Adding negative numbers

Negative numbers can be confusing to some students, because they don't see them often in real life. Modeling can help.

Real life examples of using negative numbers occur on thermometers,  altitude, and ocean depth:

• If the temperature is 10*F and drops 15 degrees, that is 10 + (-15) = -5*F.   If the temperature is -5*F and drops 7 degrees, that is -5 + (-7) = -12*F.
• If a plane flies at 45,000 feet above sea level, and descends 15,000 feet, that's 45,000 + (-15000) = 30,000 feet.
• If a submarine descends to 290 meters, and then ascends 50 meters, that's -290 + 50 = -240 meters.

If your student has trouble with these calculations, here are some hands-on models you can use to help him.

1. A number line.  Make a reusable paper number line, with numbers from -20 to 20 or farther. (Here's a link to printable number lines.   The ones with negative numbers are labeled as "integer lines.") The student can use a button, eraser, or some other marker to show what happens in the equation.  Add positive numbers by moving to the right, add negative numbers by moving to the left.
2. Hopscotch number line.  Use some chalk and a sidewalk to make a number line, and the student becomes the marker.  
   
   To model 5 + -7, the student stands on 5 and hops backward 7 places to stand on -2.
3. Beads on a string.  Make a "numberless line" of beads on a string, with the center bead being "zero."  (If you'd prefer to have the numbers labeled, have the student make clay polymer beads and write the numbers on with a permanent marker after baking.)  To model -3 + -4,  hold the -3 bead and move 4 beads to the left to land on -7. 

4.  Poker chips.  A classic manipulative for teaching negatives in many math classes  is a set of poker chips.   Positive numbers are represented by one color, and negative numbers with a different color. The two colors cancel each other out.  So to model -8 + 5, you might place 8 red chips on the table and then 5 blue chips.  Match up all the red and blue pairs that you can,  and you are left with 3 red chips, or -3.  So, -8 + 5 = -3.

NOTE:  Most students will catch on that adding negative numbers is like subtraction.  When they make that discovery, introduce this idea:

Subtraction = adding the opposite.

The opposite of every positive number is its negative counterpart.  This will be an important concept when it comes to understanding how subtracting negatives works.   More on that later in a future post, but for now, if you have to model the subtraction of negative numbers, think of taking away debt.  For instance,  -10 - (-3) can be thought of as "Bob owes me $10, but I'm taking away $3 of that debt because he helped me wash my dog.  So now he owes me $7."  So you could have 10 index cards each labeled "PAY ME $1.00"  to represent -10.  Then take away three of them to represent - (-3).  There will be seven PAY ME  cards left, representing a total of -7.

Or, using the poker chips, you can set out 10 red (negative) chips and subtract (take away) three of them, leaving 7 negative chips.