Subtracting Integers By Adding The Opposite

By middle school, students have developed confidence in working with addition and subtraction. They understand that subtraction means to take something away from a given amount. Using a visual tool such as a number line, they know that addition is shown by moving to the right and subtraction is shown by moving to the left. They know that 5-3=2 and no longer need models to support that understanding.

Subtracting by Adding the Opposite

After being introduced to integers, students will also recognize that a value with a negative sign is another way of moving left on the number line. 4 means 4 intervals to the right of 0 and –4 means 4 intervals to the left of 0. 

4 and –4 are opposites because they are the same distance from 0, but in opposite directions. 

Addition of a Negative

Using the number line, we can demonstrate that 5 + (–3) results in the same outcome as 5 – 3.  The number line shows how some of the positive values are essentially cancelled by the negative values and we end up at a smaller number, just as we did with subtraction.

In other words, 5 minus 3 is equivalent to 5 plus negative 3, or subtraction is equivalent to addition of the opposite.

We can also demonstrate the concept with counters.

Subtraction of a Positive

Adding integers begins to make intuitive sense fairly quickly. Using visual models, students understand how positive and negative values essentially cancel each other out. Subtraction with integers is less intuitive and visual models help develop confidence that subtraction is equivalent to the addition of the opposite.

–7 – 4 is the same as –7 plus the opposite of 4, so –7 + (–4). 

Think of this as removing 4 from negative 7. Removing 4 would mean moving to the left on the number line. It is the same as adding –4, which also results in moving to the left on the number line.

Integer counters show that in order to subtract 4, we have to first add 4 zero-sum pairs. This gives us 4 positive counters to subtract. The result is that we have –7 + –4. 

Subtraction of a Negative

Subtraction of a negative number makes even less intuitive sense. 

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–4 – (– 8) is the same as –4 plus the opposite of –8, so –4 plus 8.

Think of this as removing 8 negatives from –4. This would increase our value as we are removing more negatives than we have. Increasing value is shown on the number line by moving to the right. 

Integer counters show that in order to subtract –8, we have to first add 4 zero-sum pairs. This gives us 8 negative counters to subtract. The result is that we have –4 –( –8)=4. 

Changing a subtraction problem to the addition of the opposite shouldn’t be a trick that students memorise. Visual support will help them understand why the rule makes sense. Ultimately, they should develop confidence in moving between subtraction and addition and positive and negative numbers in ways that fit the problem they are solving.

Also read: How To Add Integers: Using Counters To Add Integers

Frequently Asked Questions on Subtracting Integers By Adding The Opposite