Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- Work out −7 + 12.
Answer
5. - Work out 2 − 9.
Answer
−7. - What is 6 × 8? What is 48 ÷ 6?
Answer
48, and 8. Division undoes multiplication.
Words used in this lesson
- product
- the answer to a multiplication. For example: the product of −3 and 4 is −12.
- quotient
- the answer to a division. For example: −20 ÷ 4 has quotient −5.
- factor
- one of the numbers being multiplied. For example: −2 is a factor in (−2) × 7.
- same signs
- both positive or both negative. For example: (−6) and (−5).
- pattern
- a rule that repeats, so you can predict the next step. For example: −9, −6, −3, 0, 3.
Follow the pattern past zero
You already know that 3 × (−2) = −6: it is three groups of −2, so −2 + (−2) + (−2). But what is −3 × (−2)? You cannot have “minus three groups” of anything.
So let a pattern decide. In the figure, a = −3 and the rows multiply it by 3, 2, 1, 0 and on down. Look at the first four rows and say how the product changes each time, then follow the pattern past zero.
With a = −3, each row down adds 3. The products climb through zero, so −3 × (−2) = 6.
Show me the steps
- a = 3: 3 × 3 = 9, 3 × 2 = 6, 3 × 1 = 3, 3 × 0 = 0. Each row the product falls by 3, so 3 × (−1) = −3 and 3 × (−2) = −6.
- a = −3: −3 × 3 = −9, −3 × 2 = −6, −3 × 1 = −3, −3 × 0 = 0. Each row the product rises by 3.
- Keep going: −3 × (−1) = 3, −3 × (−2) = 6, −3 × (−3) = 9. A negative times a negative is positive.
- a = 0: every product is 0, and every bar has zero length.
Multiplying or dividing two numbers: the same signs give a positive answer, different signs give a negative answer. (−6) × (−5) = 30 and (−7) × 4 = −28.
With several numbers, count the negative signs: an even number of them gives a positive answer, an odd number gives a negative answer. Anything multiplied by 0 is 0.
Division undoes multiplication
Since 5 × (−4) = −20, dividing −20 by −4 must give 5. So (−20) ÷ (−4) = 5: the same signs give a positive answer, exactly as for multiplication. Check any division by multiplying back.
“Two negatives make a positive” is about multiplying and dividing only. −3 + (−5) is −8, not 8: adding two negatives moves further left.
Worked examples
What does a negative times a negative mean in real life? Kofi walks down a hill steadily, losing 3 m of height every minute: a change of −3 m per minute. Where was he 4 minutes ago, compared with now?
- 4 minutes ago is a time of −4 minutes.
- Height change = rate × time = (−3) × (−4) = +12 m: Kofi was 12 m higher. The positive answer makes sense, because going back in time on a descent takes you up.
Your go. The first step is given. Work out (−48) ÷ (−6) and check it.
- Same signs, so the answer is positive.
- Finish it, then check by multiplying back.
Check the answer
48 ÷ 6 = 8, so the answer is 8. Check: 8 × (−6) = −48.
Check before you go on
1. Which of these has a positive answer?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. Work out (−6) × 4.
3. Work out (−7) × (−3).
Level 2 · Building
4. Work out (−36) ÷ 9.
5. Work out 45 ÷ (−5).
6. Aiyana, a diver, goes down steadily, changing depth by −2 m every minute. How many metres higher than now was she 5 minutes ago? (Work out (−2) × (−5).)
Level 3 · Stretch
7. Work out (−2) × (−3) × (−5).
8. Two integers multiply to give −24 and add up to 2. What is the smaller of the two?
Something to think about
Why must (−1) × (−1) equal 1? Start from (−1) × 0 = 0, and write the 0 as 1 + (−1).
One way to think about it
(−1) × (1 + (−1)) = 0. Multiplying out the bracket gives (−1) × 1 + (−1) × (−1) = 0, that is −1 + (−1) × (−1) = 0. The only number that makes this true is (−1) × (−1) = 1.
Exit ticket
9. Rosa says (−8) ÷ (−2) = −4, “because two negatives make a negative”. What is the right answer?