Using the Mean as a Measure of Center.

Using the Mean as a Measure of Center

Lesson Overview (1 of 6) Lesson Objective Lesson Description
Lesson Objective: SWBAT describe a data distribution in terms of its mean and recognize how the mean, as a measure of center, responds to changes in the number and magnitude of data values. Student- Friendly Objective: SWBAT find the mean 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 engage in kinesthetic learning to discover the mean conceptually. Much of this launch time will consist of students constructing stacks of cubes to represent data points, and then redistributing the cubes to make all of the stacks even. During the explore time, students move from using a physical model to find the mean to using the algorithm to calculate the mean. Much of the launch and explore time is conducted using a think-pair-share where students discuss the questions with a partner before reporting out to the class. During the practice time, students will practice finding the mean of different sets of data. They will also evaluate how the mean is affected when changes are made to a set of data. During this practice time, students are expected to work individually, while also regularly checking in with a nearby partner. Following the practice, students will share their answers and strategies with the class. This share-out will serve as an additional summary of the lesson. An exit ticket will be used to both assess student understanding of the concept of mean and ability to calculate the mean and also to get an overall sense of whether students recognize how the mean responds to changes in the number and magnitude of data values.

Lesson Overview (2 of 6) Lesson Vocabulary Materials Scaffolding
Mean: the balance point or average of a data set. It represents the value that each data point would take on if the total of the data values were redistributed equally. To find the mean: 1) Add up all of the values in the data set. 2) Divide the sum by how many values there are in the data set. Materials 1) Cubes (24 for every pair of students) 2) Mean class work handout 3) Exit ticket 4) Mean homework 5) Notes for struggling students Scaffolding Scaffolding buttons throughout the lesson provide additional supports and hints to help students make important connections. Cubes can be used throughout the lesson rather than just during the launch time. Handout on how to find the mean is provided for struggling students.

Lesson Overview (3 of 6) Enrichment Advanced Objective:
SWBAT identify real world situations where one would choose to use the mean to describe a set of data instead of the median. Ask students to brainstorm real life applications of this concept. To support students in doing this, a copy of a newspaper may provide some ideas. Online Resources for Absent Students Notes: (ignore the examples with negative numbers) Practice:

Lesson Overview (4 of 6) Common Core State Standard Before and After
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. 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. In Grade 5, for example, students “might find the amount of liquid each cylinder would contain if the total amount in all the cylinders were redistributed equally,” but it is not until Grade 6 that “students are able to view the amount in each cylinder after redistribution as equal to the mean of the five original amounts” (Progressions documents by McCallum: ). 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 two lessons related to statistics. The first lesson was on statistical questions and the second was on using the median as a measure of center. This lesson is the third 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 and are familiar with the median as a measure of center, they will explore the mean as a measure of center.

Lesson Overview (5 of 6) Before and After Continued
The word mean will be a new term for most students. However, some students may have some prior knowledge based on their understanding of average. By the end of this lesson, students should be able to describe a set of data using the mean. By the end of this cluster of lessons, 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 both measures of center and measures of variation to describe data distributions. The overarching 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.

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

Warm Up OBJECTIVE: SWBAT understand the mean as a number that “evens out” or “balances” a distribution and recognize how the mean reacts to changes made to a data set. Language Objective: SWBAT write about what happens to the mean when a data set changes. The line plot below shows the number of hits made by each of 10 players on a baseball team during batting practice. Median: The middle number in an ordered set of data What was the median number of hits made by the players? Show or explain how you got your answer. 2. Two additional players arrived at practice. The median number of hits increased. What could be the number of hits made by each of the two additional players? Show or explain how you got your answer. (4 min) 0 – 4 In-Class Notes Review answers as a class before clicking to show answers. When reviewing Question 1, ask students to explain what the median means in the context of the problem. They should be able to explain that half of the players made less than 10.5 hits and half of the players made more than 10.5 hits. When reviewing Question 2, ask for several different answers to ensure that students understand there is more than one possible solution to the problem. Ask students to explain why adding two values greater than 10 will increase the median number of hits. Scaffolding: Show the definition of median. Preparation Notes This warm-up was designed to activate prior knowledge about median. Based on the previous lesson in this unit, students should be able to find the median given a set of data. They should also recognize how the median reacts to changes made to a data set. It is important that students are able to complete these tasks comfortably, as they will be comparing the median to the mean throughout the lesson. This slide is connected to Math PS 2 - Reason Abstractly and Quantitatively: Attend to the meaning of quantities, not just how to compute them. 10.5 hits 12 hits and 13 hits Agenda

