Download SFUSD Unit S.2 Statistics Random Processes

1
SFUSD Mathematics Core Curriculum Development Project
2014–2015
Creating meaningful transformation in mathematics education
Developing learners who are independent, assertive constructors of their own understanding
SFUSD Mathematics Core Curriculum, Algebra 2, Unit S.2: Statistics: Random Processes, 2014–2015
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Algebra 2
S.2 Statistics: Random Processes
Number
of Days
Lesson
Reproducibles
Number of
Copies
Materials
2
Entry Task
Random Rectangles (2 pages)
1 per student
8
Lesson Series 1
CPM CCA Lesson C.1.1 (4 pages)
CPM CCA Lesson C.1.2 (4 pages)
Lesson C.1.2A Resource Page
CPM CCA Lesson C.1.3 (5 pages)
CPM CCA2 Lesson 9.3.1 (4 pages)
CPM CCA2 Lesson 9.3.2 (4 pages)
CPM CCA2 Lesson 9.3.3 (6 pages)
The Normal Distribution (13 pages)
1 per pair
1 per pair
1 per pair
1 per pair
1 per pair
1 per pair
1 per pair
1 per pair
Graphing calculators or software
2
Apprentice Task
Where There Enough Blue Ones?
1 per pair
Large jar of M&Ms, beans, or beads
Paper cups
5
Lesson Series 2
CPM CCA2 Lesson 9.1.1 (4 pages)
CPM CCA2 Lesson 9.1.1 HW (2 pages)
CPM CCA2 Lesson 9.1.2 (4 pages)
CPM CCA2 Lesson 9.1.2 HW (2 pages)
CPM CCA2 Lesson 9.1.3 (4 pages)
CPM CCA2 Lesson 9.2.1 (4 pages)
CPM CCA2 Lesson 9.2.2 (4 pages)
1 per pair
1 per student
1 per pair
1 per student
1 per pair
1 per pair
1 per pair
Roll of pennies
Pipettes (eye droppers), one per pair
Paper cups
Vinegar
Liquid dish soap
Poster paper or graphing calculators
2
Expert Task
3
Lesson Series 3
CPM CCA2 Lesson 11.2.1 (3 pages)
CPM CCA2 Lesson 11.2.2 (6 pages)
1 per pair
1 per pair
2
Milestone Task
The Marble Jar
CPM CCA2 CL 11-100
1 per student
1 per student
Index cards
Computer and projector
SFUSD Mathematics Core Curriculum, Algebra 2, Unit S.2: Statistics: Random Processes, 2014–2015
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Unit Overview
Big Idea
Students will collect and analyze data from a sample or simulation to make inferences about a population.
Unit Objectives
● Students will build upon their understanding of basic statistical measures (mean, median, mode) and graphs (histograms, dot plots, etc.) to analyze
●
●
data sets, including learning to measure variability
Students will differentiate between different sampling and selection methods and understand the benefits and drawbacks of each, including learning
methods to generate random samples and understand that these are the most representative
Students will use simulations to model surveys and experiments in order calculate confidence intervals and conduct significance tests
Unit Description
Through hands-on sampling experiments that are based in or model real-life situations, students review or learn statistical analysis tools such as data displays
and numerical summaries. They use these tools to explore and summarize data sets, including by finding measures of center and spread. Students learn
random sampling methods (especially those employing technology) and the drawbacks of non-random sampling. They see real-life examples of the Normal
distribution and learn to calculate and interpret areas under it. Students combine this learning with repeated technology-based simulations to develop an
intuitive understanding of margin of error and statistical significance.
CCSS-M Content Standards
* All Statistics and Probability Standards are Modeling Standards.
Interpreting Categorical and Quantitative Data
Summarize, represent, and interpret data on a single count or measurement variable
S.ID.4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are
data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
Making Inferences and Justifying Conclusions
Understand and evaluate random processes underlying statistical experiments
S.IC.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
S.IC.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a
spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?
SFUSD Mathematics Core Curriculum, Algebra 2, Unit S.2: Statistics: Random Processes, 2014–2015
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Make inferences and justify conclusions from sample surveys, experiments, and observational studies
S.IC.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
S.IC.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random
sampling.
S.IC.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
S.IC.6 Evaluate reports based on data.
