Chapter 2 of 8

All 8 chapters
  1. 1Below zero, readunderstood
  2. 2Calculating with negative numbers, readunderstood
  3. 3Multiples and factors, readunderstood
  4. 4Quick tests for dividing, readunderstood
  5. 5Prime numbers, readunderstood
  6. 6Prime factors, common factors and multiples, readunderstood
  7. 7Squares, cubes and roots, readunderstood
  8. 8Powers of ten, readunderstood

M3.1.2 Chapter 2

Calculating with negative numbers

It is −3 °C at night and 5 °C at noon. How many degrees did it rise?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Find the rise in temperature from −3 °C to 5 °C
  • Work out −4 + 7 and 2 − 6 on a number line
  • Multiply and divide with negatives, like −3 × −4

At night the temperature is −3 °C. At noon it is 5 °C. The rise is how many degrees it went up. How big is the rise?

Draw it first

Count the rise from −3 to 5

  1. A number line of temperatures from −5 to 5. Step 1: a dot at −3 with the label −3. Step 2: a hop from −3 up to 0 with the label +3 °C. Step 3: a second hop from 0 up to 5 with the label +5 °C.−5−4−3−2−1012345−3

    1 Put a dot on −3. This is the night temperature.

  2. A number line of temperatures from −5 to 5. Step 1: a dot at −3 with the label −3. Step 2: a hop from −3 up to 0 with the label +3 °C. Step 3: a second hop from 0 up to 5 with the label +5 °C.−5−4−3−2−1012345+3 °C

    2 Hop up to 0. That is a hop of 3.

  3. A number line of temperatures from −5 to 5. Step 1: a dot at −3 with the label −3. Step 2: a hop from −3 up to 0 with the label +3 °C. Step 3: a second hop from 0 up to 5 with the label +5 °C.−5−4−3−2−1012345+3 °C+5 °C

    3 Hop on from 0 up to 5. That is a hop of 5. The whole rise is 3 + 5 = 8 degrees.

Worked example

The temperature rises from −3 °C to 5 °C. How many degrees is the rise?

  1. From −3 up to 0 is 3 degrees.
  2. From 0 up to 5 is 5 degrees.
  3. Add the two hops: 3 + 5 = 8.

Answer: 8 degrees

You can add and take away on a number line. To add a positive number, hop to the right. To take away a positive number, hop to the left. Adding a negative number moves you left, so 3 + −6 is the same as 3 − 6.

A number line from −5 to 5. A hop from −4 to 3 has the label +7.−5−4−3−2−1012345+7
Start at −4 and hop 7 to the right. You land on 3.

Worked example

Work out −4 + 7.

  1. Start at −4. Adding 7 means 7 hops to the right.
  2. The first 4 hops reach 0. Then 3 more hops reach 3.
  3. So −4 + 7 = 3.

Answer: −4 + 7 = 3

A number line from −5 to 5. A hop from 2 to −4 has the label −6.−5−4−3−2−1012345−6
Start at 2 and hop 6 to the left. You land on −4.

Worked example

Work out 2 − 6.

  1. Start at 2. Taking away 6 means 6 hops to the left.
  2. Two of the hops take you from 2 to 0.
  3. The other four hops go on from 0 to −4.
  4. So 2 − 6 = −4.

Answer: 2 − 6 = −4

Now multiplying. 3 × −4 means 3 lots of −4. That is −4 + −4 + −4, which is −12. A positive number times a negative number gives a negative answer.

What about −3 × −4? You cannot lay out "minus 3 lots" with beans. So look for a pattern, and let it carry on.

A table with the head sum and answer. The rows are 3 × −4 with answer −12, 2 × −4 with answer −8, 1 × −4 with answer −4, 0 × −4 with answer 0, −1 × −4 with answer 4, −2 × −4 with answer 8, and −3 × −4 with answer 12. The last answer is shaded.sumanswer3 × −4−122 × −4−81 × −4−40 × −40−1 × −44−2 × −48−3 × −412
Each row has one less lot of −4. Each answer goes up by 4.

Look down the answers: −12, −8, −4, 0. They go up by 4 each time. The pattern cannot stop at 0. The next answers are 4, 8 and 12. So −3 × −4 = 12.

The rule: two numbers with the same sign give a positive answer. Two numbers with different signs give a negative answer. This holds for the product of any two numbers, and for dividing too.

Dividing undoes multiplying. We know 3 × −4 = −12. So −12 ÷ 3 = −4, and −12 ÷ −4 = 3. The same sign rule works: same signs give a positive answer, and different signs give a negative answer.

