Module Focus: Statistics and Probability Grades 6-11 Sequence of Sessions

Overarching Objectives of this November 2013 Network Team Institute Module Focus sessions for K-5 will follow the sequence of the Concept Development component of the specified modules, using this narrative as a tool

for achieving deep understanding of mathematical concepts. Relevant examples of Fluency, Application, and Student Debrief will be highlighted in order to examine the ways in which these elements contribute to and enhance conceptual understanding.

High-Level Purpose of this Session Participants will understand the coherence of the statistics work that is woven throughout the curriculum across the grade

levels.

Related Learning Experiences This session is part of a sequence of Module Focus sessions examining the Statistics and Probability curriculum, A Story of Ratios and A Story of

Functions.

Key Points Mathematical Thinking

- Explain patterns- Often a deterministic way of thinking

Statistical Thinking- Search for patterns in the presence of variability- Acknowledge role of chance variation (distinguish “signal” from “noise”)

Three overarching themes of statistics and probability in Grades 6-11 include:- Variability- Learning from Data (The Investigative Process)- Probability

Session Outcomes

What do we want participants to be able to do as a result of this session?

How will we know that they are able to do this?

Focus.  Participants will be able to identify the major work of each grade using the Curriculum Overview document as a resource in

Participants will be able to articulate the key points listed above.

preparation for teaching these modules. Coherence: P-5.  Participants will draw connections between the

progression documents and the careful sequence of mathematical concepts that develop within each module, thereby enabling participants to enact cross- grade coherence in their classrooms and support their colleagues to do the same .  (Specific progression document to be determined as appropriate for each grade level and module being presented.)

Standards alignment.  Participants will be able to articulate how the topics and lessons promote mastery of the focus standards and how the module addresses the major work of the grade in order to fully implement the curriculum.

Implementation.  Participants will be prepared to implement the modules and to make appropriate instructional choices to meet the needs of their students while maintaining the balance of rigor that is built into the curriculum.

Instructional supports.  Participants will be prepared to utilize models appropriately in promoting conceptual understanding throughout A Story of Units.

Session Overview

Section Time Overview Prepared Resources Facilitator Preparation

Statistical Thinking vs Mathematical Thinking

12 Minutes

Establish the distinction between statistical and mathematical thinking.

Statistics and Probablity in Grades 6-11 PPT

Grade 6 – 11 Statistics and Probability—Overarching Themes

23 Minutes

Explore the themes that connect statistics and probability across the grades within A Story of Ratios and A Story of Functions.

Statistics and Probablity in Grades 6-11 PPT

Linking the Two Statistics Themes

5 Minutes

Establish that the theme of learning from data through the investigative process is inherently tied to the theme of variability

Statistics and Probablity in Grades 6-11 PPT

Example Trajectory 26 Minutes

Provide examples from the lessons of grades 6, 7 and 11 that show

Statistics and Probablity in Grades 6-11 PPT

how students develop and mature in their understanding over one of the overarching themes.

Other Developmental Dimensions

29 Minutes

Describe other developmental trajectories in the statistics content in order to show how students are expected to develop deeper understanding as they move from grade 6 to grade 11.

Statistics and Probablity in Grades 6-11 PPT

Questions and Discussion

0 MinutesAnswer questions and facilitate discussion.

Session Roadmap

Section: Statistical Thinking vs Mathematical Thinking Time: 12 minutes

[12 minutes] In this section, you will… Distinguish between mathematical and statistical thinking

Materials used include:

TimeSlide #

Slide #/ Pic of Slide Script/ Activity directions GROUP

1 1 NOTE THAT THIS SESSION IS DESIGNED TO BE 90 MINUTES IN LENGTH.Welcome! In this session, we will examine the study of statistics and probability across grades 6-11 in A Story of Ratios and A Story of Functions.

1 2 Introduce the objectives for this session: Distinguish between mathematical and statistical thinking See how statistical thinking is developed in the statistics

content of Grades 6 – 11 Because the statistics content is spread over multiple grades,

it is important for teachers to understand the overarching themes that provide coherence to the statistics curriculum.

Examples from the lessons will be used to illustrate conceptual development across grades.

1 3 Go over the agenda for the session. The session is divided into 5 parts, followed by time for questions and discussion. Advise participants that even though there will be time for questions at the end, they can ask questions as you go through the presentation. There will be several activities and participants are encouraged to participate in those activities. The activities are based on classroom activities from the lessons, but have been modified in anticipation of a group much larger than a typical class.