SFUSD Mathematics Core Curriculum, Algebra 2, Unit S.2: Statistics: Random Processes, 2014–2015
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Progression of Mathematical Ideas
Prior Supporting Mathematics
Current Essential Mathematics
Future Mathematics
The unit before covers probability. This correlates
well with the idea of how probable is any particular
sample statistic drawn from a population.
Students calculate mean, median, mode, and use
technology to find measures of spread, like the
standard deviation. They learn to draw relative
frequency histograms and connect the area under a
distribution with a proportion of individuals. They
learn to calculate z-scores and use tables or
technology to find areas under the Normal
distribution. This unit requires only basic algebra
skills for the calculations. Much of the material in the
unit is language-based as many of the ideas in
statistics involve new concepts and vocabulary. This
unit relies heavily on technology to create
simulations that help students develop an intuitive
understanding of statistical inference.
The big ideas regarding statistics in Algebra 2 are
essentially the same as that for Stats and
Probability or AP Stats but in concept only. By
feeling comfortable with the big ideas, students will
be able to digest the rigor of the AP class.
In Algebra the students are introduced to the
concepts of mean and standard deviation. Our
Entry Task begins with these ideas.
SFUSD Mathematics Core Curriculum, Algebra 2, Unit S.2: Statistics: Random Processes, 2014–2015
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Unit Design
All SFUSD Mathematics Core Curriculum Units are developed with a combination of rich tasks and lessons series. The tasks are
formative assessments of student learning. The tasks are designed to address four central questions:
2 days
8 days
2 days
5 days
2 days
Lesson Series 3
Milestone
Task
Lesson Series 2
Expert
Task
Lesson Series 1
What do you already know?
What sense are you making of what you are learning?
How can you apply what you have learned so far to a new situation?
Did you learn what was expected of you from this unit?
Apprentice
Task
Entry Task
Entry Task:
Apprentice Task:
Expert Task:
Milestone Task:
3 days
2 days
Total: 24 days
SFUSD Mathematics Core Curriculum, Algebra 2, Unit S.2: Statistics: Random Processes, 2014–2015
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Entry Task
Apprentice Task
Expert Task
Milestone Task
Random Rectangles
Were There Enough Blue Ones?
Using a Simulation to Evaluate an
Experiment
The Marble Jar
CCSS-M
Standards
S.IC.1, S.IC.4
S.IC.1, S.IC.4
S.IC.3, S.IC.4, S.IC.5, S.IC
S.IC.4, S.IC.5, S.IC.4, S.IC.1, S.IC.
Brief
Description
of Task
Illustrative Math, Why
Randomize?
Sampling distribution of a proportion
– estimating the number of objects
of a certain color in a collection
(blue M&M’s, for example)
● Survey
● Sampling from population
with known proportion
● Survey the class and then
aggregate data with
simulations using
technology
● Where does the class data
rank within the distribution?
Students conduct an experiment to
answer a question of interest.
● For example, is there a
difference between heart
rates for people who are
standing still versus for
people who are walking in
place?
● They will use their results for
the difference between
treatment and control to judge
whether we think there is a
significant difference in
parameters by conducting a
simulation that generates fake
data based on just
randomness (versus
treatment effect).
This task includes two problems
involving possible simulations
(though the data can be given
instead), asking students to
analyze the results and make
inferential conclusions.
Source
http://www.uwlax.edu/faculty/h
asenbank/archived/mth126sp
10/Notes/Project2%20%20Random%20Rectangles
%20option.pdf
CPM Core Connections Algebra 2
Lesson 11.1.3
http://blog.mrwaddell.net/archives/782
Illustrative Math, The Marble Jar
AND
CL 11-100 from CPM Core
Connections Algebra 2 Chapter 11
Closure
Also uses great customizable
simulation resource
http://lock5stat.com/statkey/
SFUSD Mathematics Core Curriculum, Algebra 2, Unit S.2: Statistics: Random Processes, 2014–2015
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Lesson Series 1
Lesson Series 2
Lesson Series 3
CCSS-M
Standards
S.IC.3, S.IC.4, S.IC.5
S.IC.4
S.IC.5, S.IC.6, S.IC.4
Brief
Description
of Lessons
Survey/observational study vs. experiment
Possible: Review/teaching of mean, median,
IQR, standard deviation, histograms, stemplots,
dot plots, and boxplots. CPM Appendix C
(partial?)
