Primary & Middle School Mathematics · Grades 1–9

16Multiplication

Three bags of seven marbles: you could add $7 + 7 + 7$ , but there is a faster way — multiplication, the operation of repeated addition. This chapter builds the multiplication tables from pictures, then teaches the column method.

16.1 What multiplying means

Definition 16.1 (Multiplication)

The product $3 \times 7$ means “three times seven”:

3 \times 7 = 7 + 7 + 7 = 21 .

On a picture, $3 \times 7$ is a rectangle of dots: $3$ rows of $7$ dots.

The same 21 dots, counted by rows or by columns: the order of a product does not matter. Knowing 3 × 7 gives 7 × 3 for free — half the tables to learn! — The same $21$ dots, counted by rows or by columns: the order of a product does not matter. Knowing $3 \times 7$ gives $7 \times 3$ for free — half the tables to learn!

Proposition 16.2 (Rules that help)

The order does not matter: $a \times b = b \times a$ .
Multiplying by $1$ changes nothing; multiplying by $0$ gives $0$ .
Multiplying by $10$ appends a zero: $34 \times 10 = 340$ ; by $100$ , two zeros.
A table can be rebuilt from its neighbors: $6 \times 8 = 5 \times 8 + 8 = 40 + 8 = 48$ .

Proof. Admitted at this level. ∎

Example 16.3 (The table square)

All the tables fit in one square. To read $6 \times 7$ : row $6$ , column $7$ .

$\times$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$
$1$	1	2	3	4	5	6	7	8	9
$2$	2	4	6	8	10	12	14	16	18
$3$	3	6	9	12	15	18	21	24	27
$4$	4	8	12	16	20	24	28	32	36
$5$	5	10	15	20	25	30	35	40	45
$6$	6	12	18	24	30	36	42	48	54
$7$	7	14	21	28	35	42	49	56	63
$8$	8	16	24	32	40	48	56	64	72
$9$	9	18	27	36	45	54	63	72	81

The square is symmetric across its diagonal — that is rule 1 made visible.

16.2 Column multiplication

Method 16.4 (Multiplying by a one-digit number)

To compute $234 \times 6$ :

multiply the units: $4 \times 6 = 24$ : write $4$ , carry $2$ ;
multiply the tens and add the carry: $3 \times 6 + 2 = 20$ : write $0$ , carry $2$ ;
multiply the hundreds and add the carry: $2 \times 6 + 2 = 14$ : write $14$ .

\begin{array}{r} 2\,3\,4 \\ \times \quad 6 \\ \hline 1\,4\,0\,4 \end{array}

Estimate to check: $234$ is close to $200$ , and $200 \times 6 = 1\,200$ : the answer $1\,404$ is plausible.

Example 16.5 (Splitting to multiply mentally)

The column method secretly splits the number by places. Mentally, do the same:

23 \times 4 = (20 + 3) \times 4 = 20 \times 4 + 3 \times 4 = 80 + 12 = 92 .

And with a friendly neighbor: $99 \times 5 = 100 \times 5 - 5 = 495$ .

Why splitting works: a 23 × 4 rectangle of unit squares is a 20 × 4 part plus a 3 × 4 part — 80 + 12 = 92 squares. — Why splitting works: a $23 \times 4$ rectangle of unit squares is a $20 \times 4$ part plus a $3 \times 4$ part — $80 + 12 = 92$ squares.

16.3 Multiplication in problems

Example 16.6

A minibus carries $9$ passengers. How many passengers can $27$ minibuses carry?

It is a multiplication: $27$ groups of $9$ .
$27 \times 9 = 243$ (columns, or $27 \times 9 = 27 \times 10 - 27 = 270 - 27$ ).
Sentence: the minibuses can carry $243$ passengers.

16.4 Exercises

Exercise 16.1 ★

Write as a multiplication, then compute: $8 + 8 + 8 + 8$ ; $5 + 5 + 5$ ; $20 + 20 + 20 + 20 + 20$ .

Solution

Solution of Exercise 16.1.

$8 + 8 + 8 + 8 = 4 \times 8 = 32$ ; $5 + 5 + 5 = 3 \times 5 = 15$ ; $20 \times 5 = 100$ .

Exercise 16.2 ★

Recite and complete: $6 \times 7$ ; $8 \times 4$ ; $9 \times 9$ ; $7 \times 8$ ; $6 \times 6$ .

Solution

Solution of Exercise 16.2.

$42$ ; $32$ ; $81$ ; $56$ ; $36$ .

Exercise 16.3 ★

Draw a dot rectangle for $4 \times 6$ , then one for $6 \times 4$ . What do you notice about the two counts?

Solution

Solution of Exercise 16.3.

Both rectangles contain the same $24$ dots — one is the other turned on its side: $4 \times 6 = 6 \times 4$ .

Exercise 16.4 ★

Compute using the rules of Proposition 16.2: $56 \times 10$ ; $8 \times 100$ ; $73 \times 0$ ; $45 \times 1$ ; $30 \times 10$ .

Solution

Solution of Exercise 16.4.

$560$ ; $800$ ; $0$ ; $45$ ; $300$ .

