Download Montana Curriculum Organizer: High School Mathematics Statistics

Montana Curriculum Organizer
Statistics and Probability
Mathematics
November 2012
Materials adapted from Arizona, Delaware and Ohio Departments of Education
Montana Curriculum Organizer: Statistics and Probability Mathematics
This document is a curriculum organizer adapted from other states to be used for planning scope and sequence, units, pacing and other materials that support a focused,
coherent, and rigorous study of mathematics K-12.
Page 3
TABLE OF CONTENTS
How to Use the Montana Curriculum Organizer
Page 4
Standards for Mathematical Practice: Grades 9-12 Statistics and Probability
Page 7
Statistics and Probability: Interpreting Categorical and Quantitative Data
9-12.S.ID.1, 9-12.S.ID.2, 9-12.S.ID.3, 9-12.S.ID.4, 9-12.S.ID.5, 9-12.S.ID.6a-c, 9-12.S.ID.7, 9-12.S.ID.8,
9-12.S.ID.9
Clusters:
Summarize, represent, and interpret data on a single count or measurement variable.
Summarize, represent, and interpret data on two categorical and quantitative variables.
Interpret liner models
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Statistics and Probability: Making Inferences and Justifying Conclusions
9-12.S.IC.1, 9-12.S.IC.2, 9-12.S.IC.3, 9-12.S.IC.4, 9-12.S.IC.5, 9-12.S.IC.6
Clusters:
Understand and evaluate random processes underlying statistical experiments.
Make inferences and justify conclusions from sample surveys, experiments, and observational studies.
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Statistics and Probability: Conditional Probability and the Rules of Probability
9-12.S.CP.1, 9-12.S.CP.2, 9-12.S.CP.3, 9-12.S.CP.4, 9-12.S.CP.5, 9-12.S.CP.6, 9-12.S.CP.7
Clusters:
Understand independence and conditional probability and use them to interpret data.
Use the rules of probability to compute probabilities of compound events in a uniform probability model.
Page 18
November 2012
References
Page 2
Montana Curriculum Organizer: Statistics and Probability Mathematics
This document is a curriculum organizer adapted from other states to be used for planning scope and sequence, units, pacing and other materials that support a focused,
coherent, and rigorous study of mathematics K-12.
HOW TO USE THE MONTANA CURRICULUM ORGANIZER
The Montana Curriculum Organizer supports curriculum development and instructional planning. The Montana Guide to
Curriculum Development, which outlines the curriculum development process, is another resource to assemble a complete
curriculum including scope and sequence, units, pacing guides, outline for use of appropriate materials and resources and
assessments.
Pages 4 and 5 of this document is important for planning curriculum, instruction and assessment. It contains the
Standards for Mathematical Practice grade-level explanations and examples that describe ways in which students ought
to engage with the subject matter as they grow in mathematical maturity and expertise. Therefore, a copy of these pages
for easy access may help increase rigor by integrating the Mathematical Practices into all planning and instruction and
help increase focus of instructional time on the big ideas for that grade level.
Pages 7 through 18 consists of tables organized into learning progressions that can function as units. The table for each
learning progression, unit, includes 1) domains, clusters and standards organized to describe what students will Know,
Understand, and Do (KUD), 2) key terms or academic vocabulary, 3) instructional strategies and resources by cluster to
address instruction for all students, 4) connections to provide coherence, and 5) the specific standards for mathematical
practice as a reminder of the importance to include them in daily instruction.
Description of each table:
LEARNING PROGRESSION
STANDARDS IN LEARNING PROGRESSION
Name of this learning progression, often this correlates
Standards covered in this learning progression.
with a domain, however in some cases domains are split
or combined.
UNDERSTAND:
What students need to understand by the end of this learning progression.
KNOW:
What students need to know by the end
of this learning progression.
DO:
What students need to be able to do by the end of this learning progression,
organized by cluster and standard.
KEY TERMS FOR THIS PROGRESSION:
Mathematically proficient students acquire precision in the use of mathematical language by engaging in discussion with
others and by giving voice to their own reasoning. By the time they reach high school they have learned to examine
claims, formulate definitions, and make explicit use of those definitions. The terms students should learn to use with
increasing precision in this unit are listed here.
INSTRUCTIONAL STRATEGIES AND RESOURCES:
Cluster: Title
Strategies for this cluster
Instructional Resources/Tools
Resources and tools for this cluster
STANDARDS FOR MATHEMATICAL PRACTICE:
A quick reference guide to the eight standards for mathematical practice is listed here.
November 2012
Page 3
Montana Curriculum Organizer: Statistics and Probability Mathematics
This document is a curriculum organizer adapted from other states to be used for planning scope and sequence, units, pacing and other materials that support a focused,
coherent, and rigorous study of mathematics K-12.
