1. Using the Median as a Measure of Center

2. Lesson Overview (1 of 6) Lesson Objective
SWBAT describe a data distribution in terms of its median and recognize how the median, as a measure of center, responds to changes in the number and magnitude of data values.
Student- Friendly Objective: SWBAT find the median of a set of data and explain how it reacts to change.
Lesson Description
The lesson begins with students engaging in a whole-class review of how to find the median. Following the review, students are asked to think about the median more deeply through questioning. Much of this launch time will be conducted using a think-pair-share where students discuss the questions with a partner before reporting out to the class. During the explore time, students are expected to work individually, while also regularly checking in with a nearby partner. Following the exploration, students will share their answers and strategies with the class. This share-out will serve as the summary of the lesson. An exit ticket will be used to both assess student understanding of the concept of median and ability to calculate the median and also to get an overall sense of whether students recognize how the median responds to changes in the number and magnitude of data values.

3. Lesson Overview (2 of 6) Lesson Vocabulary
Median: the number that marks the middle of an ordered set of data. Half of the values lie at or below the median and half of the values lie at or above the median.
Materials
1) Index cards or blank paper
2) Median class work handout
3) Exit ticket
4) Median homework
5) Number squares for struggling students
Scaffolding
Throughout the Explore portion of the lesson, manipulatives (number squares) will be provided for students. Scaffolding buttons throughout the lesson provide additional supports and hints to help students make important connections. A handout on how to find the median is provided for struggling students

4. Lesson Overview (3 of 6) Enrichment Advanced Objective:
SWBAT identify real world situations where one would choose to use the median to describe a set of data instead of the mean.
Ask students to brainstorm real life applications of this concept
To support students in doing this, give students the definition of mean.
A copy of a newspaper may give students some ideas.
Online Resources for Absent Students
Common Core
State Standard
Notes:
Practice:
6.SP.2: Understand that a set of data collected to answer a
statistical question has a distribution which can be
described by its center, spread, and overall shape.

5. Lesson Overview (4 of 6) Before and After
While Grades 4 and 5 provide students with an opportunity to do the pre-work necessary to understand measures of center, measures of center, and statistics in general, are not formally introduced until grade 6. Therefore, the concepts covered in this lesson will be new to students at the sixth grade level.
Aside from the basic groundwork laid out in grades 4 and 5, coming into this lesson, students will have had one lesson related to statistics. The introductory lesson for the unit focused on what makes a question statistical.
This lesson is the second lesson in the unit and is designed to pick up where the last lesson left off. Now that students have an understanding of what a statistical question is, they will be introduced to various ways to analyze the data collected by asking a statistical question.
Throughout the unit, students learn that data can be analyzed using center, spread, and shape. The first cluster of lessons in the unit look at measures of center as ways to describe a data distribution. This lesson in particular focuses on the median as a measure of center.

6. Lesson Overview (5 of 6) Before and After Continued
By the end of this lesson, students should be able to describe a set of data using the median. By the end of this cluster of lesson, students should be able to describe a set of data using both the median and mean. They should also be able to determine which measure of center is appropriate to use to describe a particular set of data. Ultimately students should be able to use measures of center and measures of variation to describe data distributions.
The goal is for students to see that the data collected in response to a statistical question have certain attributes (center, spread, overall shape). In Grade 7, when students expand their study of statistics to work with samples, students will see that these attributes relate important information about the sample from which the data were collected.
The idea of choosing median vs. mean is relevant in the real world. Often times home prices are reported in the context of the median home price. Salaries for sports players might also be expressed in relation to the median salary. Grades, on the other hand, and sports statistics, are often reported using the mean. Students should understand the difference between the two measures of center to understand given statistics and also to become more informed consumers.