Exercise 16.5 ★

Compute in columns: $132 \times 3$ ; $217 \times 4$ ; $408 \times 7$ .

Solution

Solution of Exercise 16.5.

$132 \times 3 = 396$ ; $217 \times 4 = 868$ ; $408 \times 7 = 2\,856$ .

Exercise 16.6 ★

Compute mentally by splitting (as in Example 16.5): $31 \times 5$ ; $42 \times 3$ ; $25 \times 6$ .

Solution

Solution of Exercise 16.6.

$31 \times 5 = 30 \times 5 + 5 = 155$ ; $42 \times 3 = 120 + 6 = 126$ ; $25 \times 6 = 20 \times 6 + 5 \times 6 = 120 + 30 = 150$ .

Exercise 16.7 ★

Compute with a friendly neighbor: $99 \times 7$ ; $101 \times 6$ ; $98 \times 5$ .

Solution

Solution of Exercise 16.7.

$99 \times 7 = 700 - 7 = 693$ ; $101 \times 6 = 600 + 6 = 606$ ; $98 \times 5 = 500 - 10 = 490$ .

Exercise 16.8 ★

Estimate first, then compute: $389 \times 5$ (estimate with $400$ ); $612 \times 8$ (estimate with $600$ ).

Solution

Solution of Exercise 16.8.

$389 \times 5$ : estimate $400 \times 5 = 2\,000$ ; exactly $1\,945$ .

$612 \times 8$ : estimate $600 \times 8 = 4\,800$ ; exactly $4\,896$ .

Exercise 16.9 ★

An egg box holds $6$ eggs. How many eggs are in $38$ boxes? Write the operation, compute, answer with a sentence.

Solution

Solution of Exercise 16.9.

$38 \times 6 = 228$ . There are $228$ eggs in the boxes.

Exercise 16.10 ★★

A classroom has $8$ rows of $4$ desks; each desk seats $2$ students. How many students can the room seat? (Two multiplications.)

Solution

Solution of Exercise 16.10.

Desks: $8 \times 4 = 32$ . Students: $32 \times 2 = 64$ . The room can seat $64$ students.

Exercise 16.11 ★★

Find all the ways of arranging $24$ chairs into equal rows (rows of $1$ , of $2$ , …). Which arrangements are possible? (Use the table square: look for $24$ inside it.)

Solution

Solution of Exercise 16.11.

$24$ appears in the table square as $1 \times 24$ , $2 \times 12$ , $3 \times 8$ , $4 \times 6$ — and the same pairs reversed. Possible arrangements: $1$ row of $24$ , $2$ rows of $12$ , $3$ rows of $8$ , $4$ rows of $6$ , $6$ rows of $4$ , $8$ rows of $3$ , $12$ rows of $2$ , $24$ rows of $1$ .

Exercise 16.12 ★★

Zoe computes $47 \times 6$ like this: $40 \times 6 = 240$ , $7 \times 6 = 42$ , and then $240 + 42 = 282$ . Léa computes $47 \times 6 = 50 \times 6 - 3 \times 6 = 300 - 18 = 282$ . Explain each method. Which do you prefer for $38 \times 4$ ? Compute it both ways.

Solution

Solution of Exercise 16.12.

Zoe splits $47$ into $40 + 7$ (the column method in her head); Léa replaces $47$ by the friendly $50$ and corrects by taking away the $3 \times 6$ counted too much. For $38 \times 4$ : Zoe’s way, $120 + 32 = 152$ ; Léa’s way, $40 \times 4 - 2 \times 4 = 160 - 8 = 152$ . Both work — Léa’s is quicker here because $38$ is close to $40$ .

$\times$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$
$1$	1	2	3	4	5	6	7	8	9
$2$	2	4	6	8	10	12	14	16	18
$3$	3	6	9	12	15	18	21	24	27
$4$	4	8	12	16	20	24	28	32	36
$5$	5	10	15	20	25	30	35	40	45
$6$	6	12	18	24	30	36	42	48	54
$7$	7	14	21	28	35	42	49	56	63
$8$	8	16	24	32	40	48	56	64	72
$9$	9	18	27	36	45	54	63	72	81

$\times$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$
$1$	1	2	3	4	5	6	7	8	9
$2$	2	4	6	8	10	12	14	16	18
$3$	3	6	9	12	15	18	21	24	27
$4$	4	8	12	16	20	24	28	32	36
$5$	5	10	15	20	25	30	35	40	45
$6$	6	12	18	24	30	36	42	48	54
$7$	7	14	21	28	35	42	49	56	63
$8$	8	16	24	32	40	48	56	64	72
$9$	9	18	27	36	45	54	63	72	81

$\times$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$
$1$	1	2	3	4	5	6	7	8	9
$2$	2	4	6	8	10	12	14	16	18
$3$	3	6	9	12	15	18	21	24	27
$4$	4	8	12	16	20	24	28	32	36
$5$	5	10	15	20	25	30	35	40	45
$6$	6	12	18	24	30	36	42	48	54
$7$	7	14	21	28	35	42	49	56	63
$8$	8	16	24	32	40	48	56	64	72
$9$	9	18	27	36	45	54	63	72	81