Physical Manipulative: Additive Inverse Number Line
Introduction
Studies show that young students often lack a basic understanding of the additive inverse, which is, in my opinion, a real shame.
I particularly appreciate Paul Lockhart's description of the necessity for identities when performing arithmetic. From page 85 of The Mending of Broken Bones:
"The maneuvers that any artist or craftsman performs are those suited to the medium. Whether it is tying knots, performing ballet, carving wood, or doing algebra, the medium—be it rope, the human body, a piece of wood, or numbers—dictates what can be done to it and how hard it can be pushed before the system breaks down...So, just as woodcarvers don't put too much stress on the wood lest it crack or break, we can't go around asking numbers to obey our whims....The point is that we need to familiarize ourselves with what our numerical entities and operations actually do and also be aware of what they do not."
One thing our numerical entities actually do is additively interact with the quantity zero. For every value in the set complex numbers, there is another value that negates it. When added together, these values make nothing. This is an abstract concept that extends beyond numbers: Our muscles flex and stretch and return to a comfortable resting position, Newton's third law of physics requires each action to have an equal and opposite reaction, and Jedi and Sith balance The Force.
This simple interactive number line is a tool for students to use during arithmetic activities to move independently from procedures of "taking away" towards more generalizable additive structures. In simpler terms, students who are comfortable with subtraction of counting numbers using this tool will adapt smoother to subtracting negative integers in middle school.
The Boring Stuff (Massachusetts State Content and Practice Standards)
Massachusetts Content Standards | 1.OA, 2.OA, 3.OA, 4.OA, 5.OOA, 6.NS, 7.NS, 8.NS |
Massachusetts Standards for Mathematical Practice | 2, 4, 5, 6, 7, 8 |
Materials
A piece of sturdy paper or cardboard (I used a broken down box)
Pencil
Scissors
Tape or glue
Either of these two "number line" options:
Option 1:
Two flexible measuring tapes
Permanent marker
Option 2:
This ruler printable and this reversed ruler printable from printable-ruler.net
Colored printer paper
Assembly
Glue or tape your number lines together.
Using flexible measuring tapes and a permanent marker to mark the negative values is better for mathematicians who want access to values farther from zero or those who want a more durable tool. 
Using paper is better for mathematicians who like a cleaner visual or do not have access to flexible measuring tapes. I printed this ruler on green cardstock and this reverse ruler on pink. Make the "zero-ing" window.
Cut a small rectangle (about 1"x 1".) Mark the height of the number line with a pencil.
Cut two small rectangles to form the window. Draw an arrow for precision. Note: yours will probably be cleaner than mine because I ripped the center bar and had to use some tape to reinforce.
Feed the number line through the window.
Feed the flexible tape or paper through the back of the left cutout, over the center bar, and through the front of the right cutout.
Suggested Use
The interactive number line is useful for modeling expressions involving subtraction by first considering the additive inverse. For example, the expression 5 - 3 could be phrased verbally as "five and the additive inverse of three." To perform this operation with my interactive number line, I can pull the number line three to the left to represent - 3. When I do so, I have magically transformed 3 to 0. That's what - 3 does after all. The rest of the values go with it, including 5 which is now in the 2 position. Instead of taking 3 away from 5, I've manipulated the whole number line so that I can count new distances impacted by setting 3 to 0.
This feels complicated, I know. Is this that "new math" everyone is complaining about? Let's see the generalizability in action. What if I'd like to try -4 - 3? Once again, pulling the line three to the left puts -4 in the -7 position. Why not pull it 4 to the left instead and look where -3 is? What happens when we pull it 3 to the left and then 4 to the left? Now that's what I call doing math!
How about 2 + 1? That's 2 and the additive inverse of -1, which is +1. Let's pull it one to the right, 2 is now magically in the 3 position. How neat is that?
Not only does this feel magical and generalizable, it also introduces students to my personal favorite method of subtraction for competition and money handling alike: counting up. If I am ringing up an item that is $8 and a customer hands me a $20, and then I'm distracted by the power going out, and then I realize I'm hungry for lunch, it's challenging to subtract by taking away. It's much easier for my brain and most other brains to count from $8 to $10 ($2), then to $20 ($12). This tool models why counting up works.
Have fun out there in the wild world of numbers!
Comments