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Concept 5 of 8

Symbol Equations: Shapes Stand for Numbers

1

Today's big idea

Solve the line that shows only one shape first, then substitute its value into the rest.

Same shape means same number, so one solved value unlocks every other line.

2

How it works

👀 What's going on?

A star or a circle can stand in for a number you do not know yet. Two or three short lines are enough to pin that number down. This is algebra wearing a costume, and you can already do it.

💡 Here's the trick

Solve the easy one first★ + ★ = 8⟵ ONE shape — start here↓ solve: ★ = 4★ + △ = 10↓ substitute ★ = 44 + △ = 10 → △ = 6Same shape = same number, every time

Look at the boxes above. The top line holds only one kind of shape, so it gives up its value at once. Once a shape has a value, it keeps that value in every line below.

🪄 How to do it

  1. 1.Hunt for the line in the boxes above that holds one shape only.
  2. 2.Share that total out to find what a single shape is worth.
  3. 3.Put that value into the next line and solve for the new shape.

✍️ Try it

■ + ■ + ■ + ■ = 20. What is ■ worth? Then, if ■ + ▲ = 13, what is ▲?

Four of the same shape make 20, so ■ is 20 shared by 4, which is 5. Then 5 + ▲ = 13, so ▲ must be 8.

⚠️ Watch out

You might be thinking you can guess a pair that makes the first line work. Try ★ + ● = 10 on its own and you could pick 1 and 9, or 4 and 6. A line with two shapes has many answers, so it can never be your starting point. Begin where only one shape appears.

🌟 Why this matters

Shopkeepers, cooks and builders all work out an unknown amount from what they already know. You do it too when you split a bill or work back from a total.
Exam tip: Circle the line with one shape in it before you read anything else.

3

Watch me solve one

Quanta solves it

“Watch how I think through one of these — then you try.”

THE PROBLEM

“★ + ★ + ★ = 21 and ★ + ● = 12. Find ●.”

  1. 1

    Solve the single-shape equation

    3★ = 21 → ★ = 7.

  2. 2

    Substitute

    7 + ● = 12

  3. 3

    Solve for ●

    ● = 12 − 7 = 5.

  4. 4

    Answer ✓

    ● = 5

Now you try one — same steps, different story 👇

4

See it in action

⚡ Time to play!

Solve the equation with only ● first, then put its value into the other one.

SYMBOL EQUATION · ROUND 1 OF 3

● + ● = 10

● + ■ = 13

5

Quick check

✅ VERIFICATION

If ◆ + ◆ = 14 and ◆ + ★ = 20, what is ★?

⚡ Finish the activity above first to answer.

Keep going!

Up next: Letter Arrangement

Category + all letters used once → one real word.

Concept 6 of 8 →