Year 8 · Unit 3 Integers · Lesson 1 of 6

Who decided where zero is?

An integer says how far you are from a zero that somebody chose, and which way. Move the zero and watch every number change while nothing else moves.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. Put these in order, smallest first: 305, 35, 350, 53.
    Answer35, 53, 305, 350. Compare the hundreds first, then the tens.
  2. A thermometer reads 3 °C. What does it read after the temperature falls by 4 degrees?
    Answer−1 °C. Count down 3 steps to zero, then 1 more step below it.
  3. Which number is halfway between 20 and 30 on a number line?
    Answer25, the same distance (5) from each end.
Words used in this lesson
integer
a whole number, positive, negative or zero. For example: −4, 0, 17.
positive
more than zero: above it, or to its right. For example: +5 m, above sea level.
negative
less than zero: below it, or to its left. For example: −5 m, below sea level.
opposite
the same distance from zero, on the other side. For example: −6 and 6.
greater than, >
further right, or higher. For example: 3 > −2.
less than, <
further left, or lower. For example: −9 < −4.
sea level
the zero that heights are measured from. For example: 0 m.

A sign is a direction

Sea level is a zero that people chose. The shore of the Dead Sea is about 430 m below it, so its height is written −430 m. The top of Kilimanjaro is about 5895 m above it: +5895 m. Without the sign, “430 m” tells you how far but not which way.

The figure is a harbour cliff. Every place on it has a height, but the number you give each place depends on where you put zero. Predict first: if zero moves up to the harbour wall, what will the diver's number be? Then move the slider and see.

sea level
+18lighthouse lamp, m
−7diver, m
−15seabed, m

Zero is at sea level. The lamp is 18 m above it, so +18, and the diver is 7 m below it, so −7.

Show me the steps
  1. Zero at sea level: lamp +18, harbour wall +4, sea level 0, diver −7, seabed −15.
  2. Zero at the harbour wall: lamp +14, harbour wall 0, sea level −4, diver −11, seabed −19.
  3. Zero at the diver: lamp +25, harbour wall +11, sea level +7, diver 0, seabed −8.
  4. Zero at the seabed: lamp +33, harbour wall +19, sea level +15, diver +8, seabed 0. Every number is positive.
  5. Zero at the lamp: lamp 0, harbour wall −14, sea level −18, diver −25, seabed −33. Every number is negative.
  6. In every case the order from top to bottom is the same, and the lamp is always 33 m above the seabed.

Integers and opposites

The integers are the whole numbers and their negatives: …, −3, −2, −1, 0, 1, 2, 3, … They go on for ever in both directions. Zero is neither positive nor negative: it is the place you measure from.

Two numbers are opposites when they are the same distance from zero, on different sides. The opposite of 6 is −6, and the opposite of −6 is 6. Zero is its own opposite.

On a number line, numbers increase to the right (or upwards). So the number further left (or lower) is the smaller one, even when it looks bigger: −8 < −1, because −8 is further left. A temperature of −8 °C is colder than −1 °C.

The symbol < means “is less than” and > means “is greater than”. The wide end always faces the greater number. Try it: put a and b anywhere, and watch the comparison and the opposite of a.

−7
−2
−7 < −2comparison
7opposite of a

−7 is further left than −2, so −7 < −2, even though 7 is more than 2.

Show me the steps
  1. a = −7 and b = −2. −7 is further left, so −7 < −2. The opposite of a is 7.
  2. a = −7 and b = 3. −7 is further left, so −7 < 3. Every negative number is less than every positive number.
  3. a = 5 and b = −9. 5 is further right, so 5 > −9. The opposite of a is −5.
  4. a = 0 and b = −1. 0 is further right, so 0 > −1. Zero is its own opposite.

A common mistake: “−10 is bigger than −3 because 10 is bigger than 3.” The 10 tells you how far from zero, and the minus sign tells you it is on the left. Further left means smaller, so −10 < −3.

Worked example: ordering

Order these night-time temperatures, coldest first: 3 °C, −8 °C, 0 °C, −1 °C, −12 °C.

  1. The negatives are the coldest, so they come first. Among them, the one furthest from zero is the furthest left: −12, then −8, then −1.
  2. Then zero, then the positive number, 3.
  3. Written with <: −12 < −8 < −1 < 0 < 3.

Your go. The first two steps are given. Order these heights, lowest first. They are all about right: the shore of Lake Assal in Djibouti, −155 m; Schiphol airport near Amsterdam, −3 m; the shore of the Caspian Sea, −28 m; the floor of Death Valley, −86 m; the top of Mount Kenya, 5199 m.

  1. Lowest is the negative furthest from zero: Lake Assal, −155 m.
  2. Next: Death Valley, −86 m.
  3. Now finish the list yourself, then check.
Check the full order

−155 < −86 < −28 < −3 < 5199: Lake Assal, Death Valley, the Caspian Sea, Schiphol, Mount Kenya.

Check before you go on

1. Which of these numbers is the greatest?

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. What is the opposite of −9?

3. One morning it is −6 °C in Ulaanbaatar and −2 °C in Oslo. Which city is warmer?

Level 2 · Building

4. How many integers are there between −4 and 3, not counting −4 and 3 themselves?

5. In the harbour figure, Mei the diver is 7 m below sea level. If zero is moved up to the top of the harbour wall, 4 m above sea level, what number does the diver get? (Check with the slider afterwards.)

6. Which list is in order from smallest to largest?

Level 3 · Stretch

7. I am an integer. I am greater than −7 and less than −3. I am even, and I am further from zero than −4 is. What number am I?

8. With zero at sea level, a bench on a cliff path is at +12 m and a rock pool is at −5 m. Where should zero be put, in metres above sea level, so that the bench and the rock pool become opposites? Give your answer to one decimal place.

Something to think about

Is there a largest negative integer? Is there a smallest one? Decide, then explain how you know.

One way to think about it

−1 is the largest: every other negative integer is further left. There is no smallest: whatever negative integer you name, one less than it is smaller still, so the integers go on for ever to the left.

Exit ticket

9. Amira says −10 is greater than −3 because 10 is greater than 3. Which reply is right?