We talk about how you feel if someone gives you a positive thing, or if someone takes one away. With the students, we brainstorm on things that are POSITIVE and things that are NEGATIVE. I have a big number line ($^-10$ to $10$, say) above or along the top of my whiteboard. "I believe that adding and subtracting with negative numbers makes sense. Thanks to Alan Mesfin who suggested an alternative of tying on helium balloons (as in the film "Up") instead of adding puffs of hot air to represent adding a positive number. My balloon ends up at height +5.Įventually, we want students to read the calculation as "Four add negative two, subtract positive five, subtract negative one, add positive seven" (or replacing the operation words add/subtract with plus/minus, but always insisting on positive and negative for the signs accompanying the numbers), and think to themselves "Four, down two, down five, up one, up I add two sandbags (down two), subtract five puffs of hot air (down five), subtract one sandbag (up one), then add seven puffs of hot air (up seven). We can now describe a calculation such as 4 + (-2) - (+5) - (-1) + (+7) in the following way: In this model, we represent positive numbers as 'puffs' of hot air, and negative numbers as sandbags. The first model we offer is the hot air balloon, as seen in the game Up, Down, Flying Around. There are four possibilities that we need to be able to understand with our models: We will make suggestions about how to use language precisely in order to support the understanding of the distinction between operations and directed numbers. The models for teaching addition and subtraction of positive and negative numbers that we share in this article are designed to lead to understanding. For example, we have all heard students say things like "minus four minus two equals six, because two minuses make a plus!" We are often frustrated when we hear students say "Two minuses make a plus", because it shows a rote-learned phrase that is often misapplied. The original article is contained within this version. This article is an expanded version of one published on NRICH in 2008.
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