Today's teenager is a savvy consumer of digital music and the constantly-evolving technology that plays it. Ask a typical student what they know about iTunes versus Pandora versus Spotify—most of them will have an opinion on the "best" service for listening to songs. This lesson links students' existing interest in music with the mathematical topics of frequency and relative frequency.
The activity assumes that students know what Shuffle Mode does when they listen to digital music. Shuffle Mode is a function on digital music players that "shuffles" or randomly rearranges the order of a list of songs. Each time a person presses Shuffle Mode, the playlist is rearranged. If we assume a music player's Shuffle Mode is truly random, the chances of any particular song being played would equal 1 divided by the total number of songs (1/total #). This is analogous to rolling a fair die; each number on the die has an equal probability of being rolled (1/6 or 16.7%).
Lesson Plan Template: General Lesson Plan
Learning Objectives: What should students know and be able to do as a result of this lesson?
Students will summarize and interpret data generated by two imaginary digital music players set on the Shuffle function where one player is "fair" and one is "unfair."
Specifically, students will:
- Find the frequency of each song from a six-song playlist getting played as the first track after 30 trials, and fill in the appropriate boxes on a two-way frequency table.
- Calculate the relative frequency each song getting played as the first track after 30 trials, and fill in the appropriate boxes on a two-way relative frequency table. (Answers should be in percents.)
- Analyze the data for associations or trends, and justify your answer to the Guiding Question about whether each music player is fair or unfair by using evidence from the frequency tables.
Prior Knowledge: What prior knowledge should students have for this lesson?
- Convert fractions to percents
- Distinguish between independent and dependent events (probability)
- Know vocabulary at beginning of the lesson (on Entrance Ticket): categorical data, frequency, relative frequency
- Understand how to tally data and fill in a two-way frequency table
- Fluently calculate relative frequency (converting a fraction to a percent)
Guiding Questions: What are the guiding questions for this lesson?
Given two random digital music players, is each player fair or unfair?
Teaching Phase: How will the teacher present the concept or skill to students?
- Project the Powerpoint file "Lesson Overview" onto screen. Introduce the lesson title, and have a student read aloud the state standard that aligns with the lesson.
- Move to slide 2, and pass out Entrance Tickets. Students should independently define each word. Allow no more than 3 to 5 minutes.
- Choose Option A or B. (Teachers with access to a SmartBoard may want to write the definitions in blanks provided on slide 2.)
- Option A: collect the Entrance Tickets to grade later. Discuss definitions as a group.
- Option B: Discuss definitions as a group. Have students tape the Entrance Tickets into their notes for later review.
- Transition to the slides by discussing the recent popularity of buying music online through sites like iTunes. Ask students to share briefly their favorite technologies for listening to and storing digital music files.
- Continue that after customers purchase digital music online, they can organize the songs into playlists, which can then be randomized by pressing a "shuffle mode" function.
- Lead into the Guiding Question by noting that numerous articles online debate whether the "shuffle mode" is actually random. Ask students to think of human factors that could affect the randomness of a song playlist's order. (Possible answers: Does the listener press "shuffle mode" repeatedly and end up hearing the same songs multiple times? Are the songs being played according to popularity rankings?)
- Transition to the day's Guiding Questions: "Given two random digital music players, is each player fair or unfair? How can you justify your answer using evidence from two-way relative frequency tables?"
- Tell students they'll be viewing some simulated data from two imaginary music players. They should imagine that a student creates an experiment to test whether two randomly chosen digital music players with "shuffle mode" are fair or unfair. (Remind students that "fair" means any outcome is equally likely to occur.) The student picks 6 songs and presses "shuffle mode" 30 times in a row. The student then records which song played during that trial by marking an X underneath that song's title.
- Today’s task is to summarize and interpret the data, then to answer the Guiding Question using conclusions drawn from the two-way frequency tables the students will fill in.
- Demonstrate with a similar example. (Put sample data and blank frequency tables up on the projector screen.) Walk students through the process of tallying the number of plays and writing those numbers in the appropriate column of the Frequency Table. Then review how to find relative frequencies, turning a fraction into a percent. Then write the percents in the appropriate columns of the Relative Frequency Table.
Guided Practice: What activities or exercises will the students complete with teacher guidance?
- Pass out the In-Class Worksheets and seat students in small groups (if they aren't already assigned to small groups).
- The teacher should walk around as students sort through the data and first tally frequencies and then calculate relative frequencies. Assist individuals and groups when necessary. Encourage them to show their work when calculating relative frequencies, and then to continue on to the analysis questions when they complete the tables.
Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the lesson?
After students complete the In-Class Worksheets, they should move on to complete the Independent Practice worksheets. As the name suggests, these are meant to be completed alone and submitted for an individual grade. Students should create two dot plots, one for each music player, and then answer the two questions on the back.
Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
Close the lesson by having students return to the Guiding Question about the fairness or unfairness of each music player. Ask them to share their findings from the data, and to reflect on whether 30 trials is really enough to determine with certainty whether a music player is fair or not. (See the questions on the Independent Practice worksheet for a similar question.) Ask how many trials they would require before they'd be convinced that a player is fair or not, and ask them to explain that opinion.
Students will complete two forms of summative assessment:
- In-class Worksheet that students will complete during the lesson.
- Independent Practice Assignment to be completed as homework.
Teachers can monitor students' progress at several points through the lesson.
- At the start of class, students will complete an Entrance Ticket before working in small groups. They will write definitions to vocabulary words found in the standard that are central to this lesson's activities.
- During the lesson, teachers can monitor students in small groups for their level of fluency calculating relative frequency for the given data.
- After students calculate relative frequencies, teachers can evaluate students' ability to create two different two-way frequency tables.
- At the end of the lesson, teachers can evaluate students' ability to analyze data to find associations or trends and apply that evidence to justify their answers to the Guiding Question.
Feedback to Students
- Feedback from the teacher to the whole class (discussing answers to Entrance Ticket); questions to ask:
- What is the definition for each vocabulary word?
- Can you list or draw an example of the word?
- Which words are hard for you to remember?
- How can we remember what each word means?
- Feedback from the teacher to students in small groups (while calculating relative frequencies and completing two-way frequency tables); questions to ask:
- Compare frequency and relative frequency. How are they similar and different?
- How can we find the relative frequency of a song after thirty trials?
- Feedback from the teacher to individual students (written feedback about their answers to the Guiding Question); questions to ask:
- Did the student draw the correct conclusions from the data?
- Did they include data in their answer to the Guiding Question?
Accommodations & Recommendations
Mathematical practice standards addressed:
MAFS.K12.M.1.1: Make sense of problems and persevere in solving them.
MAFS.K12.M.2.1: Reason abstractly and quantitatively.
MAFS.K12.M.4.1: Model with mathematics.
Source and Access Information
Name of Author/Source: Helen Exner
District/Organization of Contributor(s): Broward
Access Privileges: Public
* Please note that examples of resources are not intended as complete curriculum.