Significance and confidence intervals via
simulations
Sources
Choosing a representative sample
Sampling methods (convenience, etc)
bias, etc
Treatment, placebo, etc.
Conclusions from data (causation, etc)
Relative Frequency Histograms
Introduce idea of (though not calculation of?)
margin of error
Normal Distributions (CPM: The Normal
Probability Density Function)
Proportions (area under curve)
Percentiles
CPM Core Connections Algebra 2, Appendix C
CPM Core Connections Algebra 2, 9.3
CPM Core Connections Algebra 2, 9.1 and 9.2
CPM Core Connections 11.2.1, 11.2.2
SFUSD Mathematics Core Curriculum, Algebra 2, Unit S.2: Statistics: Random Processes, 2014–2015
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Entry Task
Random Rectangles
What will students do?
Mathematics Objectives and Standards
Math Objectives:
● Students will be able to estimate the population mean from a sample
distribution.
●
Students will be able to understand bias in sampling.
CCSS-M Standards Addressed: S.IC.1, S.IC.4
Potential Misconceptions:
● You may have to review dot plots.
Framing Student Experience
Launch:
Introduce the question of how to find the mean of a population that is
uncountable. How can we use samples to accomplish this?
During:
Students work through the 100 Random Rectangles activity.
Closure/Extension:
Discuss the idea of bias in sampling. How did the mean of the random sample
distribution compare to the mean of the judgment sample distribution?
SFUSD Mathematics Core Curriculum, Algebra 2, Unit S.2: Statistics: Random Processes, 2014–2015
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Random Rectangles
How will students do this?
Focus Standards for Mathematical Practice:
Structures for Student Learning:
Academic Language Support:
Vocabulary: dot plot, bias, sample, sample size, population
Sentence frames:
Differentiation Strategies:
Participation Structures (group, partners, individual, other):
SFUSD Mathematics Core Curriculum, Algebra 2, Unit S.2: Statistics: Random Processes, 2014–2015
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Lesson Series #1
Lesson Series Overview: This lesson series formally introduces the normal distribution, including how to compute proportions using the area under the curve
(with the empirical rule, tables, and technology). It also includes a series of lessons reviewing (or teaching, for those students without a common core
background yet) shape, center, spread, and data displays (histograms, box plots, dot plots).
CCSS-M Standards Addressed: S.ID.4
Time: 8 days
Lesson Overview – Days 1 and 2
Description of Lesson:
This lesson involves investigating data, graphical and numerical representation of data,
comparing data, and measuring variability. Read the CPM Teacher Notes for details.
Resources
CPM Core Connections Algebra 2, Appendix C: C.1.1 and C.1.2
Suggested Problems: C-3, C-11, C-20, C-21, C-23, C-26, C-35
Notes:
•
•
A class set of graphing calculators is required.
•
Methods and Meanings on p. 757 in CPM Core Connections Appendix C is a
good resource for students and instructors.
This lesson can be omitted if students already have a strong background from
previous classes.
Lesson Overview – Day 3
Resources
Description of Lesson:
This lesson involves measuring variability through the standard deviation. Read the
CPM Teacher Notes for details.
CPM Core Connections Algebra 2, Appendix C: C.1.3
Suggested Problems: C-44, C-46, C-49, C-51, C-56
Notes:
•
•
A lass set of graphing calculators is required.
This lesson can be omitted if students already have a strong background from
previous classes.
SFUSD Mathematics Core Curriculum, Algebra 2, Unit S.2: Statistics: Random Processes, 2014–2015
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Lesson Overview – Days 4 and 5
Description of Lesson:
This lesson involves relative frequency histograms and the normal probability density
function. Read the CPM Teacher Notes for details.
Resources
CPM Core Connections Algebra 2, Chapter 9: 9.3.1 and 9.3.2
Suggested Problems: 9-70, 9-71, 9-72, 9-73, 9-84, 9-85, 9-86, 9-88
Notes:
This lesson can be omitted if students already have a strong background from previous
classes.
Lesson Overview – Day 6
Resources
Description of Lesson:
This lesson involves measuring location in a distribution using percentiles. Read the
CPM Teacher Notes for details.
