Year 4: Negative numbers: counting back through zero
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Negative numbers: counting back through zero
It can be so cold that the temperature goes below zero. A temperature of three degrees below zero is written −3°C. Numbers below zero are called negative numbers. They have a minus sign in front. Look at the thermometer: the liquid is below the 0 line.
On a number line, zero is in the middle. Numbers to the right are positive and get bigger. Numbers to the left are negative. The further left a number is, the smaller it is. So −9 is smaller than −2.
Counting back through zero. If you count back in 3s from 9, you say 6, 3, 0, −3, −6… After 0 the numbers keep going down, but they are negative. When a jump crosses zero, split it. To work out 2 − 5, go down 2 to reach 0, then 3 more to reach −3.
The gap between two temperatures. To find how many degrees from −4°C to 5°C, count up to 0 (4 degrees) and then up to 5 (5 more degrees): 4 + 5 = 9 degrees.
Key words. A negative number is a number less than zero. Degrees Celsius (°C) measure temperature. Zero is neither positive nor negative.
Key ideas
- Negative numbers are less than zero and have a minus sign: −1, −2, −3…
- On a number line, the further left, the smaller the number.
- Counting back through zero: … 2, 1, 0, −1, −2 …
- To cross zero, split the jump into the part that reaches 0 and the part that goes past it.
- The gap between a negative and a positive number: add the two distances from zero.
Key words. A negative number is less than zero. It has a minus sign in front, like −4. Think of temperatures below freezing.
Method. Counting back in 3s means taking away 3 each time. A number line helps when we go through zero.
- Draw a number line. Numbers to the left of 0 are negative (below zero). Numbers to the right are positive.
- Jump back 3 each time, starting from 12. When we pass zero the numbers become negative.12 − 3 = 9
9 − 3 = 6
6 − 3 = 3
3 − 3 = 0
0 − 3: go down 0 to reach 0, then 3 more below zero → −3
−3 − 3: start below zero and go down 3 more → −6 - Write the numbers in order. 9, 6, 3, 0, −3, −6
Check. Go the other way from the last number: −6 + 3 = −3. ✓
Watch out. After 0 comes −1, not 1. The numbers below zero count down: −1, −2, −3...
Method. Draw a number line, start at 4 and jump down 7. When you pass 0 the numbers change between positive and negative.
- Show the start on a number line. This lets us see where zero is.
- Go down 7. Split the jump into two parts: the part that reaches 0, then the part that goes past it.4 − 7: go down 4 to reach 0, then 3 more below zero → −3
- Write the answer. −3°C
Check. Do the opposite: −3 + 7 = 4. ✓
Watch out. Going down past zero gives a negative number. Going up from a negative number makes it closer to zero.
Method. On a number line, numbers further to the left are smaller. Mark each number, then read them off from left to right.
- Put the numbers on a number line. The line shows which are below zero.
- Read from left to right. Left is smallest, right is biggest.−8, −5, 0, 2, 3
- Write them smallest first. −8, −5, 0, 2, 3
Check. The numbers below zero are the smallest. Of those, the one with the biggest digit (like −9) is the furthest from zero, so it is the smallest. ✓
Watch out. −9 is smaller than −2. The further below zero, the smaller the number.
Method. The difference between two numbers is how far apart they are on a number line.
- Mark both numbers on a number line. Seeing the gap helps.
- Count the gap. One number is below zero and one is above. Count up to zero first, then on to the other number.−4 up to 0 is 4.
0 up to 5 is 5.
4 + 5 = 9 - Write the answer. 9°C
Check. Start at −4 and go up 9: −4 + 9 = 5. ✓
Watch out. Do not just subtract the digits when one number is negative. Go through zero: −4 and 5 are 9 apart, not 1.
Method. Find the numbered mark nearest the top of the red liquid, then count the small marks to the top.
- Look at the numbers on the scale. The scale is numbered every 5 degrees. The small marks are 1 degree each.Big numbers: ..., −5, 0, 5, 10, ...
- Find the numbered mark just below the top of the red. The top of the red is between −5 and 0.The mark below is −5.
- Count the small marks above it. Each small mark is 1 degree.2 small marks above −5: −5 + 2 = −3
Check. Count from 0 instead: the top of the red is 3 small marks below 0, so the temperature is −3°C. ✓
Watch out. Going up the scale the numbers get bigger. Below zero, the number nearest the bottom is the smallest.
