Negative Numbers

According the the National Council of Teachers of Mathematics (NCTM), math should not be learned through the memorization of meaningless steps. Math should be learned as problem-solving with real life situations, reasoning with patterns, and understanding.

This page overviews a set of lesson plans (that I have used) to present negatives more in line with the standards of the NCTM. These lessons were used in an urban seventh grade and should be suitable for all introductions to negatives. The purpose of this presentation is to show that "rules" never have to be taught when introducing negatives. The "rules" follow naturally from the concepts and the very nature of mathematics itself.

Written 2003

Formatted 2009

 


Demonstrating Negatives Using Measurement

Negatives are strongly associated with measurement concepts, but not really associated with counting concepts. It is more instructive then to teach negatives with measurement, and to reduce dependence of examples that are based on counting.

Introduction: What types of measurement are available to us? Examples:
  • height & altitude
  • distance
  • speed
  • temperature
  • time
  • weight & mass
  • volume
We will now ask the questions: "How far?" and "In what direction?" "How far?" will tell us the magnitude; "In what direction?" will tell us the sign. Note: this is a good precursor for the ideas of vectors.

Negatives are used primarily to represent three types of situations:

Once students can express the ideas below they have demonstrated an understanding of the uses for negatives. The questions "Why do we need to know this?" and "How will we ever use this?" have been confronted.

[a] Change in the downward or backwards direction.

What does a negative sign mean for each of these?
  • altitude - falling, descending
  • distance - moving left, south, backwards
  • speed - slowing down, reversing
  • temperature - getting colder
  • time - going back in time- science fiction only
  • weight & mass - getting lighter
  • volume - shrinking

[b] Below a reference Point
  • height & altitude - below Sea level
  • distance - south of the Equator
  • speed - going backwards (stopped = 0)
  • temperature - colder than freezing
  • time - BC, or before present
  • weight & mass -rarely used
  • volume - rarely used

[c] Ordered Comparisons

These can be the most confusing for students. Emphasize the language as suggested.
  • height & altitude - below
  • distance - behind, closer than
  • speed - slower than
  • temperature - colder than
  • time - before
  • weight & mass - lighter than
  • volume - smaller than


Introductory Activities

Addition: Put up a number line that is spaced about 1 step per number. Have a student come up and walk 5 steps then 3 more steps. Ask: "Where did he end up," and "What operation did he demonstrate?" Show: 5+3=8. Have another student walk backwards 5 steps then 3 more steps. Ask the same questions along with, "How do we represent walking backwards." Show: -5+(-3)=-8. Next, have another student walk forward 5 steps then backwards 3. Ask the same questions along with, "If walking then walking again was represented by addition before, how should we represent it this time?" Show: 5+(-3)=2. Have student make up their own operational "rules" to describe the relationships, they have just seen. Compare their rules to each other, and to traditional rules for addition with signs.

Subtraction: Have two students stand below the number line, the first, "A," at 3, and the second, "B," at 8. Ask: "How far is it to B from A?" Make a point that the question was asked in the order "to - from." Tell A to point to B and ask, "What direction is A pointing?" Show: 8-3=+5. Have A and B trade places and ask the same questions. Show: 3-8=-5. Ask: "What does the negative represent?" Next repeat these steps starting with A at 2 and B at -6. Have student make up their own operational "rules" to describe the relationships, they have just seen. Compare their rules to each other, and to traditional rules for addition with signs.

Other Negatives Lessons

 

Teacher Help

 


Continuing the Discussion:

Demonstrate to students examples of word expressions that imply negatives. Have the students write their own word expressions, and draw their own diagrams to demonstrate negative numbers. Compare and contrast similar expressions. Be sure they practice with different types of measurements and the three different uses as shown above.

Examples:

[a] An enemy airplane is flying at 12000 feet. You are flying at 8000 feet. How far must your missile rise to defend yourself?

[b] The enemy is flying at 12000 feet. You are flying at 8000 feet. How far will his bomb drop when he tries to hit you?

12000 - 8000 = 40000

8000 - 12000 = -40000

Notice the difference in signs between the two answers implies that one missile rises and the other descends.

[c] The enemy airplane is at 12000 feet. You are in a submarine 1000 feet below sea level. How far must your missile rise to defend your self?

Note: each problem is subtraction structured in the form: (goes to) - (comes from) = answer

12000 - (-1000) = 13000

Discuss: What is similar and different about each problem? Why?

Comments:

  • Problem [a] does not demonstrate a negative, but reminds students how subtraction relates to distance problems.
  • Problem [b] shows how a negative is used to represent a comparison or a change in altitudes.
  • Problem [c] show how sea level is the reference point for altitude and below it is negative.
 
 

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