Subtraction, Minus, Less Than, or Adding in the Opposite Direction

Big Topic or  perhaps not.  In all the years I taught Algebra, this operation
caused more problems for my students then any other.  You would think that the  "–"
sign would be easily understood by high school.  If I could recall some of the comments
made by my students I would list them for you.  But I cannot.  What I can do is give you 
an idea of what some were like.


1. Some called it a minus, while others argued that  the "–" sign meant negative.
2. Those that declared it to mean negative said that it was not minus.
3. They would not read it as an opposite of a positive.
4. They would question the operation of a problem like 5 – 3 = ? 


Lets try something: 1. We are going to put the integers –20 to +100 on the wall starting 
                                    in KINDERGARTEN.
                               2. We will call –11, "the opposite of 11"
                               
           If you teach 15 + 11 = ?  You can go to the integers on your wall, start at 15 and
                               go 11 to the right.  The students will get a great visual as you move to
                               the right, and your words will help the audio learners.  Then you 
                               have individuals show the class like you did it.


           If you teach 15 – 11 = ?  This is 15 + (–11) so you can again go to your integers 
                                on your wall, start at 15 and go 11 to the left.  Here again, the 
                                power of your moving to the left will be of great benefit.  And 
                                of course all those future actors can do like you did for the class.  


We are, "adding in the opposite direction." 


So 5 – 3 = 2  The "–" sign means I am going to my left from where ever I am.


     5 + 3 = 8  The "+" sign means I am going to my right from where ever I am.


I seldom see any the integers in the primary grades, but they should be there
for more reasons than just what I am speaking to here.  When students are first
told about the integers, in 5th grade, they become confused.  They now see a 
sign "–" in front of a number like "–5" and if it is added to 8, what are talking
about?            –5 + 8 = ?


Now think about what we did above.  Start at –5 and go to the right 8.  We 
get to the number 3.  We start at the "opposite of 5".  


What about –5 – 8 = ? 
We start at  –5 and we go to our left  –5 – 8 = –13.
                                                         +(–5) +(–8) = – 13


I contend that the more familiar your students are with numbers the easier
it will be for them to work with them.  In that last problem, we are simply
adding to our left as we have added to our right.   


 WE ARE ADDING IN THE OPPOSITE
                DIRECTION!