The Area Model: Grid Multiplication

Today's big idea

Slice one hard multiplication into a four-room rectangle and add the rooms.

24 x 36 splits into 30 + 6 and 20 + 4: rooms 600, 120, 120 and 24 make 864.

How it works

👀 What's going on?

Nobody works out 24 x 36 in their head in one go. There are too many pieces to hold at once, and one of them always slips. But 30 x 20 and 6 x 4 you can already do without thinking. A rectangle is the tool that turns the hard sum into easy ones like those.

💡 Here's the trick

A rectangle is a multiplication. A floor 6 tiles across and 4 tiles down holds 6 x 4 = 24 tiles. You can count them one by one to check. So 24 x 36 is just a much bigger floor, 36 tiles across and 24 tiles down. That is far too many to count, so cut the floor up instead. In the picture above the top edge is cut into 30 + 6, and the left edge into 20 + 4. That makes four rooms. Cutting a floor moves no tiles, so the four rooms together still hold everything the whole floor held. And now every room is a fact you already know. The rooms are 30 x 20 = 600, 6 x 20 = 120, 30 x 4 = 120 and 6 x 4 = 24. Add those four and you get 864.

🪄 How to do it

1.Draw the rectangle and cut each side into its tens and its ones, the way 36 becomes 30 + 6 in the picture above.
2.Multiply the two numbers that meet inside each room, and write that answer in the room.
3.Add the four room answers together.

✍️ Try it

Work out 27 x 15. Cut 27 into 20 + 7 and 15 into 10 + 5, then find what the four rooms come to.

The rooms are 20 x 10 = 200, 7 x 10 = 70, 20 x 5 = 100 and 7 x 5 = 35. Adding all four gives 405. Notice how alike 70 and 100 look once they are on the page. Write each answer inside its own room before you start adding.

⚠️ Watch out

You might be thinking you only need the two big corner rooms, 20 x 10 and 7 x 5. That gives 235, which is far too small. Every room is a real part of the floor with real tiles in it. Leaving a room out means leaving its tiles behind. Two cuts always make four rooms, so look for four every time.

🌟 Why this matters

Working out 24 packets at 36 rupees each is this exact sum. So is working out how many tiles a floor needs. It is also the picture hiding behind the column method you will use for years.
Exam tip: Sketching the four rooms takes ten seconds and catches the slip that loses the mark.

Watch me solve one

Quanta solves it

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

THE PROBLEM

“Solve 32 × 21 using the area model.”

1
Slice the sides into friendly pieces
21 → 20 + 1 (top). 32 → 30 + 2 (side). Now 4 little rooms.
2
Find each room's area
30×20 = 600. 2×20 = 40. 30×1 = 30. 2×1 = 2.
3
Add all 4 rooms
600 + 40 + 30 + 2
4
Total ✓
32 × 21 = 672

Now you try one — same steps, different story 👇

See it in action

⚡ Time to play!

THE PROBLEM

26 × 47

Picture a rectangle 26 tall × 47 wide — its AREA is the answer.

Big multiplications feel hard because your brain juggles too many things at once. The Area Model fixes that — we'll slice this rectangle into 4 easy rooms.

Quick check

✅ VERIFICATION

Using the area model for 23 x 14, what is the area of the room where 20 meets 10?

⚡ Finish the activity above first to answer.

Keep going!

Up next: The Column Method

The column method, with the mystery 0 explained.

Concept 3 of 10 →