1. Lesson 4A – Using Mean to Find the Missing Value
Finding the Missing Value
Lesson 4A – Using Mean to Find the Missing Value

2. Warm Up
OBJECTIVE: Given a data set, SWBAT find a missing value to produce a specific mean.
Language Objective: SWBAT explain orally or in writing how to find a missing value of a set of data.
The line plots below show two different distributions with the same mean. Use the line plots to answer the questions below.
Amount of Goals Scored X X
Key:
X – one game X
Amount of Goals Scored X X
Goals
Goals
How many games are there in each situation?
What is the total number of goals in each situation?
How do these facts relate to the mean in each case?
Challenge: How many goals would have to be scored in
the next game to have a mean of 5 goals?
(4 min) 0 – 4
In-Class Notes
SWBAT = Student Will Be Able To
Students should answer these questions independently on paper.
Scaffolding: Show the definition of mean.
Review answers as a class before clicking to show answers.
When reviewing Question 1, ask a student to come to the board to prove that there are 4 games (student should point to the four X’s on either, or each, line plot).
When reviewing Question 2, ask students to explain how they know that there are 12 goals in each situation.
When reviewing Question 3, before a student answers the question, ask a student to explain what the mean is (a definition in his/her own words would suffice). It may be helpful to first ask a student to explain the concept of mean using stacks of cubes (which were used in the previous lesson).
The challenge is a true challenge in that very few students will be able to solve it. Many students will use guess and check as a way to try to solve it, but there is a formal strategy to use. The challenge problem is essentially the type of question that this lesson is designed to answer.
If a student solves the challenge problem with a clear strategy, ask that student to write out the strategy on chart paper while the class reviews the warm-up problems. Then as lesson progresses ‘Johnny’s’ strategy can be referenced and refined as necessary.
Agenda
Scaffolding

3. Warm Up
OBJECTIVE: Given a data set, SWBAT find a missing value to produce a specific mean.
Language Objective: SWBAT explain orally or in writing how to find a missing value of a set of data.
The line plots below show two different distributions with the same mean. Use the line plots to answer the questions below.
Amount of Goals Scored X X
Key:
X – one game X
Amount of Goals Scored X X
Goals
Goals
How many games are there in each situation?
What is the total number of goals in each situation?
How do these facts relate to the mean in each case?
Challenge: How many goals would have to be scored in
the next game to have a mean of 5 goals?
(4 min) 0 – 4
In-Class Notes
Students should answer these questions independently on paper.
Scaffolding: Show the definition of mean.
Review answers as a class before clicking to show answers.
When reviewing Question 1, ask a student to come to the board to prove that there are 4 games (student should point to the four X’s on either, or each, line plot).
When reviewing Question 2, ask students to explain how they know that there are 12 goals in each situation.
When reviewing Question 3, before a student answers the question, ask a student to explain what the mean is (a definition in his/her own words would suffice). It may be helpful to first ask a student to explain the concept of mean using stacks of cubes (which were used in the previous lesson).
The challenge is a true challenge in that very few students will be able to solve it. Many students will use guess and check as a way to try to solve it, but there is a formal strategy to use. The challenge problem is essentially the type of question that this lesson is designed to answer.
If a student solves the challenge problem with a clear strategy, ask that student to write out the strategy on chart paper while the class reviews the warm-up problems. Then as lesson progresses ‘Johnny’s’ strategy can be referenced and refined as necessary.
Mean: the balance point or average of a data set
Add up all of the #’s
Divide by how many #’s there are
Agenda

4. Warm Up
OBJECTIVE: Given a data set, SWBAT find a missing value to produce a specific mean.
Language Objective: SWBAT explain orally or in writing how to find a missing value of a set of data.
The line plots below show two different distributions with the same mean. Use the line plots to answer the questions below.
Amount of Goals Scored X X
Key:
X – one game X
Amount of Goals Scored X X
Goals
Goals
4 games
How many games are there in each situation?
What is the total number of goals in each situation?
How do these facts relate to the mean in each case?
Challenge: How many goals would have to be scored in the next game to have a mean of 5 goals?
(4 min) 0 – 4
In-Class Notes
Students should answer these questions independently on paper.
Scaffolding: Show the definition of mean.
Review answers as a class before clicking to show answers.
When reviewing Question 1, ask a student to come to the board to prove that there are 4 games (student should point to the four X’s on either, or each, line plot).
When reviewing Question 2, ask students to explain how they know that there are 12 goals in each situation.
When reviewing Question 3, before a student answers the question, ask a student to explain what the mean is (a definition in his/her own words would suffice). It may be helpful to first ask a student to explain the concept of mean using stacks of cubes (which were used in the previous lesson).
The challenge is a true challenge in that very few students will be able to solve it. Many students will use guess and check as a way to try to solve it, but there is a formal strategy to use. The challenge problem is essentially the type of question that this lesson is designed to answer.
If a student solves the challenge problem with a clear strategy, ask that student to write out the strategy on chart paper while the class reviews the warm-up problems. Then as lesson progresses ‘Johnny’s’ strategy can be referenced and refined as necessary.
Agenda