CPM Core Connections Algebra 2, Chapter 9: 9.3.3
Suggested Problems: 9-97, 9-98, 9-99, 9-100, 9-101, 9-102, 9-103, 9104, 9-105
Notes:
Choose an appropriate sized sample of the suggested problems.
Lesson Overview – Days 7 and 8
Description of Lesson:
This lesson involves measuring location in a distribution using z-scores.
Resources
Statistics in Action, 2nd Edition (Watkins, Shaeaffer, Cobb)
Lesson 2.5, The Normal Distribution, pp. 83–95
Notes:
This is a textbook summary of some topics from earlier in the lesson series but also
includes a description of using z-scores.
SFUSD Mathematics Core Curriculum, Algebra 2, Unit S.2: Statistics: Random Processes, 2014–2015
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Apprentice Task
Were There Enough Blue Ones?
What will students do?
Mathematics Objectives and Standards
Math Objectives:
● Students will be able to use sampling distribution of a proportion,
estimating the number of objects of a certain color in a collection
(blue M&M’s, for example).
Framing Student Experience
Launch:
Calculate the proportion of boys and girls in the class. Is this a good estimate of
the proportion of boys and girls in the whole school? What about the whole city?
During:
Students will work through Problem 11-27 from CPM Core Connections Algebra 2.
CCSS-M Standards Addressed: S.IC.1, S.IC.4
Potential Misconceptions:
● Students may confuse the sample proportion with the population
proportion.
Closure/Extension:
Ask students to interpret a given confidence interval. For example, “A campaign
poll states we are 95% confident that 47% of the population plans to vote for
candidate A with a margin of error of +/-5%.”
SFUSD Mathematics Core Curriculum, Algebra 2, Unit S.2: Statistics: Random Processes, 2014–2015
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Were There Enough Blue Ones?
How will students do this?
Focus Standards for Mathematical Practice:
Structures for Student Learning:
Academic Language Support:
Vocabulary: sample, population, variability, interval, confidence interval, sample distribution
Sentence frames:
Differentiation Strategies:
Participation Structures (group, partners, individual, other):
Students work in small groups.
SFUSD Mathematics Core Curriculum, Algebra 2, Unit S.2: Statistics: Random Processes, 2014–2015
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Lesson Series #2
Lesson Series Overview: In this lesson series, students will learn some basic techniques of performing opinion surveys along with their limitations and pitfalls.
Students learn why randomness is a cornerstone of statistical studies. In the last section, students will create a histogram with percentages called a relative
frequency histogram. Students will learn a new way to describe the shape of a distribution and use it to model certain distributions.
CCSS-M Standards Addressed: S.ID.4
Time: 5 days
Lesson Overview – Day 1
Resources
Description of Lesson:
Students will write research questions and consider issues of bias as they write survey
items to investigate those questions.
CPM Core Connections Algebra 2, Chapter 9: 9.1.1
Notes:
•
•
Read the CPM teacher notes.
HW: Problems 9-7 and 9-8.
Lesson Overview – Day 2
Resources
Description of Lesson:
Students will compare the representative nature of samples selected using intentional
choice with those selected randomly. They will learn that random sampling produces
samples that, in the aggregate, represent populations better than samples intentionally
selected by humans.
CPM Core Connections Algebra 2, Chapter 9: 9.1.2
Notes:
•
•
•
Read the CPM teacher notes.
Classwork: Problems 9, 16, 20, 21
HW: Problems 22 and 23.
SFUSD Mathematics Core Curriculum, Algebra 2, Unit S.2: Statistics: Random Processes, 2014–2015
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Lesson Overview – Day 3
Resources
Description of Lesson:
CPM Core Connections Algebra 2, Chapter 9: 9.1.3
Students consider populations represented by particular convenience samples.
Notes:
Read the CPM teacher notes.
Lesson Overview – Day 4
Resources
Description of Lesson:
CPM Core Connections Algebra 2, Chapter 9: 9.2.1
Students discover the importance of randomization in an experiment.
Notes:
Read the CPM teacher notes.
Lesson Overview – Day 5
Resources
Description of Lesson:
CPM Core Connections Algebra 2, Chapter 9: 9.2.2
Students think critically about and analyze the conclusions of various fictitious
studies.
