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Teaching Positive and Negative Integers
Positive and Negative Integers
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.
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
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"
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.
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
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:
Have each student make up three problems and circle the correct
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
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://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,