Watch out
The number with the bigger digit is not always the bigger number. −9 is smaller than −2, because it is further below zero. And after 0 comes −1, not 1. A number line helps you check.Exercise 1E Negative numbers
Non-calculatorMethod. Find the numbered mark nearest the top of the red liquid, then count the small marks to the top.
- Look at the numbers on the scale. The scale is numbered every 5 degrees. The small marks are 1 degree each.Big numbers: ..., −5, 0, 5, 10, ...
- Find the numbered mark just below the top of the red. The top of the red is between −5 and 0.The mark below is −5.
- Count the small marks above it. Each small mark is 1 degree.2 small marks above −5: −5 + 2 = −3
Check. Count from 0 instead: the top of the red is 3 small marks below 0, so the temperature is −3°C. ✓
Watch out. Going up the scale the numbers get bigger. Below zero, the number nearest the bottom is the smallest.
Method. Each small step on the line is worth 1. Count from a number you can read.
- Check what each step is worth. The labelled numbers are 5 apart and there are 5 gaps between them, so each step is 1.5 ÷ 5 = 1
- Work out each missing number. Start at the nearest labelled number. Counting up makes the number bigger. Counting down makes it smaller.−8: start at −10 and count 2 steps up → −8
−3: start at −5 and count 2 steps up → −3
2: start at 0 and count 2 steps up → 2
7: start at 5 and count 2 steps up → 7 - Write the numbers from left to right. −8, −3, 2, 7
Check. The numbers go up by 1 each time, including 0 in the middle: −10, −9 … 0 … 9, 10. ✓
Watch out. Left of zero the numbers get smaller as you go left. Do not count the wrong way!
Key words. A negative number is less than zero. It has a minus sign in front, like −4. Think of temperatures below freezing.
Method. Counting back in 2s means taking away 2 each time. A number line helps when we go through zero.
- Draw a number line. Numbers to the left of 0 are negative (below zero). Numbers to the right are positive.
- Jump back 2 each time, starting from 7. When we pass zero the numbers become negative.7 − 2 = 5
5 − 2 = 3
3 − 2 = 1
1 − 2: go down 1 to reach 0, then 1 more below zero → −1
−1 − 2: start below zero and go down 2 more → −3 - Write the numbers in order. 5, 3, 1, −1, −3
Check. Go the other way from the last number: −3 + 2 = −1. ✓
Watch out. After 0 comes −1, not 1. The numbers below zero count down: −1, −2, −3...
Key words. A negative number is less than zero. It has a minus sign in front, like −4. Think of temperatures below freezing.
Method. Counting back in 5s means taking away 5 each time. A number line helps when we go through zero.
- Draw a number line. Numbers to the left of 0 are negative (below zero). Numbers to the right are positive.
- Jump back 5 each time, starting from 20. When we pass zero the numbers become negative.20 − 5 = 15
15 − 5 = 10
10 − 5 = 5
5 − 5 = 0
0 − 5: go down 0 to reach 0, then 5 more below zero → −5 - Write the numbers in order. 15, 10, 5, 0, −5
Check. Go the other way from the last number: −5 + 5 = 0. ✓
Watch out. After 0 comes −1, not 1. The numbers below zero count down: −1, −2, −3...
Key words. A negative number is less than zero. It has a minus sign in front, like −4. Think of temperatures below freezing.
Method. Counting on in 3s means adding 3 each time. A number line helps when we go through zero.
- Draw a number line. Numbers to the left of 0 are negative (below zero). Numbers to the right are positive.
- Jump on 3 each time, starting from −10. When we pass zero the numbers become negative.−10 + 3: start below zero and go up 3 → −7
−7 + 3: start below zero and go up 3 → −4
−4 + 3: start below zero and go up 3 → −1
−1 + 3: go up 1 to reach 0, then 2 more above zero → 2
2 + 3 = 5 - Write the numbers in order. −7, −4, −1, 2, 5
Check. Go the other way from the last number: 5 − 3 = 2. ✓
Watch out. After 0 comes −1, not 1. The numbers below zero count down: −1, −2, −3...
Key words. A negative number is less than zero. It has a minus sign in front, like −4. Think of temperatures below freezing.
Method. Counting back in 10s means taking away 10 each time. A number line helps when we go through zero.
- Draw a number line. Numbers to the left of 0 are negative (below zero). Numbers to the right are positive.
- Jump back 10 each time, starting from 40. When we pass zero the numbers become negative.40 − 10 = 30
30 − 10 = 20
20 − 10 = 10
10 − 10 = 0
0 − 10: go down 0 to reach 0, then 10 more below zero → −10
−10 − 10: start below zero and go down 10 more → −20 - Write the numbers in order. 30, 20, 10, 0, −10, −20
Check. Go the other way from the last number: −20 + 10 = −10. ✓
Watch out. After 0 comes −1, not 1. The numbers below zero count down: −1, −2, −3...
