Concept 5 of 7
The Missing Piece: Algebraic Detectives
Today's big idea
Ask whether the whole or a part is missing, then add or subtract to match.
When things were taken from the mystery number, it is the whole, so add them back.
How it works
👀 What's going on?
Your friend asks: 'I started with 10,000 candies. After my party I had 6,000 left. How many did I share?' Nobody has told you the answer, and there is no sum written down. That is what makes these feel harder than they are. Mark the hidden number with an X, then work like a detective.
💡 Here's the trick
Look at the picture above. A missing-piece puzzle is a balance. Both sides of the = sign hold the same amount. So the whole is always the parts added together, and every piece you know brings you closer to the one you do not.
🪄 How to do it
- 1.Draw the whole and its parts, the way the picture above does. Mark the hidden one X.
- 2.Ask which one is missing: the whole, or one of the parts?
- 3.If a part is missing, subtract. If the whole is missing, add the parts back together.
✍️ Try it
Solve 10,000 − X = 6,000. Then solve X − 3,000 = 5,000. They look almost the same. Do you use the same move for both?
In the first, 10,000 is the whole and X is a part, so subtract: X = 4,000. In the second, X is the whole, because things were taken from it. So add the parts back: 3,000 + 5,000, giving X = 8,000.
⚠️ Watch out
You might be thinking you can always find the gap between the two numbers. That works only when the whole is one of them. In X − 3,000 = 5,000 the gap is 2,000, and 2,000 is not the answer. Always ask what is missing before you choose a move.
🌟 Why this matters
Every time you work out how much more you need to save, you are finding a missing piece. So is working out how many days are left until a trip. It is also the first real step into algebra.
Exam tip: Costs, scores and time-left questions are missing-piece problems in disguise.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“9,000 + ? = 21,000 — find the mystery piece.”
- 1
Spot what you know
Total is 21,000. One piece is 9,000. The other piece is hiding. 🕵️
- 2
Find the gap
Use the opposite operation — subtract.
- 3
Do the subtraction
21,000 − 9,000 = 12,000
- 4
Mystery piece found ✓
? = 12,000 Solved!
Now you try one — same steps, different story 👇
See it in action
⚡ Time to play!
The [?] is a mystery weight. Drag it until both sides of the scale balance.
PUZZLE 1 OF 2
Left side too light — drag more →
Left pan: 6,000 · Right pan: 10,000
Mystery weight: 0
Quick check
✅ VERIFICATION
Solve X − 2,000 = 7,000.
⚡ Finish the activity above first to answer.
Keep going!
Up next: The Rewind Button
Use the opposite operation to prove your answer is right.