A table with the head times fact and divide fact. The rows are 3 × −4 = −12 and −12 ÷ 3 = −4; 3 × −4 = −12 and −12 ÷ −4 = 3; −3 × −4 = 12 and 12 ÷ −3 = −4; −3 × −4 = 12 and 12 ÷ −4 = −3.times factdivide fact3 × −4 = −12−12 ÷ 3 = −43 × −4 = −12−12 ÷ −4 = 3−3 × −4 = 1212 ÷ −3 = −4−3 × −4 = 1212 ÷ −4 = −3
Each times fact gives two divide facts. The same sign rule works for both.

See it three ways

With things

At 3 °C the temperature starts to fall. It falls 2 degrees each hour for 3 hours. What is the total change, and where does it end?

As a picture

A number line of temperatures from −5 to 5. Three hops each go 2 degrees to the left, from 3 to 1, from 1 to −1 and from −1 to −3. Each hop has the label −2 °C.−5−4−3−2−1012345−2 °C−2 °C−2 °C

In symbols

3 × −2 = −6. The change is −6, so the temperature ends at 3 + −6 = −3.

Try it

Carry on a pattern

You need: a sheet of paper, a pencil.

  1. Write 3 × −2 = −6 at the top of the page.
  2. Write 2 × −2 under it. Work out the answer by taking the answer above and adding 2.
  3. Go on down to −3 × −2. Each time the first number goes down by 1.
  4. Look at the answers. Do they go up or down?

What you will see: The answers are −6, −4, −2, 0, 2, 4, 6. They go up by 2 each time, so −3 × −2 = 6.

If you cannot try it, this is what happens.
A table with the head sum and answer. The rows are 3 × −2 with answer −6, 2 × −2 with answer −4, 1 × −2 with answer −2, 0 × −2 with answer 0, −1 × −2 with answer 2, −2 × −2 with answer 4, and −3 × −2 with answer 6.sumanswer3 × −2−62 × −2−41 × −2−20 × −20−1 × −22−2 × −24−3 × −26

A wrong idea many people have

A negative times a negative is a negative, so −3 × −4 = −12.

Why it feels right: Two minus signs feel twice as negative. And 3 × −4 really is −12, so the minus sign seems to win every time.

The test that breaks it: Look at the pattern table. 0 × −4 is 0. The next row has one less lot of −4, and the answer goes up by 4, not down. So −1 × −4 is 4.

What is true: Two numbers with the same sign give a positive answer. So −3 × −4 = 12.

Key words

rise
how many degrees the temperature goes up.
product
the answer when you multiply numbers.

Back to your guess

The answer is B: The rise is 8 degrees. It goes from −3 up to 0, and then on up to 5.

First the temperature climbs 3 degrees to reach 0. Then it climbs 5 more. 3 + 5 = 8 degrees. You count across zero. You do not take 3 away from 5.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. At night it is −6 °C. By noon it is 4 °C. How many degrees did it rise?

    Tap your answer.

    1. Not this one. You may have taken 4 away from 6. When one temperature is below zero, count up to zero first and then on to the other. Go to M3.1.2. Try another.

    2. Not this one. You read the noon temperature. The rise is how far the temperature went up, starting from −6. Go to M3.1.2. Try another.

    3. Yes, that is right. From −6 up to 0 is 6 degrees. From 0 up to 4 is 4 more degrees. 6 + 4 = 10.

    See the answer

    10 degrees

    From −6 up to 0 is 6 degrees. From 0 up to 4 is 4 more degrees. 6 + 4 = 10.

    • If you chose A: You may have taken 4 away from 6. When one temperature is below zero, count up to zero first and then on to the other. Go to M3.1.2.
    • If you chose B: You read the noon temperature. The rise is how far the temperature went up, starting from −6. Go to M3.1.2.

  2. Work out −8 + 5.

    Tap your answer.

    1. Not this one. You may have added 8 and 5 and kept the minus. Adding 5 moves you to the right, towards zero, not further below it. Go to M3.1.2. Try another.

    2. Yes, that is right. Start at −8 and hop 5 to the right. You land on −3.

    3. Not this one. You started at −8 and hopped 5 to the right. You have not reached 0, so the answer is still below zero and needs a minus sign. Go to M3.1.2. Try another.

    See the answer

    −3

    Start at −8 and hop 5 to the right. You land on −3.

