Q&A #561

Teachers' Lounge Discussion: Borrowing in mathematics

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From: Pat Ballew

To: Teacher2Teacher Public Discussion
Date: 2000062214:03:56
Subject: Re: Subtracton - borrowing from 10's

Borrowing is one of those words we hope elementary teachers will be
using less with students these days, but I know we have several
generations who grew up on the term (I always wondered when the tens
or hundreds column was going to come along and ask to be repaid.)  I
like the term the Oriental teachers use, decomposition.  each "one" in
the a column can be decomposed into ten of the items in the next
column to the right (a ten is ten ones, a hundred is ten tens, etc. 
To go the other way (what you probably grew up calling a "carry", they
call "composing".

Whichever language you use, try several approaches to show the student
"why" this makes sense with real quantities.. If Hanna is really
having trouble, ignore the algorithm and talk about the ideas.  What
does  it mean to subtract, how would you do it if you had a real task
involving removing 45 from a group of 61 (or finding out how many to
add to 45 to make 61, which is the same question).  Ask Hanna to see
how she thinks about subtraction, then try to lead her to see that
both ideas really produce the same answer. For each problem in the
home work, let her describe the situation in real language as removing
something from something, or how much must be added to something to
get something (which ever makes sense to her).  Don't worry about the
"short" way to get the answer until she has some real grasp of what
the answer means.....

  Then if she still has problems, try drawing pictures.  If she sees
subtraction as "take away" as many young people do, then draw a
picture of the first value, and cross out the ones that are to be
removed.  You could use X for tens and I for ones so the picture of 32
- 17 would look like: 
      XXX II    and then we need to erase 17, We can erase the ten
enough, but there are only two "ones".  How shall we subtract the 7? 
Now the need for "borrowing" or "decomposing" has a meaning.  We can
make change for one of the X to turn it into IIIII IIIII  and with the
two we have we now need to take seven ones from the 12 that are there
leaving five.

so our original looked like     XXX II
and then we took away ten to get XX II
and decomposed a ten to make      X II IIIII IIIII
and now removing seven gives      X          IIIII

so the answer must be fifteen.  I bet that if Hanna is as clever as
Grandmother, she will catch on quick. Then you can show her the SHORT
way to do these that you learned as a girl in school, and whatever
words you use to describe the process, she will know what it means...

good luck, and thanks to grandparents everywhere for the help they
give kids with homework.

 -Pat Ballew, for the Teacher2Teacher service

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