Mathematics is a human endeavor with scientific, social, and cultural relevance. Relevant context creates an opportunity
for student ownership of the study of mathematics. In Montana, the Constitution pursuant to Article X Sect 1(2) and
statutes §20-1-501 and §20-9-309 2(c) MCA, calls for mathematics instruction that incorporates the distinct and unique
cultural heritage of Montana American Indians. Cultural context and the Standards for Mathematical Practices together
provide opportunities to engage students in culturally relevant learning of mathematics and create criteria to increase
accuracy and authenticity of resources. Both mathematics and culture are found everywhere, therefore, the incorporation
of contextually relevant mathematics allows for the application of mathematical skills and understandings that makes
sense for all students.
Standards for Mathematical Practice: Grades 9-12 Explanations and Examples
Standards
Students are expected to:
Explanations and Examples
The Standards for Mathematical Practice describe ways in which students ought to engage with the subject matter as they
grow in mathematical maturity and expertise.
HS.MP.1. Make sense of
problems and persevere
in solving them.
High school students start to examine problems by explaining to themselves the meaning of a problem and looking for entry
points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form
and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They
consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into
its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on
the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to
get the information they need. By high school, students can explain correspondences between equations, verbal
descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for
regularity or trends. They check their answers to problems using different methods and continually ask themselves, “Does
this make sense?” They can understand the approaches of others to solving complex problems and identify
correspondences between different approaches.
High school students seek to make sense of quantities and their relationships in problem situations. They abstract a given
situation and represent it symbolically, manipulate the representing symbols, and pause as needed during the manipulation
process in order to probe into the referents for the symbols involved. Students use quantitative reasoning to create
coherent representations of the problem at hand; consider the units involved; attend to the meaning of quantities, not just
how to compute them; and know and flexibly use different properties of operations and objects.
High school students understand and use stated assumptions, definitions, and previously established results in constructing
arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures.
They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify
their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about
data, making plausible arguments that take into account the context from which the data arose. High school students are
also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which
is flawed, and—if there is a flaw in an argument—explain what it is. High school students learn to determine domains to
which an argument applies, listen or read the arguments of others, decide whether they make sense, and ask useful
questions to clarify or improve the arguments.
High school students can apply the mathematics they know to solve problems arising in everyday life, society, and the
workplace. By high school, a student might use geometry to solve a design problem or use a function to describe how one
quantity of interest depends on another. High school students make assumptions and approximations to simplify a
complicated situation, realizing that these may need revision later. They are able to identify important quantities in a
practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and
formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their
mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the
model if it has not served its purpose.
High school students consider the available tools when solving a mathematical problem. These tools might include pencil
and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical
package, or dynamic geometry software. High school students should be sufficiently familiar with tools appropriate for
their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the
insight to be gained and their limitations. For example, high school students analyze graphs of functions and solutions
generated using a graphing calculator. They detect possible errors by strategically using estimation and other
mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize
the results of varying assumptions, explore consequences, and compare predictions with data. They are able to identify
relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve
problems. They are able to use technological tools to explore and deepen their understanding of concepts.
HS.MP.2. Reason
abstractly and
quantitatively.
HS.MP.3. Construct
viable arguments and
critique the reasoning of
others.
HS.MP.4. Model with
mathematics.
HS.MP.5. Use
appropriate tools
strategically.
November 2012
Page 4
Montana Curriculum Organizer: Statistics and Probability Mathematics
This document is a curriculum organizer adapted from other states to be used for planning scope and sequence, units, pacing and other materials that support a focused,
coherent, and rigorous study of mathematics K-12.
HS.MP.6. Attend to
precision.
HS.MP.7. Look for and
make use of structure.
HS.MP.8. Look for and
express regularity in
repeated reasoning.
November 2012
High school students try to communicate precisely to others by using clear definitions in discussion with others and in their
own reasoning. They state the meaning of the symbols they choose, specifying units of measure, and labeling axes to
clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical
answers with a degree of precision appropriate for the problem context. By the time they reach high school they have
learned to examine claims and make explicit use of definitions.
By high school, students look closely to discern a pattern or structure. In the expression x2 + 9x + 14, older students can see
the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the
strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective.
They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several
objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its
value cannot be more than 5 for any real numbers x and y. High school students use these patterns to create equivalent
expressions, factor and solve equations, and compose functions, and transform figures.
High school students notice if calculations are repeated, and look both for general methods and for shortcuts. Noticing the
regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might
lead them to the general formula for the sum of a geometric series. As they work to solve a problem, derive formulas or
make generalizations, high school students maintain oversight of the process, while attending to the details. They
continually evaluate the reasonableness of their intermediate results.
