Lesson 9

Mean

9.1: Close to Four (5 minutes)

Warm-up

The purpose of this warm-up is to prepare students to find the mean of a data set. While the goal of the activity is for students to create an expression with a value close to 4, the discussions should focus on the reasoning and strategies students used in creating their expression. Students should notice that they would get the same result if they divided the value of the entire expression in the numerator by 4 as they would if they divided each number in the numerator by 4 because there are 4 numbers in the numerator. 

During the partner discussions, identify students with different strategies for creating an expression with a value of 4. Ask them to share during the whole-class discussion.

Launch

Arrange students in groups of 2. Give students 1 minute of quiet work time, and then 2 minutes to share their response with a partner. Follow with a whole-class discussion. If students do not interpret the directions of “close to 4” to mean that their expression can have an exact value of 4, tell them it can be exactly 4.

Student Facing

Use the digits 0–9 to write an expression with a value as close as possible to 4. Each digit can be used only one time in the expression.

\(\displaystyle \left(\boxed{\phantom{TEST}}+ \boxed{\phantom{TEST}}+ \boxed{\phantom{TEST}}+ \boxed{\phantom{TEST}} \right) \div 4\)

Student Response

For access, consult one of our IM Certified Partners.

Anticipated Misconceptions

Some students may think they need to use all of the digits from 0 to 9. Tell them that only 4 digits need to be used, although they are welcome to try finding a good solution using 2 digit numbers if they want.

Activity Synthesis

Poll the class on whether the value of their expression is exactly 4 or is close to 4. Ask selected students to share their strategy for creating an expression with a value of 4. Record and display their responses for all to see.

As students share their reasoning, consider asking some of the following questions:

  • “How did you decide on the value of the numerator?”
  • “How did the denominator affect your strategy?”
  • “How might your strategy change if the denominator was a different number, say, 6 or 10?”
  • “How might your strategy change if the numerator had more numbers or fewer numbers?”

9.2: Spread Out and Share (15 minutes)

Activity

This activity introduces students to the concept of mean or average in terms of equal distribution or fair share. The two contexts chosen are simple and accessible, and include both discrete and continuous values. Diagrams are used to help students visualize the distribution of values into equal amounts.

The first set of problems (about cats in crates) can be made even more concrete by providing students with blocks or snap cubes that they can physically distribute into piles or containers.  Students using the digital activities will engage with an applet that allows students to sort cats. For the second set of problems (about hours of work), students are prompted to draw two representations of the number of hours of work before and after they are redistributed, creating a visual representation of fair shares or quantities being leveled out.

As students work, identify those with very different ways of arranging cats into crates to obtain a mean of 6 cats. Also look for students who determine the redistributed work hours differently. For example, some students may do so by moving the number of hours bit by bit, from a server with the most hours to the one with the fewest hours, and continue to adjust until all servers have the same number. Others may add all the hours and divide the sum by the number of servers. 

Launch

Arrange students in groups of 2. Provide access to straightedges. Also consider providing snap cubes for students who might want to use them to physically show redistribution of data values. If using the digital lesson, students will have access to an applet that will allow them to sort cats, snap cubes may not be necessary but can be provided. 

Give students 3–4 minutes of quiet work time to complete the first set of questions and 1–2 minutes to share their responses with a partner. Since there are many possible correct responses to the question about the crates in a second room, consider asking students to convince their partner that the distribution that they came up with indeed has an average of 3 kittens per crate. Then, give students 4–5 min to work on the second set of questions together.

Representation: Develop Language and Symbols. Ensure access to virtual or concrete manipulatives to connect symbols to concrete objects or values. Provide students with blocks or snap cubes that they can physically distribute into piles or containers.
Supports accessibility for: Conceptual processing

Student Facing

  1. The kittens in a room at an animal shelter are arranged in five crates, as shown.
    Five squares represent 5 crates.  The number of cats pictured in each crate is 2, 1, 4, 3, 0.
    1. The manager of the shelter wants the kittens distributed equally among the crates. How might that be done? How many kittens will end up in each crate?

    2. The number of kittens in each crate after they are equally distributed is called the mean number of kittens per crate, or the average number of kittens per crate.

      Explain how the expression \(10 \div 5\) is related to the average.

    3. Another room in the shelter has 6 crates. No two crates contain the same number of kittens, and there is an average of 3 kittens per crate.

      Draw or describe at least two different arrangements of kittens that match this description. You may choose to use the applet to help.

