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Q&A #6433


Subtracting integers

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From: Jeanne (for Teacher2Teacher Service)
Date: May 31, 2001 at 19:14:14
Subject: Re: Subtracting integers

Hi Jackie,

I agree with Ralph that the rules for adding and subtracting integers can be 
very confusing for many students. So, I like to teach the rules of addition 
and subtraction of integers in conjunction with manipulatives/pictures.  
Let's look at some simpler problems before going on to the problems you 
mention in your inquiry.

I use a manipulative called "tile spacers" to help me teach addition and
subtraction of integers. They are shaped like plus signs and can be purchased 
from a home improvement store that sells ceramic tiles. To make negative signs 
just take a pair of scissors and snip two "legs" off.  They have the advantage 
of being relatively inexpensive AND quiet.  

Anyway, I begin by showing my students that a "+" and a "-" make a zero and
that "++" and "--" make zero and that "+++" and "---" make zero...etc.

Teaching hint:  Give your son a sheet of colored paper to use as a work 
surface. This will make more sense in a bit.

I have my students put 5 "+'s" and 5 "-'s" on their work surface.  Extra 
tiles are kept off to the side and placed on (or removed from) the work 
surface as needed.  I make sure they know they've "created" zero.

Example:  3 + -4  "3 plus -4"   This means we put in 3 and we put in -4

  -------------       -------------        -------------
 | +++++       |     | +++++  +++  |      | +++++  +++  |
 | -----       | --->| -----       |  --->| -----  ---- | ---> Value = -1
 |             |     |             |      |             |
  -------------       -------------        -------------
   Value = 0           Put in 3              Put in -4


Example:  3 - -4  "3 minus -4"   This means we put in 3 and we TAKE OUT -4

  -------------       -------------        -------------
 | +++++       |     | +++++  +++  |      | +++++  +++  |
 | -----       | --->| -----       |  --->| -           | ---> Value = 7
 |             |     |             |      |             |
  -------------       -------------        -------------
   Value = 0           Put in 3              Take out -4

After we have done several examples using the tile spacers and I see my
students are feeling pretty confident, we have the discussion of "3 - (-4)"
and "3 + 4."  They are different problems that have the same result.  The
latter problem, "3 + 4" is easier to compute than "3 - (-4)" so many people
change "3 - (-4)" to "3 + (+4)."

Now for the problems you've mentioned in your letter...
+11 - +12  means we put in +11 and then TAKE OUT +12  

  -------------       -------------        -------------
 | +++++       |     | +++++  +++++|      | ++++        |
 | -----       | --->| -----  +++++|  --->| -----       | ---> Value = -1
 |             |     |        +    |      |             |      which matches
  -------------       -------------        -------------       your son's
   Value = 0           Put in +11           Take out +12       answer.


+11 + -12 means we put in +11 and then PUT IN -12

  -------------       -------------        -------------
 | +++++       |     | +++++  +++++|      | +++++  +++++|
 | -----       | --->| -----       |  --->| -----  -----|---> Value = -1
 |             |     | +++++  +    |      | +++++  +    |     which matches
 |             |     |             |      | -----  --   |     the previous
  -------------       -------------        -------------      problem.  So, 
   Value = 0           Put in +11           Put in -12        yes, these are
                                                              equivalent.

*****
-10 - -7 means we put in -10 and TAKE OUT -7

  -------------       -------------        -------------
 | +++++       |     | +++++  -----|      | +++++ ---   |
 | -----       | --->| -----  -----|  --->| -----       | ---> Value = -3 
|              |     |             |      |             |      which matches
  -------------       -------------        -------------       your son's
   Value = 0           Put in -10           Take out -7        answer.


-10 + +7 = means we put in -10 and then PUT IN +7.

  -------------       -------------        -------------
 | +++++       |     | +++++  -----|      | +++++  -----|
 | -----       | --->| -----       |  --->| -----  +++++|---> Value = -3
 |             |     |        -----|      |        -----|     which matches
 |             |     |             |      |        ++   |     the previous
  -------------       -------------        -------------      problem.  So, 
   Value = 0           Put in -11           Put in +7         yes, these are
                                                              equivalent.

Since these pairs of problems are equivalent (result in the same answer), 
AND most students find addition easier, many students are taught simply to 
change every subtraction problem to an addition problem.
     So,  11 - +12 becomes 11 + -12  which equals -1.
     And -10 - -7 becomes -10 + + 7 which equals -3.


> So why wouldn't +11 - +12 = +1?
Well, going back to my pictures...+11 - +12 means we put in +11 and then 
TAKE OUT +12

  -------------       -------------        -------------
 | +++++       |     | +++++  +++++|      | ++++        |
 | -----       | --->| -----  +++++|  --->| -----       |---> Value = -1
 |             |     |        +    |      |             |     not +1     
 |             |     |             |      |             |                 
  -------------       -------------        -------------                    
   Value = 0           Put in +11           Take out +12     

Hope this helps!

 -Jeanne, for the T2T service

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