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# Multiplying and dividing integers

Number Foundations · Integers

<!-- Speaker note: 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). About 45 minutes (Predict and Run 8 min · Investigate 15 min · Modify 12 min · Practise and Make 10 min). -->

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## Today

**We are learning to** a × b is a jumps of b from zero. A negative number of jumps goes the opposite way, so a negative times a negative is positive. Division undoes it with the same signs.

- 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.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need Adding and subtracting integers (subtracting goes the opposite way), which this lesson echoes: a negative number of jumps goes the opposite way. Two justifications are offered: the pattern (3 × (−2), 2 × (−2), … goes up by 2 each time, so it keeps going past zero) and the line. Use whichever lands; the debt model (taking away 3 debts of $2 leaves you $6 better off) is a good third in class. The common mistake is to use the addition rule for multiplication ("two negatives make a bigger negative"); it is planted after the method, in Kai's working, and the review question puts −4 + (−3) next to −4 × (−3). This lesson is Year 8 (AC9M8N04). It sits in Number Foundations so that Expanding brackets and the Year 8 Linear relations lessons can rely on it. -->

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## Why it matters

> 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.

<!-- Speaker note: You can add and subtract integers by moving along a number line. Now: multiplying and dividing. $3 \times 2$ is 3 jumps of 2 from zero, and $3 \times (-2)$ is 3 jumps of $-2$. But what is $-3 \times (-2)$? A negative number of jumps? In this lesson you find out, first from a pattern, then on the line. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

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

- $-5$
- $1$
- $-6$
- $6$

<!-- Speaker 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. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands. Students who chose −6 usually say 'two negatives make a negative'. Do not correct it yet. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fnumber%2Fmultiplying-integers%2Fstep-2&code=widget%3A+numberline%0Arange%3A+-12%2C+12%0Alabels%3A+2%0Aproduct%3A+a+%C3%97+b%0Aslider+a%3A+-4%2C+4%2C+3%0Aslider+b%3A+-4%2C+4%2C+-2%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/number/multiplying-integers/?full=step-2#step-2)

<!-- Speaker note: $a$ is the number of jumps and $b$ the size of each jump, from zero. Say each answer before you press −: $-6$, $-4$, $-2$, $0$, … What comes next? -->

<!-- Speaker note: Teacher: have the class say each answer aloud before the click: −6, −4, −2, 0, … and then 2. Ask: why would it suddenly stop going up? -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* 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.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## The pattern

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

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

* Answer: 0, 2, 4, 6

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

<!-- Speaker note: Teacher: model the worked example live. Write the pattern down the board at writing pace, then draw 3 jumps of −2 and the reverse for −3 × (−2). Reveal the prepared lines afterwards to compare. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Signs from the pattern

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

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

$$3 \times (-2) = -6$$

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

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write 3 × (-2) = -6 and say "3 jumps of -2: left 6." Line 1: write -3 × (-2) = 6 and say "A negative number of jumps goes the opposite way: right 6." Line 2: write same signs + and say "3 × 2 = 6 and -3 × (-2) = 6." Line 3: write different signs - and say "3 × (-2) = -6 and -3 × 2 = -6." Line 4: write -12 ÷ 3 = -4 and say "Dividing undoes multiplying: 3 × (-4) = -12, so the same rule holds." Line 5: write -12 ÷ (-3) = 4 and say "Same signs, positive: check -3 × 4 = -12." -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Signs from the pattern

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

- $-4$
- $-9$
- $4$

* Answer: $-4$

<!-- Speaker note: −12 ÷ 3 = −4, because 3 × (−4) = −12. Different signs give a negative. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## 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.
```

* Answer: line 3

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

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## 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.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/number/multiplying-integers/?full=step-4#step-4)

<!-- Speaker note: What this step is for: One change at a time: positive times negative, negative times positive, negative times negative, the edge case of zero, then division with each pair of signs. Hints, in order: 4 jumps of −3. / −4 jumps of 3: 4 jumps of 3 would go right, so these go left. / 4 jumps of −3 would go left to −12. A negative number of jumps goes the opposite way. For students who need more support: Use the board for every product: set a and b and read R. A printed sign table (+ × + = +, and so on) can come out after the pattern, not before. For early finishers: Work out (−2)⁴ and (−2)⁵. Then: what is the sign of the product of 7 negative numbers? Why? When students get stuck: A student says −4 × (−3) = −12 (Kai's mistake). Ask: What is −4 + (−3)? Is multiplying the same as adding? Fix: Go back to the pattern: 2 × (−3), 1 × (−3), 0 × (−3), −1 × (−3). Each step adds 3. / A student gets the size right but always the wrong sign for division. Ask: What would you multiply to check? Fix: Check with multiplication: if −12 ÷ 3 = 4, then 3 × 4 should be −12. It is 12, so the answer is −4. / A student adds instead of multiplying (−6 × −5 = −11). Ask: How many jumps, and how big is each one? Fix: Set it on the board: a = −6 is off the line, so use −3 × (−5) first, then double. -->

<!-- Speaker note: Teacher: each task changes one sign. Ask for the sign before the number. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Multiply

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

* Answer: 30

<!-- Speaker note: Same signs: 30. -->

<!-- Speaker note: Teacher: (−1)³ is a stretch for some; let them multiply two at a time. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Divide

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

* Answer: -6

<!-- Speaker note: Different signs: −6. Check: −3 × (−6) = 18. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Powers of −1

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

- $-1$
- $-3$
- $1$

* Answer: $-1$

<!-- Speaker note: (−1) × (−1) = 1, then × (−1) again: −1. An odd number of negatives gives a negative. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Review: adding is different

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

* Answer: -7

<!-- Speaker note: Adding a negative moves left: −7. Compare: −4 × (−3) = 12. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## 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 $-12$.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/number/multiplying-integers/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Choosing two negatives whose product is a given positive reverses the lesson: from the answer back to the signs. Hints, in order: Which pairs of whole numbers multiply to 12? / Make both of them negative. -->

<!-- Speaker note: Teacher: collect pairs. Ask: could one of them be positive? (Then the product would be negative.) -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

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

* 28, −4 and 1.

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this leads: Next in Number: squares, powers and primes (AC9M7N01, AC9M7N02), then fractions. Negative fractions and decimals (node 2) close the fractions unit. Where it leads: expanding brackets with negatives (−2(x − 5) = −2x + 10), gradients and rules with negative values in Linear relations, and powers of negative numbers. -->