Page 5
Montana Curriculum Organizer: Statistics and Probability Mathematics
This document is a curriculum organizer adapted from other states to be used for planning scope and sequence, units, pacing and other materials that support a focused,
coherent, and rigorous study of mathematics K-12.
November 2012
Page 6
Montana Curriculum Organizer: Statistics and Probability Mathematics
This document is a curriculum organizer adapted from other states to be used for planning scope and sequence, units, pacing and other materials that support a focused,
coherent, and rigorous study of mathematics K-12.
LEARNING PROGRESSION
Statistics and Probability: Interpreting Categorical and
Quantitative Data
STANDARDS IN LEARNING PROGRESSION
9-12.S.ID.1, 9-12.S.ID.2, 9-12.S.ID.3, 9-12.S.ID.4, 912.S.ID.5, 9-12.S.ID.6a-c, 9-12.S.ID.7, 9-12.S.ID.8, 912.S.ID.9
UNDERSTAND:
Data are gathered, displayed, summarized, examined, and interpreted to discover patterns and deviations from patterns.
Which statistics to compare, which plots to use, and what the results of a comparison might mean, depend on the
question to be investigated and the real-life actions to be taken.
When making statistical models, technology is valuable for varying assumptions, exploring consequences and comparing
predictions with data.
Causation implies correlation yet correlation does not imply causation.
KNOW:
DO:
Quantitative data can be described in terms of
Summarize, represent, and interpret data on a single count or
key characteristics: measures of shape, center, measurement variable.
9-12.S.ID.1 Represent data with plots on the real-number line (dot
and spread.
plots, histograms, and box plots).
The shape of a data distribution might be
9-12.S.ID.2 Use statistics appropriate to the shape of the data
described as symmetric, skewed, flat, or bell
distribution to compare center (median, mean) and spread
shaped, and it might be summarized by a
(interquartile range, standard deviation) of two or more different
statistic measuring center (such as mean or
data sets.
median) and a statistic measuring spread (such 9-12.S.ID.3 Interpret differences in shape, center, and spread in the
as standard deviation or interquartile range).
context of the data sets, accounting for possible effects of
extreme data points (outliers).
Strategies for fitting a function to a data display 9-12.S.ID.4 Use the mean and standard deviation of a data set to fit it
and informally assessing the fit.
to a normal distribution and to estimate population percentages.
Recognize that there are data sets for which such a procedure is
What the slope and intercept of the linear
not appropriate. Use calculators, spreadsheets, and tables, and
model mean within the context of the data.
Montana American Indian data sources to estimate areas under
the normal curve.
Summarize, represent, and interpret data on two categorical and
quantitative variables.
9-12.S.ID.5 Summarize categorical data for two categories in twoway frequency tables. Interpret relative frequencies in the context
of the data (including joint, marginal, and conditional relative
frequencies). Recognize possible associations and trends in the
data.
9-12.S.ID.6 Represent data on two quantitative variables on a scatter
plot, and describe how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve
problems in the context of the data. Use given functions or
choose a function suggested by the context. Emphasize
linear, quadratic, and exponential models.
b. Informally assess the fit of a function by plotting and
analyzing residuals.
c. Fit a linear function for a scatter plot that suggests a linear
association.
Interpret linear models.
9-12.S.ID.7 Interpret the slope (rate of change) and the intercept
(constant term) of a linear model in the context of the data.
9-12.S.ID.8 Compute (using technology) and interpret the correlation
coefficient of a linear fit.
9-12.S.ID.9 Distinguish between correlation and causation.
November 2012
Page 7
Montana Curriculum Organizer: Statistics and Probability Mathematics
This document is a curriculum organizer adapted from other states to be used for planning scope and sequence, units, pacing and other materials that support a focused,
coherent, and rigorous study of mathematics K-12.
KEY TERMS FOR THIS PROGRESSION:
Dot plot, Histogram, Box plot, Scatter plot, Inter-quartile range, Standard deviation, Measure of center, Outliers,
Normal distribution, Skewed distribution, Correlation coefficient, Two-way frequency table
INSTRUCTIONAL STRATEGIES AND RESOURCES:
Cluster: Summarize, represent, and interpret data on a single count or measurement variable.
It is helpful for students to understand that a statistical process is a problem-solving process consisting of four steps:
1) formulating a question that can be answered by data;
2) designing and implementing a plan that collects appropriate data;
3) analyzing the data by graphical and/or numerical methods; and
4) interpreting the analysis in the context of the original question.