7. Lesson Overview (6 of 6) Topic Background Research Based Strategies:
Turn and Talk/Think-Pair-Share: “Various researchers (e.g. Douglas Reeves, Richard Allington, Vygotsky) have linked academic success with the capacity to engage in conversation and to ask and answer questions in full sentences.
One of the most powerful and easy to implement moves is called: Turn and talk, or think, pair, share, or partner talk. All of these are variations of a practice that has far reaching benefits for students. Simply defined, “turn and talk” is a teacher offered opportunity for students to turn to another student and talk something through for a very brief period of time before whole group discussion or lecture resumes.”
-Lucy West & Antonia Cameron
Metamorphosis Teaching Learning Communities

8. Warm Up
OBJECTIVE: SWBAT recognize how the median reacts to changes made to a data set.
Language Objective: SWBAT write about what happens to the median when a data set changes.
Determine whether the following questions are statistical or not.
Explain how you know.
1) How old am I?
A = Not statistical
2) How tall are the students in my class?
A = Statistical
(4 min) 0 – 4
In-Class Notes
When reviewing the answers, for questions that are deemed not statistical, ask students to rephrase them so that they are statistical.
1) How old am I?
Anticipate that some students will reason that Question 1 is statistical since it can be answered with a number.
2) Do you exercise?
Anticipate that some students will say Question 3 is not statistical only because it is a yes/no question.
Scaffolding: Show the definition of a statistical question.
Preparation Notes
This warm-up was designed to activate prior knowledge about statistical questions. Based on the previous lesson in this unit, students should be able to differentiate between statistical questions and those that are not (6.SP.1: Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers). In their explanations, students should reason that B is a statistical question, because it allows for variability by providing a variety of possible responses. A and C are not statistical questions, because there is only one response.
3) Do you exercise?
A = Not statistical
Scaffolding
Agenda

9. Agenda: Warm Up – Review of Statistical ???’s (Individual)
OBJECTIVE: SWBAT recognize how the median reacts to changes
made to a data set.
Language Objective: SWBAT write about what happens to the median when a data set changes.
Warm Up – Review of Statistical ???’s (Individual)
Launch – Review of Median (Whole class)
Launch – Class Line-Up (Whole Class)
Explore – Name Lengths – Partner
Explore – Apartment Costs Class Work (Ind)
Summary – Stability of the Median (Whole Class)
Assessment – Exit Ticket (Ind)
(1 min) 4 – 5
In-Class Notes
Briefly review today’s objective and agenda, pointing out that students will do some exploring as a class, have an opportunity to do some partner practice and some individual work, and then be asked to show what they’ve learned on an exit ticket.
Preparation Notes
The structure of today’s lesson is designed to provide an in-depth study of the median and its usefulness in describing data distributions. Sixth grade students will not have had experience working with measures of center including the median in earlier grades. However, based on years experience of organizing data, students should grasp the concept fairly quickly. Once students understand what the median represents, the focus shifts from how to find the median to its application and real-world use. By the end of the lesson, students should understand that the median is one measure that can be used to describe a set of data. Students should also recognize that the median remains stable when data is removed or added to a data set.

10. Launch (A) – Review of Last Class
OK, we have learned how to form a statistical question. Now what?!
But then what do we do with the data we collect?
Then what?
Ask a question and collect data!
Organize it!
(2 min) 5 – 7
In-Class Notes
Summarize the warm-up before beginning. For example, “Based on our warm-up, we know a statistical question will have answers that vary from one person to the next.”
Ask students to brainstorm ways to organize data (least to greatest, tables, bar graphs, line plots, etc.).
Preparation Notes
Providing this brief review of how to conduct a data investigation should serve as a summary of Lesson 1 in this unit. It is setting up the connection between the data collection process (specifically data analysis) and measures of center. The goal of Lesson 1 was to recognize a statistical question. Now the progression goes from asking statistical questions to collecting data and analyzing it. The median is introduced as a way to describe the data as part of the analysis.
Examine/analyze the data!
Agenda

11. Launch (A) Turn and Talk (30 sec)
number of toppings students like on their pizzas
When we analyze data, what are we looking for?
Today!
Median
Center
Mean
Spread
(2 min) 7 – 9
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 median to describe data distributions. However, it is important for students to realize that the median is just one way to describe a data distribution; there are multiple ways to describe a set of data. The next lesson will focus on using the mean (another measure of center) to describe data distributions. Then spread and shape will be looked at as ways to describe sets of data. After this lesson and the two lessons following it, students should recognize three different ways to describe data (center, spread, shape). In response to the prompt, students most likely will not specifically identify center, spread, and shape as things we look for when we analyze data. However, their ideas will probably relate to the concepts as their answers will most likely focus on similarities and differences in data. The vocabulary of center, spread, and shape should remain consistent throughout the unit.
Shape
Agenda