Method. Put both numbers on a number line. The number further to the right is greater.
- Mark both numbers on a number line. Negative numbers go to the left of zero.
- Find which is further to the right. Further right means bigger.2 is to the right of −3.
- Write the symbol. −3 < 2
Check. In temperatures, −3°C is colder than 2°C. ✓
Watch out. A negative number with a big digit is a small number. −10 is less than −2.
Method. Put both numbers on a number line. The number further to the right is greater.
- Mark both numbers on a number line. Negative numbers go to the left of zero.
- Find which is further to the right. Further right means bigger.−4 is to the right of −7.
- Write the symbol. −7 < −4
Check. In temperatures, −7°C is colder than −4°C. ✓
Watch out. A negative number with a big digit is a small number. −10 is less than −2.
Method. Put both numbers on a number line. The number further to the right is greater.
- Mark both numbers on a number line. Negative numbers go to the left of zero.
- Find which is further to the right. Further right means bigger.0 is to the right of −1.
- Write the symbol. 0 > −1
Check. In temperatures, −1°C is colder than 0°C. ✓
Watch out. A negative number with a big digit is a small number. −10 is less than −2.
Method. Put both numbers on a number line. The number further to the right is greater.
- Mark both numbers on a number line. Negative numbers go to the left of zero.
- Find which is further to the right. Further right means bigger.−10 is to the right of −12.
- Write the symbol. −10 > −12
Check. In temperatures, −12°C is colder than −10°C. ✓
Watch out. A negative number with a big digit is a small number. −10 is less than −2.
Method. Put both numbers on a number line. The number further to the right is greater.
- Mark both numbers on a number line. Negative numbers go to the left of zero.
- Find which is further to the right. Further right means bigger.5 is to the right of −5.
- Write the symbol. −5 < 5
Check. In temperatures, −5°C is colder than 5°C. ✓
Watch out. A negative number with a big digit is a small number. −10 is less than −2.
Method. Put both numbers on a number line. The number further to the right is greater.
- Mark both numbers on a number line. Negative numbers go to the left of zero.
- Find which is further to the right. Further right means bigger.−8 is to the right of −18.
- Write the symbol. −8 > −18
Check. In temperatures, −18°C is colder than −8°C. ✓
Watch out. A negative number with a big digit is a small number. −10 is less than −2.
Method. On a number line, numbers further to the left are smaller. Mark each number, then read them off from left to right.
- Put the numbers on a number line. The line shows which are below zero.
- Read from left to right. Left is smallest, right is biggest.−8, −5, 0, 2, 3
- Write them smallest first. −8, −5, 0, 2, 3
Check. The numbers below zero are the smallest. Of those, the one with the biggest digit (like −9) is the furthest from zero, so it is the smallest. ✓
Watch out. −9 is smaller than −2. The further below zero, the smaller the number.
Method. On a number line, numbers further to the left are smaller. Mark each number, then read them off from left to right.
- Put the numbers on a number line. The line shows which are below zero.
- Read from left to right. Left is smallest, right is biggest.−12, −7, −2, 7, 12
- Write them smallest first. −12, −7, −2, 7, 12
Check. The numbers below zero are the smallest. Of those, the one with the biggest digit (like −9) is the furthest from zero, so it is the smallest. ✓
Watch out. −9 is smaller than −2. The further below zero, the smaller the number.
Method. On a number line, numbers further to the left are smaller. Mark each number, then read them off from left to right.
- Put the numbers on a number line. The line shows which are below zero.
- Read from left to right. Left is smallest, right is biggest.−41, −14, −4, 4, 14
- Write them biggest first. 14, 4, −4, −14, −41
Check. The numbers below zero are the smallest. Of those, the one with the biggest digit (like −9) is the furthest from zero, so it is the smallest. ✓
Watch out. −9 is smaller than −2. The further below zero, the smaller the number.
Method. Draw a number line, start at 5 and jump down 8. When you pass 0 the numbers change between positive and negative.
- Show the start on a number line. This lets us see where zero is.
- Go down 8. Split the jump into two parts: the part that reaches 0, then the part that goes past it.5 − 8: go down 5 to reach 0, then 3 more below zero → −3
- Write the answer. −3°C
Check. Do the opposite: −3 + 8 = 5. ✓
Watch out. Going down past zero gives a negative number. Going up from a negative number makes it closer to zero.