Opportunities should be provided for students to work through the statistical process. In Grades 6-8, learning has
focused on parts of this process. Now is a good time to investigate a problem of interest to the students and follow it
through. The richer the question formulated, the more interesting is the process. Teachers and students should make
extensive use of resources to perfect this very important first step. Global web resources can inspire projects.
Although this domain addresses both categorical and quantitative data, there is no reference in the Standards 1-4 to
categorical data. Note that Standard 5 in the next cluster, summarize, represent, and interpret data on two categorical
and quantitative variables, addresses analysis for two categorical variables on the same subject. To prepare for
interpreting two categorical variables in Standard 5, this would be a good place to discuss graphs for one categorical
variable (bar graph, pie graph) and measure of center (mode).
Have students practice their understanding of the different types of graphs for categorical and numerical variables by
constructing statistical posters. Note that a bar graph for categorical data may have frequency on the vertical (student’s
pizza preferences) or measurement on the vertical (radish root growth over time - days).
Measures of center and spread for data sets without outliers are the mean and standard deviation, whereas median and
interquartile range are better measures for data sets with outliers.
Introduce the formula of standard deviation by reviewing the previously learned MAD (mean absolute deviation). The
MAD is very intuitive and gives a solid foundation for developing the more complicated standard deviation measure.
Informally observing the extent to which two boxplots or two dot plots overlap begins the discussion of drawing inferential
conclusions. Don’t shortcut this observation in comparing two data sets.
As histograms for various data sets are drawn, common shapes appear. To characterize the shapes, curves are
sketched through the midpoints of the tops of the histogram’s rectangles. Of particular importance is a symmetric
unimodal curve that has specific areas within one, two, and three standard deviations of its mean. It is called the Normal
distribution and students need to be able to find areas (probabilities) for various events using tables or a graphing
calculator.
Instructional Resources/Tools
TI-83/84 and TI emulator
Quantitative Literacy Exploring Data module
National Council of Teachers of Mathematics. Navigating through Data Analysis 9-12.
Printed media (e.g., almanacs, newspapers, professional reports)
Software such as TinkerPlots and Excel
Show World: This website offers data about the world that is up to date.
iearn: This website offers projects that students around the world are working on simultaneously.
Cluster: Summarize, represent, and interpret data on two categorical and quantitative variables.
In this cluster, the focus is that two categorical or two quantitative variables are being measured on the same subject.
In the categorical case, begin with two categories for each variable and represent them in a two-way table with the two
values of one variable defining the rows and the two values of the other variable defining the columns. Extending the
number of rows and columns is easily done once students become comfortable with the 2 x 2 case. The table entries are
November 2012
Page 8
Montana Curriculum Organizer: Statistics and Probability Mathematics
This document is a curriculum organizer adapted from other states to be used for planning scope and sequence, units, pacing and other materials that support a focused,
coherent, and rigorous study of mathematics K-12.
the joint frequencies of how many subjects displayed the respective cross-classified values. Row totals and column totals
constitute the marginal frequencies. Dividing joint or marginal frequencies by the total number of subjects define relative
frequencies (and percentages), respectively. Conditional relative frequencies are determined by focusing on a specific
row or column of the table. They are particularly useful in determining any associations between the two variables.
In the numerical or quantitative case, display the paired data in a scatterplot. Note that although the two variables in
general will not have the same scale, e.g., total SAT versus grade-point average, it is best to begin with variables with
the same scale such as SAT Verbal and SAT Math. Fitting functions to such data will avoid difficulties such as
interpretation of slope in the linear case in which scales differ. Once students are comfortable with the same-scale case,
introducing different scales situations will be less problematic.
Instructional Resources/Tools
TI-83/84 and TI emulator
Quantitative Literacy Exploring Data module
National Council of Teachers of Mathematics. Navigating through Data Analysis 9-12.
Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report
Software such as TinkerPlots and Excel
Cluster: Interpret linear models.
In this cluster, the key is that two quantitative variables are being measured on the same subject. The paired data should
be listed and then displayed in a scatterplot. If time is one of the variables, it usually goes on the horizontal axis. That
which is being predicted goes on the vertical; the predictor variable is on the horizontal axis.
Note that unlike a two-dimensional graph in mathematics, the scales of a scatterplot need not be the same, and even if
they are similar (such as SAT Math and SAT Verbal), they still need not have the same spacing. So, visual rendering of
slope makes no sense in most scatterplots, i.e., a 45 degree line on a scatterplot need not mean a slope of 1.
Often the interpretation of the intercept (constant term) is not meaningful in the context of the data. For example, this is
the case when the zero point on the horizontal is of considerable distance from the values of the horizontal variable, or in
some case has no meaning such as for SAT variables.
