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?

Focus. Participants will be able to identify the major work of each
How will we know that they are able to do this?
Participants will be able to articulate the key points listed above.




grade using the Curriculum Overview document as a resource in
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
Statistical Thinking
vs Mathematical
Thinking
12
Minutes
Establish the distinction between
statistical and mathematical
thinking.
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
Establish that the theme of
learning from data through the

5 Minutes
investigative process is inherently
tied to the theme of variability
Statistics and Probablity in
Grades 6-11 PPT

Statistics and Probablity in
Grades 6-11 PPT
Facilitator Preparation
26
Minutes
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.
Statistics and Probablity in
Grades 6-11 PPT
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 Minutes
Answer questions and facilitate
discussion.
Example Trajectory

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:
Time Slide 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
1
7
Script/ Activity directions
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.
GROUP
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
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.
GROUP
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…
Materials used include:
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.
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
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.
GROUP
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
Turnkey Materials Provided
●
Statistics and Probability In Grades 6-11
Additional Suggested Resources
●
Active learning
Turn and talk