teaching integers

Tracy,

I teach 6-8 resource and coop math classes. I recently had my media specialist make me a banner-type number line. I intend to use it as a visual aid and keep it displayed on one of my walls. The longer you make it (more numbers) the smaller the numbers get. Mine is 15 feet long. I’ll ask her Monday which program she used and get back to you.

Also, many lessons stress the importance of explaining “operations with integers” using the white-square/black-square models. I developed what I consider a more visually, interactive model for one of my math centers.

I use a magentice dry erase board and a couple boxes of poker chips. Get a role of magnetic tape or magnetic circles and attach a small piece to each poker chip. Develop your lessons to model addition/subtraction of integers. Make a worksheet with a series of problems and have each student model the addition/subtraction using two colors of chips.

Robert

Thre standard method is to use arrows: arrows pointing forward (left to right) are positice, and arrows pointing backward (right to left, counting downwards) are negative.

You can do a lot with this by drawing on the blackboard.
You can put a number line semi-permanently on your blackboard with water-based school paint; it washes off later but doesn’t erase. The you can draw the forward and backward arrows with coloured chalk.
If you have a whiteboard, you can put the number-line on semi-permanently with water-based but not erasable markers, and again wash off later; then arrows with erasable markers.
if you really want to fight with a box full of materials, you could cut arrows of various lengths out of paper and find a way to stick them up, but drawing is much faster and allows you to do many, many examples.

By the way, the whole presentation of “integers” in many junior high texts is at best misleading and at worst mathematically incorrect. ANY kind of number can be positive or negative, integer or fraction or decimal. After your students have some mastery of whole numbers, show them that it all works *exactly the same* for fractions and decimals. This saves huge headaches later.

Two other pitfalls:
When you first present adding a positive and a negative, some bright light is going to say that it’s “really” just subtraction. Then that works fine just long enough for people to get habituated to it and like it (the gambler’s fallacy) and then when you go to adding two negatives all H breaks loose because that doesn’t fit into the structure they have just built. When you hear this one, say firmly NO, that is an oversimplification and will not work next week. They’ll still do it but will cope better if forewarned.
Worse yet, many bright lights (and even possibly your textbook if badly written) often give the FALSE “rule” of “two negatives make a positive.” This is simply WRONG, repeat WRONG. It is a false statement and is wrong more than half the time, a pretty lousy “rule” if it gets you less than 50% correct.
To list:
negative plus negative = negative (“rule” is WRONG)
negative minus negative = EITHER negative or positive, depending on which is larger (“rule” is WRONG)
negative (opposite) of negative = positive (“rule” works in this case, but not above or below, so it misleads you)
negative times negative = positive (“rule” works in this case, but not above or below, so it misleads you)
negative divided by negative = positive (“rule” works in this case, but not above or below, so it misleads you)
negative to the exponent of negative = EITHER negative or positive, depending on whether exponent is odd or even (“rule” is WRONG)
negative to negative root: impossible if root is even, negative if root is odd (“rule” is WRONG)
Please don’t teach something that is dead wrong more than half the time.

While I’m on that issue, when you get to equations, this takes a few minutes longer to teach and to write but makes it possible for kids to get beyond chapter 1 of algebra: Teach real mathematical operations, which are called addition, subtraction, multiplication, division, square, square root, etc. Please do NOT teach “pick up a number and move it.” That belongs on kindergarten felt boards. Many people teach a quick a quick shorthand of move the number to the opposite side and change its sign. OK, that works in Chapter 1 with addition and subtraction. Then you get to Chapter 2 with division and fractions (which they’ve been badly taught and are afraid of anyway). The students learn to pick up the number on top and move it to the bottom, and of course they change its sign which you told them to, and of course they then get every division problem wrong. Not worth it to teach a mistake. I ttakes a little longer to write out real math, but the payback is that you don’t fail and spend far longer re-re-re-doing everything, and you can do advanced problems (do the same *mathematical* operation to both sides) as easily as simple ones.