Home > Math > Teaching Positive and Negative Integers Part Two

Part One
Part Two
Part Three

## Materials:

Red and blue see-through chips (such as those used in bingo) which show their colors on an overhead projector. I like to have red for negative because of the language carryover to being "in the red."

### Lesson Plan:

We can use a number line to explain negative numbers, but before there were number lines, we used numbers to count *things.* We can understand negative numbers with things, too, if we can understand that negative one (-1) is the opposite of positive one, and that two opposites, in effect, cancel each other out. (Two people pulling with equal force in opposite directions is another good analogy for this concept.)

This "opposite" idea is an important one for understanding what negative numbers mean.

Bring out the chips. Put three "positive" colored chips on the positive side. Ask the students to write that as a number. Then ask them what the opposite of that would be (-3, or three red chips). Explore the concept with further examples and discussion and real-world examples.

Proceed to adding two groups of positive numbers; the goal is to relate what the student already knows to the new domain. Negative integers don't change the rules for positive integers -- they just help us explain things that positive integers don't do a good job with. The more students can tell *you* about the operations, and demonstrate them with the chips, the better.

Keep making the conceptual connection. This 'canceling out' concept can be easily and totally lost, but it doesn't have to be. Call upon students to refresh your memory and explain how two numbers can "add up" to NOTHING.

For a verbal approach to making the shift to negative integers, you can announce that henceforth subtraction will be banned, and only "adding the negative of a number" will be permitted. That can be demonstrated with the chips. Instead of actually taking away two positive chips, adding two negative chips would mean that two of the positive ones were 'canceled out' as in the examples of football yards and money. Take whatever time is needed to make sure students see that connection and really understand the "canceling out" idea. Students can be guided to discover that "subtracting a negative number" would therefore be adding the negative of that negative number.... the positive one.

Demonstrate this with several different problems. This may be enough new information for one day; have students practice a set of simple subtraction problems that are written as addition of negative numbers. Be sure they can demonstrate the problem with symbols and with chips.

Adding negative numbers. Put three red chips on the negative side of the array. Make the connection between the abstract "red chip" or "negative number" or "minus 3" and one or more of the real-life examples.
Ask the students to tell you how that would be written in mathematician's language ("-3) Then tell them you're going to "add" something and ask how to write that... and then add four more red chips to the negative side. Have the students tell you how to write that:

-3 + -4

Again, have them make the connection between the symbols and a real-life situation. Ask them if they can solve that probelm (or, write it as one number).

Try different problems -- with the chips being the same sign -- and see if the students can come up with a rule for getting the right answer.

Okay, now for the tough stuff.

Here's what happens when the you add a bigger negative number; see if the students can discover that it's the same thing as saying 3 - 4.

(Another approach is to do that problem as a "subtraction" problem. Since you've run out of things to physically take away, you take a plus and minus, since they cancel each other out, and "add zero." Then you can take away four -- and you're left with the one negative chip.)

Mnemonic for remembering what to do with positive and negative numbers when adding and subtracting: Using the melody of “Row, row, row, your boat” give the students the ollowing words: "Same Sign, Add and keep, Different sign, subtract. Take the sign of the higher number Then it’ll be exact!" Show students how to use this to approach a problem; sing it a few times, dance around a little.

Note: Some students will hang onto the idea that "plus means add" -- so -4 + 5 is 9 -- and "minus means subtract," so -4 - 9 is 5. Sometimes a quick explanation that now that we've entered the advanced world of algebra, we need to be more flexible, can really help with this confusion.

Other ideas: As part of the University of South Carolina's Teaching Mathematics with Manipulatives Videocourse, Henri Piccioto's "Lab Gear" is used to teach a number of algebraic concepts including integers and factoring of polynomials. He uses an "upstairs" and "downstairs" analogy that works well. Piccioto has also written a Comparison and History of Algebraic Manipulatives which explains several options well.

Part Three

Part One

### References

Chinn, SJ and Ashcroft, JR (1998) 'Mathematics for Dyslexics: A Teaching Handbook' 2nd edn. London, Whurr

Dickey, E. M. Course Materials for Teaching Middle and High School Mathematics for Manipulatives. University of South Carolina, 1995. Teaching Mathematics with Manipulatives Videocoursehttp://129.252.97.21/dickey/nctm1996/sdtitle.html

O'Brien, Thomas C. and Shirley A. Casey. Children Learning Multiplication Part I. School Science and Mathematics v. 83 n. 3 3/83

Piccioto, Henri.Comparison and History of Algebraic Manipulatives http://www.picciotto.org/math-ed/manipulatives/alg-manip.html (as of 06/03/02).

Steeves, Joyce. Various presentations at International Dyslexia Association conferences, 2000-2001.