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# Sad faces and Negative Numbers

There’s some context that’s necessary for this post. So go read Nathan Kraft’s post about arithmetic with negative numbers.

What’s the mistake? Thoughts?

This piece of student work is kindly submitted by 3-act mavenĀ Chris Robinson.

## 6 replies on “Sad faces and Negative Numbers”

Mark Betnelsays:

I love that!

So I would ask the student why, using their sad-face system, they didn’t conclude that the answer was -6. They would say “Because it isn’t, it’s positive.” and I would say “maybe …. but sad faces mean negative, right? And you struck out 8 of the sad faces by subtracting 8 of the 14 sad faces, leaving 6 more behind…”

And they would say, in my dream of how these conversations go: “Maybe the subtraction should have turned the 14 sad faces into 14 happy faces?”

Why is it wrong? (Hate asking that). Leftover subtrahend gives you the opposite. 7-20, 13 to circle means -13 is the answer. If you owe 6 sads, that’s -(-6)

Although probably the student got lucky, it’s hard to say without further questioning. I’d like to see -14-(-8)

I think the student meant to draw happy faces on the right (losing something bad is like getting something good). And then got lucky by interpreting those leftover sad faces as happy faces to get +6.

This what I see happened. Student tried to use the model of integer chips (essentially). They have (-8) or 8 sad faces on the left side, and (-14) sad faces on the right side.

This student knew the answer before doing the model, so they removed sad faces until they are left with six, but this model actually shows (-22) – (-16) = (-6).

In this case I usually ask the student in person to show me (-8) on their whiteboard or with their integer chips. The first number shows us our starting value, then I would ask the student to remove 14 negative chips. At which point student looks at me with a puzzled look. I ask, “Why the puzzled look?”

“There are only 8 chips!”

“Correct! So how can we show that we are removing negatives, when we don’t have enough negatives to remove from?”

This is the tough part, and I put forward to you guys to help me. The answer is we add 0-pairs until we have enough negatives to remove. This is not natural, but how do we guide students to do that?

Great find Chris!

Personally, I like the integer chip method for adding, but have always hated it for subtracting. It’s just not intuitive and I hate trying to force it. I much prefer stressing that taking away negatives is equivalent to gaining positives (or losing positives is like gaining negatives), and then go back to the model.

I can agree with that Nathan. Zero-pairs is a pretty sophisticated “mathy” trick. I’ve struggled with this one A LOT. Trying to teach conceptually. I have liked using my army men, and talking about retreat. It makes sense to kids, however setting up the problem is the hardest part.

What I DO like about the zero-pairs is it really gets across the weirdness of integer subtraction, and why integers were not “real” numbers way back when.

I suppose I will struggle with this forever.