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Learning Math: Patterns, Functions, and Algebra

Proportional Reasoning Part A: Two Different Meanings of “More” (15 minutes)

Session 4, Part A

Problem A1

Suppose there are two classes in a school, one with 20 students and one with 25. If the first class has 10 girls and the second class has 12, which class has more girls? Note 2

 

Video Segment

In this video segment, participants share their answers to Problem A1 with Professor Cossey. Watch the segment after completing Problem A1, and compare your answer with those of the onscreen participants. If you get stuck on the problem, you can watch the video segment to help you.

What different definitions of “more” are used here?
What is your definition of “more”?

You can find this segment on the session video, approximately 3 minutes and 8 seconds after the Annenberg Media logo.

 


If you said that the second class has more girls, you’re making an absolute comparison. You probably thought that 12 is 2 more than 10, so there are more girls in the class with 12.

If you said that the first class has more girls, you’re making a relative comparison. You probably thought 10 is half of 20, and 12 is less than half of 25, so there are more girls in the class with 10.

Clearly these are two different interpretations of “more.” Although both interpretations are correct, in some cases it is more appropriate to look at relative rather than absolute comparisons. For example, compare an all-girl class of 20 students with a class of 25 students, 22 of whom are girls. In a sense, there are “more” girls in the class with 20.

 


Problem A2

Describe the numerical calculations you would do with two numbers to make an absolute comparison.

 


Problem A3

Describe the numerical calculations you would do with two numbers to make a relative comparison.

 


Problem A4

Give some examples of situations where it is more useful to make an absolute comparison. Give some examples of situations where it is more useful to make a relative comparison.

 


Problem A5

Suppose the average height of eighth-grade students is greater than the average height of seventh-grade students. Is this an absolute or relative comparison? Why?

 

Notes

Note 2

Groups: Begin Part A by discussing the meaning of “more.” Read through the exercise and share ideas about which class has “more” girls. Consider the difference between absolute and relative comparisons. Answer Problems A1-A5 in pairs or in small groups.

Solutions

Problem A1

Either answer is defendable. The first class has a higher percentage of girls — half its students are girls — while girls make up less than half of the second class. The second class, however, has a larger number of girls. Essentially, the meaning of “more” is the real issue in this problem.

 


Problem A2

An absolute comparison is done by counting: 22 is more than 20, because you count to 20 before counting to 22. Or it can be done by subtraction: 22 is more than 20 because 22 – 20 = 2, a positive number.

 


Problem A3

A relative comparison is done by finding percentages or fractions or by finding a rate. On a quiz, 22 out of 25 is worse than 18 out of 20, even though 22 is larger in absolute terms.

 


Problem A4

You might use absolute comparisons when looking at annual salaries, but a relative comparison when looking at per-hour wages. An absolute comparison might tell you that extended cable TV is more expensive than basic cable, but a relative comparison might tell you that you get more channels per dollar on extended cable. A truck may be able to travel further than a compact car before needing to be refueled (an absolute comparison), but the compact car may travel more miles per gallon (a relative comparison).

 


Problem A5

This is a relative comparison, because it compares heights “per student” by using the average. An absolute comparison might compare the total height of all eighth graders to the total height of all seventh graders.

 

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