To make some sense of Pearson’s correlation coefficient, students should recall their middle school experience with the
Quadrant Count Ratio (QCR) as a measure of relationship between two quantitative variables.
Noting that a correlated relationship between two quantitative variables is not causal (unless the variables are in an
experiment) is a very important topic and a substantial amount of time should be spent on it.
Instructional Resources/Tools
TI-83/84 and TI emulator
Quantitative Literacy Exploring Data module
NCTM Navigating through Data Analysis 9-12.
Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report
Software such as TinkerPlots and Excel
The Ohio Resource Center
The National Council of Teachers of Mathematics, Illuminations
STANDARDS FOR MATHEMATICAL PRACTICE:
1. Make sense of problems and persevere in solving them.
5. Use appropriate tools strategically.
2. Reason abstractly and quantitatively.
6. Attend to precision.
3. Construct viable arguments and critique the reasoning of
7. Look for and make use of structure.
8. Look for and express regularity in repeated
others.
4. Model with mathematics.
reasoning.
November 2012
Page 9
Montana Curriculum Organizer: Statistics and Probability Mathematics
This document is a curriculum organizer adapted from other states to be used for planning scope and sequence, units, pacing and other materials that support a focused,
coherent, and rigorous study of mathematics K-12.
November 2012
Page 10
Montana Curriculum Organizer: Statistics and Probability Mathematics
This document is a curriculum organizer adapted from other states to be used for planning scope and sequence, units, pacing and other materials that support a focused,
coherent, and rigorous study of mathematics K-12.
LEARNING PROGRESSION
STANDARDS IN LEARNING PROGRESSION
Statistics and Probability: Making Inferences and Justifying 9-12.S.IC.1, 9-12.S.IC.2, 9-12.S.IC.3, 9-12.S.IC.4, 9Conclusions
12.S.IC.5, 9-12.S.IC.6
UNDERSTAND:
The conditions under which data are collected are important in drawing conclusions from the data; in critically reviewing
uses of statistics in public media and other reports, it is important to consider the study design, how the data were
gathered, and the analyses employed as well as the data summaries and the conclusions drawn.
Collecting data from a random sample of a population makes it possible to draw valid conclusions about the whole
population, taking variability into account.
Randomly assigning individuals to different treatments allows a fair comparison of the effectiveness of those treatments.
KNOW:
DO:
Recognize the purposes of and differences
Understand and evaluate random processes underlying statistical
among sample surveys, experiments, and
experiments.
9-12.S.IC.1 Understand statistics as a process for making inferences
observational studies.
about population parameters based on a random sample from
that population.
9-12.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?
Make inferences and justify conclusions from sample surveys,
experiments, and observational studies.
9-12.S.IC.3 Recognize the purposes of and differences among sample
surveys, experiments, and observational studies; explain how
randomization relates to each.
9-12.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.
9-12.S.IC.5 Use data from a randomized experiment to compare two
treatments; use simulations to decide if differences between
parameters are significant.
9-12.S.IC.6 Evaluate reports based on data.
KEY TERMS FOR THIS PROGRESSION:
Randomization, Sample survey, Experiment, Observational study, Inferences, Population parameters,
Simulation, Population mean, Population proportion.
INSTRUCTIONAL STRATEGIES AND RESOURCES:
Cluster: Understand and evaluate random processes underlying statistical experiments.
Inferential statistics based on Normal-probability models is a topic for Advanced Placement Statistics (e.g., T-tests). The
idea here is that all students understand that statistical decisions are made about populations (parameters in particular)
based on a random sample taken from the population and the observed value of a sample statistic (note that both words
start with the letter “s”). A population parameter (note that both words start with the letter “p”) is a measure of some
characteristic in the population such as the population proportion of American voters who are in favor of some issue, or
the population mean time it takes an Alka Seltzer tablet to dissolve.
As students are mastering the statistical process, it is instructive for them to investigate questions such as “If a coin spun
five times produces five tails in a row, could one conclude that the coin is biased toward tails?” One way a student might
answer this is by building a model of 100 trials by experimentation or simulation of the number of times a truly fair coin
produces five tails in a row in five spins. If a truly fair coin produces five tails in five tosses 15 times out of 100 trials, then
there is no reason to doubt the fairness of the coin. If, however, getting five tails in five spins occurred only once in 100
trials, then one could conclude that the coin is biased toward tails (if the coin in question actually landed five tails in five
spins).
A powerful tool for developing statistical models is the use of simulations. This allows the students to visualize the model
and apply their understanding of the statistical process.
November 2012
Page 11
Montana Curriculum Organizer: Statistics and Probability Mathematics
This document is a curriculum organizer adapted from other states to be used for planning scope and sequence, units, pacing and other materials that support a focused,
coherent, and rigorous study of mathematics K-12.