Agenda: Warm Up – Review of the Median (Individual)
OBJECTIVE: SWBAT understand the mean as a number that “evens out” or “balances” a distribution and recognize how the mean reacts to changes made to a data set. Language Objective: SWBAT write about what happens to the mean when a data set changes. Warm Up – Review of the Median (Individual) Launch – Stacks of Cubes (Small Group) Explore – Calculating the Mean (Small Group) Summary – Stability of the Mean (Whole Class) Practice – Movies Watched Class Work (Individual) Assessment _ Exit Ticket (Individual) (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 introduction to the mean and its usefulness in describing data distributions. The previous lesson in the unit introduced students to the median as a measure of center. This lesson will continue to reference measures of center, with the mean being the focus as opposed to the median. By the end of the lesson, students should understand that the mean, like the median, is a measure that can be used to describe a set of data. Students should also recognize that in comparing the median to the mean, the median is a more stable measure of center, as it is not impacted by change as much.

Launch – Review Turn and Talk (30 sec)
number of toppings students like on their pizzas 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 lesson focused on using the median (another measure of center) to describe data distributions. In the next lesson, spread and shape will be looked at as ways to describe sets of data. By the end of the next lesson, students should recognize three different ways to describe data (center, spread, shape). This exact slide was part of the previous lesson (on 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

Launch Six students in a middle-school class make a stack of cubes to show the number of letters in his or her name. Paul (1 min) 7 – 8 In-Class Notes Ask students what the word vary means and where they have seen it in other lessons. Make the connection between the word vary and variability, which they learned about in the first lesson of the unit and will learn about in more detail in upcoming lessons. Students should see that the cubes represent a data distribution that was created by posing a statistical question. Ask students how many pieces of data are in this set of data (6). Ask students how the data points could be represented in a more standard form (students should see that this is simply a representation of the data set 6, 4, 3, 2, 6, 3). Preparation Notes Teacher may choose to have 3-d representations of the stacks in the front of the room to make it clear the the 2-d shapes are really models of three-dimensional figures. Al Leo Arlene Joseph Mia You can see from the stacks that the six names vary in length. Agenda

Launch - Review What is the median name length?
Remember to order the data from least to greatest! Paul (2 min) 8 – 10 In-Class Notes Give students about 30 seconds to solve this individually. They should be able to solve the question in their heads, but give them the option of solving it on paper. Scaffolding: Provide struggling students with a handout with the six stacks printed on it. The hint should be included on the handout. Scaffolding: Show hint to remind students to organize the data before finding the median. Preparation Notes If the teacher chooses to represent the data with 3-d representations, the teacher may choose to order the stacks from least to greatest to then discuss the median using a visual. Al Leo Arlene Joseph Mia Agenda

Launch - Review Median = 3.5 letters What is the median name length?
Arlene Joseph Paul Leo Al Mia (> 1 min) 8 – 10 In-Class Notes Ask students for their answers before showing them the answer on the slide. Ask students to explain what a median of 3.5 letters means in the context of this problem. Students should be able to articulate that half of the names have less than 3.5 letters and half of the names have more than 3.5 letters. Ensure that students identify the median with a label (3.5 letters instead of 3.5). 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. Agenda

Launch Partner 1) Using cubes, make stacks like the ones shown below.
2) Make the 6 stacks all the same height by moving the cubes around. (3 min) 10 – 13 In-Class Notes Distribute 24 cubes to each pair of students. Ensure that students understand that they need to have 6 stacks still standing when they are done moving the cubes around. Otherwise they are likely to build stacks based on the factors of 24 (3 stacks of 8, 4 stacks of 6, etc.). Preparation Notes Before class prepare bags containing 24 cubes each. Students are less likely to be off-task if they have only what they need in front of them. Color coordinated blocks will provide more support for students (6 blue cubes in each bag, 4 green cubes, etc.). If cubes are not available, other alternatives could include: paper squares (cut out in advance) or some sort of candy that can be stacked, such as Starburst. Arlene Paul Leo Al Joseph Mia Agenda