12. Launch (A)
Vocabulary
What is the definition of median?
The median is the number that marks the middle of an ordered set of data. Half of the values lie at or below the median and half of the values lie at or above the median.
(3 min) 9 – 12
In-Class Notes:
Teacher asks question and asks students to create a definition of median in their groups (30 seconds).
Discuss ideas about the definition of median.
Click to show formal definition.
Students should write the definition down in their notes (preferably in a vocabulary section).
Before showing the ordered set of data, ask students what an ordered set of data is.
Stress the importance of ordering a set of data from least to greatest before trying to find the median, as the data will not always come as an ordered set.
Before showing how to find the median, ask students for strategies they might use to find the middle number.
Also stress crossing out from both ends – it is very important that students are methodical, as a common error is to cross out one more value on the left side than the right, which changes the value of the median.
Connect the mathematical definition of median to the use of medians on roadways.
Extension: Ask students to develop a faster method to determine the median for a set of data (perhaps even to write an equation or rule). Another way to phrase this is: Given the number of data points in a data set, how can you determine the number of numbers below or above the median very quickly?
Preparation Notes:
Median is a new concept for students in the 6th grade. However, students tend to grasp the concept fairly quickly.
This slide is connected to Math PS 6 – Attend to Precision: Communicate precisely to others. Use clear definitions in discussion with others and in their own reasoning.
median
Agenda

13. How do you find the median?
Put the data (numbers) in order from least to greatest.
Begin with number on the far left and the far right, crossing off one on each end at a time.
Stop when you cannot cross off a number on both ends
If there is only 1 number left in the middle, this number is the median.

14. Launch (A)
How do you find the median when there is 2 numbers in the middle?
Follow steps 1-3 above
Add the 2 numbers and divide by 2
This is your median
hmmm I have two data points left, 8 and 10…what do I do?
9 = median
(1 min) 12 – 13
In-Class Notes:
Poll students for their ideas on how to find the median of a data set with an even number of data points before clicking to show answer.
Preparation Notes:
Agenda

15. Launch (B) – Class Challenge! Whole Class
1) On the index card in front of you, write the number of letters in your FIRST name.
2) Stand up with your index card in your hand.
3) Without talking, organize yourselves in a line from least to greatest.
(5 min) 13 – 18
In-Class Notes
Have one student read the directions aloud.
Preparation Notes:
The purpose of this exercise if for students to engage in kinesthetic learning. Each student represents a piece of data. By physically lining up from least to greatest, and then sitting down as each student is “crossed off” to find the middle value, students physically find the median. This activity allows for learning through discovery.
Each student will need an index card or piece of paper to write on.
This slide is connected to Math PS 4 - Model With Mathematics: Apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.
Agenda

16. Launch (B) Whole Class
1) How could we find the median number of letters in students’ names in our class?
2) What is the median number of letters in students’ names?
3) What would happen if we added another student’s name length to our data?
(5 min) 13 – 18
In-Class Notes
Once students are in line from least to greatest, discuss how they could find the median of their data (name lengths).
Once students have identified that they will need to remove data from both sides of the line until there is only one piece of data left, ask students to find the median of their data.
A suggested method for finding the median includes having students return to their seats one by one, beginning with the very first person in line, then the last person in line, then the 2nd person in line, etc. Eventually there will only be one (or two) people standing. If there is one person standing, that student’s data represents the median. If there are two people left standing, the average of those two students’ data represents the median.
Questions 3-4 serve as an introduction to the explore segment of the lesson. Explain to the students that in today’s exploration the focus will be on the stability (or lack thereof) of the median as a measure of center. They should know that instead of finding the median, they will be exploring how the median might change in certain situations (like when a person with 54 letters in his/her name is added to a data set).
Preparation Notes:
The purpose of this exercise if for students to engage in kinesthetic learning. Each student represents a piece of data. By physically lining up from least to greatest, and then sitting down as each student is “crossed off” to find the middle value, students physically find the median. This activity allows for learning through discovery.
This slide is connected to Math PS 4 - Model With Mathematics: Apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.
4) What would happen if that student’s name had 54 letters?
Agenda

17. Explore (A) Individual (Notes)
The students in Ms. Jee’s class collected data to answer the statistical question, “How many letters are in the first names of Ms. Jee’s students?” The data is displayed below.
What is the median for these data? 3 8 5 5 4 3 6 7 10 6 7
(3 min) 18 – 21
In-Class Notes
Have students work individually to find the median at their seats. Students should be completing the problem in their notebooks. Notes should be titled “Median.”
Accommodation: Print out a copy of the data and cut out the number squares. Give a set of the number squares to struggling learners who could benefit from manipulating the data with their hands.
Preparation Notes
This slide serves as a basic review of the median. Students need to be able to complete this slide in order to grasp the ideas presented throughout the lesson.
Agenda