Method. Draw a number line, start at −2 and jump up 6. When you pass 0 the numbers change between positive and negative.
- Show the start on a number line. This lets us see where zero is.
- Go up 6. Split the jump into two parts: the part that reaches 0, then the part that goes past it.−2 + 6: go up 2 to reach 0, then 4 more above zero → 4
- Write the answer. 4°C
Check. Do the opposite: 4 − 6 = −2. ✓
Watch out. Going down past zero gives a negative number. Going up from a negative number makes it closer to zero.
Method. Draw a number line, start at −4 and jump down 3. When you pass 0 the numbers change between positive and negative.
- Show the start on a number line. This lets us see where zero is.
- Go down 3. Split the jump into two parts: the part that reaches 0, then the part that goes past it.−4 − 3: start below zero and go down 3 more → −7
- Write the answer. −7°C
Check. Do the opposite: −7 + 3 = −4. ✓
Watch out. Going down past zero gives a negative number. Going up from a negative number makes it closer to zero.
Method. Draw a number line, start at −9 and jump up 9. When you pass 0 the numbers change between positive and negative.
- Show the start on a number line. This lets us see where zero is.
- Go up 9. Split the jump into two parts: the part that reaches 0, then the part that goes past it.−9 + 9: start below zero and go up 9 → 0
- Write the answer. 0°C
Check. Do the opposite: 0 − 9 = −9. ✓
Watch out. Going down past zero gives a negative number. Going up from a negative number makes it closer to zero.
Method. Find which temperature is the furthest below zero. That is the coldest.
- Read each temperature. Each thermometer shows one city.Oslo −8°C, Leeds 3°C, Moscow −14°C
- Order them from coldest to warmest. The lowest number is the coldest. Of the negative numbers, −14 is the furthest below zero.−14 < −8 < 3
- Write the answer. Moscow
Check. Moscow at −14°C is 6 degrees colder than Oslo at −8°C. ✓
Watch out. −14 is colder than −8, because it is further below zero.
Method. The difference between two numbers is how far apart they are on a number line.
- Mark both numbers on a number line. Seeing the gap helps.
- Count the gap. One number is below zero and one is above. Count up to zero first, then on to the other number.−8 up to 0 is 8.
0 up to 3 is 3.
8 + 3 = 11 - Write the answer. 11°C
Check. Start at −8 and go up 11: −8 + 11 = 3. ✓
Watch out. Do not just subtract the digits when one number is negative. Go through zero: −4 and 5 are 9 apart, not 1.
Method. The difference between two numbers is how far apart they are on a number line.
- Mark both numbers on a number line. Seeing the gap helps.
- Count the gap. Both numbers are below zero. Find how far each is from zero, then take the smaller from the bigger.−14 is 14 below zero and −8 is 8 below zero.
14 − 8 = 6 - Write the answer. 6°C
Check. Start at −14 and go up 6: −14 + 6 = −8. ✓
Watch out. Do not just subtract the digits when one number is negative. Go through zero: −4 and 5 are 9 apart, not 1.
Method. The difference between two numbers is how far apart they are on a number line.
- Mark both numbers on a number line. Seeing the gap helps.
- Count the gap. One number is below zero and one is above. Count up to zero first, then on to the other number.−2 up to 0 is 2.
0 up to 7 is 7.
2 + 7 = 9 - Write the answer. 9°C
Check. Start at −2 and go up 9: −2 + 9 = 7. ✓
Watch out. Do not just subtract the digits when one number is negative. Go through zero: −4 and 5 are 9 apart, not 1.
Method. The difference between two numbers is how far apart they are on a number line.
- Mark both numbers on a number line. Seeing the gap helps.
- Count the gap. Both numbers are below zero. Find how far each is from zero, then take the smaller from the bigger.−8 is 8 below zero and −3 is 3 below zero.
8 − 3 = 5 - Write the answer. 5°C
Check. Start at −8 and go up 5: −8 + 5 = −3. ✓
Watch out. Do not just subtract the digits when one number is negative. Go through zero: −4 and 5 are 9 apart, not 1.
Method. The difference between two numbers is how far apart they are on a number line.
- Mark both numbers on a number line. Seeing the gap helps.
- Count the gap. One number is below zero and one is above. Count up to zero first, then on to the other number.−6 up to 0 is 6.
0 up to 4 is 4.
6 + 4 = 10 - Write the answer. 10°C
Check. Start at −6 and go up 10: −6 + 10 = 4. ✓
Watch out. Do not just subtract the digits when one number is negative. Go through zero: −4 and 5 are 9 apart, not 1.