Launch Whole Class How many cubes should be in each stack so that the heights are all the same? Leo (1 min) 13 – 14 In-Class Notes Ask students to explain how they solved the problem before showing the answer on the slide. Preparation Notes The answer shown on the slide serves as a reinforcement of what students already should have done themselves and also of what they heard from the share-out. Arlene Paul Al Joseph Mia Agenda

Launch Think-Pair-Share
The stack height that you found represents the mean number of letters in the students’ names. How might you explain what the mean of a data set is? Mean = 4 letters (2 min) 14 – 16 In-Class Notes Have students complete this question by engaging in think-pair-share, a cooperative discussion strategy that promotes structured discussion and student accountability. Students should think about the question individually for about 30 seconds before engaging in 30 second conversation with a nearby peer. Then the class should discuss ideas as a whole. Ensure that students have the chance to articulate their understanding of the mean before moving onto the next slide and showing the formal definition. Preparation Notes This slide gives students the opportunity to process what they just did. It is important that students understand that when they evened out the stacks, they were finding the mean. Although students do the pre-work necessary to understand the mean in Grade 5, the mean as a measure of center is never formally introduced. The mean is a new concept for students in Grade 6. Students may have prior knowledge related to averages that can be leveraged to develop an understanding of mean. Arlene Paul Leo Al Joseph Mia Agenda

Launch Vocabulary The mean of a set of data is:
the balance point or average of a data set. It represents the value that each data point would take on if the total of the data values were redistributed equally. (2 min) 16 – 18 In-Class Notes Ask students if they have heard the word average before. Ask students for examples of where averages are used in the real world (sports, grades, etc). Make it explicit that mean and average are the same concept. Students should write the definition of mean in their notes (preferably in a vocabulary section). When writing down the definition, students should illustrate what the mean is by drawing a picture. They should draw what the 6 stacks looked like originally and what they look like now. Scaffolding: Provide a copy of the definition to struggling students. Omit key words and require the students to fill in the missing words. Preparation Notes This idea of balance is critical to students’ conceptual understanding of what the mean is. Students should see the mean as a balance point before they are taught the algorithm for finding the mean. Arlene Paul Leo Al Joseph Mia Agenda

Launch Turn and talk (30 secs)
Imagine the image below contains glasses of water. Explain to a friend how to find the mean amount of water. (2 min) 18 – 20 In-Class Notes Ask students to discuss how this problem in pairs for 30 seconds. After the turn and talk, students should share their ideas with the class. Scaffolding: Once an appropriate and concise explanation has been made, ask one or two students to repeat the answer. This gives struggling students several opportunities to hear an explanation of what the mean is. Preparation Notes This picture exemplifies the statistics progression from Grade 5 to Grade 6. In Grade 5, students might be asked to find the amount of liquid each cylinder would contain if the total amount in all the cylinders were redistributed equally. In Grade 6, students should be able to view the amount in each cylinder after redistribution as equal to the mean of the five original amounts. 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. Agenda

Launch Notes What is the mean number of letters in the set of data below? How can this cube be distributed evenly? Miriam Luis Ben (2 min) 20 – 22 In-Class Notes Students should solve this problem individually in their notes before reviewing it as a class. Anticipate that some students will say the mean is 4 letters with one letter leftover. Push them to think about how they could redistribute that one “leftover” cube. Questions to ask students: How can the mean be 4.33 letters when a “third” of a letter does not exist? How can you predict when the mean number of data set will not be a whole number? For struggling students, remind them that to find the mean, they should balance out the stacks. Preparation Notes This example is provided to show students that the mean is not always a whole number. Later on in the lesson, when students find the mean number of letters in the students’ names at their own tables, they will most likely find that they can not distribute the blocks evenly (there will be blocks leftover). This slide will prepare them for the possibility of having a mean number of letters that is not a whole number. 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

Launch: First Names Small Group
1) Create a stack of cubes to represent the letters in your own first name. 2) Line up your stack with those of your group members. 3) Find the mean number of letters in your group’s set of data by redistributing the cubes to make the stacks all the same height. (3 min) 22 – 25 In-Class Notes Ensure that when the students redistribute the cubes, they maintain the same number of stacks (if there are three people in one group, there still needs to be three towers when they are done with step 3). Question to ask: Does it matter if you line up your stacks from least to greatest? (No) Why/why not? When is is important that data be lined up from least to greatest? (When finding the median) Once groups have completed step #3, ask one group to share its data. Then have the class find the mean for that group’s set of data. The group who shared the data should assess its classmates’ answers. Preparation Notes Each student will need cubes to make a stack to represent his/her own name. The purpose of this exercise if for students to engage in kinesthetic learning. Each stack represents a piece of data. By physically evening out the towers, students physically find the mean. This activity allows for learning through discovery. Agenda