Provide opportunities for students to clearly distinguish between a population parameter which is a constant, and a
sample statistic which is a variable.
Instructional Resources/Tools
TI-83/84 and TI emulator
Quantitative Literacy The Art and Techniques of Simulation module
Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report
Software such as TinkerPlots and Fathom
Cluster: Make inferences and justify conclusions from sample surveys, experiments, and observational studies.
This cluster is designed to bring the four-step statistical process (GAISE model) to life and help students understand how
statistical decisions are made. The mastery of this cluster is fundamental to the goal of creating a statistically literate
citizenry. Students will need to use all of the data analysis, statistics, and probability concepts covered to date to develop
a deeper understanding of inferential reasoning.
Students learn to devise plans for collecting data through the three primary methods of data production: surveys,
observational studies, and experiments. Randomization plays various key roles in these methods. Emphasize that
randomization is not a haphazard procedure, and that it requires careful implementation to avoid biasing the analysis. In
surveys, the sample selected from a population needs to be representative; taking a random sample is generally what is
done to satisfy this requirement. In observational studies, the sample needs to be representative of the population as a
whole to enable generalization from sample to population. The best way to satisfy this is to use random selection in
choosing the sample. In comparative experiments between two groups, random assignment of the treatments to the
subjects is essential to avoid damaging problems when separating the effects of the treatments from the effects of some
other variable, called confounding. In many cases, it takes a lot of thought to be sure that the method of randomization
correctly produces data that will reflect that which is being analyzed. For example, in a two-treatment randomized
experiment in which it is desired to have the same number of subjects in each treatment group, having each subject toss
a coin where heads assigns the subject to treatment A and tails assigned the subject to treatment B will not produce the
desired random assignment of equal-size groups.
The advantage that experiments have over surveys and observational studies is that one can establish causality with
experiments.
Standard 4 addresses estimation of the population proportion parameter and the population mean parameter. Data need
not come from just a survey to cover this topic. A margin-of-error formula cannot be developed through simulation, but
students can discover that as the sample size is increased, the empirical distribution of the sample proportion and the
sample mean tend toward a certain shape (the Normal distribution), and the standard error of the statistics decreases
(i.e., the variation) in the models becomes smaller. The actual formulas will need to be stated.
Standard 5 addresses testing whether some characteristic of two paired or independent groups is the same or different
by the use of resampling techniques. Conclusions are based on the concept of p-value. Resampling procedures can
begin by hand but typically will require technology to gather enough observations for which a p-value calculation will be
meaningful.
Use a variety of devices as appropriate to carry out simulations: number cubes, cards, random digit tables, graphing
calculators, computer programs.
Instructional Resources/Tools
TI-83/84 and TI emulator
Quantitative Literacy The Art and Techniques of Simulation module
Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report
Making Sense of Statistical Studies
Focus in High School mathematics: Statistics and Probability
Navigating through Data Analysis in Grades 9-12
“A Sequence of Activities for Developing Statistical Concepts,” Franklin and Kader, The Statistics Teacher Network
Number 68 Winder 2006.
Software such as TinkerPlots and Fathom
November 2012
Page 12
Montana Curriculum Organizer: Statistics and Probability Mathematics
This document is a curriculum organizer adapted from other states to be used for planning scope and sequence, units, pacing and other materials that support a focused,
coherent, and rigorous study of mathematics K-12.
STANDARDS FOR MATHEMATICAL PRACTICE:
1. Make sense of problems and persevere in solving them.
5. Use appropriate tools strategically.
2. Reason abstractly and quantitatively.
6. Attend to precision.
3. Construct viable arguments and critique the reasoning of
7. Look for and make use of structure.
8. Look for and express regularity in repeated
others.
4. Model with mathematics.
reasoning.
November 2012
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Montana Curriculum Organizer: Statistics and Probability Mathematics
This document is a curriculum organizer adapted from other states to be used for planning scope and sequence, units, pacing and other materials that support a focused,
coherent, and rigorous study of mathematics K-12.
November 2012
Page 14
Montana Curriculum Organizer: Statistics and Probability Mathematics
This document is a curriculum organizer adapted from other states to be used for planning scope and sequence, units, pacing and other materials that support a focused,
coherent, and rigorous study of mathematics K-12.
LEARNING PROGRESSION
STANDARDS IN LEARNING PROGRESSION
Statistics and Probability: Conditional Probability and the
9-12.S.CP.1, 9-12.S.CP.2, 9-12.S.CP.3, 9-12.S.CP.4, 9Rules of Probability
12.S.CP.5, 9-12.S.CP.6, 9-12.S.CP.7
UNDERSTAND:
In a probability model, sample points represent outcomes and combine to make up events.
