Subtracting Negative Numbers: 4.5
Supporting materials
Indicator of Progress
Success depends on students being able to add and subtract positive and negative numbers, and to give a meaning to these operations.
The most difficult aspect is to subtract a negative number. Earlier, students can add positive and negative numbers, and subtract positive numbers, but cannot make sense of subtracting negative numbers. They interpret subtraction as 'taking away' and it makes no sense to them to 'take away' a negative number of things. Success depends on being able to give appropriate meanings to the subtraction operation.
Students who do not have a useful model for subtraction of negative numbers will have to focus on memorising rules without understanding, with consequent higher chances of forgetting. They may see mathematics as arbitrary and unreasonable.
Illustration 1: Subtraction as 'take away' and as 'difference between'
All of the arithmetic operations have several real world meanings.
For example, 25 − 8 can be interpreted as:
- take away Example: I have 25 objects, but I lose 8. Then I have 17 left.
- difference between Example: Kim is 25 and Tim is 8. There is a difference of 17 years between their ages.
If students only have the 'take away' meaning for subtraction, then subtracting a negative number will be meaningless. For example I have 25 objects, but I lose (-8) is an apparently meaningless question and is no help to work out 25 − (-8). Instead, 25 − (-8) is best interpreted as the difference between 25 and (-8).
To find out if students can interpret subtraction as 'difference between' observe their responses when they work out 103 − 95. If they count up from 95, or back from 103, they are using ‘difference between’ e.g. add 5 to get to 100, add 3 to get to 103, altogether I added 8.
If they take away 95, it is likely that they have not realised the fact that ‘difference between’ will provide them with an answer to problems like this (e.g. 103 take away 90 is 13, then take away another 5 to get 8).
Teaching Strategies
A good teaching strategy will develop meaning from a model. Models assist students by providing a tool to support their thinking. Careful links need to be made between symbolic work and work with the models. There are 5 suggestions below, but it is not intended that they should all be used. Use one thoroughly, and draw on others to assist students with persistent problems.
Activity 1: Patterns derives meaning not from a physical model but by generalising a number pattern.
Activity 2: Subtraction as 'difference between' models subtraction on a number line.
Activity 3: Postie stories and Activity 4: Pairs of Opposites and Activity 5: Walking on a number line present three additional models that can help students give meaning to 'subtracting negatives'.
Activity 6: Calculator buttons is a reminder to show students how to input negative numbers and subtraction on a calculator.
Activity 1: Patterns
Since mathematics is a language of pattern, we can often convince people that subtracting negatives has the same effect as adding a positive by showing these operations as the extension of a pattern. For example, start at 9 - 5 and gradually reduce what is being subtracted. A pattern develops that the answers are increasing. Continue this pattern into subtraction of negatives.
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Observing this pattern does not make a proof, but it can convince students of the answers. Mathematicians prove this result by logical deduction from basic properties of numbers.
It is easy to see from the table that subtracting a negative is the same as adding its inverse
e.g. 9 - (-2) = 11 = 9 + 2.
Activity 2: Subtraction as 'difference between'
In this approach, subtraction is interpreted as the difference between two numbers on a number line.
Step 1. Interpret subtraction of positive numbers on the number line as “difference between”.
The diagram above shows that the difference between 6 and 2 is 4 ie 6 – 2 = 4
Step 2. Extend the 'difference between' idea to negative numbers.
The diagram above shows that the difference between 6 and (-1) is 7 ie 6 – (-1) = 7.
Step 3. Think about whether the difference itself is positive or negative.
The difference is positive if the first number is the greater, and negative if it is smaller. So 6 - 2 = 4 and 2 - 6 = -4.
So 2 - 6 = (-4) and (-1) – 6 = (-7)
Students can observe for themselves that 6 – (-1) = 7 and 6 + 1 = 7.
This example indicates that subtracting a number is equivalent to adding its opposite (additive inverse).
Activity 3: Postie stories
In this imaginary scenario, a postie delivers cheques (positive money) and bills (negative money). This easily portrays addition of integers. Subtraction occurs when the postie shows up and takes away cheques or bills that have been wrongly delivered to you.