Explore So I need to have cubes in my pocket every time I need to find the mean? Yikes! What if Wolfe+585 joined my group! He has 590 letters in his name! That is a lot of cubes to redistribute! Adolph Blaine Charles David Earl Frederick Gerald Hubert Irvin John Kenneth Lloyd Martin Nero Oliver Paul Quincy Randolph Sherman Thomas Uncas Victor William Xerxes Yancy Zeus Wolfeschlegelsteinhausenbergerdorffvoralternwarengewissenhaftschaferswessenschafewarenwohlgepflegeundsorgfaltigkeitbeschutzenvonangreifendurchihrraubgierigfeindewelchevoralternzwolftausendjahresvorandieerscheinenwanderersteerdemenschderraumschiffgebrauchlichtalsseinursprungvonkraftgestartseinlangefahrthinzwischensternartigraumaufdersuchenachdiesternwelchegehabtbewohnbarplanetenkreisedrehensichundwohinderneurassevonverstandigmenschlichkeitkonntefortplanzenundsicherfreuenanlebenslanglichfreudeundruhemitnichteinfurchtvorangreifenvonandererintelligentgeschopfsvonhinzwischensternartigraum, Senior (1 min) 25 – 26 In-Class Notes Briefly explain who Wolfe+585 is. Ask students for the vocabulary word that would describe a data point of 590 being added to the set of data (outlier). Ask students if using cubes would be an efficient method for finding the mean if there is a very large data point like 590 in the data set or if there are many data points. They should see that cubes can be useful for finding the mean of small data sets with low-value data points. Preparation Notes Hubert Blaine Wolfeschlegelsteinhausenbergerdorff, Sr. (Wolfe+585, Senior) is the short name of a Philadelphian typesetter, who has the longest personal name ever used. “585” represents the number of additional letters in his full surname (total 590). He appeared in all editions of the Guinness Book of World Records from roughly 1975 to 1985 as having the longest personal name. This slide is included to get students thinking about the efficiency of the method they have been using to find the mean. Agenda

Explore Small Group What happens if you don’t have cubes and you are asked to find the mean? How else could you find the mean? Hints! -Count how many stacks you have at your table. -Count how many cubes you have altogether. (4 min) 26 – 30 In-Class Notes Students should discuss potential strategies within their small groups. Teacher should be circulating around the room, listening to strategies. If students think they have a strategy that works, ask them to test it out with their own data. Discuss groups’ ideas as a class. Students should have discovered and agreed upon the algorithm before moving onto the next slide. Scaffolding: Provide hints to groups only if they are struggling. Preparation Notes Although the mean is a new concept for 6th graders, some might have had exposure to it in the past. Therefore, some students might already know the algorithm. 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. Agenda

Explore: How to Find the Mean Notes
1) Add up all of the values in the data set. 6 + 4 + 3 + 2 + 6 + 3 = 24 2) Divide the sum by how many values there are in the data set. 24 ÷ 6 = 4 24 Arlene 1 3 2 6 4 5 = 4 1 2 3 4 5 6 6 Mean = 4 letters Paul 1 2 3 4 Leo 1 2 3 Mia 1 2 3 Al 1 2 (3 min) 30 – 33 In-Class Notes Before performing any calculations, ask students to recall what the mean number of letters was for this example when they solved it by redistributing the cubes (slide 12). Students should write down the steps for finding the mean in their notes. They should also include the example. When including the example, drawing the stacks is optional. If a student chooses not to draw the stacks, the data should be written in standard form (6, 4, 3, 2, 6, 3). Ask students, “What does our answer really mean?” Emphasize that the mean is 4 letters, not 4. Be very explicit that regardless of whether one finds the mean using cubes or using the algorithm, the answer will be the same. However, the algorithm is a more efficient method. Scaffolding: Provide struggling students with notes that include the stacks and the steps. They should fill in the calculations themselves. Preparation Notes Although the mean is a new concept for 6th graders, some might have had exposure to it in the past. Therefore, some students might already know the algorithm. This slide is connected to Math PS 2 - Reason Abstractly and Quantitatively: Attend to the meaning of quantities, not just how to compute them. Joseph Agenda