The probabilities of the events can be computed by applying the Addition and Multiplication Rules.
Interpreting these probabilities relies on an understanding of independence and conditional probability, which can be
approached through the analysis of two-way tables.
KNOW:
DO:
Understand that two events A and B are
Understand independence and conditional probability and use
independent if the probability of A and B
them to interpret data.
9-12.S.CP.1 Describe events as subsets of a sample space (the set
occurring together is the product of their
of outcomes) using characteristics (or categories) of the
probabilities.
outcomes, or as unions, intersections, or complements of other
Understand the conditional probability of A
events ("or," "and," "not").
given B as P(A and B)/P(B), and interpret
9-12.S.CP.2 Understand that two events A and B are independent if
independence of A and B as saying that the
the probability of A and B occurring together is the product of their
conditional probability of A given B is the same
probabilities, and use this characterization to determine if they are
as the probability of A, and the conditional
independent.
probability of B given A is the same as the
9-12.S.CP.3 Understand the conditional probability of A given B as
probability of B.
P(A and B)/P(B), and interpret independence of A and B as saying
that the conditional probability of A given B is the same as the
Construct a two-way table as a sample space.
probability of A, and the conditional probability of B given A is the
same as the probability of B.
Recognize the concepts of conditional
9-12.S.CP.4
Construct and interpret two-way frequency tables of data
probability and independence in everyday
including
information from Montana American Indian data sources
language and everyday situations.
when two categories are associated with each object being
classified. Use the two-way table as a sample space to decide if
The characteristics of any context used to
events are independent and to approximate conditional
determine when it is appropriate to apply the
probabilities. For example, collect data from a random sample of
Addition and Multiplication Rules.
students in your school on their favorite subject among math,
science, and English. Estimate the probability that a randomly
selected student from your school will favor science given that the
student is in tenth grade. Do the same for other subjects and
compare the results.
9-12.S.CP.5 Recognize and explain the concepts of conditional
probability and independence in everyday language and
everyday situations. For example, compare the chance of having
lung cancer if you are a smoker with the chance of being a
smoker if you have lung cancer.
Use the rules of probability to compute probabilities of
compound events in a uniform probability model.
9-12.S.CP.6 Find the conditional probability of A given B as the
fraction of B's outcomes that also belong to A, and interpret the
answer in terms of the model.
9-12.S.CP.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A
and B), and interpret the answer in terms of the model.
KEY TERMS FOR THIS PROGRESSION:
Conditional probability, Unions, Intersections, Complements, Independence, Sample space
INSTRUCTIONAL STRATEGIES AND RESOURCES:
November 2012
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Montana Curriculum Organizer: Statistics and Probability Mathematics
This document is a curriculum organizer adapted from other states to be used for planning scope and sequence, units, pacing and other materials that support a focused,
coherent, and rigorous study of mathematics K-12.
Cluster: Understand independence and conditional probability and use them to interpret data.
The Standard for Mathematical Practice, precision is important for working with conditional probability. Attention to the
definition of an event along with the writing and use of probability function notation are important requisites for
communication of that precision. For example: Let A: Female and B: Survivor, then P(A|B) =. The use of a vertical line for
the conditional “given” is not intuitive for students and they often confuse the events B|A and A|B. Moreover, they often
find identifying a conditional difficult when the problem is expressed in words in which the word “given” is omitted. For
example, find the probability that a female is a survivor. The standard, Make sense of problems and persevere in solving
them, also should be employed so students can look for ways to construct conditional probability by formulating their own
questions and working through them such as is suggested in Standard 4 above. Students should learn to employ the use
of Venn diagrams as a means of finding an entry into a solution to a conditional probability problem.
It will take a lot of practice to master the vocabulary of “or,” “and,” “not” with the mathematical notation of union ( ∪ ),
intersection ( ∩ ), and whatever notation is used for complement.
The independence of two events is defined in Standard 2 using the intersection. It is far more intuitive to introduce the
independence of two events in terms of conditional probability (stated in Standard 3), especially where calculations can
be performed in two-way tables.
The Standards in this cluster deliberately do not mention the use of tree diagrams, the traditional way to treat conditional
probabilities. Instead, probabilities of conditional events are to be found using a two-way table wherever possible.
However, tree diagrams may be a helpful tool for some students. The difficulty is realizing that the second sets of
branches are conditional probabilities.
Instructional Resources/Tools
The Titanic Problem is a well-known problem that revolves around conditional probability.
Focus in High School Mathematics: Reasoning and Sense Making in Statistics and Probability, NCTM.
