On the other hand, the equation c – b = a means that Zed starts at 0, walks forward c steps, then walks backwards b steps, and he ends up at a. The last two sets of steps cancel each other out, so Zed lands back at a. But we know that to walk forward c steps, he can first walk forward a steps and then walk forward b steps. If Zed wants to compute c – b, he starts at 0, walks forward c steps, and then walks backwards b steps. The equation a + b = c means Zed starts at 0, walks forward a steps, and then walks forward b steps, and he ends at c. Let’s use the measurement model to come up with another explanation. On the right, we start with quantity c and take away b things. So we’re left with just our original quantity of a. On the left, we started with a things and combined that with b things, but then we immediately take away those b things. Imagine taking away (subtracting) quantity b from both sides of this equation: a + b = c. On the other hand, suppose we know the equation a + b = c is true. On the right, we would be combining (adding) quantity a with quantity b. On the left, that mans we started with quantity c, took away b things, and then put those b things right back! Since we took away some quantity and then added back the exact same quantity, there’s no overall change. Start with this equation, and imagine adding quantity b to both sides. Means we start with quantity c and take away quantity b, and we end up with quantity a. Why It’s True, Explanation 1:įirst we’ll use the definition of the operations. Try it out!Īddition and Subtraction: Explanation 1 Arithmetic Fact:Ī + b = c and c – b = a are the same mathematical fact. In other words, we can think of every subtraction problem as a “missing addend” addition problem. Which is the case depends on the values you choose for a, b, and c!) (So either both equations are true or both are false. More generally, for any three whole numbers a, b, and c, these two equations express the same fact. These two questions are exactly the same: We defined addition as combining two quantities and subtraction as “taking away.” But in fact, these two operations are intimately tied together. Which models do you think will be useful for explaining how the operations work? Why?.Which models are useful for computing? Why?.How well do the models we discussed match up with how you usually think about whole numbers and their operations?.Of the models we discussed so far, do you prefer one of them?.Before we really dig in to thinking about the operations, discuss with a partner: Each of these models lends itself to thinking about the operation in a slightly different way.
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