1 4 Introduce the scenario described on this slide. Ask for thoughts on what is happening here. Most “math” folks will see a pattern here and attempt to come up with explanations for why the pattern exists. For example, often people suggest that it is a contagious disease that started in the county in the lower right. Others might say it is a mosquito borne disease and there is a big area of stagnant water in the lower right county. People come up with all kinds of creative explanations.However, almost never does anyone suggest that this might just be something that occurs at random and that the numbers observed in the different counties are just consistent with chance differences that would be expected as consistent with random behavior. This is the big point. IN fact, these numbers are just random digits.The statistical thinker would know to ask the question—could this just have happened by chance? Only if chance could be ruled out as a plausible explanation for the pattern should we be convinced that some else is at work here and go looking for other explanations.

1 5 Summarize this slide, referring back to the discussion from the previous slide. Emphasize that statistical thinking is not a deterministic way of thinking. Looks for pattern in the presence of variability, recognizes the role of chance.

8 6 The main point here is that statistical thinking is complex. It is a different way of thinking. Both mathematical and statistical thinking are important. Mathematical thinking is developed slowly over 12 years of school. Likewise, statistical thinking needs to be developed and nurtured. It is not something that kids will just get with a quick or superficial exposure.

Section: Grade 6 – 11 Statistics and Probability—Overarching Themes

Time: 23 Minutes

[23 minutes] In this section, you will…Explore the themes that connect statistics and probability across the grades within A Story of Ratios and A Story of Functions

Materials used include:

Time Slide #Slide #/ Pic of Slide Script/ Activity directions GROUP

1 7 This slide just transitions to the next section: The placement of statistics and probability in the grade 6 – 1 curriculum and overarching themes that provide coherence.

2 8 This curriculum overview chart shows the placement of the statistics and probability content in Grades 6 – 8.

1 9 This curriculum overview chart shows the placement of the statistics and probability content in Grades 9 - 11. Notice that there is no statistics component in Grade 10.

1 10 The way that statistics is placed in the common core presents some challenges that teachers should be aware of. Because it is spread over any grade levels, it is easy for it to be seen as bits and pieces in isolation. Without a big picture of how it all fits together, it might be perceived as just an odd collection of tools. We want to avoid a curriculum that is perceived as “Another year, another graph…”A second challenge is that there are long time gaps between statistics modules—in some cases more than 2 years (between grade 9 module 2 and grade 11 module 4). There are some connections that need to be made from grade 7 in grade 11! Teachers may find it necessary to do some reviewing of concepts, which may require teachers to be familiar with the statistics content from grade levels other than the one they regularly teach.The final challenge is that the statistics content in the common core is content that most teachers have not had to teach prior to the common core. Many secondary teachers and almost all middle school teachers may not have ever had a statistics course as part of their preparation to teach. This makes it particularly important to have good curriculum

materials for both students and teachers.

2 11 This slide introduces 3 overarching themes that provide some coherence for the statistics content over the grades. Mention the three by name here. Each of these is covered in more detail in the slides that follow.

4 12 Statistics is all about variability. If there were no variability, there would be no need for statistical methods. For example, if every voter in New York favored increasing funding for schools, you wouldn’t need Gallup to carry out a poll to estimate support. You could just ask one voter.You can think of statistics as providing methods for learning from data. The challenge is to figure out what we can learn given that there is variability in data. To carry out a successful statistical investigation, you need to anticipate variability when planning how you will collect data, describe the variability present in a data set, understand the consequences of variability in the data, and be able to draw conclusions in a way that takes variability into account.

4 13 A second overarching theme is the investigative process. Review the four steps in the investigative process that are listed on this slide. This process can be introduced very early on, beginning in grade 6. The difference between how it is handled in grade 6 and how it is handled by the time students get to grade 11 is in the types of questions that can be addressed and the level of sophistication in the way students draw conclusions from the data.

3 14 Probability is also developed in the statistics and probability modules of grades 6 – 11 (and grade 12 is really all probability). The main goal of the probability coverage in grades 6 – 11 is to develop a basic understanding of probability concepts and to develop the foundation needed for the “ruling out chance” aspects of statistical inference.

