The language of Math is used incorrectly in our society by most of us. Funny, we do not tolerate incorrect English, or improper grammar, but math seems to have its own set of rules which we seem to want to butcher. The butchering extends from us not all understanding the definitions, so we make some up. When we do, and it gets extended beyond us, whoops we have a bigger problem. Lets try to correct a couple.
"Two Negatives make a Positive"
Not quite, which of the following are we talking about?
1. The opposite of a negative integer is a positive integer
2. A negative integer multiplied by a negative integer = a positive integer
3. The sum of two negative integers is a larger negative integer
The statement in red above is confusing and incorrect. The 3 statements that follow
are correct, and keep us aware of the operation we are discussing. We can use a
pattern to see how this happens.
#1. – (–5) = 5 In this statement we are saying the opposite of negative 5
= positive 5. It is also saying "the opposite of", "the opposite
of 5" which is 5. So, we have gone from 5 to –5 and back
to 5, all in one statement.
#2. (–5) (–5) = 25 So, why is this true? Lets look at a PATTERN!
5 x 5 = 25 5 x (–1) = – 5
5 x 4 = 20 5 x (–2) = –10
5 x 3 = 15 5 x (–3) = –15
5 x 2 = 10 5 x (–4) = –20
5 x 1 = 5 5 x (–5) = –25
5 x 0 = 0 Since Multiplication is commutative!
(–5) x 5 = – 25 so (–5) x (–1) = 5
(–5) x 4 = – 20 (–5) x (–2) = 10
(–5) x 3 = –15 (–5) x (–3) = 15
(–5) x 2 = –10
(–5) x 1 = –5
(–5) x 0 = 0
As we see the product of a positive integer and a negative integer is a negative
integer. However, the product of 2 negative integers is
a positive integer.
#3. (–5) + (–5) = –10 Here we are adding to the left from 0. Thus if I add
2 negative integers I must get a larger negative integer.
We, as educators, must be clear to our students about
what is taking place, and say it correctly.
You might say: The sum of 2 negative integers is a
negative integer.
Clarify for them that a problem like, –3 – 4 = ? is a sum and not a product.
they could rewrite it as:
+(–3) + (–4) = –7
"Lets cross cancel first"
I do not know how this improper comment became a part of our lexicon, but
it has. People the correct word is Factor!!!
I would start a class out by writing:
8 ÷ 12 ⋄ 4 ÷ 5 = or 8/12 ⋄ 4/5 =
I would then reduce the fraction 8/12 to 2/3. A hand would go up, and when I call
on them they say, "you can't do that its not on a cross"
My response was always the same, "gee, is the reducing of fractions
now illegal, I did not get the memo".
The phrase "cross cancel" places into the minds of students the wrong information.
We should never use phrases that distort the true meaning of what we are intending
to teach.
Because of this phrase I have had students factor numbers across an = sign in a
proportion.
Example: They would factor the 3's in ⅓ = 3/x
Then they would say, "but its on a cross isn't it."
They had heard the wrong phrase so often, for them it was correct. If we are going
to teach Math as we teach any subject, use the correct vocabulary.
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