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Missing Numbers in Number Sentences

Maths · Y3 Patterns
Year 3
Lesson Guide
Explanation
Here's a big secret about the equals sign: it is NOT a 'here comes the answer' arrow. It's a BALANCE — a seesaw. 7 + 5 = 12 says the left side and the right side weigh exactly the same. That's why 12 = 7 + 5 is also perfectly fine, and so is 7 + 5 = 10 + 2. Balance is everything.
Your Turn
True or false: 8 = 8? And 6 + 4 = 9 + 1? Check: both TRUE — each side balances!
Watch Out
If you think '=' means 'answer next', sentences like 12 = ▢ + 4 will look weird. Think 'balance' and they make perfect sense.
Explanation
A box (or a question mark) is just a number playing hide-and-seek. 7 + ▢ = 15 asks: 7 plus WHAT makes 15? Count up from 7 to 15 — the gap is 8. Or work it with subtraction: 15 − 7 = 8. Either way, ALWAYS check by putting the number back in: 7 + 8 = 15 ✓. The check takes two seconds and catches everything.
Your Turn
Find the hidden numbers: ▢ + 6 = 14 and 9 + ▢ = 20. Check: 8 (14−6) and 11 (20−9).
Watch Out
Don't just add the two numbers you can see! 7 + ▢ = 15 is NOT asking for 7 + 15.
Explanation
Subtraction boxes have two hiding spots and they need different thinking. 12 − ▢ = 4: 'I had 12, now I have 4 — how much was taken?' The gap: 12 − 4 = 8. But ▢ − 5 = 9 asks: 'I lost 5 and have 9 left — what did I start with?' Add it back: 9 + 5 = 14. Think about the story, not just the symbols.
Your Turn
Solve: 20 − ▢ = 13 and ▢ − 8 = 7. Check: 7 (took away), and 15 (started with).
Watch Out
▢ − 5 = 9 is the tricky one — you ADD to find the start, because you're rebuilding what was there before the loss.
Explanation
Boxes appear in × and ÷ sentences too — and your fact-family triangles solve them instantly. ▢ × 4 = 24: the box is a PART, so divide the whole: 24 ÷ 4 = 6. ▢ ÷ 3 = 5: the box is the WHOLE (it got shared out), so rebuild it: 3 × 5 = 15. Missing part → divide; missing whole → multiply.
Your Turn
Solve: 5 × ▢ = 35 and ▢ ÷ 2 = 8. Check: 7 (35 ÷ 5) and 16 (2 × 8).
Watch Out
Sketch the triangle if unsure: biggest number on top. Then cover the box and read what is left.
Explanation
The fanciest puzzles have work on BOTH sides: 6 + ▢ = 10 + 3. Don't panic — the balance idea carries you through. Work out the side you can (10 + 3 = 13), and the puzzle shrinks to 6 + ▢ = 13, which you already know how to crack: the box is 7. Both sides now weigh 13. Balanced! These puzzles star in tests, and you have the exact plan. Quiz and exercises now!
Your Turn
Solve: ▢ + 4 = 9 + 6 and 12 + 3 = ▢ + 5. Check: 11 and 10.
Watch Out
Never answer with the total (13) — the box only holds the PIECE that makes its own side reach the total.
Step 1: Introduction
Step 1 of 5
= means "both sides balance" 7 + 5 12 7 + 5 = 12: both sides weigh the same — balanced! = does NOT mean "the answer comes next". It means "left side and right side are worth the same".
7 + ▢ = 15 7 + ? = 15 Ask: "7 plus WHAT makes 15?" Count up from 7 to 15 → the gap is 8. Or use subtraction: 15 − 7 = 8. Check: 7 + 8 = 15 ✓ Balanced!
Two different hiding spots 12 − ▢ = 4 (what was taken?) Started with 12, ended on 4 → took away 12 − 4 = 8. Check: 12 − 8 = 4 ✓ ▢ − 5 = 9 (what did we start with?) Ended on 9 after losing 5 → started with 9 + 5 = 14. Check: 14 − 5 = 9 ✓ Think about the STORY: what happened, and what is hiding?
Fact families to the rescue ▢ × 4 = 24 "WHAT times 4 makes 24?" Family 4 · 6 · 24 → 24 ÷ 4 = 6. Check: 6 × 4 = 24 ✓ ▢ ÷ 3 = 5 "WHAT shared by 3 gives 5 each?" Rebuild it: 3 × 5 = 15. Check: 15 ÷ 3 = 5 ✓ Same triangle trick as fact families: missing PART → divide · missing WHOLE → multiply
Both sides busy: 6 + ▢ = 10 + 3 6 + ▢ = 10 + 3 Step 1: work out the SIDE YOU CAN — 10 + 3 = 13. Step 2: now it reads 6 + ▢ = 13. Step 3: the gap from 6 to 13 is 7. 6 + 7 = 13 = 10 + 3 ✓ Balanced! NAPLAN loves these — and now you have the plan.
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