18. Explore (A) Solution
What is the median for these data?
median 3 8 5 5 4 3 6 7 10 6 7
(<1 min) 18 – 21
In-Class Notes
Ask students to explain what a median of 6 means in this context. Students should understand that if the median is 6, 50% of the students in Ms. Jee’s class have names with 6 or fewer letters and 50% of the students in Ms. Jee’s class have names with 6 or more letters.
Before going onto the next slide, remind students that the focus of today’s lesson is not on finding the median, but on determining how the median is affected by changes made to a data set.
Preparation Notes
The purpose of this slide is to show the correct solution. It only needs to be shown for about 20 seconds (the hope is that it will provide clarity to any students who might have gotten the question wrong).
This slide is connected to Math PS 2 - Reason Abstractly and Quantitatively: Attend to the meaning of quantities, not just how to compute them.
Agenda

19. Explore (A)
median
The median of the data is 6 letters. 3 3 4 5 5 6 6 7 7 8 10
Now that we know the median name length in Ms. Jee’s class is 6 letters.
(<1 min) 18 – 21
In-Class Notes
Ask students how changes could be made to the data set (add names, remove names, etc.).
Preparation Notes
The purpose of this slide is to frame the upcoming slides. By referring back to the objective, students will have more structure around the questions they will be asked in the following slides.
This slide is connected to Math PS 2 - Reason Abstractly and Quantitatively: Attend to the meaning of quantities, not just how to compute them.
Agenda

20. Explore (A) Think-Pair-Share (Notes)
The students in Ms. Jee’s class collected data to answer the statistical question, “How many letters are in the first names of Ms. Jee’s students?” The data is displayed below.
Remove two data points from the original data set so that the median decreases. 3 3 4 5 5 6 6 7 7 8 10
(2 min) 21 – 23
In-Class Notes
In their notes, have students complete the problem under the following sub-header: “Remove 2 data points – median decreases.”
Have students complete this question by engaging in think-pair-share, a cooperative discussion strategy that promotes structured discussion and student accountability.
Accommodation: Print out a copy of the data and cut out the number squares. Give a set of the number squares to struggling learners who could benefit from manipulating the data with their hands.
Before moving onto Slide 12, ask to hear from students regarding their solutions. As a class, discuss why removing data points greater than the median decrease the median. Pose follow-up questions such as, “What would happen if data points that were less than median were removed?”
Preparation Notes
In this question, students explore how responsive the median is to changes in the data values. Does it change if we add a very large or a very small value to the data? How does it react if we make other changes in the data? The idea of the median and its stability is important in making judgments about statistical data. This idea is the goal of the lesson.
This slide is included to prepare students for their class work. Students will be asked similar questions in their class work assignment.
This slide is connected to Math PS 1 - Make sense of problems and persevere in solving them: Explain to themselves the meaning of a problem and look for entry points to its solution. Analyze givens, constraints, relationships, and goals. Make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt.
Scaffolding
Agenda

21. Explore (A) Possible Solution
Add two data points to the original data set so that the median stays the same.
median 3 3 3 4 5 5 6 6 7 7 8 10 33
Woah! 33 is far away from the rest of the data!
Is there a word for a piece of data that is much smaller or much larger than the rest of the data?
(1 min) 26 – 27
In-Class Notes
This slide shows one possible solution to the question posed in the previous slide. Most students will add numbers within the range of the data to solve the problem (for example, they might add 3 and 10). One purpose of this slide is to show that there are several possible solutions to the question. More importantly, this slide illustrates that adding a very large number (an outlier) has little effect on the median.
Preparation Notes
This slide is connected to Math PS 1 - Make sense of problems and persevere in solving them: Understand the approaches of others to solving complex problems and identify correspondences between different approaches.
OUTLIER!
Agenda

