# adding and subtracting negative numbers

21 Apr 2014

Teaching negative numbers to students is often difficult due to the fact that some of the rules are counter-intuitive. The greatest sticking point comes when students are faced with calculations with touching symbols such as

The rest of the post focuses on how to introduce this concept to students.

## discs

The method I have had the most success with is similar to the approach described in getting the buggers to add up by Mike Ollerton. This pdf can be used to visualise the four options for touching symbols. To run it as a presentation press ctrl + L on Adobe Reader for Windows or cmd + L on Mac. The presentation is used as follows:

### determine the value of the starting image

Each disc has a value of $\pm 1$, so the initial image has a net value of $0$. The next image has a value of $2$, and this forms the start of the first example. The key to this method is to define the first symbol of the touching pair as an instruction: addition as drawing extra discs and subtraction as removing discs. The second of the touching symbols describes the type of disc to draw or remove.

The first example is $2 + ^ {+} 4$ and the presentation shows four positive discs being drawn. This should demonstrate

or that two $++$ symbols can be replaced by $+$.

Continuing where the last example concluded, consider $6 + ^ {-} 5$. This time we are adding, so are drawing $5$ more negative discs. This gives

or that $+-$ symbols touching can be replaced by $-$.

### subtracting positives

Now looking at $-+$. The first symbol of $-$ tells us that we are removing discs now. The given example is $1 - ^ {+} 6$, so $6$ positive discs are removed, leading to

therefore if $-+$ are touching then they can be replaced by $-$.

### subtracting negatives

Finally, the counter-intuitive calculation: $- -$. The example is $-5 - ^ {-} 8$. By now the students should be able to explain that this means removing $8$ negative discs. The presentation shows this is

and hence $--$ symbols touching can be replaced by $+$.

### notes

Here are the four examples again with the final table I ask students to record in their books:

which is often written as

## other methods

When I have asked colleagues about their method for teaching this concept there have been a variety of other methods used:

• sand castles and holes: a variation of the method described above. There is an excellent PowerPoint on mrbartonmaths.com.
• profit and debt: students are given slips of paper with various amounts of money on. Removing some of the debt increases the amount of money a student has. I feel that discussing applications to money is most beneficial once students have a firm understanding of how to deal with touching symbols
• finally there is another good discussion of the problem of touching symbols on nrich.