Math Detours · Grade 5 · Sharpen · 3

Why long multiplication works

23 × 47 is the area of a rectangle cut into four pieces. The method you were taught is those four pieces, added up.

Name ____________________Date ____________

Tonight's idea

The area of a rectangle is its length times its width. So \(23 \times 47\) is the area of a rectangle 23 tall and 47 wide.

Cut it to make it easy. Split 47 into 40 and 7. Split 23 into 20 and 3. Now there are four easy pieces.

407 203 20 × 40 = 800 140 3 × 40 = 120 21 47 × 23 141 = 120 + 21 940 = 800 + 140 1081
23 × 47 as an area, beside the long multiplication.

The four pieces are 800, 140, 120 and 21. Together they make the whole rectangle: \(800 + 140 + 120 + 21 = 1081\).

Now look at the long multiplication beside it. The first line is \(47 \times 3 = 141\). That is the bottom strip: \(120 + 21\). The second line is \(47 \times 20 = 940\). That is the top strip: \(800 + 140\). Then you add the two strips.

Long multiplication is the four pieces of the rectangle, added two at a time.

Watch one: \(36 \times 54\)

Split 36 into 30 and 6. Split 54 into 50 and 4. The four pieces are

\[30 \times 50 = 1500, \qquad 30 \times 4 = 120, \qquad 6 \times 50 = 300, \qquad 6 \times 4 = 24.\]

Add them: \(1500 + 120 + 300 + 24 = 1944\).

Long multiplication gives the same pieces in two lines. \(54 \times 6 = 324\), which is \(300 + 24\). \(54 \times 30 = 1620\), which is \(1500 + 120\). And \(324 + 1620 = 1944\).

Pick any two numbers from 10 to 99 and see the rectangle and the working side by side: mathdetours.com/demos/areamodel.html.

Your turn

  1. Sketch a rectangle for \(14 \times 12\). Cut it into four pieces, write the area in each piece, and add them.
  2. Find the four pieces for \(38 \times 26\) and add them. A rough sketch helps. It does not need to be the right size.
  3. The long multiplication for \(62 \times 45\) has the two lines 310 and 2480. (a) Which two pieces add to 310? (b) Which two pieces add to 2480? (c) What is \(62 \times 45\)?
  4. Jo works out \(23 \times 47\) like this: "\(20 \times 40 = 800\) and \(3 \times 7 = 21\), so the answer is 821." (a) Which pieces of the rectangle did Jo leave out? (b) How much too small is Jo's answer?

Stretch: a bigger rectangle

Cut a rectangle for \(123 \times 45\) into hundreds, tens and ones. How many pieces are there? Find them all and add. How many pieces would \(123 \times 456\) need?

Answers

  1. \(10 \times 10 = 100\), \(10 \times 2 = 20\), \(4 \times 10 = 40\), \(4 \times 2 = 8\). Total 168.
  2. \(30 \times 20 = 600\), \(30 \times 6 = 180\), \(8 \times 20 = 160\), \(8 \times 6 = 48\). Total 988.
  3. (a) \(62 \times 5\) is \(60 \times 5 = 300\) and \(2 \times 5 = 10\). (b) \(62 \times 40\) is \(60 \times 40 = 2400\) and \(2 \times 40 = 80\). (c) \(310 + 2480 = 2790\).
  4. (a) The two pieces off the corners: \(20 \times 7 = 140\) and \(3 \times 40 = 120\). (b) \(140 + 120 = 260\) too small. Check: \(821 + 260 = 1081\).

Stretch. Three parts across (100, 20, 3) and two down (40, 5) make \(3 \times 2 = 6\) pieces: 4000, 800, 120, 500, 100 and 15. Total 5535. For \(123 \times 456\) there are \(3 \times 3 = 9\) pieces.

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