Notes:
Read the CPM teacher notes.
SFUSD Mathematics Core Curriculum, Algebra 2, Unit S.2: Statistics: Random Processes, 2014–2015
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Expert Task
Using a Simulation to Evaluate an Experiment
What will students do?
Mathematics Objectives and Standards
Math Objectives:
● Students will be able to conduct an experiment using random
assignment to determine a treatment effect.
● Students will be able to use technology to simulate the results of the
same experiment if there were NO treatment effect.
● Students will be able to use the results of the simulation to introduce
an intuitive notion of statistical significance.
CCSS-M Standards Addressed: S.IC.4, S.IC.5, S.IC.6
Framing Student Experience
This activity comes from: http://blog.mrwaddell.net/archives/782
Students conduct an experiment to answer a question of interest.
● For example, is there a difference between heart rates for people who
are standing still versus for people who are walking in place?
● Note that there are many alternatives for this experiment. A commenter
on the blog post linked for this task posted an example of an alternative
version where students compare playing paper basketball blindfolded
versus not.
(https://docs.google.com/a/byron.k12.mn.us/document/d/1mJrav
RxsjkV3FKKE0BHfc6bRTpia78jT1Ba2jYvJx1g/edit) Another
Potential Misconceptions:
●
●
example is card throwing using dominant versus non-dominant hands
(http://roughlynormal.com/2013/07/05/sss-card-tossing-experimentpart-2-is-the-evidence-convincing/).
They will use their results for the difference between treatment and
control to judge whether they think there is a significant difference in
parameters by conducting a simulation that generates fake data based
on just randomness (versus treatment effect).
Launch:
Ask the class whether they think that walking in place affects heart rate. Then
have a class discussion about how they would prove whether it does. This can
also be a place to practice being specific about who they are focusing on-population, sample, parameter, statistic, etc.
During:
See the blog post or the PDF version of it in the Resource files.
Closure/Extension:
Review the logic of being convinced. Preview the idea of quantifying how
unlikely an outcome needs to be to be considered significant.
SFUSD Mathematics Core Curriculum, Algebra 2, Unit S.2: Statistics: Random Processes, 2014–2015
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Using a Simulation to Evaluate an Experiment
How will students do this?
Focus Standards for Mathematical Practice:
4. Model with mathematics
5. Use appropriate tools strategically.
Structures for Student Learning:
Academic Language Support:
Vocabulary: treatment, control, parameter, statistic, distribution, random assignment
Sentence frames:
Differentiation Strategies:
Participation Structures (group, partners, individual, other):
• The experiment should be done in pairs or small groups.
• Individual students can be assigned specific roles, such as data recorder, timekeeper, etc.
• The data aggregation is done as a whole class, as is the technology-based simulation.
SFUSD Mathematics Core Curriculum, Algebra 2, Unit S.2: Statistics: Random Processes, 2014–2015
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Lesson Series #3
Lesson Series Overview: Students review significance and confidence intervals via simulations, and they make conclusions from data (causation, etc.).
CCSS-M Standards Addressed: S.IC.5, S.IC.6, S.IC.4
Time: 3 days
Lesson Overview - Day 1
Description of Lesson:
This lesson is made up of Problems 11-37, 11-38, and 11-39.
Resources
CPM Core Connections Algebra 2, Chapter 11: 11.2.1
Students will be conducting a hypothesis test. Although it is not necessary for students to fully
understand this new vocabulary, the vocabulary should nonetheless be used. Hold a brief discussion to
establish that a claim has been made (“more than 50% of the students support a dance”). Students will
be determining whether there is convincing evidence to support that claim. The survey indicates that
60% of the sample of students supports the new dance, but because of the small sample size, we
expect a large sample-to-sample variability. Thus, we cannot immediately jump to the conclusion that
more than 50% of the students support the dance. Indeed, it is plausible that only 50% (or less) of all
the students at the school support the dance, even though the small sample showed 60% support.
Notes:
• If students are not proficient at creating their own simulations, a discussion about and the
creation of a simulation could be done as a whole class activity.
