ANOTHER LOOK AT INTEGERS

I think what sparked my reason to look at this again was a

resent visit to my bank. I asked them for our balance after

a deposit, and then came home. I gave that information to

my incredibly copious spouse, who has been the guardian

of our check book for the last 40 years. The gap between

their amounts was more than I could understand. Somehow,

had a positive become a negative somewhere, and if so, this

could be a big mess.

Imagine you had a deposit of $500, but in your check book

recording you subtracted. You would be off by $1000. The

simple idea of adding integers seams to be such a problem

for so many. It is all about getting back to zero. If I earn

$200, but have $200 in bills to pay, when I am done I have

no money at all.

All we need ask now is which came first, my earnings or

my bills.

$200 + (–$200) = 0 or

(–$200) + $200 = 0

Why is this confusing?

Students know that 5 + 4 = 4 + 5, don't they?

What are we not being clear about?

Perhaps we just do not explain ourselves. Let me show

you what I mean.

We start with our digits 0 – 9 and we have hundreds

of worksheets with 7 + 2 = __ and 5 + 1 = __ .

Most of our students fill in the blanks, with little or

no retention, because all they do is stare and write

a number. Where it goes in those fine minds is

anyone's guess.

Try having them write the entire problem and the

answer with the = sign. Now their minds have the

whole story to place somewhere. Even better,

7 + (+ 2 ) = __ Read as: 7 + a positive 2 = 9

Then when they get to SUBTRACTION

7 + (–2 ) = __ Read as 7 + "the opposite of" 2 = 5

On the other side of zero

–7 + (– 2) = __ Read as "the opposite of" 7

+ "the opposite of" 2 = – 9

and with SUBTRACTION

–7 – (– 2) = __ Read as "the opposite of" 7

"the opposite of the opposite of 2 = – 5

When we add we go away fro zero, and when we

subtract we head back to zero, from either side.

I believe it is truly important that students see

integers from Kindergarten. They need not be

a surprise. We do not have to work with them,

but letting them know they are there will save

a great deal of anxiety.

SUBTRACTION WITH ZEROS

We have all been in class when the following problem was put on the board.

5003

– 1468

We remember that we cannot take 8 away from 3, so we have to borrow, or

REGROUP in todays vernacular. Thus, our 0's become a 9, our 5 is reduced

to 4 and off we go. But, why do we do that? Our 0's become 9's, really?

That's cool, we get 9 out of nothing. Oh, hold on there cowboy!

Look at this a different way.

5000 + 0 + 0 + 3

– 1000 + 400 + 60 + 8

Now the 3 needs an extra 10 to be 13, but 5000 can only hand out

1000's. Thus, we will drop off 900 in the hundreds place, 90 in the

tens place and the ones place only gets 10 leaving the following.

4000 + 900 + 90 + 13

– 1000 + 400 + 60 + 8

As we see, by expanding the numbers we get a clearer understanding

of what is happening. I hope you feel better about those 9's now.