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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Statistics and Probability
in Grades 6 - 11
A Story of Ratios
A Story of Functions
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Session Objectives
•
•
•
•
•
Explore the distinction between mathematical thinking and
statistical thinking.
Examine the development of the statistics and probability
content over grades 6 – 11.
Introduce overarching themes that provide coherence in the
statistics and probability content across the grades.
Illustrate development of statistical thinking across the grades
with a trajectory of lesson activities.
Explore other dimensions in the development of statistical
thinking across the grades.
2
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Agenda
Statistical Thinking vs Mathematical Thinking
Grade 6 – 11 Statistics and Probability—Overarching Themes
Linking the Two Statistics Themes
Example Trajectory
Other Developmental Dimensions
Questions and Discussion
3
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Consider the map of counties shown below. The number
in each county is last month’s incidence rate for a disease
in cases per 100,000 population. What do you think is going on?
(Dick Schaeffer, 2005)
4
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Statistical Thinking Versus Mathematical Thinking
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”)
5
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Statistical Thinking
Statistical thinking is HARD!
Needs to be developed and nurtured over time, much like
Mathematical thinking.
Is a skill that is critical to making informed decisions based
on empirical evidence.
6
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Agenda
Statistical Thinking vs Mathematical Thinking
Grade 6 – 11 Statistics and Probability—Overarching Themes
Linking the Two Statistics Themes
Example Trajectory
Other Developmental Dimensions
Questions and Discussion
7
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Developing Statistical Thinking Across the Grades
8
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Developing Statistical Thinking Across the Grades
9
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Developing Statistical Thinking Across the Grades
Common Core Challenges:
• Achieving coherence across the grades.
• Long time gaps—many necessary connections skip
years!
• New content for most teachers.
10
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Three Overarching Themes
• Variability
• Learning from Data (The Investigative Process)
• Probability
11
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Variability
•
•
•
•
Anticipating variability
Describing variability
Understanding variability
Drawing conclusions in the presence of variability
12
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Learning from Data
The Investigative Process
• Pose a statistical question
A statistical question is one that can be answered by
collecting data and where there will be variability in
the data.
• Collect data
• Summarize and describe the data distribution
• Draw conclusions based on data
13
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Probability
• Understanding probability
• Foundation for “ruling out chance”
14
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Agenda
Statistical Thinking vs Mathematical Thinking
Grade 6 – 11 Statistics and Probability—Overarching Themes
Linking the Two Statistics Themes
Example Trajectory
Other Developmental Dimensions
Questions and Discussion
15
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Linking the Two Statistics Themes
Step
Learning from Data
The Role of Variability
1
Pose a statistical question
Anticipating variability (in the data)
2
Collect data
Planning for variability (recognizing the
need for a representative sample or
comparable experimental groups)
3
Summarize and describe the data
distribution
Understanding variability (looking at the
data distribution using graphical and
numerical summaries)
4
Draw conclusions based on data
Taking variability into account when
drawing conclusions (sampling variability)
16
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Themes Across the Grades
Grade 6: Focus on Steps 1 and 3 (for census data), Step 4 in an informal way
Grade 7: Continue with Step 1, adds Step 2 (selecting a sample) and Step 3 (for
sample data)
Grade 8: Focus on Step 3 in the context of relationships between two variables
Grade 9: Focus moves to Step 4 in an informal way
Grade 11: Formalizes Step 4.
17
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
The Coherence Challenge
Keeping the overarching themes in mind given
• Spread over grade levels
• Long gaps between statistics modules
18
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Agenda
Statistical Thinking vs Mathematical Thinking
Grade 6 – 11 Statistics and Probability—Overarching Themes
Linking the Two Statistics Themes
Example Trajectory
Other Developmental Dimensions
Questions and Discussion
19
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Example Trajectory: Learning About a Population
Grade 6: Describing and comparing populations
Grade 7: Sampling from a population, concept of sampling
variability
Grade 11: Drawing conclusions about a population based on data
from a sample.
20
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Grade 6 —Example 1 (Lesson 19)
In 2012, Major League Baseball was comprised of two leagues: an American League
of 14 teams, and a National League of 16 teams. It is believed that the American
League teams would generally have higher values of certain offensive statistics such
as "batting average" and "on-base percentage." (Teams want to have high values of
these statistics.) Use the following side-by-side box plots to investigate these
claims.
21
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Grade 6 Example 1 Continued
1. Was the highest American League team "batting average" very different from the highest
National League team "batting average"? If so, approximately how large was the difference
and which league had the higher maximum value?
2. Was the range of American League team "batting averages" very different or only slightly
different from the range of National League team "batting averages"?
3. Which league had the higher median team "batting average"? Given the scale of the graph
and the range of the datasets, does the difference between the median values for the two
leagues seem to be small or large? Explain why you think it is large or small.
22
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Grade 6 Example 1 Continued
4.
5.
6.
Based on the box plots below for "on-base percentage," which 3 summary values (from the
5-number summary) appear to be the same or virtually the same for both leagues?
Which league's data set appears to have less variability? Explain.
Respond to the original statement: "It is believed that the American League teams would
generally have higher values of … 'on-base percentage.'" Do you agree or disagree based on
the graphs above? Explain.
23
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Grade 6 —Example 2 (Lesson 10)
Decision making by comparing distributions is an important function of statistics. Recall that
Robert is trying to decide whether to move to New York City or to San Francisco based on
temperature. Comparing the center, spread, and shape for the two temperature distributions
could help him decide.
Which city should he choose if he loves hot weather and really dislikes cold weather? What
measure of the data would justify your decision? Why did you choose that measure?
24
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Grade 6 Examples
• Both activities require students to consider the role of
variability in trying to answer the questions posed.
