One of the frustrating things about teaching math occurs when students encounter unnecessarily confusing math symbols. This week the 6th graders have been struggling with understanding how to use the subtraction operation with negative numbers. Subtraction and negative numbers share the identical notation but are very different concepts.
Subtraction is an action, what mathematicians call an operation. For instance 6 - 2 = 4 means start with the number 6 then do the action of removing 2 objects. The remaining number is equal to 4.
This is represented by the symbol -.
A negative number is the opposite of a number in the sense that under addition the two cancel each other exactly. A negative number is a thing, a place on the number line, not an action. The opposite of 3 is -3 because 3 + -3 = 0.
Negative numbers are distinguished from positive with the symbol -. See the problem?
Subtraction is a verb. Negative numbers are nouns. But the math language is the same. Imagine how awful the English language would be if nouns and verbs of related concepts were the same words and we all had to figure out which one was in use by the context of the sentence. For example what if "baker" and "baking" were combined into one word "bakering". "John is baking" would be indistinguishable from "John is a baker" because both would read "John is bakering". It would drive everyone nuts.
Our poor 6th graders have to confront this noun/verb conflation in expressions such as 6 - (-2). 6 - (-2) means "start with 6, now take away (minus sign) the opposite of 2 (-2)". Taking away the opposite of 2 can be a tricky idea.
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| Black chips are positive 1s and red chips are negative 1s. The circled red chips represent physically removing or subtracting -4. |
I have found that one great way of teaching tricky ideas is by using manipulatives - physical objects that kids can touch and, well, manipulate that stand in for mathematical ideas. This week we have been using poker chips and drawings of circles with + and - in the middle to visualize this relatively complex combination of concepts. The picture above is an illustration of 2 - (-4) = 6. The black chips are positive 1s the red negative 1s. A black chip and a red chip together make a "zero pair". So the picture starts with 4 zero pairs of black chips and red with 2 unpaired black chips left over. Hence, the number 2.
Taking away (-4) is represented by physically removing 4 red chips. Now there are no more zero pairs, just 6 black chips by their lonesome.
By Friday I got the sense that the ideas were beginning to click.
Now on to 6 x (-2) and how that can be understood as repeated addition: (-2) + (-2) + (-2) + (-2) + (-2) + (-2) = -12.
On Edit: Look out 7th and 8th graders because this exact problem is found in square roots. There is an action called taking a square root and there is a number concept with the same name and notation. Nice job mathematicians, nice job.
