Paper Together: sit a paper in a room with your friends.See how it works
This page sets out the method for negative numbers in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Negative numbers topic page.
When you see + or − with negative numbers, first tidy any double signs, then move along a number line.
When you see × or ÷ with negatives, deal with the numbers and the sign separately.
Zero is the marker. Positives go right, negatives go left.
−1 is one step left of zero. −4 is four steps left.
−8 is smaller than −3 because it is further left.
The digit looks bigger, but the sign and the direction decide.
Adding moves right. −2 + 5 starts at −2 and takes 5 steps right, landing on 3.
You can think of it as the gap from −2 up to 0 (which is 2) and then 3 more.
Subtracting moves left. 1 − 4 starts at 1 and takes 4 steps left, landing on −3.
Going past zero is allowed. Do not stop at zero unless the sum says so.
Adding a negative is a move left. 4 + (−3) is the same move as 4 − 3.
The brackets show the negative is one number, not a subtraction sign by itself.
The difference between −2 and 3 is the number of steps from one to the other: 5.
Count the steps. Do not subtract the faces and hope the sign works out.
Draw or imagine the line with zero in the middle. Mark the positives to the right in the usual order: 1, 2, 3. Mark the negatives to the left, and notice that the labels grow as you walk away from zero: -1 is next to zero, then -2, then -3. Walking left, the numbers get smaller even though the digits get bigger.
To put -3, 1, -8 and 0 in order, start at the leftmost. -8 is furthest left, then -3, then 0, then 1. Smallest first is -8, -3, 0, 1. Largest first is the same list reversed. A question that says smallest first scores the list in that direction only.
Words follow the line. Colder means smaller for temperatures in degrees Celsius. A floor labelled -2 in a car park is below floor 0. A diver at -6 metres is deeper than a diver at -2 metres if down is negative. Compare the positions, then answer in the words the question used.
Method: 1. Put zero in the middle of a number line. 2. Positives go to the right and negatives go to the left. 3. The further left a number is, the smaller it is.
Use the line for a short calculation until the sign pattern is familiar. -7 + 12 starts at -7 and takes 12 steps right. Seven of those steps reach zero, and the other five land on 5. So -7 + 12 = 5. A rise in temperature is this kind of move. A fall is a move left: -6 - 3 starts at -6 and goes three steps further left, to -9.
Brackets around a negative show that the sign belongs to the number. 4 + (-9) means start at 4 and add a move of -9, which is nine steps left, landing on -5. It is the same calculation as 4 - 9. 4 - (-9) means take away a move of -9. Taking away a leftward move is a rightward move of 9, so 4 - (-9) = 13.
Do a chain from the left, one operation at a time. -9 - (-4) + 6: first -9 - (-4) = -9 + 4 = -5, then -5 + 6 = 1. Do not add all the digits and stick one sign on the front. The signs tell you the direction of each move. A check is to land on the line and count.
Method: 1. Start on the first number on the number line. 2. Adding moves right, and subtracting moves left. 3. Subtracting a negative is the same move as adding the positive.
Multiplication of a negative is repeated addition of that negative, so 3 × (-4) = (-4) + (-4) + (-4) = -12. The factors have different signs, and the product is negative. The same pattern runs the other way: (-3) × 4 = -12. Division follows the multiplication that would undo it. (-12) ÷ 4 = -3 and (-12) ÷ (-4) = 3.
Two negatives give a positive. (-3) × (-8) = 24. One way to see it: -3 times 8 would be -24, and the extra negative sign reverses that direction, so the product is 24. The short rule is enough in a question: same signs positive, different signs negative. Work out the number part with ordinary times tables, then attach the sign.
The rule is about the two factors, not about how many minus signs you can see in a longer expression. In (-3) × (-8) + (-5) × 4, deal with each product on its own. (-3) × (-8) = 24 and (-5) × 4 = -20. The expression becomes 24 + (-20), which is 4. A later subtraction of a negative is a separate step.
Method: 1. Same signs in a product or quotient give a positive answer. 2. Different signs give a negative answer. 3. Work out the size first, then attach the sign.
The difference between two temperatures is the number of degrees from one to the other. From -8^°C to 6^°C, count 8 degrees from -8 up to 0, then 6 degrees from 0 up to 6. The gap is 8 + 6 = 14 degrees. The calculation 6 - (-8) does the same job, because subtracting a negative adds.
