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Teaching Positive and Negative Integers - Part Three
Positive and Negative Integers
Multiplying Positive and Negative Integers
You might want to spend a week reviewing adding and subtracting
and understanding the connections between negative numbers
and real-world problems before even approaching multiplication.
You can also take this time to review the concept of multiplication.
In O'Brien and Casey's (3/83) study, students proficient at
computing multiplication failed dismally in attempts to construct
a multiplication problem. 37% of fourth and, surprisingly,
44% of fifth graders constructed additive problems, e.g.,
"Johnny had five apples and Mary had 3 apples. How many
would they have together?" These students could calculate
accurately, but the gap between "math class calculation"
and solving math problems in the real world was already formed.
Teach the language: The reason those things are called
"times tables" and why we say "four times three"
is because we are doing the same thing a certain number of
times, or we have a certain number of the same
thing. When you are trying to "translate" a math
problem from words into numbers, knowing that "of"
means "times" can be very helpful. It shouldn't
be a recipe to replace understanding -but it's a tool that
Realistic use of multiplying positive and negative numbers
include regular withdrawals from bank accounts (positive times
negative). How can you connect negative times negative, though?
The National Council of Teachers of Mathematics website describes
a lesson in which students walk forward and backward and videotape
themselves, and predict which way the person will appear to
be moving. Making the students predict the outcome, and expressing
each possible outcome mathematically before showing the videos,
is an excellent way of helping the students truly own this
knowledge that going backwards backwards is the same as going
forwards. See A Videotaping
Project to Explore the Multiplication of Integers (http://illuminations.nctm.org/lessonplans/6-8/videotaping/)
Chinn and Ashcroft's Mathematics fro Dyslexics: A Teaching
Handbook (Whurr Publishers,
second ed. 1998) suggest the following (I've changed the monetary
Imagine that future time is counted in positive years, then
time in the past should be counted in negative years. Imagine
that the cost of building a certain kind of house increases
by $2000 a year, and that the value of a car decreases by
$500 a year.
In five years' time, the house will cost 10,000 more because
+5 x +2000=+10,000
In Two years tie, the car will be worth $1000 less because
+2 x (-500)=-1000
Four years ago, the house would have cost $8000 less because
(-4) x (+2000)=-8000
And Three years ago, the car would have been worth $1500
(-3) x (-500)=1500.
Multiplication as repeated addition can be shown with the
As with the addition and subtraction, take these concepts
slowly and provide lots of student interaction. Make connections
and clarify the mathematical symbols. Don't assume that students
know that 3 + -2 is the same as 3 + (-2).
Teaching to automaticity
Students should get lots of practice with these operations.
Much has been written about the dubious value of having students
perform meaningless calculations ad nauseam. However, as with
all things, a bit of balance is in order. A worksheet or timed
drill with 100 integer-addition problems is a lousy idea.
However, there is much less wrong with doing five of those
problems every day and incorporating review and practice all
year. As in reading, it is when students are fluent with working
with the symbols that they are free to understand what those
Joyce Steeves has presented many workshops on teaching math
to students with learning disabilities and recommends beginning
each day with activities such as having the student invent
problems with the answer "-3" and awarding one point
for each addition, two for subtraction, three four multiplication,
and four for division. (This need not be a competitive venture
except to compete with oneself.)
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://22.214.171.124/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,
There are several online resources about teaching positive
and negative integers. An excellent activity would be to have
students explain these web pages to other students. Many include
online practice with calculations.
to Signed Integers Somewhat language-intensive, but good
visual-symbolic introduction to integers as well as operations.
Another good introduction to integers.
Glosser's Math Goodies
Introduction to Integers (includes some online exercises)
Videotaping Project to Explore the Multiplication of Integers
An excellent idea for getting kids to "own" the
ideas of multiplication of positive and negative numbers.
Lesson using students to "be" positive and negative
numebrs on the number line.
http://www.aaamath.com/grade6.html This has many, many very
basic online lessons for practice. The colors may make you