Chapter 1 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

M3.1.1 Chapter 1

Below zero

The weather report says minus 4. How can a temperature be below zero?

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

  • Read temperatures in °C above and below zero on a thermometer
  • Order numbers like −5, 2, −1 and 0 on a number line

Amina looks at the thermometer outside her door on a cold morning. The liquid is below the 0 mark. The reading is minus 4.

A thermometer measures how hot or cold something is. In a tube thermometer the liquid rises when it is warm and falls when it is cold.

We count temperature in degrees Celsius, written °C. Pure water freezes at 0 °C. Marks above 0 are warmer. Marks below 0 are colder.

Draw it first

Read a thermometer from the freezing mark

  1. A thermometer with a scale from −10 to 10 degrees. The freezing mark at 0 is labelled freezing. In step 1 the liquid is at 0 and the caption is 0 °C. In step 2 the liquid is above 0 and the caption is 6 °C. In step 3 the liquid is below 0 and the caption is −4 °C.−10−50510°Cfreezing0 °C

    1 Find the mark 0. Pure water freezes at this temperature.

  2. A thermometer with a scale from −10 to 10 degrees. The freezing mark at 0 is labelled freezing. In step 1 the liquid is at 0 and the caption is 0 °C. In step 2 the liquid is above 0 and the caption is 6 °C. In step 3 the liquid is below 0 and the caption is −4 °C.−10−50510°Cfreezing6 °C

    2 Liquid above 0 shows a day warmer than freezing. This liquid stops at 6 °C.

  3. A thermometer with a scale from −10 to 10 degrees. The freezing mark at 0 is labelled freezing. In step 1 the liquid is at 0 and the caption is 0 °C. In step 2 the liquid is above 0 and the caption is 6 °C. In step 3 the liquid is below 0 and the caption is −4 °C.−10−50510°Cfreezing−4 °C

    3 Liquid below 0 shows a day colder than freezing. This liquid stops 4 marks below 0, at −4 °C.

A thermometer with a scale from −10 to 10 and the freezing mark at 0 labelled freezing. The liquid stops 6 marks below the 0 mark.−10−50510°Cfreezing
The liquid stands 6 marks below the 0 mark. Each mark is 1 degree.

Worked example

The liquid stands 6 marks below the 0 mark. Each mark is 1 degree. What does the thermometer read?

  1. Each mark is 1 degree, so 6 marks is 6 degrees.
  2. The liquid is below 0, so the number has a minus sign.
  3. 6 marks below 0 is −6 °C.

Answer: −6 °C

A number above zero is a positive number. A number below zero is a negative number. We write a minus sign in front, as in −4. Say "minus four" or "negative four". Zero is neither positive nor negative.

A number line from −5 to 5. Dots mark −3 and 3, with the labels −3 and 3. The number −3 is left of 0 and the number 3 is right of 0.−5−4−3−2−1012345−33
Negative numbers are left of 0. Positive numbers are right of 0.

A thermometer is a number line stood on its end. Up is the same as right, and down is the same as left. The further left a number is, the smaller it is. So −5 < −1 < 0 < 2.

A number line from −5 to 5 with four dots, at −5, −1, 0 and 2.−5−4−3−2−1012345
The four numbers from the worked example, marked on a number line.

Worked example

Put −5, 2, −1, 0 in order, smallest first.

  1. Mark the four numbers on a number line. The one furthest left is −5.
  2. The next one to the right is −1. Then comes 0, and last, on the right, is 2.
  3. Read them from left to right: −5, −1, 0, 2.

Answer: −5, −1, 0, 2

See it three ways

With things

A lift in a tall building has buttons for floors above and below the ground floor. The ground floor is 0. Press 2 to go up two floors. Press −2 to go down two floors.

As a picture

A number line from −5 to 5. A dot at 2 has the label up 2. A dot at −2 has the label down 2.−5−4−3−2−1012345up 2down 2

In symbols

2 is above 0. The number −2 is below 0. So −2 < 0 < 2.

Try it

Make a number line

You need: a sheet of paper, a pencil, two small stones or buttons.

  1. Draw a straight line across the paper.
  2. Mark 0 near the middle.
  3. Mark 1, 2, 3, 4 and 5 to the right of 0. Leave the same gap each time.
  4. Mark −1, −2, −3, −4 and −5 to the left of 0, with the same gap.
  5. Put one stone on 2 and one stone on −3. Which stone is on the smaller number?

What you will see: The stone on −3 is to the left of the stone on 2. The number on the left is always the smaller one.

If you cannot try it, this is what happens.
A number line from −5 to 5 with a dot at −3 labelled −3 and a dot at 2 labelled 2.−5−4−3−2−1012345−32

A wrong idea many people have

Minus 7 is bigger than minus 2, because 7 is bigger than 2.

Why it feels right: When you count 1, 2, 3, a bigger digit means a bigger number. The minus sign looks like a small mark in front.

The test that breaks it: Put both on a thermometer. Find −7 °C and −2 °C. Which one is lower down? Which one is colder?

What is true: Below zero, a bigger digit means further below zero. So −7 is smaller than −2.

Key words

thermometer
a tool that measures temperature.
degrees Celsius
the units of temperature, written °C. Water freezes at 0 °C.
positive number
a number above zero.
negative number
a number below zero, written with a minus sign.

Back to your guess

The answer is B: A temperature of −4 °C is colder than 0 °C.

Below 0 °C means colder than 0 °C. There is still heat. Zero is only the mark where we start counting, and the numbers go on below it.