Section: Linking the Two Statistics Themes Time: 5 Minutes

[5 minutes] In this section, you will…Establish that the theme of learning from data through the investigative process is inherently tied to the theme of variability

Materials used include:

Time Slide # Slide #/ Pic of Slide Script/ Activity directions GROUP

2 15 This slide is just a transition slide to the next section—Linking the two statistics overarching themes.

1 16 This slide shows how the theme of learning from data through the investigative process is inherently tied to the theme of variability. Spend a few minutes talking about the linkages shown in the table.

1 17 This slide shows where each grade is focused in terms of the investigative process. Notice that while the focus of each grade level is slightly different, even at grade 6 students are able to deal with most aspects of the process and will use data to answer questions.

1 18 Use this opportunity to wrap up this section and to discuss the unique challenges that the placement of the statistics content present. Emphasize that have a sense of these overarching themes will help teachers understand what they are doing at their individual grade level contributes to the overall goals of the statistics content.

Section: Example Trajectory Time: 26 Minutes

[26 minutes] In this section, you will…Provide examples from the lessons of grades 6, 7 and 11 that show how students develop and mature in their understanding over one of the overarching themes.

Materials used include:

Time Slide # Slide #/ Pic of Slide Script/ Activity directions GROUP

2 19 This slide transitions to the next section—Example trajectory.

1 20 The purpose of this section is to provide examples from the lessons of grades 6, 7 and 11 that show how students develop and mature in their understanding over one of the overarching themes. (learning from data). Spend a minute talking about the progression in student thinking described on this slide for the three grades.

3 21 Introduce this activity from Grade 6. Make sure that participants are familiar with the context and that they understand the boxplots. Point out that this is population data. Students don’t’ have to grapple with issues of sampling until grade 7.

2 22 Spend a minute talking about the answers to the first two questions. Possible answers to these questions appear below. Point out that the first two questions are making sure that students understand how to extract information from a box plot.The interesting question is question number 3, which asks students to make a judgment. We aren’t really looking for a “right” answer here, but are interested in how the student justifies the answer and ties the justification to the box plot.Possible answers:No, the highest batting averages for both leagues appear to be around .274. (Allow for estimation by students.)They appear to be only slightly different with the AL range being slightly higher. AL minimum (.234) is slightly lower than the NL minimum (.236) and from above; both leagues appear to have the same maximum. The AL has the higher median batting average at roughly .258 while the median batting average for the NL is roughly .252. Students could state that this .006 difference is significant based on several reasons, e.g., the difference of .006 is roughly 1/6 of the NL range, the AL median is close to the NL Q3, visually, the difference appears to be about the same as the difference between Q1 and the median for the NL data set, and so on.

2 23 Point out that these are different box plots than on the previous slide—they are looking at the distribution of a different variable. Again the interesting question is question 6, which asks students to use data from two populations to make a comparison and draw a conclusion.Possible answers:The Q1, median, and maximum appear to be roughly the same.

The NL data set appears to have less variability as it has a smaller IQR and smaller range.

A student might disagree with the statement given the similar medians and the other similar summary measures. Also the AL data set has a lower minimum. However, a student might agree with the

statement in that the AL data set has a higher Q3 than the AL data set.

2 24 This is a second example from grade 6. Point out that this example also deals with population data—data is available for all 12 months for each city. This question asks students to understand the graphs and to draw conclusions based on the graphs. Interest is centered on the students reasoning about the graphs.

2 25 Wrap up the discussion of this activity by making the two points on this slide. The conceptual leap referred to in the second bullet is being able to view a data distribution as something in its own right. When students encounter a data distribution and are asked to describe it, they often start to describe each individual point—there is one month that had a temperature of..., another month that had a temperature of …, and so on. Getting them to view the distribution as a whole and to think of it in terms of center, variability and shape is something that doesn’t come naturally to students at this grade level.

3 26 This slide introduces an activity from Grade 7. If possible, have the participants do part of this activity. It has been adapted a bit to accommodate the large groups that are sometimes expected in professional development settings, but even thought they won’t be doing exactly what the students do, it will still illustrate what this activity is about.The Casey at the Bat handout has two pages. On one page you will find the poem as intended and on the other page you will find the poem broken up into 29 lines of 20 words each.Start by having participants look at the poem in its intended form. Tell them you are interested in the average length of the words in the poem. Ask them to circle 8 words that they think are “representative” of words in the poem and then have them calculate the average length of the words they circled.

Discuss the answer to the first question on this slide, making sure that it is understood that the population of interest here is the population of aloof the words in the poem.

