Home > Math > Teaching Positive and Negative Integers

Part One
Part Two
Part Three

## Materials:

A very large thermometer or a picture of one, which includes temperatures below zero and only uses one temperature scale.

A picture (enlarged) of an oven thermometer.

### Lesson Plan:

Negative numbers are a pretty abstract concept. They're also a topic that is high-risk for memorizing rules for without understanding the reasoning behind the rules. Please don't settle for this!

Big Idea: One of the important "big ideas" of mathematics is that we can learn more advanced mathematical ideas by starting with ideas that we understand and extending those ideas. For integers, start with these big ideas:

• The concept of opposites and balance. As part of a videocourse in using manipulatives to teach secondary mathematics, Bettye Forte suggests an introductory activity where students are given a card with a word on it and and instructed to find the student with the card with the opposite on it, first with common concrete terms such as "hot" and "cold," then with more abstract mathematical ideas such as "above" and "below," "3 more than" and "three less than," "plus" and "minus." (Dickey, 1995)

• "Regular" adding and subtracting, including the idea that you "can't" subtract a larger number from a smaller number because you can't take away what you don't have. Discuss why this is true in most cases, and challenge students to come up with a real-life scenario in which they would have to do just that.

Some other "real life" reasons to use negative numbers are: elevators in large buildings with above and below ground levels (a pun-infested 'concrete reference'), football gains and losses (if you gain 5 yards but get a 5 yard penalty, you're where? Back where you started. That 5 yard penalty is "negative 5" because it cancels out 5 yards.) Money found and spent was another reference -- if you find 10 cents and then spend it, where are you? Back where you started -- wherever that was.

Point to your large thermometer . Explain briefly how thermometers work (or ask the students) -- that mercury shrinks when it gets colder, always the same amount, and gets bigger as it gets warmer. Ask: What would happen if it were 20 degrees outside (have a student show you where the temperature would be on the thermometer.), and the temperature got 10 degrees cooler. That mercury would shrink down...what would the new temperature be? (Have another student hop up and show where that would be). You could also, of course, wax dramatic and describe an Arctic exploration and the clothing you would have to don and the importance of exercise to avoid hypothermia. There is NO reason math should be dull! For a cross-curricular connection, consider playing the song "The Frozen Logger," with a lesson on tall tales and exaggeration.

Now try the same thing with harder numbers just to make sure the students get beyond the intuitive. On your other side is an oven thermometer. (Big idea: mathematicians take the obvious and figure out how to make it useful for more complicated situations.)

Teach the language of math: Many students need to be shown the connection between the "obvious" ten degree "difference" and the less obvious "find the difference" that they read in math problems. Show the similarities in concept and language.

Go back to your arctic expedition. It may have been 400 degrees in the oven, but it is still 10 degrees outside. What would happen if it got ten degrees colder? What would the temperature be then? Have a student show you where the temperature would be on the thermometer.

Note: Give students "wait time." Some students will have been doing the thinking before the answer, and will have gotten it. Others will need to think back after the answer is given. Encourage students who knew the answer to think of even harder problems while they're waiting -- or to try to think about where you are going and what point you are going to make. You may want to perform the same thinking process with different smaller numbers: start at 32 degrees (freezing point) and take off ten degrees. Then have a true cold snap come through and subtract 20 degrees. Be aware of how much change you can make; you may want to stick to big, round numbers and subtract from 40 to 30. Many students get to middle and high school unable to perform operations such as "50 - 10" mentally, much less "32 - 10."

Now it's zero degrees out there. What would happen if it got ten degrees COLDER???
You could call it "Ten degrees colder than zero degrees."
You could shorten that to "Ten below zero, " right?
Mathematicians are even lazier than that. They borrow that subtraction sign that means "less than" and call this number "minus ten" or "negative ten" or -10.

Then pose a more difficult question, using the thermometer as a reference: Which is colder, ten below zero or zero? First ask verbally, showing the numbers on the thermometer, then write the numbers on the board. Then make the "language switch" to Math Problem-ville: "Which number is greater, zero or ten below zero?" Ask this same question comparing 0 to negative numbers until it is clear to students that even though the number looks bigger, it stands for less.

STOP HERE.

One of the most common mistakes we make is doing too much too soon. Stop while it makes sense. THE NEXT DAY; see if another student can explain what you taught the day before. You can give appropriately differentiated math practice now, so that the student who has been chomping at the bit all this time for challenge can get some, and the student who desperately needs review can get some.

### Independent Practice :

Have students compare temperatures to say which one is warmer, then compare numbers to say which is greater. Have a copy of the thermometer available for all of them, but encourage them to try to figure out the answer without looking at the thermometer first, then to use the thermometer to check themselves. Remind the students that they shouldn't forget what they already know; so you're including some comparisons that don't go below zero.

Teach the language:Point out that a number without a sign is a "positive" number, and that it can either be written with a + sign in front of it, or with no sign at all. Remind students that language is supposed to *mean* something, and encourage them to think of how each temperature listed would feel if the room were that temperature.

Circle the colder temperature:

0 -2
0 2
0 10
0 -10
-10 10
5 -5
50 40
90 100
0 50
0 -8
0 -80
0 -25
-1 -25
-3 -25
25 0
10 0
1 0
-19 0
-10 -1
-10 -9

Have each student make up three problems and circle the correct answers.

When students are comfortable with working with temperatures with a thermometer present, take the thermometer away. A possible transition is removing all of the numbers except the zero from the thermometer. Encourage the students to use their visual memories -- remind them that often the difference between the student who seems brilliant and the one who is still confused is simply knowing what mental tricks to try.

Review this idea. Don't make this any more complicated until this part is easy.

Next step: show students a number line and ask them to compare it to the thermometer. Introduce other concrete applications of negative numbers: elevators going below the ground floor money being owed.

Cooperative learning option: This is a good topic for students to work in groups to present explanations of what negative numbers mean to the rest of the class. They can have the option of expanding on what has been taught or sticking to the basics.

On To Part Two

### 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. More information: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.