Home > Math > Teaching Positive and Negative Integers - Part Three

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## 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 works.

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 units):

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 more because
(-3) x (-500)=1500.

Multiplication as repeated addition can be shown with the chips, too:

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 symbols mean.

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.)

REFERENCES:

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

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.

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.

Math.com: Introduction to Signed Integers Somewhat language-intensive, but good visual-symbolic introduction to integers as well as operations.

Another good introduction to integers.

Mrs. Glosser's Math Goodies
http://www.mathgoodies.com/lessons/vol5/intro_integers.html
Introduction to Integers (includes some online exercises)

A Videotaping Project to Explore the Multiplication of Integers
http://illuminations.nctm.org/lessonplans/6-8/videotaping/
An excellent idea for getting kids to "own" the ideas of multiplication of positive and negative numbers.

Human Number Line http://www.iit.edu/~smile/ma9201.html
Lesson using students to "be" positive and negative numebrs on the number line.

AAA Math
http://www.aaamath.com/grade6.html This has many, many very basic online lessons for practice. The colors may make you swoon, though!