KS3 · Maths

Multiplying and dividing positive and negative integers

Four groups of six red counters makes −24. So what could 'negative five times three' mean, and why is (−3) × (−4) × (−2) × (−7) positive before you multiply?

Maths · Multiplying and dividing integers

Positive or negative? Decide before you calculate

Picture a table of two-colour counters: red counters are negative and yellow counters are positive. Four groups of −6 is 24 red counters, so 4 × −6 = −24. For each calculation below, commit to whether the answer is positive or negative, then read the reason.

Still to sort

Positive (0)

The answer is above zero.

Negative (0)

The answer is below zero.

10 of 10 still to sort.

Predict, then flip

Here is the key move. To make 'the negative of' something, flip every counter over. So −5 × 3 is the negative of five groups of 3: make five groups of 3 yellow counters (15 yellow), flip them all, and you have 15 red counters, which is −15.

Now use the same move on −3 × −5. Three groups of −5 is 15 red counters, and −3 × −5 is the negative of those three groups. What is −3 × −5?

Why is it positive?

Which explanation is closest to yours?

Marcus works out −3 × −4 and gets 12. His teacher asks, 'Good, but why is it 12?'

Which of these is closest to what you would say right now?

Two negatives make a positive. That is just the rule.You take the minus signs off and multiply 3 × 4.−3 × −4 is the negative of three groups of −4. Three groups of −4 is −12, and taking the negative flips every counter, so it becomes 12.Three groups of −4 is 12 counters, so the answer is 12.

How sure are you?

I'm sureI think soI'm guessing

Two negatives make a positive. That is just the rule.

This is a common misconception

That phrase is a slogan, not a reason, and it leaks. Try it on −3 + −4: two negatives, and the answer is −7, not a positive. What really happens in −3 × −4 is that you take the negative of three groups of −4, and taking the negative flips every counter.

You take the minus signs off and multiply 3 × 4.

This is a common misconception

Taking the signs off does give the right size, 12, which is the absolute value of the product. But the same trick would give 3 × 4 = 12 for −3 × 4 too, and that one is −12. The signs decide whether the answer is positive or negative, so they cannot just be dropped.

−3 × −4 is the negative of three groups of −4. Three groups of −4 is −12, and taking the negative flips every counter, so it becomes 12.

This matches the evidence

Yes. Notice that this explains the sign instead of reciting it, and it works every time: groups of, then the negative of, then read off the answer.

Three groups of −4 is 12 counters, so the answer is 12.

This is a common misconception

Careful: those 12 counters are red, so they are worth −12. Counting counters without seeing that they are negative is an easy slip. Three groups of −4 is −12, and −3 × −4 is the negative of that, which is where 12 comes from.

A positive answer from two negatives is not magic. It comes from structure: multiplying by a negative takes the negative of the groups.

Now the size: the area model

Multiply with a grid

Any multiplication can be seen as the area of a rectangle. Split 13 into 10 + 3 and 25 into 20 + 5. Each box is its row times its column. Fill in all four boxes, then add them up for the total area. The grid does not need to be drawn to scale.

Grid: each cell is its row times its column
×	20	5
10
3

Total area, so 13 × 25 =

Type x² as x^2 if you cannot type ². Spaces do not matter.

Smart methods for the size

Problem

Work out 24 × 15, 52 ÷ 4 and 6000 ÷ 200 without a calculator.

Multiples of 10: how big is the answer?

Make one number smaller, the other bigger

Start from 7 × 3 = 21. Now work out 0.00007 × 30 000. Where does the answer land on this line?

Your estimate

Start from 7 × 3 = 21. Now work out 0.00007 × 30 000. Where does the answer land on this line? Estimate between 0 and 30.

030

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Decide the sign from the structure, then find the size with smart methods.

What you need to know

4 × −6 means four groups of −6, which is −24. −24 ÷ 4 means sharing 24 negative counters into four groups: −6 in each group.
To make 'the negative of' something, flip every counter. −3 × −5 is the negative of three groups of −5, so −15 flips to 15.
Same signs: the answer is the absolute value of the product (−3 × −4 = 12). Different signs: the answer is the negative of the absolute value (−3 × 4 = −12). Division follows the same rule because it is the inverse of multiplication.
With several integers, count the negatives: an even number gives a positive product and an odd number gives a negative one.
For the size, partition into factors, use the area model (and run it backwards to divide), split the dividend, or use multiples of 10.

