Exploring Rate In The Context Of Music
In this lesson, students are introduced to rate in the context of music. They will explore beats per minute and compare rates using mathematical representations including graphs and double number lines.
Beats per minute is a rate. Musicians often count the number of beats per measure to determine the tempo of a song. A fast tempo produces music that seems to be racing, whereas a slow tempo results in music that is more relaxing. When graphed, sets with more beats per minute have smaller intervals on the double number line and steeper lines on the graph.
Goals and Learning Objectives
- Investigate rate in music.
- Find beats per minute by counting beats in music.
- Represent beats per minute on a double number line and a graph.
Math and Music
As a class, discuss the question “How is math connected to music?”
If students are struggling with this discussion, see if there are any musicians in class who might have ideas to share or might be willing to demonstrate something they have learned in music class. Some possible topics for discussion are as follows:
- Note lengths are measured in fractions, such as a quarter note, a half note, an eighth note, and a sixteenth note.
- In string instruments ratios are used to determine the tones created by varying string lengths. Changing the length of the string that is allowed to vibrate changes the tone it creates.
- The volume of an instrument affects the sound the instrument makes. Portions of instruments that can have varied volumes include the length of a flute, the body of a guitar, the trumpet of a horn, the body of a drum, and more.
- The rhythm in music is actually a pattern of sounds. The scales on a piano are an example of a repeating pattern.
- Musicians often count the number of beats per measure. This count determines the tempo of a song. A fast tempo produces music that seems to be racing, whereas a slow tempo results in music that is more relaxing.
Math and Music
Give some examples of how math is connected to music.
- When might a musician use counting?
- Can the volume of an instrument change its sound?
- If you know how to read music, how are notes measured?
Have students listen to the sample track and tap the beat as a class.
Watch for any students who have difficulty keeping the beat. Pair these students with those who tap the beat with confidence.
Before moving on to Work Time, make sure students understand that they will need to count the number of beats per minute in each of the sample tracks they listen to during Work Time.
Make sure these points come out in the class discussion:
- What makes music sound as if it belongs together? (Mostly it is tempo; students might also mention the key or other features of music.)
- What is tempo? (Speed, or beats per minute.)
- How can you measure tempo? (The usual measure is number of beats per minute. Help students understand that they need to measure not just one but two aspects of the situation: number of beats and amount of time.)
- Can you change tempo? (Yes, by playing faster or slower or by changing the tempo electronically.)
Technically, beats per minute is not exactly the same as tempo. However, beats per minute and tempo are synonymous in the context of this lesson. If students are interested, they can read more about beats per minute and tempo on the Internet. Students should also know that legally mixing samples of other people’s work and presenting the mix as a new song requires getting permission from the copyright holders of the original songs.
SWD: Though this discussion of mathematics is informal, consider it as an introduction or preview for students with special needs. Be sure to emphasize key information and critical concepts that will be introduced. This will support students as they work to determine salient information throughout the lesson.
- Listen to the music as a class and tap out the beat with your classmates.
Tap the Beats
Pair students who had difficulty keeping the beat during the Opening with students who are strong in this area. You might have the stronger student tap on the screen while the other partner simply taps his or her foot. The sample track in Task 2 has a strong beat to help students who are struggling. In this case, it is not important that the counting of the beats is an exact match to the actual beat.
Look for students who are able to tap out a beat at a constant rate, creating a graph that is essentially a straight line and a double number line that has points spaced in equal intervals or interval patterns. You may need to select a graph and double number line to share with the class during Ways of Thinking if not enough students are creating graphs and number lines with the desired traits.
Listen and look for the following types of student thinking to highlight during the Ways of Thinking discussion.
Look for students who:
- Get different results for beats per minute.
- Do not have a straight line on the graph or are missing points.
- Have interesting explanations of their work.
SWD: Be prepared to support students who are struggling to use the interactive appropriately. Anticipate difficulties and coach students so that they are comfortable enough with the interactive to successfully attempt the task.
ELL: For the Work Time activity, make sure to demonstrate and orally explain the activity step by step to ensure that ELLs understand what they are being asked to do.
Student does not understand the visual representation of the double number line.
- Where do you see the time on the double number line?
- What unit is shown for time?
- Where do you see the number of beats on the double number line?
- What does the first point represent on the double number line?
- Look at the points on the double number line. What do you notice about the distance between the points?
