5.3 Number Sentences & Missing Values

Paper D (Year 6) lists 'number sentences with all four operations' among its tested skills, and these questions are the doorway to the algebra that dominates Papers E and F. A number sentence is simply an equation written with numbers, like 7 + 5 = 12. ICAS turns it into a problem by hiding something — a number, or even the operation itself — and asking you to find it. No calculator appears for these questions, so everything below is built on mental arithmetic and inverse operations.

The Equals Sign Is a Balance

The single most important idea: = means 'has the same value as', not 'the answer is'. The sentence 8 + 4 = 5 + 7 is true because both sides equal 12. Think of a balance scale — if both pans hold the same weight, the scale is level.

This balance view matters because ICAS writes questions like:

6 + 9 = □ + 8

Many students write 15 in the box because 6 + 9 = 15 — but then the right side is 15 + 8 = 23 and the scale tips. The correct approach: the left side is 15, so the right side must also be 15, meaning □ = 15 − 8 = 7. The wrong answer 15 is almost always one of the multiple-choice options — it is the classic distractor.

Inverse Operations

Inverse operations undo each other:

Worked example (Paper B–C level): □ − 17 = 25. Addition undoes subtraction: □ = 25 + 17 = 42. Check: 42 − 17 = 25 ✓. Always substitute your answer back — it takes five seconds and catches slips.

Worked example (Paper D level): 84 ÷ □ = 12. This one needs care because the unknown is the divisor. Ask: '84 divided by what gives 12?' Use the multiplication fact: 12 × □ = 84, and since 12 × 7 = 84, □ = 7. Turning a division into the related multiplication fact is the reliable move.

Balancing Both Sides

When a sentence has working on both sides, do the same thing to each side to keep it balanced.

3 × □ = 9 + 12

1. Work out the complete side: 9 + 12 = 21.
2. Now solve 3 × □ = 21 with the inverse: □ = 21 ÷ 3 = 7.

□ + 14 = 50 − 8

1. Right side: 50 − 8 = 42.
2. □ = 42 − 14 = 28.

The habit is: simplify whichever side you can, then use one inverse operation. Trying to do both at once is where errors creep in.

Missing Operations

Sometimes the missing thing is the operation sign:

24 ◯ 6 = 18

Test each operation mentally: 24 + 6 = 30 ✗, 24 − 6 = 18 ✓. The answer is −. Testing in the order +, −, ×, ÷ is quick because each test is a single mental calculation. For a harder one like 7 ◯ 8 = 56, addition and subtraction fail fast, and 7 × 8 = 56 ✓ — knowing times tables cold is what makes these questions nearly free marks.

Symbols as Unknowns — the Bridge to Algebra

In later papers the box becomes a letter or shape, but the thinking is identical:

• □ + 9 = 23 is the same problem as x + 9 = 23 → x = 14
• 4 × △ = 36 is 4y = 36 → y = 9
• If ★ + ★ + ★ = 24, then ★ = 8 (three equal parts of 24)

Papers E–F extend this to linear equations with the unknown on both sides, like 5x − 3 = 2x + 9. The balance idea still rules: subtract 2x from both sides to get 3x − 3 = 9, add 3 to both sides to get 3x = 12, then divide by 3: x = 4. Every step is the same 'keep the scale level' logic from the box problems.

Three Difficulty Levels, One Method

Note the Paper D trap: when the unknown follows a subtraction, students often add instead of subtracting. Say it in words — '45 take away something leaves 18' — and the something is clearly 45 − 18.

Typical Distractor Traps

• The unfinished side: answering 15 for 6 + 9 = □ + 8 (the correct value is 7).
• The wrong inverse: adding when you should subtract, as in 45 − □ = 18.
• The reversed division: for □ ÷ 4 = 9, answering 36 ÷ 4 style slips — the answer is 9 × 4 = 36, and 36 ÷ 4 = 9 is the check, not the method confusion.
• Partial balancing: in 2 × □ + 3 = 17, forgetting the order — first subtract 3 (2 × □ = 14), then divide (□ = 7). Undo operations in reverse order, exactly like a function machine run backwards.

Since these questions are designed for mental arithmetic, estimation is your safety net: if □ × 8 = 240, the answer must be around 30 (because 3 × 8 = 24), so options like 3 or 300 can be eliminated instantly.