5. Warm Up
OBJECTIVE: Given a data set, SWBAT find a missing value to produce a specific mean.
Language Objective: SWBAT explain orally or in writing how to find a missing value of a set of data.
The line plots below show two different distributions with the same mean. Use the line plots to answer the questions below.
Amount of Goals Scored X X
Key:
X – one game X
Amount of Goals Scored X X
Goals
Goals
4 games
How many games are there in each situation?
What is the total number of goals in each situation?
How do these facts relate to the mean in each case?
Challenge: How many goals would have to be scored in the next game to have a mean of 5 goals?
(4 min) 0 – 4
In-Class Notes
Students should answer these questions independently on paper.
Scaffolding: Show the definition of mean.
Review answers as a class before clicking to show answers.
When reviewing Question 1, ask a student to come to the board to prove that there are 4 games (student should point to the four X’s on either, or each, line plot).
When reviewing Question 2, ask students to explain how they know that there are 12 goals in each situation.
When reviewing Question 3, before a student answers the question, ask a student to explain what the mean is (a definition in his/her own words would suffice). It may be helpful to first ask a student to explain the concept of mean using stacks of cubes (which were used in the previous lesson).
The challenge is a true challenge in that very few students will be able to solve it. Many students will use guess and check as a way to try to solve it, but there is a formal strategy to use. The challenge problem is essentially the type of question that this lesson is designed to answer.
If a student solves the challenge problem with a clear strategy, ask that student to write out the strategy on chart paper while the class reviews the warm-up problems. Then as lesson progresses ‘Johnny’s’ strategy can be referenced and refined as necessary.
12 goals
Agenda

6. Warm Up
OBJECTIVE: Given a data set, SWBAT find a missing value to produce a specific mean.
Language Objective: SWBAT explain orally or in writing how to find a missing value of a set of data.
The line plots below show two different distributions with the same mean. Use the line plots to answer the questions below.
Amount of Goals Scored X X
Key:
X – one game X
Amount of Goals Scored X X
Goals
Goals
How many games are there in each situation?
What is the total number of goals in each situation?
How do these facts relate to the mean in each case?
Challenge: How many goals would have to be scored in the next game to have a mean of 5 goals?
(4 min) 0 – 4
In-Class Notes
Students should answer these questions independently on paper.
Scaffolding: Show the definition of mean.
Review answers as a class before clicking to show answers.
When reviewing Question 1, ask a student to come to the board to prove that there are 4 games (student should point to the four X’s on either, or each, line plot).
When reviewing Question 2, ask students to explain how they know that there are 12 goals in each situation.
When reviewing Question 3, before a student answers the question, ask a student to explain what the mean is (a definition in his/her own words would suffice). It may be helpful to first ask a student to explain the concept of mean using stacks of cubes (which were used in the previous lesson).
The challenge is a true challenge in that very few students will be able to solve it. Many students will use guess and check as a way to try to solve it, but there is a formal strategy to use. The challenge problem is essentially the type of question that this lesson is designed to answer.
If a student solves the challenge problem with a clear strategy, ask that student to write out the strategy on chart paper while the class reviews the warm-up problems. Then as lesson progresses ‘Johnny’s’ strategy can be referenced and refined as necessary.
12 goals
The distributions look different and the data values are different. However, they both show 4 games with a total of 12 goals. Thus, they both have a mean of 3 goals.
Challenge
Agenda

7. Warm Up
OBJECTIVE: Given a data set, SWBAT find a missing value to produce a specific mean.
Language Objective: SWBAT explain orally or in writing how to find a missing value of a set of data.
The line plots below show two different distributions with the same mean. Use the line plots to answer the questions below.
Amount of Goals Scored X X
Key:
X – one game X
Amount of Goals Scored X X
Goals
Goals
How many games are there in each situation?
What is the total number of goals in each situation?
How do these facts relate to the mean in each case?
Challenge: How many goals would have to be scored in the next game to have a mean of 5 goals?
(4 min) 0 – 4
In-Class Notes
Students should answer these questions independently on paper.
Scaffolding: Show the definition of mean.
Review answers as a class before clicking to show answers.
When reviewing Question 1, ask a student to come to the board to prove that there are 4 games (student should point to the four X’s on either, or each, line plot).
When reviewing Question 2, ask students to explain how they know that there are 12 goals in each situation.
When reviewing Question 3, before a student answers the question, ask a student to explain what the mean is (a definition in his/her own words would suffice). It may be helpful to first ask a student to explain the concept of mean using stacks of cubes (which were used in the previous lesson).
The challenge is a true challenge in that very few students will be able to solve it. Many students will use guess and check as a way to try to solve it, but there is a formal strategy to use. The challenge problem is essentially the type of question that this lesson is designed to answer.
If a student solves the challenge problem with a clear strategy, ask that student to write out the strategy on chart paper while the class reviews the warm-up problems. Then as lesson progresses ‘Johnny’s’ strategy can be referenced and refined as necessary.
Thirteen goals (12 so far + 13 more = 25 goals ÷ 5 = 5)
Agenda