Navigating through Probability in Grades 9-12, NCTM.
Cluster: Use the rules of probability to compute probabilities of compound events in a uniform probability
model.
Identifying that a probability is conditional when the word “given” is not stated can be very difficult for students. For
example, if a balanced tetrahedron with faces 1, 2, 3, 4 is rolled twice, what is the probability that the sum is prime (A) of
those that show a 3 on at least one roll (B)? Whether what is asked for is P(A and B), P(A or B), or P(A|B) can be
problematic for students. Showing the outcomes in a Venn Diagram may be useful. The calculation to find the probability
that the sum is prime (A) given at least one roll shows 3 (B) is to count the elements of B by listing them if possible,
namely in this example, there are 7 paired outcomes (31, 32, 33, 34, 13, 23, 43). Of those 7 there are 4 whose sum is
prime (32, 34, 23, 43). Hence in the long run, 4 out of 7 times of rolling a fair tetrahedron twice, the sum of the two rolls
will be a prime number under the condition that at least one of its rolls shows the digit 3.
Note that if listing outcomes is not possible, then counting the outcomes may require a computation technique involving
permutations or combinations, which is a STEM topic.
In the above example, if the question asked were what is the probability that the sum of two rolls of a fair tetrahedron is
prime (A) or at least one of the rolls is a 3 (B), then what is being asked for is P(A or B) which is denoted as P(AB) in set
notation. Again, it is often useful to appeal to a Venn Diagram in which A consists of the pairs: 11, 12, 14, 21, 23, 32, 34,
41, 43; and B consists of 13, 23, 33, 43, 31, 32, 34. Adding P(A) and P(B) is a problem as there are duplicates in the two
th
events, namely 23, 32, 34, and 43. So P(A or B) is 9/16 + 7/16 – 4/16 = 12/16 or 3/4, so 3/4 of the time, the result of
rolling a fair tetrahedron twice will result in the sum being prime, or at least one of the rolls showing a 3, or perhaps both
will occur. 
It should be noted that the Multiplication Rule in Standard 8 is designated as STEM when it is connected to the
discussion of independence in Standard 2 of the previous S-CP cluster. The formula P(A and B) = P(A)P(B|A) is best
illustrated in a two-stage setting in which A denotes the outcome of the first stage, and B, the second. For example,
suppose a jar contains 7 red and 3 green chips. If one draws two chips without replacement from the jar, the probability
of getting a red followed by a green is P(red on first and green on second) = P(red on first)P(green on second given a
red on first) = (7/10)(3/9) = 21/90. Demonstrated on a tree diagram indicates that the conditional probabilities are on the
second set of branches.
November 2012
Page 16
Montana Curriculum Organizer: Statistics and Probability Mathematics
This document is a curriculum organizer adapted from other states to be used for planning scope and sequence, units, pacing and other materials that support a focused,
coherent, and rigorous study of mathematics K-12.
Instructional Resources/Tools
NCTM Navigating through Probability 9-12.
The National Council of Teachers of Mathematics, Illuminations
TI-83/84 and TI emulator (for permutations and combinations calculations)
STANDARDS FOR MATHEMATICAL PRACTICE:
1. Make sense of problems and persevere in solving them.
5. Use appropriate tools strategically.
2. Reason abstractly and quantitatively.
6. Attend to precision.
3. Construct viable arguments and critique the reasoning of
7. Look for and make use of structure.
others.
8. Look for and express regularity in repeated
4. Model with mathematics.
reasoning.
November 2012
Page 17
Montana Curriculum Organizer: Statistics and Probability Mathematics
This document is a curriculum organizer adapted from other states to be used for planning scope and sequence, units, pacing and other materials that support a focused,
coherent, and rigorous study of mathematics K-12.
November 2012
Page 18
Montana Curriculum Organizer: Statistics and Probability Mathematics
This document is a curriculum organizer adapted from other states to be used for planning scope and sequence, units, pacing and other materials that support a focused,
coherent, and rigorous study of mathematics K-12.
This Curriculum Organizer was created using the following materials:
ARIZONA - STANDARDS FOR MATHEMATICAL PRACTICE EXPLANATIONS AND EXAMPLES
http://www.azed.gov/standards-practices/mathematics-standards/
DELAWARE – LEARNING PROGRESSIONS
http://www.doe.k12.de.us/infosuites/staff/ci/content_areas/math.shtml
OHIO – INSTRUCTIONAL STRATEGIES AND RESOURCES (FROM MODEL CURRICULUM)
http://education.ohio.gov/GD/Templates/Pages/ODE/ODEDetail.aspx?Page=3&TopicRelationID=1704&Content=134773
November 2012
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