Student does not understand the visual representation of the graph.
- What is the scale of the horizontal axis?
- What is the scale of the vertical axis?
- How far right of 0 is the first point? How far above 0 is the first point? What does this point represent?
- Look at the points on the graph. What do they form? Explain why.
Student has thoughtful answers for the questions.
- Where do you see the number of beats per minute on the double number line? On the graph?
ELL: Some of the words in the questions and prompts can be somewhat difficult for ELLs to follow. If necessary, rephrase using words you know students can understand to allow ELLs to fully participate and to have a fair chance to answer the questions.
Mathematical Practice 1: Make sense of problems and persevere in solving them.
- Make note of any students who struggle but persevere in making sense of the rate beats per minute as represented on a graph and on a double number line.
Mathematical Practice 2: Reason abstractly and quantitatively.
- As students work, identify several students who can show how each line of the double number line represents an axis of the graph and how a point on the graph is the intersection of two corresponding points on the double number line. Look for students who can explain the meaning of the quantities as represented in each model. These students are showing the ability to reason abstractly and quantitatively.
Mathematical Practice 4: Model with mathematics.
- Look for students who use the mathematical models of rate on a graph (straight line) and on a double number line (evenly spaced points) to conclude that beats per minute is a constant rate.
- Answer will vary. Possible answer: The graph of the beats is a straight line so the beat must be constant.
- The beats are expressed as points on the top line of the double number line.
- The beats are expressed as points on the coordinate plane. Each point represents the total number of beats in relation to the number of seconds of song that has played.
- Both the double number line and the graph show that the beats are at a constant rate. On the double number line, the constant rate is expressed by the even spacing of the dots. On the graph, the constant rate is expressed by the straight line of the graph.
- Answers will vary. Students will repeat the process with a new track in order to verify the conclusions they made concerning the first track they listened too.
Tap the Beats
Open the interactive and play one of the sound tracks. Follow the instructions and tap the beat as you listen to the songs.
After listening to a track, discuss these questions with your partner.
- Is the number of beats per minute constant? How do you know?
- How are the beats expressed on the double number line?
- Choose the graph option. How are the beats expressed on the graph?
- Compare the graph and the double number line. How are they alike? How are they different?
- Play another track and verify your findings about the questions.
INTERACTIVE: Tap the Beats
Think about what a point on the graph means.
Have students share their graphs and double number lines and explain what they represent.
By the end of the discussion, all students should understand one another’s representations, as well as the common mistakes and misconceptions outlined in the Mathematics section.
ELL: Encourage students to use the academic vocabulary that they have learned. As they participate in the discussion, be sure to monitor for knowledge of the topic. Stay alert to follow up on interventions that seem unclear or ambiguous. When ELLs contribute, focus on content, and don’t allow grammatical difficulties to distract you from understanding the meaning. Help ELLs who make grammatical mistakes by rephrasing, but do it only when your rephrasing will not become an interruption or interfere with their thinking.
Have students look at different groups’ work, and then ask what makes each set look different. Students should understand that sets with more beats per minute have smaller intervals on the double number line and steeper lines on the graph.
Discuss answers to the previous task, in which students compared the graph and the double number line. All students should realize that beats per minute represents a constant rate. They should recognize that the intervals on the double number line are equal and that the points on the graph form a regular, linear pattern.
Facilitate the discussion to help students understand the mathematics of the lesson. Ask questions such as the following:
- Why is the tempo of a song considered a rate?
- [Name]’s graph represents the first sample track. [Name]’s graph represents the second sample track. How are the graphs the same? How are the graphs different?
- What would a double number line of [Name]’s graph look like?
- How can you tell by looking at a graph or a double number line if one sample track has a faster tempo than another sample track?
Ways of Thinking: Make Connections
Take notes on the beats per minute and how they are represented on the double number line and the graph.
- What is meant by constant rate ?
- How does your double number line show beats per minute as a constant rate?
- How does your graph show beats per minute as a constant rate?
Reflect On Your Work
Have each student do a quick reflection before the end of class. Review the reflections.
If any reflections look interesting enough to pursue later, save them to share with the class when appropriate.
Write a reflection about the ideas discussed in class today. Use one of the the sentence starters below if you find it to be helpful.
- Something I have learned about rates is …
- Something I still wonder about rates is...