22. Explore (A) Individual
Mr. Nunez drives a bus. The line plot below shows the number of passengers that were on his bus for each of the last 10 trips he made.
Bus Trips
(4 min)
In-Class Notes
By the sixth grade, students should know how to read a line plot. Before posing the question, “What is the median for these data?” it may be helpful to review what the X’s represent (frequency – in this case, the number of trips). Teachers can assess students’ understanding of line plots by asking questions like: What does the x axis represent? What do the numbers 1-9 represent in terms of the data? What do the X’s represent in terms of the data? What do the three X’s above 6 mean? How many trips were made with 8 passengers? 5 passengers? 1 passenger? What was the most frequent number of riders on the bus trips?
As a class, review two strategies for finding the median when given a line plot.
1) Methodically cross out the X’S on the actual line plot (one on the left, one on the right, left, right, etc.)
2) Record the data from the line plot from least to greatest and then find the middle value.
Review all of these ideas before showing the actual question (What is the median for these data).
If students need additional review, ask students to make statements about the line plot. For example, The most passengers on a bus trip was 9.
Accommodation: Provide handout with line plot on a paper for students who need to be able to mark up the shape in order to find the median.
As a class, discuss what happens when there are an even number of data points vs. an odd number of data points.
Once students determine that 6 is the median, ask them what that means. They should know it means 6 passengers and they should also understand that 50% of the bus trips had 6 or less passengers and 50% of the bus trips has 6 or more passengers.
Preparation Notes
This slide is included to get students thinking about the median in a different context rather than just looking at ordered sets of data. It also serves as review of line plots, a fourth (4.MD.4) and fifth (5.MD.2) grade concept that was reviewed in Lesson 1 of the Unit.
This slide is connected to Math PS 2 - Reason Abstractly and Quantitatively: Attend to the meaning of quantities, not just how to compute them.
What is the median for these data?
Be prepared to share your strategy.
A = 6 passengers
Agenda

23. Explore (A) Turn-and-Talk (30 secs)
Mr. Nunez drives a bus. The line plot below shows the number of passengers that were on his bus for each of the last 10 trips he made.
There are 72 passengers on the bus for Mr. Nunez’s 11th trip. How does the median of the original data set change?
Bus Trips
A = The median does not change! The median number of passengers remains at 6.
(1 min)
In-Class Notes
Before having students answer this question, ask students how this question is different from the others (where they had to add/take away data points to change the median). Students should make the connection to previous learning. They should see that instead of providing a data point for a given change, now they are being asked for the change given a new data point. Ask students to recall what happened to the median in previous examples when they added data points to a data set.
Have students discuss this question with a peer. Then discuss as a whole class.
If you feel students could benefit from scaffolding for this question, before posing the actual question, first ask students how the median would change if the 11th trip has 10 passengers.
Students are not expected to manually solve the problem – they are just expected to reason through it by talking to a peer.
If students are asked to manually solve the problem and share out, click to show the answer to reaffirm the idea(s) from the share out.
Review the word outlier here.
Preparation Notes
This slide is intended to get students thinking about how an outlier might affect the median of a data set. Students will have time to explore the effects of an outlier on a median throughout the lesson.
In this question, students explore how responsive the median is to changes in the data values. Does it change if we add a very large or a very small value to the data? How does it react if we make other changes in the data? The idea of the median and its stability is important in making judgments about statistical data. This idea is the goal of the lesson.
This slide is connected to Math PS 3 – Construct Viable Arguments and Critique the Reasoning of Others: Understand and use stated assumptions, definitions, and previously established results in constructing arguments.
This slide is connected to Math PS 1 - Make sense of problems and persevere in solving them: Consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution.
Agenda

24. Classwork Summary – Sharing Question #1A
A) What is the median monthly rent charged for the apartments?
hmmm I have 2 data points left, 900 and 950…what do I do?
$925
(1 min) 42 – 43
In-Class Notes
Ask to hear what students did when they were left with two data points in the middle and then post it on the board or PPT.
Ask students what a median of 925 means in this situation. They should understand that half of the apartments cost less than $925 per month and half of the apartments cost more than $925 per month.
Remind students to follow along on their handout, which they use to take notes if they do not already have similar answers recorded from their Explore time.
This slide reviews #1A from the Class Work handout.
Preparation Notes
This slide is connected to Math PS 2 - Reason Abstractly and Quantitatively: Attend to the meaning of quantities, not just how to compute them.
This slide serves as reinforcement of the idea of the median as the middle of an ordered set of data.
750
800
900
900
950
950
1,000
1,250
Agenda