Yesterday, I started reteaching adding and subtracting signed numbers to my Pre-Algebra group. Most of them failed the test on the topic and it struck me that they didn't really understand the concept associated with this topic.
Most of the time, this topic is taught basically with the idea that if the signs are the same, you add the numbers and the answer has the same sign. If the signs are different, the numbers are subtracted and the sign of the answer matches the sign of the larger number.
They have difficulty with this idea because they don't understand that the signed numbers represent distance and the signs refer to the direction. So today, I retaught this concept talking about distance and showing how it works on a number line.
I actually saw a few light bulbs flick on during the lesson. At the end, each student created a number line they could use to answer some simple problems such as 3 - 8.
I showed them how you move 3 units to the right starting at 0 to represent the first number, then I moved left 8 units showing the -8. I repeated this demonstration with two or three more problems. Once they had a better grasp on it, I had them use a number line to practice it. It was amazing how many students wanted to know if they "had" to use the number line. Couldn't they just put the answer down.
I pointed out that they kept coming up with the wrong answers so that told me they really didn't understand the process and they needed to number to help them do better. It was great that no one argued with me about that. It was wonderful that several students actually accomplished more in the time than they had when I tried teaching it the other way.
As a side note: When I taught square roots in Algebra I, I drew a picture on the coordinate plane in Quadrant 1 to represent the positive area. Most students only show the positive values of a square root. I then drew a square in Quadrant III where you can have two negative values which still results in a positive area. My students saw the reason why taking a square root results in two answers.
Because of yesterdays lesson, I will be teaching the addition and subtraction of integers using distance rather than trying to teach it the usual way.