Do not subtract the digits and hope. 6 - 8 = -2 is the wrong gap. It drops the minus sign on -8, so the count never crosses zero. The readings are on opposite sides of zero, so both sides of the gap count. A difference of temperatures is positive unless the question asks for a signed change.
A signed change is a different question. 'Falls by 4 degrees' from -5^°C is -5 - 4 = -9^°C. 'Rises by 4 degrees' is -5 + 4 = -1^°C. 'How much warmer is 6 than -8?' is the positive gap, 14 degrees. Read whether the answer is a new temperature or a difference before you start.
Method: 1. Count the degrees from the lower temperature up to zero. 2. Then count from zero up to the higher temperature. 3. Add those two parts to get the difference.
Exam questions write negative numbers in brackets, such as (-4) × (-3) or 6 × (-5). Treat the sign and the size separately. Two numbers with the same sign multiply or divide to give a positive answer. Two numbers with different signs give a negative answer. So (-4) × (-3) = 12, but 6 × (-5) = -30 and (-48) ÷ 8 = -6.
When there are several operations in one line, follow the order of operations: brackets, then multiplication and division, then addition and subtraction. In 8 - 3 × (-5) the multiplication comes first, 3 × (-5) = -15, and then 8 - (-15) = 8 + 15 = 23. Rewrite each double sign, such as minus a negative, as a single sign before you add up.
Method: 1. Circle the multiplication and division and do those first. 2. Decide the sign of each product from the sign rule, then work out its size. 3. Rewrite "minus a negative" as plus, and "plus a negative" as minus. 4. Finish the addition and subtraction using a number line if it helps.
Tip: Say each sign out loud: "minus times minus is plus". Saying the rule slows you down enough to avoid a slip.
To order a list that includes negatives, picture a number line. The further left a number is, the smaller it is. That means -9 is smaller than -6, even though 9 is bigger than 6. A question might say "Start with the coldest temperature", which means start with the smallest number, the most negative one.
Lists may also have decimals, such as -2.1, 0.4, -5 and 1. Put the negatives first and order them by size, remembering that -5 is less than -2.1. Then add the zero if there is one, and order the positives. A quick check is to tick off each number from your answer as you read the question. The final list must contain exactly the numbers you were given, and none should be left out.
Method: 1. Sort the numbers into negatives, zero and positives. 2. Order the negatives, with the biggest-looking one first. 3. Then write zero, if there is one, and the positives in increasing order. 4. Count the numbers to check you have not dropped one.
Questions on change use the words "rises", "falls", "difference" and "how much colder". A fall of 11 degrees from 4^°C is 4 - 11 = -7. The difference between two temperatures is found by subtracting the smaller from the larger: the difference between 4^°C and -10^°C is 4 - (-10) = 14 degrees. Read the question carefully to see whether it asks for a final value or for a difference.
A sketch of a vertical number line, like a thermometer, shows the steps. Count up from the lower number to zero, then up to the higher number: from -10 to 0 is 10 and from 0 to 4 is 4, so the gap is 14. The same method works for heights above and below sea level, floors in a building that includes basement levels, and bank balances that are overdrawn. Whichever context it is, finish by writing the unit, such as degrees, metres or pounds.
Method: 1. Draw a quick vertical number line and mark each value. 2. Write each change as an addition for a rise and a subtraction for a fall. 3. For a difference, count from the lower value up to zero, then up to the higher value. 4. Give the answer with its unit and sign where the sign matters.
Tip: A difference is a distance, so it is never negative. A final temperature can be.
A sequence that decreases by the same amount continues past zero into the negatives. For 14, 8, 2, -4 the rule is "subtract 6", so the next terms are -4 - 6 = -10 and -10 - 6 = -16. Do not stop at zero: the pattern is exactly the same on the other side. Write each new term down before you work out the next one, so a slip is easy to spot.
A missing number question looks like square - (-4) = 9. Work backwards with the inverse operation. Subtracting a negative is the same as adding, so square + 4 = 9 and the missing number is 5. Always put your answer back into the question to check that it works: 5 - (-4) = 5 + 4 = 9. This takes only a few seconds and catches the sign mistakes that cost the accuracy mark.
Method: 1. Find the gap between neighbouring terms, including its sign. 2. Apply the same gap again to get the next terms. 3. For a missing number, write the question as an equation and use the inverse. 4. Put the answer back into the original line to check.
AQA-style and Edexcel-style questions. Yumio is not affiliated with, endorsed by or connected to AQA, Pearson Edexcel or any other exam board.