Subtraction Interaction

It is time to expose the secret double life of subtraction.

It all comes down to that foundational number sense we work with students to develop as they grow as elementary school mathematicians. We give students the opportunity to build ways of looking at numbers and operations that begin so thoughtfully and then become intuitive. So intuitive, in fact, that it barely registers. An automaticity easy to take for granted, but let’s try to put our minds in slow motion and really discern what is happening when we think about a couple of subtraction problems.

Take a moment to look at the numbers you are working with here. Much like we would do in class, notice what is the same about the two expressions and what is different. Now, take yourself through a solution. Don’t jump to the algorithm, you don’t need to! You will have a much better time if you let the values in these problems guide you to an efficient mental strategy. Try to either mentally hang onto or quickly annotate your thought process.

Okay. Now, can you articulate how you approached each problem? Do a same/different analysis on your processes.

Let’s start by thinking about 23-4. There are many ways you may have thought about it, but let’s zoom in on two typical models. As we do in class, I am going to illustrate a thought process on a number line. That doesn’t mean a number line was used (or harmed) in the working out of this solution, but it is a very legible way to indicate the steps moved through in order to get from point a (question) to point b (solution).

Did you solve it like this? We would call this a distance or difference strategy. This involves the mathematician seeing the two values in a subtraction problem as bookends between which a difference or distance exists. You may have added up from the lesser value or subtracted from the larger value. You may have travelled 19 units in one fell swoop, or in a few swoops of a decomposed 19. Either way, the distance between the two points is 19.

Or did your strategy more closely resemble this one? This we call a removal strategy. You are quite literally removing 4 from 23. The resulting amount is 19. Same answer generated, approach quite different.

Look at both number lines and locate all parts of the number sentence. Where is the 23? Where is the 4? Where is the 19?

Think about your answer. If you were asked to articulate why, what would you say?

Now, let’s think about 71-68. What was your original approach? Look at those values. Scrutinize them. Let them talk to you. Do you want to stick with what you did originally, or do you want to rethink your approach and try something new?

Why?