Explore Notes 1) Add up all of the values in the data set. 6 + 4 + 3 = 13 2) Divide the sum by how many values there are in the data set. 13 ÷ 3 1 2 3 4 5 6 13 3 1 2 3 4 1 2 3 (1 min) 33 – 34 In-Class Notes Before performing any calculations, ask students to recall what the mean number of letters was for this example when they solved it by redistributing the cubes (slide 15). Ask students to calculate the mean individually using the algorithm before showing the answer. Students should add this example to their notes. They should have already drawn the stacks in their notes from the earlier example where they found the mean by redistributing the cubes (slide 15). Now they will add to their notes by including the calculations. Ask students, “What does our answer really mean?” Emphasize that the mean is 4 1/3 letters, rather than just 4 1/3 Be very explicit that regardless of whether one finds the mean using cubes or using the algorithm, the answer will be the same. However, the algorithm is a more efficient method. Scaffolding: Provide struggling students with notes that include the towers and the steps. They should fill in the calculations themselves. 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. Miriam Luis Ben Agenda

Explore – Class Challenge! Whole Class
How would we calculate the mean number of letters in students’ names in our class? 1. Add up the number of letters in every student’s name 2. Divide the sum by the number of students in our class (4 min) 34 – 38 In-Class Notes Ask students to generate the steps for finding the mean in this situation before showing Step #1 and Step #2. Poll the students on what they think the average number of letters will be. Push them to support their estimates. When using the applet, have each student go around the room and say the number of letters in his/her name. Teacher should be filling in this data as the students report it. Ask students to explain the “behind the scenes” steps that were taken to find the mean using the online tool. In other words, how did the website go from a list of data to producing the actual mean? Ask students to compare the mean and median number of letters once these measures of center are produced on the website. Questions to ask: Which measure of center represents the “typical” number of letters in the first name of a student in our class? If a new student joined our class today, how many letters would you predict to be in his/her first name? Do not exit the website when done. The data will be required for the next slide. Before moving onto the next slide, record the mean and median number of letters for the class. These numbers will be used as points of comparison for the next two slides. Preparation Notes Practice using the online tool before class to ensure it works properly. The website includes mode as a measure of center. In the 2011 framework, mode is not specified as a measure of central tendency and as such is not a focus of instruction. This concept can be introduced during instruction as needed. Calculate Agenda

Explore Think-Pair-Share
What if Wolfe really did join our class?! How would that piece of data affect the average number of letters in students’ names? Adolph Blaine Charles David Earl Frederick Gerald Hubert Irvin John Kenneth Lloyd Martin Nero Oliver Paul Quincy Randolph Sherman Thomas Uncas Victor William Xerxes Yancy Zeus Wolfeschlegelsteinhausenbergerdorffvoralternwarengewissenhaftschaferswessenschafewarenwohlgepflegeundsorgfaltigkeitbeschutzenvonangreifendurchihrraubgierigfeindewelchevoralternzwolftausendjahresvorandieerscheinenwanderersteerdemenschderraumschiffgebrauchlichtalsseinursprungvonkraftgestartseinlangefahrthinzwischensternartigraumaufdersuchenachdiesternwelchegehabtbewohnbarplanetenkreisedrehensichundwohinderneurassevonverstandigmenschlichkeitkonntefortplanzenundsicherfreuenanlebenslanglichfreudeundruhemitnichteinfurchtvorangreifenvonandererintelligentgeschopfsvonhinzwischensternartigraum, Senior (1 min) 38 – 39 In-Class Notes After students engage in a think-pair-share, go back to the online tool. The class data should still be there. Add 590 as a data point. Ask students to compare the new mean to the original mean and the new median to the original median. Ask students to compare the new mean to the new median. Do not exit the website when done. The data will be required for the next slide. Before moving onto the next slide, record the mean and median number of letters for this example. These numbers will be used as points of comparison for the next slide. Preparation Notes The applet includes mode as a measure of center. In the 2011 framework, mode is not specified as a measure of central tendency and as such is not a focus of instruction. This concept can be introduced during instruction as needed. 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. Calculate Agenda

