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# What Makes A Good Resource - Subtracting Negative Numbers

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Created on 08 September 2009 by ncetm_administrator
Updated on 03 September 2010 by ncetm_administrator

# Subtracting Negative Numbers

Resource:

Teacher comment:
I met this idea first when, on a course, we had a set of cards with a mixture of positive and negative numbers on them. We were considering the question “what happens to the total when one of the cards is taken away”. I liked the clarity of modelling subtraction as removing a card and that you paid less attention to what you were taking away, and more to what was left. Whether you took away a positive or negative card, all you needed to do was add up the remaining cards.

While experimenting with macros, I was reminded of the card activity and decided to try and re-create it using Excel.

What I did:
I wanted to begin by building some confidence in adding a mixture of positive and negative numbers. I used the spreadsheet to display four random numbers and asked “What would the total of these numbers be?” We worked together on describing different strategies for doing this; some would take pairs of differently signed numbers and combine these using a kind of “cancelling out” strategy. Others would choose one number (usually the largest positive number) and add to this each number in turn by essentially adding (for positive) and subtracting (for negative) a number until all four numbers were used. Yet another, strategy (often the quickest and most efficient) was to total the positives, total the negatives and find the difference between the two totals.

I then showed the facility for taking away on the spreadsheet. I found it best to start by taking away a positive number. Students seemed very happy that taking away a positive number made the total smaller. I did this a couple of times, and then took away a negative number. Most students did as I had expected; they were unworried by what I had taken away, and just added up what was left. Consequently, most correctly wrote on their mini-whiteboards a total that was bigger than before.

I decided to go with a bit of ‘classroom theatre’ for the next bit. On getting them to show their answers, I expressed surprise, saying that surely the answer answer should be smaller because I had taken away. A few students immediately rubbed out their answer, but most stuck to their guns. I continued with my surprised expression and asked “how can the total increase when we have subtracted a number?!”. I found that most student descriptions included the words ‘taking away a negative number’, which really pleased me.

Reflection:
I have found that when moving to problems like 8 – (-4) the students do struggle. The way in which the symbols in this equation speak to them seems to make them want to say “4”. To try to overcome this I now very deliberately record the calculation alongside the revealing of the numbers from the spreadsheet. So when the total of the original five numbers is revealed, I write this number on the board. When I take away one of the negative numbers (say, for example, -4) I write “– (-4)” next to the total whilst saying “subtract negative 4” This certainly seemed to help forge a link between what they were doing in the activity and the sort of written calculation you would find in a text book.

I now always use this activity when introducing negative number arithmetic, and I am convinced that it has improved my students’ understanding of the concept.

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