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Teaching Positive and Negative Integers Part Two
Positive and Negative Integers
Negative and Positive Numbers: Presentation Two
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."
An overhead projector
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,
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 itll
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
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.
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://18.104.22.168/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
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,