8. Center Spread Shape Launch – Review Turn and Talk (30 sec)
When we analyze data, what are we looking for?
Median
Center
Today!
Mean
Spread
(2 min) 5 – 7
In-Class Notes
Ask students to discuss the question, “What are we looking for when we analyze data?” in pairs or small groups for 30 seconds.
After students have shared their own answers, click to show desired answers.
Preparation Notes
The focus of today’s lesson is using the mean to describe data distributions. However, it is important for students to realize that the mean is just one way to describe a data distribution; there are multiple ways to describe a set of data. The previous lessons focused on using the mean and median (another measure of center) to describe data distributions. In the upcoming lessons, spread and shape will be looked at as ways to describe sets of data. By the end of the unit, students should be able to explain three different ways to describe data (center, spread, shape). This exact slide was part of the previous lessons (on mean and median). In response to the prompt, students may have trouble remembering the exact terminology, but they should be able to explain what the words mean based on their introduction in the previous lesson.
Shape
Agenda

9. Launch Turn-and-talk
The mean number of letters in students’ first names for eight students is 6 letters.
What would this look like if we modeled it with cubes?
(1 min) 7 – 8
In-Class Notes
Give students about 20 seconds to think about this question individually before talking to a partner. Students should be able to visualize the image in their heads, but give them the option of drawing a picture on paper.
Ask student(s) to share out before moving onto the next slide.
Preparation Notes
Provide cubes to students so that those who learn best with manipulatives have the option of using them.
Agenda

10. Launch Number of Letters
The mean number of letters in students’ first names for eight students is 6 letters.
What would this look like if we modeled it with cubes?
(1 min) 8 – 9
In-Class Notes
Ask students why some stacks of cubes have more than one color in them (they should see that the stacks with more than one color originally had fewer than 6 cubes, so they needed cubes from the other towers to be balanced out).
Number of Letters
Agenda

11. Launch Number of Letters
The mean number of letters in students’ first names for eight students is 6 letters.
How could we figure out the total number of letters in their names?
(1 min) 9 – 10
In-Class Notes
Ask for student ideas before going over answers (on the next slides).
For struggling students, it may be helpful to remind students that one cube represents one letter.
Number of Letters
Agenda

12. Launch Number of Letters
The mean number of letters in students’ first names for eight students is 6 letters.
How could we figure out the total number of letters in their names?
Method 1: Count all of the cubes 5 4 3 2 1 6 11 10 9 8 7 12 17 16 15 14 13 18 22 21 20 23 19 24 29 28 27 26 25 30 33 32 35 34 36 31 41 42 39 38 40 37 45 44 47 46 48 43
(1 min) 10 – 11
In-Class Notes
After presenting this strategy, point out that while it works, it is not the most efficient strategy. In mathematics, we are always looking for a faster way of doing things.
Then ask students if there is a more efficient strategy for finding the total number of letters in the students’ names.
Once a student has suggested multiplication, move onto the next slide.
Number of Letters
Agenda

13. 1 2 3 4 5 6 7 8 Launch Number of Letters 8 X 6 = 48 letters
The mean number of letters in students’ first names for eight students is 6 letters.
How could we figure out the total number of letters in their names?
Method 2: Multiply
8 X 6
= letters 5 4 3 2 1 6 5 4 3 2 1 6 5 4 3 2 1 6 5 4 3 2 1 6 5 4 3 2 1 6 5 4 3 2 1 6 5 4 3 2 1 6 5 4 3 2 1 6
(1 min) 11 – 12
In-Class Notes
Make a connection here to the E/LA strategy of annotation of text.
Point out that annotation does not only mean underline! One could put squares around numbers, circles around direction words, underline the question, etc. This could be an opportunity to develop a formal annotation policy for the math classroom.
Stress the importance of annotation of word problems. 1
Number of Letters 2 3 4 5 6 7 8
Agenda

14. Launch
The mean number of people in five students’
families is 3 people per family. How could we
figure out the total number of people in the five
families?
5 X 3
= people 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1
(2 min) 12 – 14
In-Class Notes
Read the first sentence and review the word “per” before reading the question – relate it to multiplication and ratios/rates.
It may be helpful to cross out the word “per” and replace it with for every (on the board where this lesson is being projected).
Make a connection again to the E/LA strategy of annotation of text.
Stress the importance of annotation of word problems.
Give students about 20 seconds to think about this question individually before talking to a partner. Students should be able to visualize the image in their heads, but give them the option of drawing a representation on paper.
Scaffolding: Show the hint.
Additional Scaffolding: Show the five stacks of cubes.
Review students’ ideas before clicking to show the answer.
Preparation Notes
Provide cubes to students so that those who learn best with manipulatives have the option of using them. 1 2 3 4 5
Hint
Agenda
Scaffolding

15. Launch The mean number of people in five students’
families is 3 people per family. How could we
figure out the total number of people in the five
families?
Hint: Think about what this would look like if we modeled it with cubes. 3
(2 min) 12 – 14
In-Class Notes
Read the first sentence and review the word “per” before reading the question – relate it to multiplication and ratios/rates.
It may be helpful to cross out the word “per” and replace it with for every (on the board where this lesson is being projected).
Make a connection again to the E/LA strategy of annotation of text.
Stress the importance of annotation of word problems.
Give students about 20 seconds to think about this question individually before talking to a partner. Students should be able to visualize the image in their heads, but give them the option of drawing a representation on paper.
Scaffolding: Show the hint.
Additional Scaffolding: Show the five stacks of cubes.
Review students’ ideas before clicking to show the answer.
Preparation Notes
Provide cubes to students so that those who learn best with manipulatives have the option of using them.
Hint
Agenda
Scaffolding

