Negative Sign vs. Subtraction

I had an amazing conversation with my 2^nd hour 8^th graders. They were very confused about why you would subtract the 9 in an inequality like this: 9 – 3x > 42.

When I said you had to look at the sign in front of the 9, not all of them understood. So I showed them that if you add 9, then you have 9 + 9 on the left, which results in 18. But we want 0. So you have to do whatever it takes to get to 0.

Then you bring the -3x down. This is where it got interesting. I said negative three, and they said to me – it’s not negative, that’s a subtraction sign! So I said that I know they think there’s a difference between a subtraction sign and a negative sign, but really they’re the same thing. Think about changing a subtraction sign into “plus a negative.” That’s saying that they’re the same.

One student in particular was tenacious with her questions. She knew exactly what she didn’t understand and she stuck with it. Another seemed stuck in the mindset of “But you didn’t teach this to us” (especially the part about subtracting 9 instead of adding it).

So I put the following problems on the board.

5 + 3x = 24

5 – 3x = 24

3x + 5 = 24

3x – 5 = 24

We talked about the first two and why you subtract the 5.

The one student started to see patterns between them and she tried to say what they were. She said that when there’s addition, you always subtract the number. But then I showed her a new problem where that’s not always true.

-5 + 3x = 24

We also talked about how in the second problem on the board, 5 – 3x = 24, you have to bring the negative with the 3x. I know it looks like a subtraction sign, and it is, but it’s also a negative 3x. They could think of it as changing the subtraction sign to plus a negative, and then voila, it’s negative.

What I loved about this conversation is that it’s a real mind-bender for students. It shows their conceptual understanding maturing from what they learned in elementary school, where a negative is different than subtraction. I told them this – that sometimes learning is confusing. And I had seen them making many mistakes on this and was very glad to be having this conversation.

This conversation worked because they were ready for it. It reminds me of when I tried to teach them about dividing fractions in a conceptual way, rather than the procedure, and they were so confused and didn’t want to hear it. Then later one student was actually curious about it. Why does dividing by a decimal (or a fraction) result in a bigger number? Usually dividing results in a smaller number. His curiosity about this made him ready to talk about it.

I also got to tell them that it doesn’t matter to me at what point they understand this stuff. If they get it yesterday, great; if they get it today, that’s fine too. They can retake if they need to. I don’t want to limit when they can learn it.