KS3 · Maths

Negative numbers in context

Moscow is at −8 °C and Alice Springs at 29 °C. How many degrees apart is that, and what does a number below zero even mean?

Try it first

Put the numbers where they live

Zero is the fixed point. Negative numbers live to its left, positive numbers to its right. Drag each tag to where you think its number belongs, then check how close you were.

Maths · Negatives in real life

Which is colder? Put these cities in order

Order these January temperatures from coldest to warmest

1 · Coldest5 · Warmest

  1. Los Angeles, 13 °C

  2. Moscow, −8 °C

  3. Alice Springs, 29 °C

  4. New York, 2 °C

  5. Quebec, −7 °C

Watch out: Easy trap: −8 looks bigger than −7 because 8 is bigger than 7. Below zero, the bigger-looking number is the colder one.

Predict, then check

Alex is playing a game with a number line. Commit to an answer before you look.

Alex places a dot on a number line exactly five away from zero. Where could the dot be?

Maths · Differences across zero

How far apart are 3 and −9?

Picture 3 and −9 on the number line, with zero between them. A friend asks how far apart they are.

Which answer is closest to what you think right now?
How sure are you?

Maths · Balances below zero

Problem

Izzy has £1 in her account and her lunch costs £3. Sofia has £2 and spends £3.25. What is each balance?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Picture the number line and ordering, distances and overdrawn balances start to make sense.

What you need to know

  • A negative number is a value below an agreed zero: below 0 °C, below the ground floor, below sea level.
  • Numbers further to the left on the number line are smaller. Among negatives, the one closer to zero is greater.
  • An integer is any positive or negative whole number, or zero: −2, 0 and 153 are all integers.
  • To find the difference between a negative and a positive number, count to zero first, then on to the other number.
  • A negative balance means money owed.

The big picture

A negative number is a value below an agreed zero, such as a temperature below 0 °C, a car park level below the ground or a depth below sea level. On the number line, numbers further to the left are smaller, and that one picture answers three questions: which is greater, how far apart two numbers are, and what lies halfway. The same thinking handles balances that dip below zero, where a negative balance means money owed.

Key points

1Negative numbers exist because we need to describe values below an agreed zero, such as temperatures below 0 °C, a car park level three below the ground (−3) or anything below sea level.
2Further left means smaller: −7 < −5, −3 < −1 and −11 < −8. Closer to zero means greater among negatives: −9 > −10, −57 > −58 and −0.5 > −0.6.
3'Five away from zero' has two answers, −5 and 5, one each way along the line. There is not always a negative answer: three away from 10 is 7 or 13.
4The difference between 3 and −9 is 12 (9 to reach zero, then 3 more), not 6. Always go through zero.
5To find the number halfway between two numbers, find the distance between them, halve it, and move that far from either number.
6To find a balance below zero, subtract to reach zero and then subtract the rest. A negative balance shows money owed.

Worked example

Problem

Jacob has 75p in his account. He spends £2.65 on lunch and 80p on a drink. What is his new balance?

⚠ Watch out

Finding the difference between 3 and −9 by doing 9 − 3 = 6. That ignores zero. Count 9 from −9 up to zero, then 3 more: the difference is 12.

🧠

Memory hook

Ask 'which is colder?' instead of 'which is smaller?'. −8 °C is colder than −7 °C, so −8 is less than −7. And when two numbers are on either side of zero, always walk through zero.

✓

Check yourself

Cover the page and try this: how far apart are −6 and 4, and which number is exactly halfway between them?

Flashcards

(14)
What is an integer?
Any positive or negative whole number, or zero. For example −2, 0 and 153 are all integers.
Why do negative numbers exist?
They describe values below an agreed zero.
Name three places an agreed zero is used.
0 °C for temperature, the ground floor for lifts and car parks (three below is level −3), and sea level for heights and depths.
On a number line, which numbers are smaller?
The ones further to the left.
Why is −9 greater than −10?
−9 is closer to zero, so it sits further right on the number line.
What question makes comparing negatives easier?
Ask which is colder. −8 °C is colder than −7 °C, so −8 is less than −7. It works the same way when the numbers are not temperatures.
A dot is 'five away from zero'. Where could it be?
At −5 or at 5: two answers, one each way along the line. The two dots are double the distance apart.
Is there always a negative answer to 'n away from' a number?
No. Three away from 10 is 7 or 13.
How do you find the difference between a negative number and a positive number?
Start with the smaller number, find how far it is to zero, then how much further to the other number, and add them.
What is the usual wrong answer for the difference between 3 and −9, and what is the right one?
The wrong answer is 6 (taking 3 from 9 and ignoring zero). The right answer is 12.
How do you find the number exactly halfway between two numbers?
Find the distance between them, halve it, and move that far from either number.
In a list of temperatures, where is the greatest difference?
Between the warmest and the coldest.
What does a negative account balance show?
Money owed.
How do you find a new balance when you spend more than you have?
Start at the balance, subtract to reach zero, then subtract the rest.

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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