The big picture

Multiplying and dividing integers is two jobs: decide the sign, then find the size. The sign comes from structure. Multiplying by a negative means taking 'the negative of' the groups, so same signs give the absolute value of the product and different signs give the negative of it, and division follows because it is the inverse. The size comes from efficient methods: partitioning into factors, the area model, inverse operations and multiples of 10.

Key points

1The sign comes from structure, not from a slogan: multiplying by a negative takes the negative of the groups.

2The absolute value of a number is its distance from zero, so 5 and −5 both have an absolute value of 5.

3Same signs give the absolute value of the product (or quotient). Different signs give the negative of the absolute value.

4Multiplication and division are inverses, so 15 ÷ 5 = 3 tells you 15 ÷ −5 = −3.

5In a product of several integers, an even number of negatives gives a positive and an odd number gives a negative.

6A product is the result of multiplying. In a division, the dividend is divided by the divisor and the answer is the quotient.

7Write partitions out fully so no power of 10 goes missing: 20 × 50 = 2 × 5 × 10 × 10 = 1000.

8If one number is made 10 times bigger and the other 10 times smaller, the product stays the same.

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Worked example

Problem

In a product number pyramid, each brick is the product of the two bricks below it. The bottom row is −2, 3, −1. What is the top brick?

⚠ Watch out

Losing a power of 10 when multiplying. 20 × 50 is not 100. Written out fully it is 2 × 5 × 10 × 10 = 10 × 100 = 1000. Writing every partition down stops a ten going missing.

🧠

Memory hook

Red is negative, yellow is positive, and 'the negative of' means flip. Then count the minus signs: an even number leaves you positive, an odd number leaves you negative.

✓

Check yourself

Without multiplying, decide whether (−1) × 8 × (−6) × (−2) × 5 is positive or negative. Then say how −36 ÷ −9 follows from a multiplication fact.

Flashcards

(15)

In 6 ÷ 3, which number is the dividend, which is the divisor, and what is the answer called?

6 is the dividend (what we are dividing) and 3 is the divisor (what we are dividing by). The answer is the quotient. The result of a multiplication is the product.

What does 4 × −6 mean with two-colour counters?

Four groups of −6: four groups of six red counters, which is −24.

What does −24 ÷ 4 mean with counters?

Share 24 negative (red) counters into four equal groups. Each group has six red counters, so −24 ÷ 4 = −6.

How do you make 'the negative of' something with counters?

Flip every counter over. Red (negative) becomes yellow (positive), and yellow becomes red.

Why does −3 × −5 = 15?

It is the negative of three groups of −5. Three groups of −5 is 15 red counters. Flip them all and they become 15 yellow counters.

What is the absolute value of a number?

Its distance from zero. 5 and −5 both have an absolute value of 5.

What is the sign rule for the product of two integers?

Same signs: the absolute value of the product (−3 × −4 = 12). Different signs: the negative of the absolute value (−3 × 4 = −12).

Why does division follow the same sign rule as multiplication?

Division is the inverse of multiplication. 15 ÷ 5 = 3, so 15 ÷ −5 = −3.

How can you find the sign of a product of many integers without calculating?

Count the negatives. An even number gives a positive product; an odd number gives a negative product.

Why is 'two negatives make a positive' not a good explanation?

It is a slogan, not a reason. The sign comes from structure: 'the negative of' groups. The slogan also misleads for addition, where −3 + −4 = −7.

How do you work out 24 × 15 by partitioning into factors?

24 × 15 = 2 × 12 × 3 × 5. Pair 12 × 5 = 60 and 2 × 3 = 6, then 60 × 6 = 360.

How does the area model show 13 × 25?

Four rectangles: 10 × 20 = 200, 10 × 5 = 50, 3 × 20 = 60 and 3 × 5 = 15. Add them: 325.

How do you divide 52 by 4 by splitting the dividend?

Split 52 into parts that 4 goes into: 40 + 12. Then 40 ÷ 4 = 10 and 12 ÷ 4 = 3, so the answer is 13.

How can a common factor help with 6000 ÷ 200?

Write it as (60 × 100) / (2 × 100). The 100s cancel because any number over itself is 1, leaving 60 ÷ 2 = 30.

One number is made 10 times bigger and another 10 times smaller. What happens to their product?

It stays the same: 3 × 7, 0.3 × 70, 30 × 0.7 and 0.03 × 700 all equal 21.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Positive or negative? Decide before you calculate

Which explanation is closest to yours?

Multiply with a grid

Make one number smaller, the other bigger

More KS3 Maths topics