1 27 Introduce the idea of a random sample here. In the classroom activity, students will select a random sample of words using two bags of numbers. One bag contains the numbers from 1 to 29 and the other bag contains the numbers from 1 to 20. They will pick a number from bag 1 to determine a line in the poem and then a number from bag 2 to determine a word in that line. This is repeated 8 times.Since there may be a large number of participants in the professional development setting, use the next slide to assist participants in the selection of their random sample.

1 28 Have participants select a random sample of words from the poem (using the first 8 lines of the table on the slide) using the page in the handout that has the poem broken up into 29 lines of 20 words each. Have them calculate the mean for the random sample and compare this to the mean they got when they just picked the words themselves.Make sure that they understand that in the classroom setting, each student will get a different random sample.

1 29 The class then makes a table of the averages from their self-selected samples and from their random samples. What happens is that the means from the self-selected samples have a tendency to be larger than the means from the random samples (there are a lot of short words in the poem, and people tend to overlook them in favor of longer words when they self-select).Creating dot plots of the two sets of averages reveals this pattern.

1 30 Here the actual mean length of words in the population is revealed and students compare the actual population mean to the dot plots of the means from self-selected samples and the means of random samples.Students see that the means from the random samples tend to center around 4.2 and also usually tend to vary less from sample to sample that the self-selected sample means. This hopefully convinces students of the utility of random sampling.

1 31 Summarize the important concepts from this grade 7 example.

2 32 To see how the sophistication ramps up by grade 11, consider this grade 11 example of using sample data to learn about characteristics of two populations. The expected answer is shown on the next slide.

2 33 Chances are many participants will not know how to do this problem, so don’t spend too much time on the details of how the solution was obtained. Instead, make the point that students are now using sample data to reason about a population (here even to compare two populations) and recognizing sampling variability by including a margin of error.

Section: Other Developmental Dimensions Time: 29 Minutes

[29 minutes] In this section, you will…Describe other developmental trajectories in the statistics content. These dimensions show how students are expected to develop deeper understanding as they move from grade 6 to grade 11.

Materials used include:

Time Slide #Slide #/ Pic of Slide Script/ Activity directions GROUP

3 34 This slide just transitions to the next section—Other developmental dimensions.

2 35 Spend a few minutes discussing other developmental trajectories in the statistics content. These dimensions show how students are expected to develop deeper understanding as they move from grade 6 to grade 11.

3 36 This final activity is to give participants a sense of what we are working to achieve in the statistics curriculum. This is the activity that is in the final lessons of Grade 11.Use this slide to introduce the experiment that is the basis of this example.

1 37 This slide provides the data from the experiment that will be analyzed to determine if there is a treatment effect (whether the nutrient had an effect on the weight of the tomatoes. Point out that the treatment group had a higher mean and ask if this means that they can be sure that the treatment is effective. Ask if they think that if they had 10 tomatoes that did not get the treatment (i.e. 10 control tomatoes) and they randomly divided them into two groups of 5, the two group means would be equal.

1 38 Spend a minute discussing this slide. Make sure participants understand the question of interest and why we need to ask this question. You can relate this back to the opening activity on the difference between statistical thinking and mathematical thinking.

1 39 Again, make sure this reasoning is clear, as it is the basis for the rest of the activity.

1 40 If you have a small group, you can provide each person or each small group with a set of 10 index cards and have them actually do this. With a large group, just bring one set and demonstrate the random division into tow groups. Emphasize that this would be done many times, leading to the plot on the next slide

1 41 Explain that this dot plot was generated by repeating the random division into two groups of five 250 times. Emphasize that this tells us what “chance” differences would look like if there were not treatment effect.

1 42 Now take a look at the actual observed difference and locate the difference of 2.24 in the distribution on the previous slide. This could be considered as consistent with chance behavior.A result as large as what was observed only happened about 5 times out of 250 (2 % of the time) just by chance—not very often. Many people would take this as evidence of a treatment effect. Others may be more conservative and say that they aren’t convinced.

1 43 Wrap up by describing how the tomato activity illustrates these important overall themes and how they come together.

14 44 Thank the participants and use the remaining time to answer questions.

Use the following icons in the script to indicate different learning modes.

Video Reflect on a prompt Active learning Turn and talk

Turnkey Materials Provided

● Statistics and Probability In Grades 6-11

Additional Suggested Resources

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