• Both activities will be challenging for grade 6 students—this is
a new way of thinking about data that requires them to think
about data as a distribution rather than in terms of individual
points. This is a big conceptual leap.
25
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Grade 7 Example—Casey at the Bat
1.
2.
Suppose you wanted to learn about the lengths of
the words in the poem Casey at the Bat. You plan
to select a sample of words from the poem and use
these words to answer the following statistical
question: On average, how long is a word in the
poem? What is the population of interest here?
Look at the poem, Casey at the Bat by Ernest
Thayer, and select eight words you think are
representative of words in the poem. Record the
number of letters in each word you selected. Find
the mean number of letters in the words you chose.
26
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Casey at the Bat Continued
3.
4.
A random sample is a sample in which every
possible sample of the same size has an equal
chance of being chosen. Do you think the set of
words you wrote down was random? Why or why
not?
Working with a partner, follow your teacher’s
instruction for randomly choosing eight words.
Begin with the title of the poem and count a
hyphenated word as one word.
1.
2.
Record the eight words you randomly selected and find the mean
number of letters in those words.
Compare the mean of your random sample to the mean you found in
Exercise 4.
27
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Casey at the Bat Continued
28
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Casey at the Bat Continued
5.
As a class, compare the means from exercise 4 and the means from
exercise 6. Your teacher will provide a chart to compare the means.
Record your mean from exercise 4 and your mean for exercise 6 on this
chart.
29
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Casey at the Bat Continued
6.
7.
8.
9.
Do you think means from exercise 4 or the means from
exercise 6 are more representative of the mean of all of the
words in the poem? Explain your choice.
The actual mean of the words in the poem Casey At The
Bat is 4.2 letters. Based on the fact that the population
mean is 4.2 letters, are the means from Exercise 4 or means
from Exercise 6 a better representation the mean of the
population. Explain your answer.
How did population mean of 4.2 letter compare to the
mean of your random sample from exercise 6 and to the
mean you found in exercise 4?
Summarize how you would estimate the mean number of
letters in the words of another poem based on what you
learned in the above exercises.
30
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Grade 7 Example
• Illustrates the need for random sampling
• Develops the concept of sampling variability
31
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Grade 11 Example—Margin of Error
A newspaper in New York took a random sample of
500 people from New York City and found that 300
favored a certain candidate for governor of the state.
A second newspaper polled 1000 people in upstate
New York and found that 550 people favored the
opposing candidate. Explain how you would interpret
the results.
32
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Margin of Error Continued
Possible answer: In New York City, the proportion of people
who favor the candidate is 0.60 +/- 0.044% or from 0.556 to
0.644 of the people. In upstate New York, the proportion of
people who favor the candidate is 0.55 +/- 0.032 or from 0.518
to 0.592. Because the margins of error for the proportion that
favor the candidate produce intervals that overlap, you cannot
really say that the proportion who prefer this candidate is
different for people in New York City and people in upstate
New York.
33
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Agenda
Statistical Thinking vs Mathematical Thinking
Grade 6 – 11 Statistics and Probability—Overarching Themes
Linking the Two Statistics Themes
Example Trajectory
Other Developmental Dimensions
Questions and Discussion
34
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Other Developmental Dimensions
• Census of entire population (G6) → Sample from a population (G7, 9, 11)
• Data on one variable (G6, 7) → Two variables with focus on relationship
between variables (G8 (informal), 9, 11)
• One group (G6) → Comparing two groups (G7 (informal) 9, 11)
• Data from sampling (observational studies) → data from experiments (G11)
35
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Where It All Leads
Activity from last lessons of Grade 11 illustrates where we hope students will
be at the end of their Grades 6 – 11 statistics journey.
Imagine that 10 tomatoes of varying shapes and sizes have
been placed in front of you. These 10 tomatoes (all of the same
variety) have been part of a nutrient experiment where the
application of a nutrient is expected to yield larger tomatoes that
weigh more. All 10 tomatoes have been grown under similar
conditions (soil, water, sunlight, etc.) except that 5 of the tomatoes
received the additional nutrient supplement. Using the weight data
of these 10 tomatoes, you wish to examine the claim that the
nutrient yields heavier tomatoes on average.
36
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
More Tomatoes
37
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
More Tomatoes
Question of Interest:
Is the difference in the two group means
due to the nutrient, or might it just be due
to chance? That is, could it just be due to
the expected differences that would occur
just because of random assignment of
tomatoes to groups?
38
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
More Tomatoes
The reasoning:
Suppose that the nutrient had no effect. Then the big
tomato that weighed 9.1 ounces would have weighed
9.1 ounces with or without the nutrient.
If this is the case, the difference in the treatment and
the control group means is just a consequence of the
random assignment.
Is this a plausible explanation for the difference?
39
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
More Tomatoes
Let’s investigate:
If there were no treatment effect, what kind of
differences would be expected just by chance?
1.
Write the 10 weights on the cards provided.
2.
Turn the cards over, mix well, and divide them into two
groups (group A and group B).
3.
Calculate the group A mean, the group B mean, and the
difference in the two means.
40
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
More Tomatoes
If this process was repeated many times, we would have a sense of what
“chance differences” would look like if there is no treatment effect.
41
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
More Tomatoes
Drawing conclusions:
The observed difference in sample means is 2.24.
Is consistent with chance?
42
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Tomato Example
Illustrates
1. The investigative process
2. The role of variability in decision
making
3. The link between statistics and
probability
43
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Thanks
Thank you for your attention and participation.
We hope this overview of statistics and probability in Grades 6 – 11 has
provided some perspective on the content of these modules.
Questions and Discussion
44
© 2012 Common Core, Inc. All rights reserved. commoncore.org