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March 13, 2013

A Gigabyte of Music: How Much is That?

Author Mary Klein is a mathematics teacher in Urbana, Illinois

Time

One class period plus an assignment

Background

In an effort to stem the illegal downloading of music and the resulting financial losses to performers and others in the music industry, the entertainment industry has filed over 250 lawsuits against individuals. Fines can be high and since students are frequent downloaders, this issue is a concern to teens, their parents, and the schools they attend.

The mathematical methods used in this lesson are relatively new; they make conversions easier and more fun for students.

Lesson Overview

Students will learn to:

  • make unfamiliar units of measure more understandable by using a mathematical conversion technique;
  • convert conversion equations to fractions;
  • convert from one unit of measure to another;
  • solve problems involving conversions.

Procedure

1. Distribute the Online NewsHour story, “Entertainment Industry Targets Individual Downloaders” for students to read. Briefly discuss the story emphasizing how schools must protect their computer network and then focus on this paragraph from the story:

UC Berkeley is including an orientation session for incoming students that warns about the dangers of downloading copyrighted material. The university will also cut off students’ access if they transfer more than five gigabytes worth of files per week.

Ask, “How large is five gigabytes?” Mention that our main goal today will be to make less familiar units of measure more understandable by using a mathematical conversion technique.

2. Tell the students that we need the following conversions: (Write them on the board.)

1 gigabyte = 1000 megabytes
1 CD = 700 megabytes (This data item was taken from a package of CDs; other sizes may be available.)

 

We also need to remember that because “1 gigabyte = 1000 megabytes” the fractions “1 gigabyte / 1000 megabytes” and “1000 megabytes / 1 gigabyte” both equal one. Multiplying by one preserves identity (x * 1 = x), so multiplying a measurement by one of these fractions produces an equivalent measurement. Since (x*1*1*1 = 1) multiple changes can be made at once.

3. Write the problem on the board: “How many CDs does it take to hold 5 gigabytes of files?”

Show the work explaining as you write:

5 gigabytes * (1000 megabytes / 1 gigabyte) * (1 CD / 700 megabytes) =(5 gigabytes /1) * (1000 megabytes / 1 gigabyte) * (1 CD / 700megabytes) =(Note: Draw a slash through the bolded units of measure to show canceling)

(5 * 1000* 1 CDs) / (1 * 700) = 5000 CDs / 700 = 7.142857… CDs

So it would take 8 CDs to hold 5 gigabytes of files.

4. Students generally find that linking together the fractions is fun, but some people will have trouble deciding which fraction to use. Point out that we choose the first fraction so that its denominator matches the unit used in the measurement to be converted. The unit used in the numerator of the first fraction will appear in the denominator of the second fraction and so on. In general, the unit in the numerator of one fraction will match the unit in the denominator of the fraction immediately to its right. Remind students that a numerator cancels with a denominator.

5. Ask how this problem would change if we wanted to know how many songs fit in five gigabytes of files. Write these conversions on the board. Explain the information in the parenthesis as you write. Then solve the problem.

1 CD = 80 minutes of playing time (copied from the label of a package of CDs)1 song = 2.5 minutes of music (A typical song lasts from 2 to 3 minutes. Emphasize that our final answer will be only a rough estimate, but it will help us understand the size of five gigabytes.)

Solution: “How many songs fit in five gigabytes of files?”

5 gigabytes * (1000 megabytes / 1 gigabyte) * (1 CD / 700 megabytes) * (80 min. / 1 CD) *
(1 song / 2.5 min.) =5 gigabytes * (1000 megabytes / 1 gigabyte) * (1 CD / 700megabytes) * (80 min. / 1 CD) * (1 song / 2.5 min.) =(Note: Draw a slash through the bolded units of measure to show canceling)(5 * 1000 * 1 * 80 * 1 song) / (1* 700 * 1 * 2.5) = (400,000 songs / 1750) = about 228.571… songs or about 229 songs

6. Finally, remind students that rounding should occur only at the last step of any problem. A nice thing about these problems is that we can enter all the numbers into the calculator in such a way that the equal key is hit only once; there is only one chance to round.

7. Distribute worksheets and allow students to work in groups.

Extension Activity

1. Show students that this technique is also useful in making rates more understandable. For example, some people understand rates of pay more easily if the rate is given in terms of “yearly pay” and some understand “hourly pay” better. (Remember “per” translates as division.)

$18 per hour =

$18/ 1 hour = ($18 / 1 hour) * (8 hours / 1 day) * (5 days / 1 week) * (50 weeks / 1 year) = $36,000 / year or $36,000 per year

$50,000 per year = ($50,000 / 1 year) * (1 year / 50 weeks) * (1 week / 5 days) * (1 day / 8 hours) = $25 / hour or $25 per hour.

2. Investigate your high school’s policy toward downloading music on school computers.

Author Mary Klein is a mathematics teacher with experience at the high school, middle school, and junior high school level. She is the author of several Web-based lessons on mathematics and basketball. She lives with her husband and son in Urbana, Illinois.

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  • The Materials You Need

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    Common Core Standards

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    Relevant National Standards:
      Correlation to National Teachers of
      Mathematics Standards
    • Recognize and apply mathematics in contexts outside mathematics
    • Apply appropriate technique, tools, and formulas to determine measurements
    • Use the language of mathematics to express mathematical ideas precisely

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