16. Launch
The mean number of people in five students’
families is 3 people per family. How could we
figure out the total number of people in the five
families?
5 X 3
= people
Hint: Think about what this would look like if we modeled it with cubes. 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1
(2 min) 12 – 14
In-Class Notes
Read the first sentence and review the word “per” before reading the question – relate it to multiplication and ratios/rates.
It may be helpful to cross out the word “per” and replace it with for every (on the board where this lesson is being projected).
Make a connection again to the E/LA strategy of annotation of text.
Stress the importance of annotation of word problems.
Give students about 20 seconds to think about this question individually before talking to a partner. Students should be able to visualize the image in their heads, but give them the option of drawing a representation on paper.
Scaffolding: Show the hint.
Additional Scaffolding: Show the five stacks of cubes.
Review students’ ideas before clicking to show the answer.
Preparation Notes
Provide cubes to students so that those who learn best with manipulatives have the option of using them. 1 2 3 4 5
Hint
Agenda
Scaffolding

17. Launch
Zack will take a total of 3 mathematics tests this month. He wants to have a mean score of 80 at the end of the month. How many total points does Zack need in order to have a mean score of 80?
(2 min) 14 – 16
In-Class Notes
After reading the problem and before students begin to answer the question, annotate the problem as a class.
Give students about 20 seconds to think about this question individually before talking to a partner. Students should be able to visualize the image in their heads, but give them the option of drawing a representation on paper.
Have students share out answers before revealing the answer.
Once the answer appears, make the connection between the word “OF” and multiplication. This should activate students prior knowledge, as they learn multiplication as “groups of” in lower grades.
Students could feasibly argue that Zack could have more than 240 total points in order to have a mean score of 80. The question, without stating it explicitly, is asking for an exact mean of 80 points. However, since it is not explicitly stated, students who can support why an answer greater than 240 points works are correct as well (these students would be under the assumption that Zack wants a mean minimum score of 80, rather than an exact mean score of 80).
__________ stacks of ___________ = ___________ 3 80
240 points
Agenda
Hint
Scaffolding

18. Launch
Zack will take a total of 3 mathematics tests this month. He wants to have a mean score of 80 at the end of the month. How many total points does Zack need in order to have a mean score of 80?
Hint: Think about what this would look like if we modeled it with cubes.
(2 min) 14 – 16
In-Class Notes
After reading the problem and before students begin to answer the question, annotate the problem as a class.
Give students about 20 seconds to think about this question individually before talking to a partner. Students should be able to visualize the image in their heads, but give them the option of drawing a representation on paper.
Have students share out answers before revealing the answer
Once the answer appears, make the connection between the word “OF” and multiplication. This should activate students prior knowledge, as they learn multiplication as “groups of” in lower grades.
Students could feasibly argue that Zack could have more than 240 total points in order to have a mean score of 80. The question, without stating it explicitly, is asking for an exact mean of 80 points. However, since it is not explicitly stated, students who can support why an answer greater than 240 points works are correct as well (these students would be under the assumption that Zack wants a mean minimum score of 80, rather than an exact mean score of 80).
Agenda
Scaffolding

19. Launch
Zack will take a total of 3 mathematics tests this month. He wants to have a mean score of 80 at the end of the month. How many total points does Zack need in order to have a mean score of 80?
Hint: Think about what this would look like if we modeled it with cubes.
(2 min) 14 – 16
In-Class Notes
After reading the problem and before students begin to answer the question, annotate the problem as a class.
Give students about 20 seconds to think about this question individually before talking to a partner. Students should be able to visualize the image in their heads, but give them the option of drawing a representation on paper.
Have students share out answers before revealing the answer
Once the answer appears, make the connection between the word “OF” and multiplication. This should activate students prior knowledge, as they learn multiplication as “groups of” in lower grades.
Students could feasibly argue that Zack could have more than 240 total points in order to have a mean score of 80. The question, without stating it explicitly, is asking for an exact mean of 80 points. However, since it is not explicitly stated, students who can support why an answer greater than 240 points works are correct as well (these students would be under the assumption that Zack wants a mean minimum score of 80, rather than an exact mean score of 80).
__________ stacks of ___________ = ___________ 3 80
240 points
Agenda

20. Launch Think-Pair-Share
Do you think it is possible to have other sets of data about 4 games that have a mean of 3 goals that are different from the ones we explored in the warm-up?
Amount of Goals Scored X X X
Amount of Goals Scored X X
(2 min) 16 – 18
In-Class Notes
Give students about 30 seconds to think about this question individually before talking to a partner. Students might be able to visualize the image in their heads, but give them the option of drawing a representation on paper.
After students discuss the question with a peer, rather than taking student generated sets of data, ask students to show a thumbs up if they think it is possible and a thumbs down if they do not think it is possible.
Preparation Notes
Provide cubes to students so that those who learn best with manipulatives have the option of using them.
Goals
Goals
Agenda

