Multiplying and dividing integers

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.

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.

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.

Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.

Explain it to me

Multiplying is jumps from zero · Sound on for the voiceover

$a \times b$ is $a$ jumps of $b$ from zero. $3 \times (-2)$ is 3 jumps of $-2$ : left to $-6$ .

A negative number of jumps goes the opposite way: $-3 \times (-2) = 6$ . The pattern agrees: $3 \times (-2) = -6$ , $2 \times (-2) = -4$ , $1 \times (-2) = -2$ , $0 \times (-2) = 0$ , so $-1 \times (-2) = 2$ .

The rule: same signs give a positive, different signs give a negative. Dividing follows the same rule, because it undoes multiplying: $-12 \div 3 = -4$ , since $3 \times (-4) = -12$ .

$-3 \times (-2): 3 jumps of -2 would go to -6; a negative number of jumps goes the opposite way, to 6.$

-3 \times (-2)

: 3 jumps of

-2

would go to

-6

; a negative number of jumps goes the opposite way, to

6

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

−3 × (−2)

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

What is

-3 \times (-2)

-5

1

-6

6

TryTry it on the board and compare the result with your prediction.

Follow the pattern

$a$ 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?

Try it yourselfYour edits are saved in this browser.

The board shows $3 \times (-2)$ . Make $a$ smaller one step at a time until it is $-3$ . Was your prediction right?

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

The sign rules

Fill in the table. Each step down in $a$ : what happens to the answer?

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

Worked exampleBefore each step, say what you think comes next.

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

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

Read Kai's working one line at a time. Which rule did he use?

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

a	3	2	1	0	−1	−2	−3
a × (−2)	−6	−4	−2

Which is

-12 \div 3

-4

-9

4

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

ModifyChange the problem on purpose and see what happens.

One change at a time

Each task changes one sign. Use the board to check.

Need a hint?

4 jumps of −3.

Need a hint?

−4 jumps of 3: 4 jumps of 3 would go right, so these go left.

Need a hint?

4 jumps of −3 would go left to −12. A negative number of jumps goes the opposite way.

No board: use the rule and check by multiplying.

Try it yourselfYour edits are saved in this browser.

Work out $4 \times (-3)$ . Set the board to check.

Work out $-4 \times 3$ .

Work out $-4 \times (-3)$ .

Edge case: work out $-5 \times 0$ .

Work out $-20 \div (-4)$ .

PractiseA few on your own, to check you have it.

All four signs

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

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

What is

(-1)^3

, which is

(-1) \times (-1) \times (-1)

-1

-3

1

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

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

A positive from negatives

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

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

Try it yourselfYour edits are saved in this browser.

Need a hint?

Which pairs of whole numbers multiply to 12?

Another hint

Make both of them negative.

Set $a$ and $b$ so that both are negative and $a \times b = 12$ .

Use the board. This ticks when you get there.

For teachers

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.

Open the teacher guide: answers, solutions, slides and a worksheet.

Students will be able to

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.

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.

Running the lesson

Predict: take a show of hands. Students who chose −6 usually say 'two negatives make a negative'. Do not correct it yet.
Run: 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?
Investigate: 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.
Modify: each task changes one sign. Ask for the sign before the number.
Practise: (−1)³ is a stretch for some; let them multiply two at a time.
Make: collect pairs. Ask: could one of them be positive? (Then the product would be negative.)

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.

Stretch and support

Stretch: Work out (−2)⁴ and (−2)⁵. Then: what is the sign of the product of 7 negative numbers? Why?

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.

Where it 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.

What each step is for

−3 × (−2): Committing to a sign for −3 × (−2) before the board shows it tests whether they carry over the addition rule (two negatives make a bigger negative).
follow the pattern: Making a smaller one step at a time shows the answer going up by 2 each time, through zero and past it: the pattern forces a negative times a negative to be positive.
the sign rules: The pattern and the line give one rule: the same signs give a positive, different signs a negative, for multiplying and for dividing. Kai's mistake uses the addition rule for multiplication.
one change at a time: 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.
all four signs: Multiplying, dividing and powers of −1, mixed, with the addition rule as the planted mix-up; the review keeps adding negatives apart from multiplying them.
a positive from negatives: Choosing two negatives whose product is a given positive reverses the lesson: from the answer back to the signs.

Now you can

Multiply and divide integers, including two negatives.
Explain why a negative times a negative is positive, from a pattern or the number line.
Tell the rule for multiplying apart from the rule for adding negatives.

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.Next lesson4. Squares and square rootsA square number is the area of a square: 6² = 6 × 6 = 36. The square root undoes it: √36 = 6, the side of a square with area 36.

Source: content/maths/number/multiplying-integers/lesson.md