Invented Strategies for Subtraction

Date: 09/18/2002 at 18:54:51
From: Julie
Subject: Two children's invented strategies for subtraction

I need some help understanding two students' ways of understanding 
math.  

Here is how "Brent" solved for 82-35:
First he subtracted 2 from 82 and then 5 from 35, so he could take 
80-30 = 50. Then he subtracted 5 from 2 and got 3. Then he took 3 from 
50 to get 47.

Here is how "Katie" solved for 72-47:
She took 2 away from both numbers and got the problem 70-45 to make 
an easier problem, the answer being 70-40 = 30, and taking 5 away to 
get 25, so the answer was 72-46 is the same, 25.

How could I explain to these students to subtract, for example, 63-25 
and 123-67?

What do both of these students understand about subtraction to help 
them invent this unique solution strategy?

I know this is a confusing question, but I just need some sort of 
explanation or meaning. Thank you very much!

Date: 09/18/2002 at 23:16:37
From: Doctor Peterson
Subject: Re: Two children's invented strategies for subtraction

Hi, Julie.

I'm not quite sure what the question is - are you supposed to explain 
to them how to use their own methods to do the two new problems, or 
how to do it "right"??

The first method, algebraically, looks like this:

    82 - 35 = (80 + 2) - (30 + 5)

            = (80 - 30) - (5 - 2)

            = 50 - 3

            = 47

On a number line, it is

                      30   35                 80  82
    +------------------+---+-------------------+--+---
                           |<-------------------->|
                       |<--------------------->|
                       |<->|                   |<>|

"Brent" moves both numbers down to the nearest multiples of ten, then 
finds that one was moved 3 more than the other. Since it is the lower 
number that moved farther, the move increased the difference by 3, 
and we have to subtract 3 from 50 to get the answer. I can't tell you 
just how he is thinking, since he could be using algebra as far as I 
can tell, or may be picturing a number line or counters. But he 
clearly has a good grasp of numbers!

"Katie"'s method is a little more conventional:

    72 - 47 = (72 + 2) - (45 + 2)

            = 70 - 45

            = 70 - (40 + 5)

            = 70 - 40 - 5

            = (70 - 40) - 5

            = 30 - 5

            = 25

The first part uses the fact that if you add or subtract the same 
amount to/from two numbers, the difference is unchanged; on a number 
line this means sliding them the same amount:

    0                       45 47          70 72
    +------------------------+-+------------+-+
                               |<------------>|
                             |<------------>|

The rest is just subtracting one digit at a time, starting at the 
left, that is, first subtracting the 40 and then the 5.

See if you can apply these methods to your numbers.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/