21. Launch Think-Pair-Share
Do you think it is possible to have other sets of data about 4 games that have a mean of 3 goals that are different from the ones we explored in the warm-up?
Let’s work together to make a new distribution of 4 games with a mean of 3 goals.
Amount of Goals Scored X X X
Amount of Goals Scored X X
(2 min) 16 – 18
In-Class Notes
Give students about 30 seconds to think about this question individually before talking to a partner. Students might be able to visualize the image in their heads, but give them the option of drawing a representation on paper.
After students discuss the question with a peer, rather than taking student generated sets of data, ask students to show a thumbs up if they think it is possible and a thumbs down if they do not think it is possible.
Preparation Notes
Provide cubes to students so that those who learn best with manipulatives have the option of using them.
Goals
Goals
Agenda

22. Launch
Our data set must have a total of four games. Suppose that the first game had three goals. We can show these three goals by making a stack of three cubes.
(1 min) 18 – 19
In-Class Notes
Have a student read the slide.
Click for the stack of 3 cubes to appear.
Ask students if it was necessary to start with three goals. For example, “Could we have started with a different number of goals, like 5 goals, in the first game?” Students should recognize that this is just one of many ways to begin the data set.
Number of Goals
Agenda

23. Launch
Our data set must have a total of four games. Suppose that the first game had three goals. We can show these three goals by making a stack of three cubes.
How many more stacks do we need to make to show the remaining games?
(1 min) 19 – 20
In-Class Notes
Have a student read the question aloud.
Ask students to think about the answer in their heads (silently).
Then request that all students show their answer with their fingers (students should hold 3 fingers up in the air).
When all students have their answer represented with their fingers, move onto the next slide to show the answer.
Number of Goals
Agenda

24. Launch
Our data set must have a total of four games. Suppose that the first game had three goals. We can show these three goals by making a stack of three cubes.
How many more stacks do we need to make to show the remaining games? 3 stacks!
(< 1 min) 19 – 20
In-Class Notes
If everyone had three fingers up in the air, show the answer and move on.
If students had different ideas about how many more stacks are required, ask students to support their ideas until everyone understands why 3 more stacks are necessary.
Number of Goals
Agenda

25. Launch
Now, suppose another game has five goals. We can make a stack of five cubes to represent that game.
(1 min) 20 – 21
In-Class Notes
Have a student read the slide.
Click for the stack of 5 cubes to appear.
Ask students if it was necessary to include a game with five goals. For example, “Could we have chosen a different number of goals, like 2 goals, for the second game?” Students should recognize that this was just one of a few ways to continue with the data set.
Number of Goals
Agenda

26. Launch
Now, suppose another game has five goals. We can make a stack of five cubes to represent that game.
How many goals are represented with just the two stacks we’ve made? 3 2 5 4 1 3
+ 5
= 8 goals 3 2 1
(< 1 min) 20 – 21
In-Class Notes
Have a student read the question aloud.
Ask students to think about the answer in their heads (silently).
Then request that all students show their answer with their fingers (students should hold 8 fingers up in the air).
When all students have their answer represented with their fingers, click to show the answer.
Number of Goals
Agenda

27. Launch Turn-and-talk Number of Goals In order to have a mean of
3 goals for the four games,
what should we do next? 5 4 3 2 1 3
(1 min) 21 – 22
In-Class Notes
Students should think about this question independently (20 seconds) before talking to a peer (30 seconds).
As students discuss ideas, encourage them to sketch what the set of data (4 data points) could look like.
The goal is for students to see that between the next two games, there need to be 4 goals.
As students discuss their ideas with a peer, circulate and try to find someone who is articulating this idea.
Ask that person to share his/her idea with the class.
Ask other student(s) to explain the idea in their own words and/or defend why it makes sense. 2 1
Number of Goals
Agenda

28. Launch Number of Goals In order to have a mean of
The goal is to have a total of 12 goals in the 4 games. Since we have 8 goals so far, the remaining 2 games must have 4 goals altogether.
In order to have a mean of
3 goals for the four games,
what should we do next? 5 4 3 2 1 3
(1 min) 22 – 23
In-Class Notes
Ask a student to read the words in the bubble aloud (this should be a summary of what students said in the previous slide). 2 1
Number of Goals
Agenda

29. Launch
“Since we have 8 goals so far, the remaining 2 games must have 4 goals altogether.”
Make a sketch of what our remaining stacks might look like. 5 4 3 2 1 3
(1 min) 23 – 24
In-Class Notes
Students should begin by drawing the two stacks that are present. They should sketch the remaining stacks independently.
As students finish up, click to show the answer.
Ask students to raise their hand if this looks like their drawing.
Ask students if there is another way to draw the remaining stacks and still fit the criteria (students could have drawn a stack of 1 and a stack of 3 and vice versa).
Preparation Notes
Provide cubes to students so that those who learn best with manipulatives have the option of using them. 2 2 1 2 1 1
Number of Goals
Agenda

