Primary & Middle School Mathematics · Grades 1–9

10Multiplication: First Steps

Four boxes of three cakes: rather than $3 + 3 + 3 + 3$ , we write $4 \times 3$ . Multiplication is the shortcut for adding the same number many times — and its best picture is a neat array of rows and columns.

10.1 Adding the same number

Definition 10.1 (Multiplication)

$4 \times 3$ means “four times three”:

4 \times 3 = 3 + 3 + 3 + 3 = 12 .

The sign is $\times$ , read “times”. The answer is the product (multiplication grows up in Chapter 16).

An array: 4 rows of 3. Counting by rows gives 4 × 3; the same dots by columns give 3 × 4 — so the order does not matter. — An *array*: $4$ rows of $3$ . Counting by rows gives $4 \times 3$ ; the same dots by columns give $3 \times 4$ — so the order does not matter.

Example 10.2 (Order does not matter)

$4 \times 3$ and $3 \times 4$ count the same $12$ dots, one by rows, one by columns. Turning the array on its side proves it. So learning one product gives its twin for free.

10.2 Skip counting and the easy tables

Method 10.3 (Building a table by skip counting)

To find $5 \times 4$ , count by fours five times: $4, 8, 12, 16, 20$ . Skip counting on the number line is the table.

Table of $2$ : $2, 4, 6, 8, 10, \dots$ (the doubles);
table of $5$ : $5, 10, 15, 20, 25, \dots$ (ends in $5$ or $0$ );
table of $10$ : $10, 20, 30, \dots$ (just add a zero).

The table of 2 as hops on the line: 2, 4, 6, 8, 10. Five hops of 2 reach 10, so 5 × 2 = 10. — The table of $2$ as hops on the line: $2, 4, 6, 8, 10$ . Five hops of $2$ reach $10$ , so $5 \times 2 = 10$ .

Example 10.4 (Doubles and halves)

Multiplying by $2$ is doubling: $2 \times 7 = 14$ . The reverse, halving, undoes it: half of $14$ is $7$ . Doubles known by heart make the whole table of $2$ instant — and prepare division (Chapter 17).

10.3 Multiplying in problems

Example 10.5

“A box holds $5$ eggs. How many eggs in $4$ boxes?” Four groups of five: $4 \times 5 = 20$ (skip count: $5, 10, 15, 20$ ). There are $20$ eggs.

10.4 Exercises

Exercise 10.1 ★

Write as a multiplication, then compute: $2 + 2 + 2$ ; $5 + 5 + 5 + 5$ ; $10 + 10 + 10$ .

Solution

Solution of Exercise 10.1.

$2 + 2 + 2 = 3 \times 2 = 6$ ; $5 + 5 + 5 + 5 = 4 \times 5 = 20$ ; $10 + 10 + 10 = 3 \times 10 = 30$ .

Exercise 10.2 ★

Draw an array of $3$ rows of $5$ dots. What multiplication does it show? And turned on its side?

Solution

Solution of Exercise 10.2.

$3$ rows of $5$ : $3 \times 5 = 15$ . Turned on its side, $5$ rows of $3$ : $5 \times 3 = 15$ — the same $15$ .

Exercise 10.3 ★

Skip count to fill the table of $2$ up to $2 \times 6$ ; the table of $5$ up to $5 \times 6$ .

Solution

Solution of Exercise 10.3.

Table of $2$ : $2, 4, 6, 8, 10, 12$ . Table of $5$ : $5, 10, 15, 20, 25, 30$ .

Exercise 10.4 ★

Compute: $2 \times 7$ ; $5 \times 3$ ; $10 \times 4$ ; $2 \times 9$ ; $5 \times 6$ .

Solution

Solution of Exercise 10.4.

$2 \times 7 = 14$ ; $5 \times 3 = 15$ ; $10 \times 4 = 40$ ; $2 \times 9 = 18$ ; $5 \times 6 = 30$ .

Exercise 10.5 ★

Compute the doubles: double of $6$ ; double of $8$ ; double of $10$ . Then the halves: half of $12$ ; half of $20$ .

Solution

Solution of Exercise 10.5.

Doubles: $12$ ; $16$ ; $20$ . Halves: half of $12$ is $6$ ; half of $20$ is $10$ .

Exercise 10.6 ★

Compute the table of $3$ by skip counting: $3, 6, 9, ?, ?, ?$ . What is $3 \times 5$ ?

Solution

Solution of Exercise 10.6.

$3, 6, 9, 12, 15, 18$ . So $3 \times 5 = 15$ .

Exercise 10.7 ★

“There are $4$ tables with $5$ chairs each. How many chairs?” Write the multiplication and answer.

Solution

Solution of Exercise 10.7.

$4 \times 5 = 20$ chairs.

Exercise 10.8 ★

A spider has $8$ legs. How many legs have $3$ spiders? (Skip count by $8$ : $8, 16, \dots$ )

Solution

Solution of Exercise 10.8.

$3 \times 8 = 24$ legs ( $8, 16, 24$ ).

Exercise 10.9 ★

Two ways to arrange $12$ dots in equal rows: draw $2$ rows of $6$ , and $3$ rows of $4$ . Write the multiplication for each.

Solution

Solution of Exercise 10.9.

$2$ rows of $6$ : $2 \times 6 = 12$ . $3$ rows of $4$ : $3 \times 4 = 12$ .

Exercise 10.10 ★★

Cakes come in boxes of $5$ . Mia needs $18$ cakes for a party. How many boxes must she buy? (Skip count by $5$ until you pass $18$ .)

Solution

Solution of Exercise 10.10.

Skip count by $5$ : $5, 10, 15, 20$ — three boxes give only $15$ (too few), four boxes give $20$ . Mia must buy $4$ boxes.