  2. Five servers were scheduled to work the number of hours shown. They decided to share the workload, so each one would work equal hours.
    • Server A: 3
    • Server B: 6
    • Server C: 11
    • Server D: 7
    • Server E: 4
    1. On the grid on the left, draw 5 bars whose heights represent the hours worked by servers A, B, C, D, and E.
      Two identical coordinate grids are indicated. Each grid is 22 units horizontally and 13 units vertically. The vertical axis has the numbers 0 through 12, in increments of 2, indicated.
    2. Think about how you would rearrange the hours so that each server gets a fair share. Then, on the grid on the right, draw a new graph to represent the rearranged hours. Be prepared to explain your reasoning.
    3. Based on your second drawing, what is the average or mean number of hours that the servers will work?
    4. Explain why we can also find the mean by finding the value of \(31 \div 5\).
    5. Which server will see the biggest change to work hours? Which server will see the least change?

Student Response

For access, consult one of our IM Certified Partners.

Launch

Arrange students in groups of 2. Provide access to straightedges. Also consider providing snap cubes for students who might want to use them to physically show redistribution of data values. If using the digital lesson, students will have access to an applet that will allow them to sort cats, snap cubes may not be necessary but can be provided. 

Give students 3–4 minutes of quiet work time to complete the first set of questions and 1–2 minutes to share their responses with a partner. Since there are many possible correct responses to the question about the crates in a second room, consider asking students to convince their partner that the distribution that they came up with indeed has an average of 3 kittens per crate. Then, give students 4–5 min to work on the second set of questions together.

Representation: Develop Language and Symbols. Ensure access to virtual or concrete manipulatives to connect symbols to concrete objects or values. Provide students with blocks or snap cubes that they can physically distribute into piles or containers.
Supports accessibility for: Conceptual processing

Student Facing

  1. The kittens in a room at an animal shelter are placed in 5 crates.
    Five squares represent 5 crates.  The number of cats pictured in each crate is 2, 1, 4, 3, 0.
    1. The manager of the shelter wants the kittens distributed equally among the crates. How might that be done? How many kittens will end up in each crate?

    2. The number of kittens in each crate after they are equally distributed is called the mean number of kittens per crate, or the average number of kittens per crate. Explain how the expression \(10 \div 5\) is related to the average.
    3. Another room in the shelter has 6 crates. No two crates has the same number of kittens, and there is an average of 3 kittens per crate. Draw or describe at least two different arrangements of kittens that match this description.

  2. Five servers were scheduled to work the number of hours shown. They decided to share the workload, so each one would work equal hours.

    server A: 3

    server B: 6

    server C: 11

    server D: 7

    server E: 4

    1. On the grid on the left, draw 5 bars whose heights represent the hours worked by servers A, B, C, D, and E.
      Two identical coordinate grids are indicated. Each grid is 22 units horizontally and 13 units vertically. The vertical axis has the numbers 0 through 12, in increments of 2, indicated.
    2. Think about how you would rearrange the hours so that each server gets a fair share. Then, on the grid on the right, draw a new graph to represent the rearranged hours. Be prepared to explain your reasoning.
    3. Based on your second drawing, what is the average or mean number of hours that the servers will work?
    4. Explain why we can also find the mean by finding the value of the expression \(31 \div 5\).

    5. Which server will see the biggest change to work hours? Which server will see the least change?

Student Response

For access, consult one of our IM Certified Partners.

Student Facing

Are you ready for more?

Server F, working 7 hours, offers to join the group of five servers, sharing their workload. If server F joins, will the mean number of hours worked increase or decrease? Explain how you know.

Student Response

For access, consult one of our IM Certified Partners.

Anticipated Misconceptions

In the first room, to get each crate to have the same number of cats, some students might add new cats, not realizing that to “distribute equally” means to rearrange and reallocate existing quantities, rather than adding new quantities. Clarify the meaning of the phrase for these students.

Some students may not recognize that the hours for the servers could be divided so as to not be whole numbers. For example, some may try to give 4 servers 6 hours and 1 server has 7 hours. In this case, the time spent working is still not really divided equally, so ask the student to think of dividing the hours among the servers more evenly if possible.

Activity Synthesis

Invite several students with different arrangements of cats in the second room with 6 crates to share their solutions and how they know the mean number of cats for their solutions is 3. Make sure everyone understands that their arrangement is correct as long as it had a total of 18 kittens and 6 crates and no two crates have the same number of cats. Show that the correct arrangements could redistribute the 18 cats such that there are 3 cats per crate.

Then, select previously identified students to share how they found the redistributed work hours if the workers were to spread the workload equally. Start with students who reallocated the hours incrementally (from one server to another server) until the hours level out, and then those who added the work hours and dividing the sum by 5. 

Students should see that the mean can be interpreted as what each member of the group would get if everything is distributed equally, without changing the sum of values.  