30. How can we be sure that the mean is 3 goals?
Launch
How can we be sure that the mean is 3 goals? 5 4 3 2 1 3
(1 min) 24 – 25
In-Class Notes
Have a student read the question aloud.
Scaffolding: Show hint.
Ask students for their ideas.
Desired answer: Redistribute the cubes to see if each stack has 3 cubes in it. In other words, balance the stacks so that each has 3.
Click to show the cubes being redistributed.
Preparation Notes
Provide cubes to students so that those who learn best with manipulatives have the option of using them. 2 2 2 1 1 1
Number of Goals
Agenda
Hint

31. How can we be sure that the mean is 3 goals?
Launch
How can we be sure that the mean is 3 goals?
Could we balance out the stacks so that each has three cubes? 5 4 3 2 1 3
(1 min) 24 – 25
In-Class Notes
Have a student read the question aloud.
Scaffolding: Show hint.
Ask students for their ideas.
Desired answer: Redistribute the cubes to see if each stack has 3 cubes in it. In other words, balance the stacks so that each has 3.
Click to show the cubes being redistributed.
Preparation Notes
Provide cubes to students so that those who learn best with manipulatives have the option of using them. 2 2 2 1 1 1
Number of Goals
Agenda
Hint

32. Launch – Summary Turn-and-talk
What is a general strategy that can be used to come up with different sets of data of 4 games with a mean of 3 goals?
(2 min) 25 – 27
In-Class Notes
Students should think about this question independently (20 seconds) before discussing with a peer (40 seconds).
Then students should share out strategies with the class.
The general strategy is the following:
Calculate how many goals are needed in total (4 X 3 = 12)
Find four numbers that when added together equal 12
In addressing the first step of the strategy, one could make a connection to mathematical vocabulary with the word factor. Both 3 and 4 are factors of 12.
Once the general strategy has been established, it is recommended that the teacher (or a reliable student) write it on an anchor poster and hang it in the classroom.
Agenda

33. Explore: Guided Practice
Edward is training for 3 weeks to prepare for a road race. He has run for 2 weeks so far. The distances he ran during his first 2 weeks are shown in the table below.
Week
Distance (in miles) 1 8 2 13 3 ?
(1 min) 27 – 28
In-Class Notes
Have a student read the question aloud.
Have the students read the question a second time independently.
Optional: Annotate the question.
Ask students to identify how this question is similar to and different than the previous question.
Similarities:
There is a desired mean stated.
We are looking for a data point that will fit the given parameters.
Differences:
We are given two specific data points instead of creating all of our own.
There is only one data point that will work (only one correct answer).
Ask students if they can identify the difference between this type of problem and the problems they solved in the previous lesson (in the previous lesson, students were given the data points and had to calculate the mean – now they are working backwards – they have the mean, but need to calculate a data point).
What distance must Edward run in his 3rd week in order to have a mean running distance of 10 miles for his 3 weeks of running?
Agenda

34. Explore: Guided Practice Turn-and-talk
What distance must Edward run in his 3rd week in order to have a mean running distance of 10 miles for his 3 weeks of running?
How could we model a mean
distance of 10 miles for 3 weeks of running using cubes?
Remember that the total number of miles is:
10 X 3 = 30 miles
(1 min) 28 – 29
In-Class Notes
Scaffolding: Show hint.
Students should think about this question independently (20 seconds) before sharing ideas with a peer for 20 seconds.
Students should be able to visualize the image in their heads, but give them the option of drawing a picture on paper.
Have students share ideas before moving onto the answer in the next slide. If students drew pictures, ask them to hold them up.
Preparation Notes
Provide cubes to students so that those who learn best with manipulatives have the option of using them.
Agenda
Hint

35. Explore: Guided Practice Turn-and-talk
What distance must Edward run in his 3rd week in order to have a mean running distance of 10 miles for his 3 weeks of running?
How could we model a mean
distance of 10 miles for 3 weeks of running using cubes?
Remember that the total number of miles is:
10 X 3 = 30 miles
(1 min) 28 – 29
In-Class Notes
Scaffolding: Show hint.
Students should think about this question independently (20 seconds) before sharing ideas with a peer for 20 seconds.
Students should be able to visualize the image in their heads, but give them the option of drawing a picture on paper.
Have students share ideas before moving onto the answer in the next slide. If students drew pictures, ask them to hold them up.
Preparation Notes
Provide cubes to students so that those who learn best with manipulatives have the option of using them.
= 30
Desired Mean
Agenda

36. Explore: Guided Practice
How could we model Edward’s actual miles run so far using cubes?
Week
Distance (in miles) 1 8 2 13 3 ?
(< 1 min) 28 – 29
In-Class Notes
Show solution and move on if students do not have any questions.
= 30
Desired Mean
Agenda
Hint
Answer

37. Explore: Guided Practice
How could we model Edward’s actual miles run so far using cubes?
Week
Distance (in miles) 1 8 2 13 3 ?
Actual Miles
(< 1 min) 28 – 29
In-Class Notes
Show solution and move on if students do not have any questions.
= 30
Desired Mean
Agenda
Answer

38. Explore: Guided Practice
How could we model Edward’s actual miles run so far using cubes?
Week
Distance (in miles) 1 8 2 13 3 ?
Actual Miles
(< 1 min) 28 – 29
In-Class Notes
Show solution and move on if students do not have any questions.
= 30
Actual Distances
Desired Mean
Agenda

