They'd freeze. They'd guess. They'd get the strict rules of multiplication confused with the much looser rules of addition and subtraction. I'd get a lot of -19's and 5's. The problem was that the signs are opposite. With same signs, we just add and keep the sign. With opposite signs, it can feel like we're in a whole other ballgame.

We find 7, we find -12, and we figure out which integer is farther from zero. Since -12 is farther, our answer will be negative.

Next comes the fun part. Once we identify that our answer will be negative, we fold the numberline in half.

And then we count the spaces between the two numbers to get 5. This, combined with our previous step, gives us 7 - 12 = -5.

My students' ability to work with negative integers, even after the manipulative was removed, improved by 62% and my thesis was [eventually, after many, many edits] accepted in May 2011.

When I'm at the board and a problem like "7 - 12" comes up, I don't always have the time to stop and show students on the ruler how we get -5. So, I ask this string of questions:

**"[In 7 - 12] Which number is farther from zero?"**

**-12**

**"OK, so our answer will be negative. How much farther?"**

**5**

**"OK, so our answer will be?..."**

**-5**

These questions works really well in the moment. When students are working on classwork and there is more time, I like to show them the problem on the ruler if they are having trouble. This usually happens on a 1-to-1 basis. Because manipulatives are so hands-on, they allow students to see and feel the numbers, which always seems to stick with them better.

If you find that your students are struggling with integer operations, this manipulative is available for download here.

For a fun integer operations activity, you may like this integers pennant. Students evaluate three problems on each pennant. The three problems are related to keep the focus on those pesky signs.

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