Check yourself

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

  1. Kofi reads this thermometer. What temperature does it show?

    A thermometer with a scale from −10 to 10 and the freezing mark at 0 labelled freezing. The liquid is below the 0 mark.−10−50510°Cfreezing
    Tap your answer.

    1. Not this one. The liquid is below the 0 mark, so the number needs a minus sign. A reading without one is above zero. Go to M3.1.1. Try another.

    2. Not this one. You may have counted the 0 mark as one of the marks. Start counting at the first mark under 0. Go to M3.1.1. Try another.

    3. Yes, that is right. The liquid is 3 marks below the 0 mark. Each mark is one degree, so the reading is −3 °C.

    See the answer

    −3 °C

    The liquid is 3 marks below the 0 mark. Each mark is one degree, so the reading is −3 °C.

    • If you chose A: The liquid is below the 0 mark, so the number needs a minus sign. A reading without one is above zero. Go to M3.1.1.
    • If you chose B: You may have counted the 0 mark as one of the marks. Start counting at the first mark under 0. Go to M3.1.1.

  2. Which list has the numbers 3, −2, −7, 0, −1 in order, smallest first?

    Tap your answer.

    1. Not this one. You may think −1 is smaller than −2 because 1 is smaller than 2. Below zero it is the other way round. Check on a number line. Go to M3.1.1. Try another.

    2. Yes, that is right. On a number line the smallest number is furthest left. The order from left to right is −7, −2, −1, 0, 3.

    3. Not this one. This list goes from the largest to the smallest. The smallest number goes first, on the left of the number line. Go to M3.1.1. Try another.

    See the answer

    −7, −2, −1, 0, 3

    On a number line the smallest number is furthest left. The order from left to right is −7, −2, −1, 0, 3.

    • If you chose A: You may think −1 is smaller than −2 because 1 is smaller than 2. Below zero it is the other way round. Check on a number line. Go to M3.1.1.
    • If you chose C: This list goes from the largest to the smallest. The smallest number goes first, on the left of the number line. Go to M3.1.1.

  3. Which temperature is colder, −8 °C or −3 °C?

    Tap your answer.

    1. Yes, that is right. A reading of −8 °C is 8 marks below zero and −3 °C is only 3 marks below. The one further down is colder, and it is the smaller number.

    2. Not this one. You may be thinking that a smaller digit means colder. Check on a thermometer. The number −8 is further below zero than −3, so it is colder. Go to M3.1.1. Try another.

    3. Not this one. A temperature of 8 °C has no minus sign. It is above zero, so it is warmer than both of the others. Go to M3.1.1. Try another.

    See the answer

    −8 °C

    A reading of −8 °C is 8 marks below zero and −3 °C is only 3 marks below. The one further down is colder, and it is the smaller number.

    • If you chose B: You may be thinking that a smaller digit means colder. Check on a thermometer. The number −8 is further below zero than −3, so it is colder. Go to M3.1.1.
    • If you chose C: A temperature of 8 °C has no minus sign. It is above zero, so it is warmer than both of the others. Go to M3.1.1.

  4. Which of these numbers is smaller than −2?

    Tap your answer.

    1. Not this one. You may think 1 is smaller than 2, so −1 is smaller than −2. On the number line −1 is to the right of −2, so it is bigger. Go to M3.1.1. Try another.

    2. Not this one. Zero is to the right of every negative number, so it is bigger than −2. Go to M3.1.1. Try another.

    3. Yes, that is right. A number is smaller when it is further left. The number −5 is left of −2. The numbers −1 and 0 are to the right of it.

    See the answer

    −5

    A number is smaller when it is further left. The number −5 is left of −2. The numbers −1 and 0 are to the right of it.

    • If you chose A: You may think 1 is smaller than 2, so −1 is smaller than −2. On the number line −1 is to the right of −2, so it is bigger. Go to M3.1.1.
    • If you chose B: Zero is to the right of every negative number, so it is bigger than −2. Go to M3.1.1.

  5. Which of these temperatures is below the freezing point of water?

    Tap your answer.

    1. Not this one. Water freezes at 0 °C. A reading of 0 °C is at the freezing point, not below it. Go to M3.1.1. Try another.

    2. Yes, that is right. Below freezing means less than 0 °C, so the number must be negative. Only −1 °C is below 0.

    3. Not this one. A temperature of 5 °C is above 0 °C. It is warmer than the freezing point. Go to M3.1.1. Try another.

    See the answer

    −1 °C

    Below freezing means less than 0 °C, so the number must be negative. Only −1 °C is below 0.

    • If you chose A: Water freezes at 0 °C. A reading of 0 °C is at the freezing point, not below it. Go to M3.1.1.
    • If you chose C: A temperature of 5 °C is above 0 °C. It is warmer than the freezing point. Go to M3.1.1.

  6. Mei stands on the number 1. She walks 3 steps to the left. Which number is she on?

    See the answer

    −2

    Left means smaller numbers. From 1 she steps to 0, then −1, then −2.

  7. Which is colder, −3 °C or −6 °C? Type the colder temperature.

    See the answer

    −6 °C

    The one further below zero is colder. It has the bigger number after the minus sign.

Keep it

Copy this into your exercise book.

A number below zero is negative. Negative numbers sit to the left of 0 on a number line. The further left a number is, the smaller it is. So −5 < −1 < 0 < 2.

Draw: A number line from −5 to 5 with 0 in the middle. Mark −5, −1, 0 and 2, and write −5 < −1 < 0 < 2 under it.

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.