---
title: "Multiplying and dividing integers"
subtitle: "Number Foundations · Integers · Lesson 3"
---

# Multiplying and dividing integers

**Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}}

> **Key question:** A tank loses 3 litres an hour. Three hours ago, did it hold more or less than now? A negative rate times a negative time: the answer is a positive change.


:::teacher-note

**In one sentence:** Students find the sign rules for multiplying and dividing integers from the pattern a × (−2) and from jumps on a number line (a negative number of jumps goes the opposite way), and apply them, including powers of −1 (AC9M8N04, node 1).

**Students should be able to say:** I can multiply and divide integers, including two negatives. I can explain why a negative times a negative is positive, from a pattern or the number line. I can tell the rule for multiplying apart from the rule for adding negatives.

**Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet.

:::


## Predict

Think about the pattern $3 \times (-2)$, $2 \times (-2)$, $1 \times (-2)$, $0 \times (-2)$. What happens each time?

:::keep-together

**What is $-3 \times (-2)$?**

☐ $-5$

☐ $1$

☐ $-6$

☐ $6$

My prediction, and why:

{{lines-3}}

:::


:::teacher-note

**What happens:** 6. 3 jumps of −2 would go left to −6; a negative number of jumps goes the opposite way, to 6. The pattern 3 × (−2) = −6, 2 × (−2) = −4, 1 × (−2) = −2, 0 × (−2) = 0 goes up by 2 each time, so −1 × (−2) = 2 and −3 × (−2) = 6.

:::


## Run

Try it on the board in the tutorial. Was your prediction right? What was different?

{{lines-2}}

## The sign rules


:::keep-together

### The pattern

**Fill in the pattern for $a \times (-2)$.**

{{rows 1}}

| a | 3 | 2 | 1 | 0 | −1 | −2 | −3 |
|---|---|---|---|---|---|---|---|
| a × (−2) | −6 | −4 | −2 |   |   |   |   |

:::


:::teacher-note

**Answer:** 0, 2, 4, 6

Each step down in a adds 2 to the answer, so past zero the answers are positive: −3 × (−2) = 6.

:::


:::keep-together

### Signs from the pattern

Here is how the sign rules come from the pattern and the line.

1. $3 \times (-2) = -6$ — 3 jumps of $-2$: left 6.
2. $-3 \times (-2) = 6$ — A negative number of jumps goes the opposite way: right 6.
3. $\text{same signs} \to +$ — $3 \times 2 = 6$ and $-3 \times (-2) = 6$.
4. $\text{different signs} \to -$ — $3 \times (-2) = -6$ and $-3 \times 2 = -6$.
5. $-12 \div 3 = -4$ — Dividing undoes multiplying: $3 \times (-4) = -12$, so the same rule holds.
6. $-12 \div (-3) = 4$ — Same signs, positive: check $-3 \times 4 = -12$.

Then: which is $-12 \div 3$?

**Which is $-12 \div 3$?**

☐ $-4$

☐ $-9$

☐ $4$

:::


:::teacher-note

**Answer:** $-4$

−12 ÷ 3 = −4, because 3 × (−4) = −12. Different signs give a negative.

:::


:::keep-together

### Find the mistake

**Kai works out $-4 \times (-3)$. Click the line where his reasoning goes wrong.**

```
 1  Work out −4 × −3.
 2  4 × 3 = 12.
 3  Both numbers are negative, so the answer is negative, like −4 + −3.
 4  So −4 × −3 = −12.
```

Line: {{field-15}}

Why is it wrong?

{{lines-2}}

:::


:::teacher-note

**Answer:** line 3

Line 3: that is the rule for adding (−4 + −3 = −7). For multiplying, two negatives give a positive: −4 × (−3) = 12.

:::


:::typical

## One change at a time


**Positive times negative.** Each task changes one sign. Use the board to check.

**Negative times positive.** 

**Negative times negative.** 

**Times zero.** 

**Dividing.** No board: use the rule and check by multiplying.

Work in the tutorial. Plan first: what will you change, and how will you test it?

{{box-4}}

:::


## All four signs


:::keep-together

### Multiply

**Work out $-6 \times (-5)$.**

Answer: {{field-15}}

:::


:::teacher-note

**Answer:** 30

Same signs: 30.

:::


:::keep-together

### Divide

**Work out $18 \div (-3)$.**

Answer: {{field-15}}

:::


:::teacher-note

**Answer:** -6

Different signs: −6. Check: −3 × (−6) = 18.

:::


:::keep-together

### Powers of −1

**What is $(-1)^3$, which is $(-1) \times (-1) \times (-1)$?**

☐ $-1$

☐ $-3$

☐ $1$

:::


:::teacher-note

**Answer:** $-1$

(−1) × (−1) = 1, then × (−1) again: −1. An odd number of negatives gives a negative.

:::


:::keep-together

### Review: adding is different

**Review: work out $-4 + (-3)$.**

Answer: {{field-15}}

:::


:::teacher-note

**Answer:** -7

Adding a negative moves left: −7. Compare: −4 × (−3) = 12.

:::


## Make: a positive from negatives

Find two negative numbers that multiply to 12. Then find another pair.

Stretch: find three negative numbers that multiply to $-12$.

Work in the tutorial. Plan first: what will you change, and how will you test it?

{{box-4}}

## Exit ticket

:::keep-together

**Work out −7 × (−4), 24 ÷ (−6) and (−1)⁴.**

{{lines-3}}

:::


:::teacher-note

**Answer:** 28, −4 and 1.

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