number?

Work backwards: Start from 20.

Undo −4: 20 + 4 = 24

Undo ×3: 24 ÷ 3 = 8

2

Sam has $5 more than Lily. Together they have $41. How much does each have?

Step 1: Remove the extra $5: $41 − $5 = $36.

Step 2: Divide equally: $36 ÷ 2 = $18.

Answer: Lily has $18, Sam has $18 + $5 = $23.

3

What two numbers multiply to give 36 and add to give 13?

Step 1: List factor pairs of 36: 1×36, 2×18, 3×12, 4×9, 6×6

Step 2: Check sums: 1+36=37, 2+18=20, 3+12=15, 4+9=13

Answer: 4 and 9

Knowledge Check

Select the correct answer for each question.

Question 1

I think of a number, add 8, then double it. The answer is 30. What was my number?

Question 2

Which is the first step when solving a word problem?

Question 3

Two numbers add to 20 and multiply to 96. What are they?

Question 4

A shop sells pens for $3 and notebooks for $5. Aisha spent exactly $29 and bought 7 items. How many pens did she buy?

Question 5

I think of a number, subtract 6, then multiply by 4. The answer is 32. What was my number?

Key Concepts Summary

Number Patterns Mass & Capacity