---
title: "Multiplication: First Steps"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 10
exercises: 10
source: https://one-course.com/books/math/1/en/chapter/10-multiplication-first-steps
---

# Chapter 10 — Multiplication: 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](https://one-course.com/books/math/1/en/chapter/16-multiplication#ch-g3-mult)).

![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.](https://one-course.com/images/onecourse/chapters/math-1/g2-mult/fig-19a815f5b5e6.svg)

*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](#def-g2-mult-def) 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](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-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.](https://one-course.com/images/onecourse/chapters/math-1/g2-mult/fig-7684467f1666.svg)

*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](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#ch-g3-division)).

## 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 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 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 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 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 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 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 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 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 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 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.