Elite Learners

Home›11+›Maths›Number and place value›Number line dynamics and negative numbers

Number line dynamics and negative numbers

Interval gaps, midpoints, and temperature steps across zero.

Video coming soon The written summary and downloads below are complete and ready to use.

A number line is like a ruler for numbers. Smaller numbers are on the left and bigger numbers are on the right.

In this lesson you will learn:

  • how to read any number line by working out the gaps
  • how to find the number halfway between two numbers
  • what negative numbers are, and how to put them in order
  • how temperatures change when they go across zero

1. Reading a number line

Key idea Count the gaps, not the marks. Every gap on a number line is worth the same amount.

How to read a number line

  1. Find two numbers that are written on the line
  2. Work out the difference between them
  3. Count the gaps between them
  4. Share the difference between the gaps: difference ÷ gaps
    That is what each gap is worth. Now count on in those steps.

Example: what number is the arrow pointing to?

200225250275300+25+25+25+25
  • The difference is 300 − 200 = 100.
  • There are 4 gaps. 100 ÷ 4 = 25, so each gap is worth 25.
  • Count on: 200, 225, 250, 275.
Answer275
Watch out Between 200 and 300 there are 5 marks but only 4 gaps. Always count the gaps.

2. Halfway between two numbers

Key idea The number exactly halfway is the midpoint. Find the gap, halve it, and add it to the smaller number.

Example 1: halfway between 16 and 30

  1. The gap is 30 − 16 = 14
  2. Half of 14 is 7
  3. Add it to the smaller number: 16 + 7 = 23
Answer23

Check: 23 is 7 more than 16, and 7 less than 30.

Example 2: halfway between 2.4 and 3

  1. The gap is 0.6. Half of it is 0.3.
  2. 2.4 + 0.3 = 2.7
Answer2.7

Another way: add the two numbers and halve. (16 + 30) ÷ 2 = 46 ÷ 2 = 23.

3. Numbers below zero

Key idea Numbers below zero are negative numbers. We write them with a minus sign: −5 is said “minus five” or “negative five”.
  • On a number line, negative numbers are to the left of 0.
  • The further left a number is, the smaller it is.
  • Temperatures below freezing are negative. A freezer is about −18 °C.
Watch out −8 is smaller than −3, even though 8 is bigger than 3. −8 °C is colder than −3 °C: it is further below zero.

4. Temperature changes across zero

Key idea When a jump crosses zero, split it into two: the jump to zero, then the jump from zero.

Example 1: it is −6 °C in the morning and 5 °C at lunchtime. How much warmer is it?

  1. From −6 up to 0 is 6 degrees
  2. From 0 up to 5 is 5 degrees
  3. Add the jumps: 6 + 5 = 11
Answer11 degrees warmer

Example 2: it is 4 °C. The temperature falls by 9 degrees. What is it now?

  1. Falling 4 degrees takes it down to 0
  2. That leaves 9 − 4 = 5 more degrees to fall
  3. 5 below zero is −5
Answer−5 °C
Watch out The difference between −6 and 5 is 11, not 1. Don’t just take one number away from the other: count the jumps on the line.

The mistakes that cost marks

  1. Counting the marks instead of the gaps. count the spaces between the marks.
  2. Guessing that each gap is worth 1, or 10. work it out: difference ÷ number of gaps.
  3. Thinking −8 is bigger than −3. remember the further left, the smaller. −8 is colder.
  4. Taking one number from the other across zero. The difference between −6 and 5 is not 1. jump to zero, then on: 6 + 5 = 11.
  5. Forgetting the minus sign. 5 degrees below zero is not 5 °C. write −5 °C.
  6. Halving the wrong thing for halfway. halve the gap, then add it to the smaller number.

Quick checklist

Tick each one when you can do it without help.

Download

Print these and work through them away from the screen.

Year 4 practice paper PDF · 11+ style, with answer sheet and answers Year 5 practice paper PDF · 11+ style, with answer sheet and answers Year 6 practice paper PDF · 11+ style, with answer sheet and answers

All of this is free

Every 11+ maths resource on this site is free, with no account needed. Taught sessions are currently GCSE maths only — if you would like a group opened for this subject, join the waiting list and it helps me decide what to run next.

Join the waiting list →