Multiplying Bigger Numbers

The idea in one sentence

Nobody knows \(27 \times 34\) by heart, so you split both numbers by place value and multiply the pieces you do know.

The area model

Draw a rectangle 27 wide and 34 tall. Its area is the answer. Now cut it: 27 into \(20 + 7\), and 34 into \(30 + 4\).

That makes four smaller rectangles:

Add all four: \(600 + 210 + 80 + 28 = 918\).

Every one of those four is a times-table fact with zeros attached. \(30 \times 20\) is \(3 \times 2\) with two zeros. The hard problem became four easy ones plus an addition.

Why splitting is allowed

Because multiplication distributes over addition:

You have to multiply every part of the first by every part of the second, which is why there are four products and not two. Multiplying only the tens and only the ones — the \(600 + 28\) mistake — misses two whole rectangles and lands far short.

That same rule reappears years later as expanding brackets in algebra: \((x + 7)(x + 4)\) produces four terms for exactly the same reason. The grid you draw for \(27 \times 34\) is the grid you’ll draw for quadratics.

The column method

The traditional layout does the same four products, just compressed:

    27
  × 34
  ----
   108   ← 27 × 4
   810   ← 27 × 30
  ----
   918

The 108 is \(7 \times 4\) and \(20 \times 4\) combined; the 810 is the other two. The zero on the second line is the whole trick — you’re multiplying by 30, not 3, so the answer must be ten times bigger. Forgetting it is the single most common error in long multiplication, and it produces an answer roughly a quarter of the right size.

Use whichever method you like, but be able to estimate either way: \(27 \times 34\) is about \(30 \times 30 = 900\), so 918 is believable and 198 obviously isn’t.

Common mix-ups

• Missing the zero on the second row.
• Multiplying only the matching digits. All four products, every time.
• Losing a carry in the addition at the end.
• Skipping the estimate. A quick round-and-multiply catches place-value disasters.

Check yourself

Work out \(43 \times 26\) with the area model, then estimate to check.

\(40 \times 20 = 800\), \(40 \times 6 = 240\), \(3 \times 20 = 60\), \(3 \times 6 = 18\), totalling 1,118. The estimate \(40 \times 25 = 1000\) says that’s the right size.