Math Core

A store orders 423 boxes of markers, and each box holds 36 markers. That is too many to count, so you multiply. In 4th grade you multiplied two-digit numbers. Now you will use the same steps with bigger numbers, so you can multiply any whole numbers you meet.

Start with place value

Every multi-digit product is built from smaller products. Multiplying by tens or hundreds is the first step.

$423 \times 6 = 2{,}538$
$423 \times 30 = 12{,}690$ , because $423 \times 3 = 1{,}269$ and multiplying by 10 adds a zero.
$423 \times 200 = 84{,}600$ , because $423 \times 2 = 846$ and multiplying by 100 adds two zeros.

The area model

Split the numbers by place value. Multiply each part of one number by each part of the other. These answers are called partial products.

Worked example: An area model

Find $423 \times 36$ .

Write $423 = 400 + 20 + 3$ and $36 = 30 + 6$ .

	400	20	3
30	$12{,}000$	$600$	$90$
6	$2{,}400$	$120$	$18$

Add the top row: $12{,}000 + 600 + 90 = 12{,}690$ . Add the bottom row: $2{,}400 + 120 + 18 = 2{,}538$ . Add the rows: $12{,}690 + 2{,}538 = 15{,}228$ .

The store orders 15,228 markers.

The standard algorithm

The standard algorithm is a short way to write the area model. You make one row for each digit of the bottom number.

Multiplying with the standard algorithm

Multiply the top number by the ones digit. Write that row.
Multiply by the tens digit. Write one 0 in the ones place first, because you are really multiplying by tens.
If there is a hundreds digit, multiply by it. Write two 0s first.
Add all the rows.

Worked example: Four digits times two digits

Find $2{,}457 \times 38$ .

\begin{array}{r} 2457 \\ \times\,38 \\ \hline 19656 \\ +\,73710 \\ \hline 93366 \end{array}

Row 1: $2{,}457 \times 8 = 19{,}656$ .
Row 2: $2{,}457 \times 30 = 73{,}710$ . Write the 0 first, then $2{,}457 \times 3 = 7{,}371$ in front of it.
Add: $19{,}656 + 73{,}710 = 93{,}366$ .

So $2{,}457 \times 38 = 93{,}366$ .

Common mistake

Clear out your regrouped digits before each new row. The little numbers you carried while multiplying by 8 belong only to Row 1. If you reuse them in Row 2, every digit after that will be off.

Worked example: Three digits times three digits

Find $245 \times 132$ .

\begin{array}{r} 245 \\ \times\,132 \\ \hline 490 \\ 7350 \\ +\,24500 \\ \hline 32340 \end{array}

Row 1: $245 \times 2 = 490$ .
Row 2: $245 \times 30 = 7{,}350$ .
Row 3: $245 \times 100 = 24{,}500$ .
Add: $490 + 7{,}350 + 24{,}500 = 32{,}340$ .

Check with an estimate

Round each factor to a number that is easy to multiply. Your exact answer should be close to the estimate.

For $2{,}457 \times 38$ , round to $2{,}500 \times 40 = 100{,}000$ . The answer 93,366 is close to 100,000, so it makes sense. If you had gotten 9,336, the estimate would tell you something went wrong.

Tip

When the bottom number has a 0 digit, like 205, you can skip that row. The zero row would be all zeros. Just remember to shift the next row: $312 \times 205$ is $312 \times 5$ plus $312 \times 200$ .

Practice

Practice 1

Find $300 \times 40$ .

Type a number.

Practice 2

Find $214 \times 13$ .

Type a number.

Practice 3

Find $526 \times 34$ .

Type a number.

Practice 4

Which is the best estimate for $389 \times 52$ ?

Practice 5

A bakery makes 175 muffins every day. How many muffins does it make in 28 days?

Type a number.

Practice 6

Find $3{,}046 \times 27$ .

Type a number.

Practice 7

Maya found $608 \times 45$ . Her first row was $3{,}040$ . For her second row she wrote $2{,}432$ . Then she added. What is the correct product?

Type a number.

Practice 8

Find $312 \times 205$ .

Type a number.