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Q&A #522

Teachers' Lounge Discussion: Teaching subtraction

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From: Ray M <raypublk@san.rr.com>
To: Teacher2Teacher Public Discussion
Date: 1999020501:40:48
Subject: Re: teaching substraction(sic)  & in refrence(sic) to barrowing(sic)

You don't have to know a thing about 10's to learn about borrowing. 
It is rather easy to teach most four year olds how to count in binary:
put 1 dot on your r. thumb, 2 dots on r. index, 4 dots on right
middle, etc.  Now you can uniquely represent 0 to 31 on the right
hand.  Using both hands, you can uniquely represent 0 to 1023. 
Transfering the finger notation to paper is easy: finger up =1, finger
down = 0.  Once counting in binary is second nature, you can add in
binary.  Dots can be used to visualize numbers and carrying.  There
are close parallels to all the rules in base ten.  Once you know
binary, octal and hex come almost for free and their rules parallel
those of base ten too.  So there are three useful bases other than ten
in which to learn about borrowing.  Most spreadsheets, symbolic math
programs, and scientific calculators can help you check your work.

It is simply not true that "all the numbers on the top represent 10"
and to say so will conflict with trying to teach place value.

In the example given:
 458900
-235774

the statement is made that you have to start with borrowing from the
9.  In some sense that is true, but it seems like bad advice to give
to  a second grader.  I would always give the advice to start in the
ones place, note that I need to borrow, look at the next place value
(10s), note the need to borrow, look in the next place(100s), etc. (I
kind of like Judy's story!)   But that is certainly not the only
algorithm.  I can subtract first and then clean up the answer in a
weird base if I want:
45-23=22
89-57=32
00-74=-74
Now, it would bother most people if I wrote my answer as
(22,32,-74)
but that's a nice ordered triplet and there's no rule against them. 
And there is a fairly simple rule to clean it up so that it looks like
(22,31,26)
Which is parallel to what I'd get if I did my math base 100.  And base
100 is easy to convert to base 10.  So
223126 is the answer.

In hex the problem looks like
 070094H
-0398FEH
=036796H

In octal it looks like
 01600224
-00714376
=00663626

and in binary it looks like
 111 0000 0000 1001 0100
-011 1001 1000 1111 1110
=011 0110 0111 1001 0110

While gifted 7 year olds can certainly do that, I don't recommend
trying it in an average 2nd grade classroom.  But if you, as the
teacher, will take the time to learn how to do it, you will understand
what borrowing is really about.


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