---
title: "Multiplication"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 16
exercises: 12
source: https://one-course.com/books/math/1/en/chapter/16-multiplication
---

# Chapter 16 — Multiplication

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](https://one-course.com/books/math/1/en/chapter/10-multiplication-first-steps#def-g2-mult-def) $3 \times 7$ 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!](https://one-course.com/images/onecourse/chapters/math-1/g3-mult/fig-cdd052c1de80.svg)

*The same $21$ dots, counted by rows or by columns: the order of a [product](https://one-course.com/books/math/1/en/chapter/10-multiplication-first-steps#def-g2-mult-def) does not matter. Knowing $3 \times 7$ gives $7 \times 3$ for free — half the tables to learn!*

**Proposition 16.2 (Rules that help).**

1. The order does not matter: $a \times b = b \times a$ .
2. Multiplying by $1$ changes nothing; multiplying by $0$ gives $0$ .
3. Multiplying by $10$ appends a [zero](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-zero) : $34 \times 10 = 340$ ; by $100$ , two [zeros](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-zero) .
4. 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 $234 \times 6$:

1. multiply the units: $4 \times 6 = 24$ : write $4$ , carry $2$ ;
2. multiply the tens and add the carry: $3 \times 6 + 2 = 20$ : write $0$ , carry $2$ ;
3. 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.](https://one-course.com/images/onecourse/chapters/math-1/g3-mult/fig-6ebeb59e65b5.svg)

*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?

1. It is a multiplication: $27$ groups of $9$ .
2. $27 \times 9 = 243$ (columns, or $27 \times 9 = 27 \times 10 - 27 = 270 - 27$ ).
3. 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 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 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 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](#prop-g3-mult-rules): $56 \times 10$; $8 \times 100$; $73 \times 0$; $45 \times 1$; $30 \times 10$.

**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 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](#ex-g3-mult-split)): $31 \times 5$; $42 \times 3$; $25 \times 6$.

**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 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 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 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 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 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 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 $40$.