39. Explore: Guided Practice Turn-and-talk
What distance must Edward run in his 3rd week in order to have a mean running distance of 10 miles for his 3 weeks of running?
Week
Distance
(in miles) 1 8 2 13 3 ?
(< 1 min) 28 – 29
In-Class Notes
Show solution and move on if students do not have any questions.
= 30 ?
Desired Mean
Actual Distances
Agenda
Answer

40. Explore: Guided Practice Turn-and-talk
What distance must Edward run in his 3rd week in order to have a mean running distance of 10 miles for his 3 weeks of running?
9 miles
Week
Distance
(in miles) 1 8 2 13 3 ?
Miles Needed 30
-Miles So Far 21
Run #3 9
(< 1 min) 28 – 29
In-Class Notes
Show solution and move on if students do not have any questions.
= 30 ?
Actual Distances
Desired Mean
Agenda

41. Explore: Guided Practice
Let’s prove it!
Week
Distance (in miles) 1 8 2 13 3 9
Desired Mean = 10 miles
We are correct!
(<1 min) 30 – 32
In-Class Notes
Before clicking to prove the answer is correct, ask students how the answer can be proved.
Desired answer: Redistribute the cubes so that there are 10 in each stack (balance the stacks so they are all even).
This is a long lesson, and if it is necessary to break it into 2 days, this slide serves as a good stopping point. Some sort of summary should be completed if this is the end of the lesson. The question from the next slide (Slide 50) could be used as an exit ticket. The following lesson should begin with slide 50, and students should reference back to their exit ticket for the beginning of the lesson.
Agenda

42. Explore
Edward is training for 4 weeks to prepare for a road race. He has run for 3 weeks so far. The distances he ran during his first 3 weeks are shown in the table below.
Week
Distance (in miles) 1 38 2 47 3 42 4 ?
(1 min) 32 – 33
In-Class Notes
Have a student read the question aloud.
Have the students read the question a second time independently.
Optional: Annotate the question.
Ask students to identify how this question is similar to and different than the previous question (essentially it is the same exact problem with a different set of data).
Similarities:
There is a desired mean stated.
We are looking for a data point that will fit the given parameters.
Differences:
The data points given are different (much larger than in the previous set of data).
The desired mean is different (also much larger).
A copy of this question should be distributed to students for annotation purposes and also so that students have the problem in front of them (with space to work) when they solve in small groups. The problem can be printed on a half-sheet of paper and students can use the back of the paper for the group challenge at the end of class.
What distance must Edward run in his 4th week in order to have a mean running distance of 40 miles for his 4 weeks of running?
Agenda

43. Explore Let’s use a scale instead!
How could we model a mean distance of 40 miles for 4 weeks of running using cubes?
Hmmmm. We would
need a lot of cubes!
(< 1 min) 32 – 33
In-Class Notes
Have a student read the question aloud.
Ask students to share their ideas.
Then click to show the cartoon character.
Ask students if they agree with the cartoon character.
Inform students that since stacks of cubes is not always the most efficient strategy, another strategy will be looked at.
Let’s use a scale instead!
Agenda

44. Explore Actual Distances Desired Mean
Week
Distance (in miles) 1 38 2 47 3 42 4 ?
(1 min) 33 – 34
In-Class Notes
Explain to the class that the eight boxes and the line and triangle represent a scale.
Ask them what they know needs to be true about both sides of the scale (they need to be equal).
Then ask about similarities between using a scale and stacks of cubes.
Also ask about differences between using a scale and stacks of cubes.
Once students seem to grasp the connection between the stacks of cubes and scales, move onto the next slide. However, the conversation students should have regarding the similarities and differences about scale verse cubes is important and it should not be rushed as it supports conceptual understanding.
Actual Distances
Desired Mean
How does this scale compare to using stacks of cubes? How is it similar? How is it different?
Agenda

45. Explore Small Group Actual Distances Desired Mean
Week
Distance (in miles) 1 38 2 47 3 42 4 ?
(5 min) 34 – 39
In-Class Notes
Students should work in small groups (3 – 4 students) to solve this problem.
Students should work collaboratively, but each student should be writing down his/her own work.
Give students about 4 minutes to work on the problem before reviewing the solution shown in the upcoming slides.
Before moving onto the next slide, ask a representative from one group to share what he/she put in the boxes above “Actual Distances” on the scale.
Actual Distances
Desired Mean
What distance must Edward run in his 4th week in order to have a mean running distance of 40 miles for his 4 weeks of running?
Agenda

46. 38 47 42 ? Explore Actual Distances Desired Mean
Week
Distance (in miles) 1 38 2 47 3 42 4 ? 38 47 42 ?
(< 1min) 34 – 39
In-Class Notes
This slide should reiterate the answer provided by a student.
Before moving onto the next slide, ask a representative from one group to share what he/she put in the boxes above “Desired Mean” on the scale.
Actual Distances
Desired Mean
What distance must Edward run in his 4th week in order to have a mean running distance of 40 miles for his 4 weeks of running?
Agenda