Explore Think-Pair-Share
What if Justin joined our class instead of Wolfe + 585?! How would that affect the average number of letters in students’ names? (1 min) 39 – 40 In-Class Notes After students engage in a think-pair-share, go back to the online tool. The class data should still be there. Remove 590 as a data point and enter 6 as a data point. Ask students to compare the new mean to the original mean and the new median to the original median. Ask students to compare the new mean to the new median. Preparation Notes The online tool includes mode as a measure of center. In the 2011 framework, mode is not specified as a measure of central tendency and as such is not a focus of instruction. This concept can be introduced during instruction as needed. 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. Calculate Agenda

Summary Think-Pair-Share
Previously we discussed how the median reacts to changes made to a data set. How does the mean react to changes made to a data set? Be prepared to explain your thinking. (2 min) 40 – 42 In-Class Notes Briefly review how the median reacts to changes made to a data set before students engage in discussion related to the mean. Questions for students to think about: What happened to the mean when an outlier of 590 was added to the data set? What happened to the mean when a piece of data that was close in value to the other data points was added to the data set? Can we generalize how the mean of a data set is affected by change? Push students to explain why the mean is affected by an outlier. Using the cubes and stacks to explain might be helpful (by adding 590 to the data, the mean increases because, like with the cube stacks in class, cubes need to be taken away from the stack of 590 and added to all of the other stacks, and this would increase the heights of the other stacks). Preparation Notes 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. Agenda

In 10 minutes you will be asked to stop and share your answers!
Practice Part 1 - (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 (42-52) In-Class Notes Pass out Class Work handout. 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

Practice – Complete Worksheet (10 minutes)
In-Class Notes 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. Teacher should circulate around the room to provide additional support for students who need it and also to assess student work. Extension: Provide students with a newspaper and ask them to find examples of how the mean might be a useful statistic in the real world. Preparation Notes The extension 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

Practice – Student Share Out
Part 2 – (5 Min) Students share out work. (3 min) 52 – 55 In-Class Notes The practice summary reviews some of the work that students completed independently on their class work (Questions #4a – 4c). 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 part of the 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. Preparation Notes It is possible to continue 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 Explore 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

Practice – Sharing Question #4a
What happens to the mean of a data set when you add one or more data values that are outliers? Explain. (1 min) 52 – 53 In-Class Notes Ask students to support their answer using data from the class work (Question 4a). Preparation Notes Every student might not have made it to this question during class work time. However, based on the explore and summary segments of the lesson, students should be able to answer this question without completing it on the worksheet. 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 Explore 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. 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. Scaffolding Agenda

Practice – Sharing Question #4b
What happens to the mean of a data set when you add data values that cluster near the original data set? Explain. (1 min) 53 – 54 In-Class Notes Ask students to support their answer using data from the class work (Question 4b). Preparation Notes Every student might not have made it to this question during class work time. However, based on the explore and summary segments of the lesson, students should be able to answer this question without completing it on the worksheet. 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 Explore 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. 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. Scaffolding Agenda

Practice – Sharing Question #4c
Explain why you think these changes might occur. (1 min) 54 – 55 In-Class Notes Ask students to support their explanations using examples from the launch, explore, summary, or practice portion of the lesson. Using the cubes and stacks to explain might be helpful. When an outlier is added, a lot of cubes will either need to be added to or removed from the outlier stack to even it out with the other data points. When a data point that is close in value to the other data points is added, not many cubes need to be redistributed to make the stacks even. Preparation Notes Every student might not have made it to this question during class work time. However, based on the explore and summary segments of the lesson, students should be able to answer this question without completing it on the worksheet. 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 Explore 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. 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. Agenda

Assessment – Exit Ticket
Complete and hand in the Exit Ticket before you leave! (5 min) 55 – 60 In-Class Notes Distribute Exit Tickets. Remind students to clearly show all of their thinking and work. All work should be independent! Preparation Notes The rationale of using this Exit Ticket is to (1) collect individual student data assessing their understanding of the concept of mean and ability to calculate the mean and (2) to get an overall sense of whether students recognize how the mean responds to changes in the number and magnitude of data values. Agenda

Using the Mean as a Measure of Center.

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Using the Mean as a Measure of Center.

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Presentation on theme: "Using the Mean as a Measure of Center."— Presentation transcript:

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