KS3 · Maths

Adding and subtracting positive and negative integers

Here is a sum that sounds impossible: you add something, and the answer gets smaller. It is allowed — and once you can see why, subtracting stops being scary too.

Where does the answer land?

Drag each tag to where you think its answer sits, then check. Clue: a number’s distance from zero is its absolute value — 5 and −5 are both 5 away. So which part of each sum is further from zero?

Why does that work? Counters

A yellow counter is worth +1 and a red counter is worth −1.

A pile holds 5 yellow counters and 3 red counters. Pair each red counter with a yellow one. Which integer does the pile stand for?

Turning subtractions into additions

Problem

Work out 4 − 9, then −4 − 9, then −4 − (−9).

Which idea sounds like yours?

Be honest — which of these is closest to what you thought before this lesson?
How sure are you?

Your turn: a longer calculation

Work out −3 + 7 − 4 − (−8) + (−1).

  1. First, rewrite every subtraction as an addition of the additive inverse. The subtractions here are − 4 and − (−8). The additions stay as they are.
  2. missing step
Which line is step 2?

Spot the slip

A bank balance is £75 and £134.95 is spent. What is the new balance?

A student’s answer — which line goes wrong?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Pair off the opposites, then see what is left.

What you need to know

  • Two integers that add to 0 are additive inverses, and together they make a zero pair, like a yellow and a red counter.
  • The absolute value of a number is its distance from zero, so 5 and −5 both have absolute value 5.
  • Adding a negative integer can make a number smaller; adding does not always mean bigger.
  • Any subtraction can be rewritten as adding the additive inverse, and then the addition rules decide the answer.

The big picture

A number and its additive inverse cancel to zero, and that one idea explains everything. Add two integers by pairing off opposites and seeing what is left, and turn every subtraction into an addition of the additive inverse.

Key points

1Same signs: add the absolute values and keep the sign, so −32 + (−45) = −77.
2Different signs: take the difference of the absolute values, and give the answer the sign of the integer with the greater absolute value.
3Rewrite 5 − 3 as 5 + (−3). Subtracting a negative is adding a positive: 10 − (−7) = 10 + 7.
4With several integers, total the positives and the negatives separately, then find the difference of those two totals.
5A larger amount taken from a smaller one gives a negative answer, such as a bank balance below zero.

Worked example

Problem

Work out 7 − 13, then −13 − 7, then −7 − (−13).

⚠ Watch out

Getting the size of the answer right but the sign wrong, for example saying −4 + 2 = 2. The −4 is further from zero than the 2, so the answer sits on the negative side: −2.

🧠

Memory hook

Opposites cancel — what is left is your answer. And a subtraction is just an addition of the opposite.

✓

Check yourself

Rewrite −6 − (−2) as an addition, then work it out. Before you calculate, say which part will be further from zero, and what that tells you about the sign.

Flashcards

(12)
What is the additive inverse of a number?
The number you add to it to get 0. The additive inverse of 100 is −100, and of −8 is 8.
What is a zero pair?
A number and its additive inverse, such as one yellow (+1) and one red (−1) counter. Together they make 0.
What is the absolute value of −5, and what does it measure?
5. Absolute value is distance from zero, so 5 and −5 are both 5 away.
What do yellow and red counters stand for, and how do you read the integer?
Yellow is +1 and red is −1. Pair off the zero pairs and see what is left.
Does adding always make the answer bigger?
No. Adding a negative integer can make it smaller: 3 + (−5) = −2.
How do you add two integers with the same sign?
Add the absolute values and keep the sign: −32 + (−45) = −77.
How do you add two integers with different signs?
Find the difference of the absolute values. The answer takes the sign of the integer with the greater absolute value: 32 + (−45) = −13.
How do you rewrite 5 − 3 as an addition?
5 + (−3): add the additive inverse. An addition such as 5 + 3 is left alone.
What is subtracting a negative number the same as?
Adding a positive number: 10 − (−7) = 10 + 7 = 17.
How do you work out −2 + 3 + 7 + (−4) + (−6)?
Total the positives (10) and the negatives (−12), then 10 + (−12) = −2.
A balance of £481.24 and a £575.75 bike: what is the new balance?
481.24 + (−575.75) = −£94.51. The larger amount on top gives 94.51; then remember the minus sign.
Why line up the decimal points in a column subtraction?
So digits with the same place value are combined with each other.

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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