47. 38 47 42 ? 40 40 40 40 Explore Actual Distances Desired Mean
Week
Distance (in miles) 1 38 2 47 3 42 4 ? 38 47 42 ? 40 40 40 40
(< 1min) 34 – 39
In-Class Notes
This slide should reiterate the answer provided by a student.
Before moving onto the next slide, ask a representative from one group to share what he/she did with all of the “Desired Mean” numbers
(he/she might have added them all up, or multiplied 40 X 4).
Actual Distances
Desired Mean
What distance must Edward run in his 4th week in order to have a mean running distance of 40 miles for his 4 weeks of running?
Agenda

48. 40 +40 160 40 X 4 160 38 47 42 ? 40 Explore Actual Distances
Week
Distance (in miles) 1 38 2 47 3 42 4 ? 40
X 4
160 38 47 42 ? 40
(< 1min) 34 – 39
In-Class Notes
This slide should reiterate the answer provided by a student.
Ask students what the 160 represents (the total number of miles needed).
Before moving onto the next slide, ask a representative from one group to share what he/she did with all of the “Actual Distances” numbers (he/she should have added them all up).
Actual Distances
Desired Mean
What distance must Edward run in his 4th week in order to have a mean running distance of 40 miles for his 4 weeks of running?
Agenda

49. Explore 40
+40
160
Week
Distance (in miles) 1 38 2 47 3 42 4 ? 38 47
+42 40
X 4
160 38 47 42 ? 40 40 40 40
(< 1min) 34 – 39
In-Class Notes
This slide should reiterate the answer provided by a student.
Before moving onto the next slide, ask a representative from one group to share what he/she got as the sum for the actual miles run so far.
Actual Distances
Desired Mean
What distance must Edward run in his 4th week in order to have a mean running distance of 40 miles for his 4 weeks of running?
Agenda

50. Explore 40
+40
160
Week
Distance (in miles) 1 38 2 47 3 42 4 ? 38 47
+42
127 40
X 4
160 38 47 42 ? 40 40 40 40
(< 1min) 34 – 39
In-Class Notes
This slide should reiterate the answer provided by a student.
Ask students what the 127 represents (the total number of miles run so far).
Before moving onto the next slide, ask a representative from one group to share what he/she did to calculate the number of miles necessary for Week #4.
Actual Distances
Desired Mean
What distance must Edward run in his 4th week in order to have a mean running distance of 40 miles for his 4 weeks of running?
Agenda

51. Explore 40
+40
160
Week
Distance (in miles) 1 38 2 47 3 42 4 ?
Miles Needed 160
- Miles So Far 127
Run # ? 38 47
+42
127 40
X 4
160 38 47 42 ? 40 40 40 40
(< 1min) 34 – 39
In-Class Notes
This slide should reiterate the answer provided by a student.
Before moving onto the next slide, ask a representative from one group to share what he/she calculated as the number of miles for Run #4.
Actual Distances
Desired Mean
What distance must Edward run in his 4th week in order to have a mean running distance of 40 miles for his 4 weeks of running?
Agenda

52. Explore 40
+40
160
Miles Needed 160
- Miles So Far 127
Run # 38 47
+42
127 40
X 4
160 38 47 42 ? 40 40 40 40
(< 1min) 34 – 39
In-Class Notes
This slide should reiterate the answer provided by a student.
Before moving onto the next slide, ask if students have any questions about how the problem was solved.
Have a few students explain and repeat the strategy to solve missing value problems in their own words.
Actual Distances
Desired Mean
What distance must Edward run in his 4th week in order to have a mean running distance of 40 miles for his 4 weeks of running?
Agenda

53. In 10 minutes you will be asked to stop and share your answers!
Practice
Part 2 - (10 Min)
Work independently and check in with a partner to complete your class work.
1-Worksheet
2-Share Out
Online timer link on slide - 10 min (39 – 49)
In-Class Notes
Pass out Class Work handout accordingly (there is one regular version and one modified version).
Students should complete work independently and then share with a partner to check and see if they have similar answers and/or similar strategies. The purpose of working independently is to build independence.
Click on the timer!
In 10 minutes you will be asked to stop and share your answers!
Agenda

54. Practice – Student Share Out
Part 3 – (5 Min)
Students share out work.
(6 min) 49 – 55
In-Class Notes
The practice summary reviews the work that students completed independently on their class work.
The summary should begin with students sharing their methods for solving the problems.
Display the key questions if possible while students share (using document camera, overhead projector, or another normal routine you have for sharing student work).
If a document camera, overhead projector, etc. is not accessible, possible solutions are provided in the upcoming slides.
Not every question is necessary to review. However, if there is time available, reviewing each question is an option. Reviewing the work provides students with an opportunity to assess their own work.
For each question, after students have shared their own answers, click to show desired answers.
For students who don’t need an in depth explanation, provide them with an answer key to quickly check their work and then move them onto the extension (provide them with the worksheet of challenge problems).
Preparation Notes
It is possible to conduct the summary without a student share-out. However, the thinking behind having students share their methods is that it builds in incentive for students to work for interesting solutions and ideas during the Practice time (today and in the future), it provides students a chance to take pride in their work and to practice presentation skills, and it helps with student engagement to have students listen to other students instead of the teacher’s voice when possible.
Classwork Questions
Agenda