Kids struggle with positive and negative numbers … that’s a given.
But I’ve recently hit on a reliable way to eliminate the confusion … temperature.
Using a temperature scale as a model for solving integer problems has several advantages:
1) It’s a system students already know from everyday life.
2) The relationships among positive temperatures, negative temperatures, and 0 already make sense intuitively.
3) Everything about how temperature works also works for positive and negative numbers.
Let’s see how this work. Take the problem: – 2 + 7
Tell students the first number is the temperature in the morning: – 2. The second number tells how much the temperature changes during the day; + 7 means the temperature ROSE 7 degrees. Ask the student what the new temperature will be. Most students find it easy to intuitively see that the new temperature will be + 5 degrees, meaning the answer is + 5, or just 5.
When first using this approach, provide a temperature scale with 0 degrees in the middle, the positive temperatures above 0 and the negative temperatures below 0. Students will use this as a vertical number line to figure out their answers, counting along the scale. Here’s a model of such a scale.
In time, try taking away the visual aid to see if students have internalized the temperature scale, thereby allowing them to do these kinds of problems in their mind.
Going back to examples, take another: – 2 – 5.
Tell students the first number is the temperature in the morning: – 2. The second number, once again, tells how much the temperature changes during the day; – 5 means the temperature FALLS 5 degrees. Ask the student what the new temperature will be. As long as the student has familiarity with temperatures and temps below 0, the student wills see that the new temperature will be – 7.
Once students have done a fair amount of problems, gradually lead them to see two larger patterns: 1) If both numbers have the same sign, they keep the sign and add the numbers, and 2) If the numbers have different signs — one positive, the other negative — they take the sign of the larger number and subtract the numbers. I’ve found that students have less trouble understanding these “rules” after working with temperature problems, for the rules make sense.
Here’s a set of problems that helps students learn their integer rules with temperature:
a) – 2 – 3
b) + 2 + 3
c) – 5 – 5
d) + 5 + 5
e) – 2 + 5
f) + 2 – 5
g) – 4 + 9
h) + 4 – 9
a) – 5
b) + 5
c) – 10
d) + 10
e) + 3
f) – 3
g) + 5
h) – 5