    • If you chose A: You may have added 8 and 5 and kept the minus. Adding 5 moves you to the right, towards zero, not further below it. Go to M3.1.2.
    • If you chose C: You started at −8 and hopped 5 to the right. You have not reached 0, so the answer is still below zero and needs a minus sign. Go to M3.1.2.

  3. Work out 3 − 9.

    Tap your answer.

    1. Yes, that is right. Start at 3 and hop 9 to the left. Three hops reach 0 and six more reach −6.

    2. Not this one. You took the smaller number from the bigger one. 3 − 9 hops left past zero, so the answer is below zero. Go to M3.1.2. Try another.

    3. Not this one. You added 3 and 9. This is a take away, so hop to the left. Go to M3.1.2. Try another.

    See the answer

    −6

    Start at 3 and hop 9 to the left. Three hops reach 0 and six more reach −6.

    • If you chose B: You took the smaller number from the bigger one. 3 − 9 hops left past zero, so the answer is below zero. Go to M3.1.2.
    • If you chose C: You added 3 and 9. This is a take away, so hop to the left. Go to M3.1.2.

  4. What is −5 × −2?

    Tap your answer.

    1. Not this one. You may think two minus signs make a negative answer. Same signs give a positive answer. Look at the pattern table again. Go to M3.1.2. Try another.

    2. Not this one. You may have added −5 and −2. This is a multiplying question, so work out 5 × 2 and then choose the sign. Go to M3.1.2. Try another.

    3. Yes, that is right. 5 × 2 = 10. The signs are the same, so the answer is positive.

    See the answer

    10

    5 × 2 = 10. The signs are the same, so the answer is positive.

    • If you chose A: You may think two minus signs make a negative answer. Same signs give a positive answer. Look at the pattern table again. Go to M3.1.2.
    • If you chose B: You may have added −5 and −2. This is a multiplying question, so work out 5 × 2 and then choose the sign. Go to M3.1.2.

  5. Work out −24 ÷ 6.

    Tap your answer.

    1. Not this one. You may have added 6 to −24. This is a divide question. Work out 24 ÷ 6 and then choose the sign. Go to M3.1.2. Try another.

    2. Yes, that is right. 24 ÷ 6 = 4. The signs are different, so the answer is negative. Check: 6 × −4 = −24.

    3. Not this one. The signs are different, one minus and one plus, so the answer is negative. Check: 6 × 4 is 24, not −24. Go to M3.1.2. Try another.

    See the answer

    −4

    24 ÷ 6 = 4. The signs are different, so the answer is negative. Check: 6 × −4 = −24.

    • If you chose A: You may have added 6 to −24. This is a divide question. Work out 24 ÷ 6 and then choose the sign. Go to M3.1.2.
    • If you chose C: The signs are different, one minus and one plus, so the answer is negative. Check: 6 × 4 is 24, not −24. Go to M3.1.2.

  6. At night it is −2 °C. By noon it is 7 °C. How many degrees did it rise?

    See the answer

    9 degrees

    From −2 up to 0 is 2 degrees. From 0 up to 7 is 7 more degrees. 2 + 7 = 9.

  7. Work out −7 × 3.

    See the answer

    −21

    7 × 3 = 21. The signs are different, so the answer is negative.

  8. From an earlier chapter

    Which number is the smallest: 4, −2, −7 or 0?

    Tap your answer.

    1. Not this one. Zero is bigger than every negative number. The smallest number is the one furthest to the left. Go to M3.1.1. Try another.

    2. Yes, that is right. The smallest number is the furthest left on a number line. That number is −7. It is left of −2, 0 and 4.

    3. Not this one. You may think −2 is the smallest because 2 is the smallest digit. A number further left is smaller, and −7 is further left than −2. Go to M3.1.1. Try another.

    See the answer

    −7

    The smallest number is the furthest left on a number line. That number is −7. It is left of −2, 0 and 4.

    • If you chose A: Zero is bigger than every negative number. The smallest number is the one furthest to the left. Go to M3.1.1.
    • If you chose C: You may think −2 is the smallest because 2 is the smallest digit. A number further left is smaller, and −7 is further left than −2. Go to M3.1.1.

Keep it

Copy this into your exercise book.

Hop right to add a positive number and left to take one away. Adding a negative moves you left. Same signs multiply to a positive answer. Different signs give a negative answer. Dividing follows the same rule.

Draw: A number line from −5 to 5 with a hop from −4 to 3 labelled +7. Under it write −4 + 7 = 3.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (805 KB). It opens on any phone.