Representing: MLR7 Compare and Connect. After the selected students share their solutions, invite students to discuss “What is the same and what is different?” about the approaches. Call students’ attention to the connections between approaches by asking “How is the approach ‘divide the sum of work hours by 5’ represented visually in the new bar graph?” This will help students make sense of and use different representations of the mean of a set of data.
Design Principle(s): Maximize meta-awareness; Support sense-making

9.3: Getting to School (15 minutes)

Activity

In this activity, students calculate the mean of a data set and interpret it in the context of the given situation. The first data set students see here has a dozen values, discouraging students from redistributing the values incrementally and encouraging them to use a more efficient method. In the second question, students analyze the values in data sets and use the structure (MP7) to decide whether or not it makes sense that a given mean would match the data set. 

As students work and discuss, notice the reasons they give for why the data sets in the second question could or could not be Tyler's data set. Identify students who recognize that the mean of a data set cannot be expected to be higher or lower than most of the values of the data set, and that a fair-share value would have a value that is roughly in the middle of data values.

Launch

Keep students in groups of 2. Give them 6–7 minutes of quiet work time, and then time to discuss their responses with their partner.  

Representation: Internalize Comprehension. Activate or supply background knowledge about determining means of data sets. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing

Student Facing

For the past 12 school days, Mai has recorded how long her bus rides to school take in minutes. The times she recorded are shown in the table.

9

8

6

9

10

7

6

12

9

8

10

8

  1. Find the mean for Mai’s data. Show your reasoning.
  2. In this situation, what does the mean tell us about Mai’s trip to school?
  3. For 5 days, Tyler has recorded how long his walks to school take in minutes. The mean for his data is 11 minutes. Without calculating, predict if each of the data sets shown could be Tyler’s. Explain your reasoning.

    • data set A: 11, 8, 7, 9, 8
    • data set B: 12, 7, 13, 9, 14
    • data set C: 11, 20, 6, 9, 10
    • data set D: 8, 10, 9, 11, 11
  4. Determine which data set is Tyler’s. Explain how you know.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

Select a couple of students to share how they found the mean of Mai's travel times. Poll the class briefly to see if others in the class found the mean the same way.

Then, focus the discussion on the second task and on what values could be reasonably expected of a data set with a particular mean. Ask students how they decided to rule out or keep certain sets of data as potentially belonging to Tyler. If not mentioned by students, highlight that the mean of a data set would be a value in the middle of the range of numbers in order for it to be a fair-share value.

Point out that, unlike hours of work or cats in crates, the times of travel here cannot actually be redistributed. The interpretation of mean translates into a thought experiment:

The mean is the travel time each day if all the travel times in the set were the same such that the combined travel time for this data set and the original data set was the same.

Speaking, Listening: MLR8 Discussion Supports. Use this routine to support students’ explanations of which data set is Tyler’s. Invite students to use the sentence frame “Data set ___ is Tyler’s because . . . .” Invite listeners to use the sentence frame “I agree/disagree, because . . . .” This will help students explain their own reasoning and make the sense of the reasoning of others.
Design Principle(s): Support sense-making; Optimize output (for justification

Lesson Synthesis

Lesson Synthesis

In this lesson, we look at finding the mean or the average of a numerical data set.

  • “Suppose that a data set contains the amounts of money in five piggy banks. What would the mean of this data set tell us?”
  • “Why might it make sense to think of the mean as a ‘fair share?’”
  • “How do we find the mean of a data set?”
  • “How can we interpret the mean of the heights of students in a class?”

9.4: Cool-down - Finding Means (5 minutes)

Cool-Down

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Student Lesson Summary

Student Facing

Sometimes a general description of a distribution does not give enough information, and a more precise way to talk about center or spread would be more useful. The mean, or average, is a number we can use to summarize a distribution.

We can think about the mean in terms of “fair share” or “leveling out.” That is, a mean can be thought of as a number that each member of a group would have if all the data values were combined and distributed equally among the members.

For example, suppose there are 5 bottles which have the following amounts of water: 1 liter, 4 liters, 2 liters, 3 liters, and 0 liters.

5 diagrams, each composed of 4 squares, some colored blue. From left to right, the number of blue squares in each diagram are 1, 4, 2, 3, 0.

To find the mean, first we add up all of the values. We can think of as putting all of the water together: \(1+4+2+3+0=10\).

  

A tape diagram partitioned into 10 equal parts. All 10 parts are shaded.

  

To find the “fair share,” we divide the 10 liters equally into the 5 containers: \(10\div 5 = 2\).

There are 5 identical tape diagrams each partitioned into 4 equal parts. Each diagram has 2 parts shaded.

Suppose the quiz scores of a student are 70, 90, 86, and 94. We can find the mean (or average) score by finding the sum of the scores \((70+90+86+94=340)\) and dividing the sum by four \((340 \div 4 = 85)\). We can then say that the student scored, on average, 85 points on the quizzes.

In general, to find the mean of a data set with \(n\) values, we add all of the values and divide the sum by \(n\).