Method. The difference between two numbers is how far apart they are on a number line.
- Mark both numbers on a number line. Seeing the gap helps.
- Count the gap. One number is below zero and one is above. Count up to zero first, then on to the other number.−9 up to 0 is 9.
0 up to 5 is 5.
9 + 5 = 14 - Write the answer. 14°C
Check. Start at −9 and go up 14: −9 + 14 = 5. ✓
Watch out. Do not just subtract the digits when one number is negative. Go through zero: −4 and 5 are 9 apart, not 1.
Method. Draw a number line, start at 2 and jump down 5. When you pass 0 the numbers change between positive and negative.
- Show the start on a number line. This lets us see where zero is.
- Go down 5. Split the jump into two parts: the part that reaches 0, then the part that goes past it.2 − 5: go down 2 to reach 0, then 3 more below zero → −3
- Write the answer. −3
Check. Do the opposite: −3 + 5 = 2. ✓
Watch out. Going down past zero gives a negative number. Going up from a negative number makes it closer to zero.
Method. Do the moves one at a time, using a number line. Going up makes the number bigger. Going down makes it smaller.
- Start position. The diver is at −12 m.−12 m
- Swim up 5 m. Going up adds 5.−12 + 5: start below zero and go up 5 → −7 m
- Dive down 9 m. Going down takes away 9.−7 − 9: start below zero and go down 9 more → −16 m
- Write the answer. −16 m
Check. Total change = +5 − 9 = −4. −12 − 4 = −16. ✓
Watch out. Take the moves in order, and watch which way each one goes.
Method. Put both numbers on a number line. The number further to the right is bigger.
- Draw a number line. Mark −9 and −2.
- Compare their positions. −9 is further to the left.−9 < −2
- Answer Priya. A number with a bigger digit is smaller if it is negative, because it is further below zero.No. −9 is less than −2.
Check. It is colder at −9°C than at −2°C. ✓
Watch out. Think of temperatures. The bigger the number after the minus sign, the colder it is.
“A two-digit negative number is smaller than a one-digit negative number.”
Method. Think about the smallest two-digit negative number you can find, then the biggest. Then compare with one-digit negative numbers.
- List the one-digit negative numbers. They run from −1 down to −9.−1, −2, … −9
- Find the biggest two-digit negative number. The closest to zero is −10.−10
- Compare. −10 is further left than −9, the smallest one-digit negative number. All the other two-digit negatives are further left still.−10 < −9, so every two-digit negative is smaller.
- Write the answer. Always
Check. For example −35 < −4, and −99 < −1. ✓
Watch out. This is only true for negative numbers. For positive numbers, a two-digit number is bigger.
Method. Colder means lower on the thermometer. Find the temperatures that are less than −6.
- Mark −6 on a number line. Anything to the left of −6 is colder.
- Check each temperature. Is it less than −6?3°C: not colder (it is warmer or equal)
−9°C: colder
−2°C: not colder (it is warmer or equal)
−6°C: not colder (it is warmer or equal)
−10°C: colder - Write the answer. −9°C and −10°C
Check. −6°C itself is not colder than −6°C. ✓
Watch out. −6 is not colder than −6. They are the same.
Method. Add 3 for each hour.
- After 1 hour. Go up 3.−4 + 3: start below zero and go up 3 → −1
- After 2 hours. Go up 3 again.−1 + 3: go up 1 to reach 0, then 2 more above zero → 2
- Write the answer. 2°C
Check. −4 + 6 = 2. ✓
Watch out. Go up 3 from the new temperature each hour.
Method. Find the first hour when the temperature is greater than 0.
- List the temperatures. Add 3 each hour.Hour 0: −4°C
Hour 1: −1°C
Hour 2: 2°C - Find the first one above 0. At hour 1 it is still below zero.Hour 2 is the first time it is above 0°C.
- Write the answer. 2 hours
Check. At 1 hour it is −1°C, which is below 0. At 2 hours it is 2°C. ✓
Watch out. 0 is not above 0. Wait until the number is positive.
Method. Find the temperature after 3 hours, then take away 5.
- Temperature after 3 hours. It was 2 after 2 hours. Go up 3 more.2 + 3 = 5
- Fall by 5. Go down 5 from 5.5 − 5 = 0
- Write the answer. 0°C
Check. Total change: +9 − 5 = +4. −4 + 4 = 0. ✓
Watch out. Zero is neither positive nor negative.