# Integers

This topic contains 5 replies, has 3 voices, and was last updated by  Bill McCallum 3 years, 3 months ago.

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• #2726

alwong
Member

Hello! I was wondering if you could point me in the direction of any research on why operations with integers were taken out of Grade 6 standards? I am a math coach and several of my teachers would like some reason or some research about why operations with integers are not taught in grade 6.

Thanks so much,

#2727

Bill McCallum
Keymaster

We looked at many sources, including national reports, well-regarded state standards, and standards of high achieving countries. There is no agreement on where to put operations with integers. Singapore puts them in Grade 7. Massachusetts started with addition and subtraction in Grade 6, excluding subtraction of negative numbers. Curriculum Focal Points puts them in Grade 7. And so on.

#2728

alwong
Member

Thanks! This is the message we are telling our teachers! I am glad I am on the same page as you! Teachers just wanted a firm “research-based” answer. But I will continue to share your answer!

#2738

Bill McCallum
Keymaster

You are very welcome! Sorry if I sounded a little bit crabby in my last answer. The term “research-based” sometimes does that to me.

#3247

lhwalker
Participant

I was just watching your Khan video for multiplying and dividing negative numbers. My remedial Algebra students gain solid traction with multiplying negative numbers like this:

-3 lost 3 friends
2(-3) lost 3 friends twice
-2(-3) the opposite of losing 3 friends twice.

The added benefit is that “opposite” connects with -x.

The emotional connections help with retention.

#3265

Bill McCallum
Keymaster

Lane, I think this is a useful way of helping students remember the rules, and that’s especially needed for remedial students. There’s a bit of sweeping under the rug going on here, because if you spell it out, you are saying that

(opposite of 2) times (opposite of 3)

is the same as

opposite of (2 times (opposite of 3)).

We can actually prove this using the distributive property, because that property tells me that

(opposite of 2) times (opposite of 3) plus 2 times (opposite of 3)

is the same as

((opposite of 2) + 2) times (opposite of 3)

which is just zero times (opposite of 3), namely zero.

But if I add two numbers and get zero, they must be opposites!

Of course, I’m not suggesting that you have to go over all this with your remedial students!

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