• Remind students that in the apprentice task we wanted to make an inference about the
population by taking a sample. However, to make a meaningful prediction about the number of
blue candy-coated chocolates, we simulated the sample-to-sample variability to determine a
margin of error for our prediction. Today we will continue to use simulation to determine
sample-to-sample variability.
• Read the teacher notes for this section. There are lots of important tips and pointers.
Lesson Overview - Day 2
Resources
Description of Lesson:
This lesson consists of Problem 11-50. Students will simulate sample-to-sample variability to determine
if it is plausible that two treatments in an experiment are truly different.
CPM Core Connections Algebra 2, Chapter 11: 11.2.2
SFUSD Mathematics Core Curriculum, Algebra 2, Unit S.2: Statistics: Random Processes, 2014–2015
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This is a lot like the previous day’s lesson, but in this case the students are looking at an experiment
situation instead of a survey. The simulation process is very similar, so should build on students’
abilities from previous tasks and lessons. In this case, they are testing for evidence that there is a
difference in the results by seeing how likely it is to get an experimental difference as large as the one
in the sample if there really were no difference.
Although they will not be expected to know the vocabulary, students are starting the process of
determining whether or not the results of an experiment are statistically significant.
For the experiment in Problem 11-50, we are not particularly concerned with what proportion of
tadpoles survived. The question of interest is whether there is a difference in the proportion that
survived when the tadpoles did not have mosquito larvae to feed on any more. The experimental
results of our two samples indicate a difference of 0.14: more tadpoles survived in our samples when
they had mosquitoes to feed on. But of course this is just one small sample from each type of frog food.
The true population difference in survival between the two kinds of food could be more or less (or even
negative).
Lesson Overview - Day 3
Description of Lesson:
This is a practice day, applying the confidence interval and hypothesis testing techniques to various
survey and experimental situations.
Resources
CPM Core Connections Algebra 2, Chapter 11: 11.2.2
and 11.2.3; Problems 11-52, 11-53, 11-54, 11-65
Notes:
You may choose to do any selection of these problems, depending on what you think your students
need practice on.
SFUSD Mathematics Core Curriculum, Algebra 2, Unit S.2: Statistics: Random Processes, 2014–2015
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Milestone Task
The Marble Jar
What will students do?
Mathematics Objectives and Standards
Framing Student Experience
Though a more traditional test could be used, this unit would end well with
more involved analysis of statistical situations. Both of these problems
address the standards of the unit. You might choose to do just one of them.
Alternatively, one could be done as a group test and one as an individual test.
The Marble Jar can be used as is. You may wish to create a real-life example of
the marble jar and have students conduct the sampling. If not, they can just be
given the data results as given in the activity.
Math Objectives:
●
CCSS-M Standards Addressed:
Potential Misconceptions:
●
CL 11-100 should be modified. Students should do (a). (b) could be omitted or
modified to ask them to describe ANY simulation they could set up (not just
using the calculator). In (c) they should create a data display and find the mean
(and possibly a measure of spread). Alternatively, you could provide the
numerical and graphical displays and have students fit a normal curve (they
could draw it on or you could provide it) and calculate the probability of getting
a value in as extreme as the one they find from the upper/lower 5% cutoff.
Students should do (d) as is. A part (e) could be added that asks students to
analyze the potential bias or errors in the sampling, or to hypothesize how the
sampling was designed.
Launch:
Provide students with the two scenarios, along with the data. Explain that this
will be the opportunity for them to put all of their analytical tools together. You
could also start class by having students do a quick review activity--maybe
creating a class concept map or doing a warm-up problem.
During:
Closure/Extension:
Students could present their data analysis results and conclusions or just write
them up in a poster or report.
SFUSD Mathematics Core Curriculum, Algebra 2, Unit S.2: Statistics: Random Processes, 2014–2015
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The Marble Jar
How will students do this?
Focus Standards for Mathematical Practice:
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
Structures for Student Learning:
Academic Language Support:
Vocabulary: bias census, convenience sample, experiment, mean, normal distribution, observational study, parameter, population, random sample,
sample standard deviation, variable, margin of error, population, random number, sample-to-sample variability, sampling distribution
Differentiation Strategies:
Participation Structures (group, partners, individual, other):
SFUSD Mathematics Core Curriculum, Algebra 2, Unit S.2: Statistics: Random Processes, 2014–2015