For example, if the postie takes away a $20 cheque that has been incorrectly delivered to you, then you will have $20 less than you thought. If you thought you had $50, once the postie takes away the $20 cheque you only have $30 (i.e. 50 - 20).
On the other hand, if the postie takes away a $20 bill that has been incorrectly delivered to you, then you will have $20 more than you thought. If you thought you had $50, once the postie takes away the $20 bill you will have $70 (i.e. 50 - -20).
Hence, subtracting a negative number (ie taking away a bill of $20) has the same effect as adding a positive number (ie receiving a cheque of $20).
Activity 4 : Pairs of Opposites
Opposites, in this context, are numbers of things that 'cancel each other out' when put together. ‘Opposites adding to zero’ is a fundamental property of the set of integers, so anything with this property models integers. Let us use black circles for positive, and white for negative; either way would do, as long as there is consistency.
Using the conventions described, the diagrams below are all representations of +2. The pairs of opposites, with value zero, have been ringed.
We cannot take away negative 3 (3 whites) from the first three, but we can from the fourth example. When we do so we are left with two black circles, or positive 2.
To work a subtraction problem like that from scratch, we need to follow a few steps:
- Add enough pairs of opposites (zeros) so that we can then take away what we have to subtract.
- Take away the ones to be subtracted.
- Notice that the effect is the same as adding the same number of the opposites (which we actually have added when adding the zeros because they are pairs of opposites).
Eventually students will perceive that it is more efficient to do the subtraction problem by just adding the opposites of what we are to subtract.
This annihilation model is good for many students because it uses the 'take away' meaning of subtraction. The original number is put into a form where (-3) (i.e. 3 whites) can be taken away.
Activity 5: Walking on a number line
This model interprets some numbers as points on a number line, and other numbers as movements along a number line. Students can establish that, for any starting number (positive or negative):
- Adding any positive number will move to the right from the first number
- Adding any negative number will move to the left from the first number
| For example, | |
| 8 + 3 = 11 | starting at point 8 and moving along 3 units to the right to reach point 11 |
| 8 + (-3) = 5 | starting at point 8 and moving along 3 units to the left to reach point 5 |
| -7 + 3 = -4 | starting at point -7 and moving along 3 units to the right to reach point -4 |
| -7 + (-3) = -10 | starting at point -7 and moving along 3 units to the left to reach point -10 |
The next step is to establish the idea that subtracting positive numbers reverses the effect of adding positive numbers. So subtracting a positive moves to the left from the first number. This can be shown neatly by having a student face right, as if going to add a positive, and then walk backwards.
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Adding +3
Subtracting +3 |
This also applies to negative numbers: subtracting them reverses the effect of adding them. This can be shown neatly by having a student face left, as if going to add a negative, and then walk backwards. The student, rather unexpectedly moves to the right! So subtracting a negative moves to the right from the first number. This is the same effect as adding a positive number. Voila!
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Adding –3
Subtracting –3 |
Activity 6: Calculator buttons
The models above aim to develop meaning. However there are two warnings about ways of avoiding the issue of meaning. Unfortunately these are still common, and in the long run are not helpful.
Firstly, subtracting a negative number is not just given meaning by reciting ‘a negative times a negative is a positive’. It is more important to ensure that there is an understanding of what is happening.
Secondly, the two signs in 2 – (–3) have quite different meanings. The first shows a subtraction and the second shows that the 3 is a negative integer. For this reason many authors write the negative as a raised sign and drop the brackets.
Graphics calculators and some others generally have two separate keys for subtraction and negative number. The 'binary minus' is subtraction: it is called binary because it works on two quantities, such as in 10 – 5 . The 'unary minus' changes the sign: it is called 'unary' because it works on one quantity, e.g. to turn 3 into –3 and also to turn –3 into –(–3) which is just 3. In the expression 5 – (–2 –4) there is one unary minus sign making the 2 into –2, and the other two are binary minus signs indicating subtraction.
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