Download ALGEBRA 1 SUMMER INSTITUTE 2014 OVERVIEW Overarching

ALGEBRA 1 SUMMER INSTITUTE 2014 OVERVIEW
Overarching Goals:
1. Provide teacher participants with details on how the algebra 1 curriculum has been
reorganized according to the CCSS as it pertains to the development of statistical
concepts and elementary notions of probability.
2. Improve participants’ knowledge of the application of the CCSS mathematical
practices in the classroom.
3. Improve algebra 1 content-knowledge and model effective instructional strategies
that promote student learning in mathematics.
4. Building capacity for sustainable lesson study teams to provide ongoing support
for increasing the rigor of algebra 1 curriculum according to CCSS.
Main Topic and Description: Statistical methods explain complex data sets through
simple numeric and graphical descriptions and the use of basic mathematical models.
This institute uses statistics and probability to describe natural variability inherent in our
surroundings.
Target Audience: Current and future middle and high school teachers of Algebra 1. In
addition, the institute will benefit district level math coaches and math supervisors.
Institute Structure:
Pre-program (online) WORKSHOP: Introduction to the use of software
UNIT 1: Data and Variability
UNIT 2: Graphical Descriptions of Data
UNIT 3: Numerical Descriptions of Data
UNIT 4: Bivariate Data
UNIT 5: Estimating Probabilities
UNIT 6: Sampling and Inference
UNIT 7: Modeling Data
Big Ideas:
1. The process of using statistics
2. Summarize and represent data: concepts and technology
3. The determination and meaning of common statistical measures, e.g., mean,
median, interquartile range.
4. The necessity and techniques of random sampling.
Unit (1)
Data and Variability:
Data collected from observations and experiments is inherently variable. Conclusions
drawn from data must take this into account. This unit introduces general categories of
data and explains simple methods and terminology needed to describe data.
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ALGEBRA 1 SUMMER INSTITUTE 2014 OVERVIEW
Essential Questions:
 What is statistics?
 What is a statistical question?
 What are some possible errors made while doing measurements?
 How to recognize bias in the wording of survey questions?
 What is simple random sample?
 Could bias occur in the sampling process?
 Does the sample size have an effect on statistical information?
 What are other methods of sampling?
Associated Mathematics Florida Standards
MAFS.6.SP.1: Develop understanding of statistical variability
1.1: Recognize that a statistics question as one that anticipates variability in the data
related to the question and accounts for it in the answers.
1.2: Understand that a set of data collected to answer a statistical question has a
distribution which can be described by its center, spread, and overall shape.
MAFS.6.SP.2: Summarize and describe distributions
2.5: Summarize numerical data sets in relation to their context, such as by:
a. Reporting the number of observations
b. Describing the nature of the attribute under investigation, including how it was
measured and its units of measurement.
MAFS.912.S-ID.1: Summarize, represent, and interpret data on a single count or
measurement variable
1.3: Interpret differences in shape, center, and spread in the context of the data sets,
accounting for possible effects of extreme points (outliers).
Learning Objectives:
Participants will:
1. Distinguish statistical questions from non-statistical ones
2. Identify population
3. Identify data to be collected
4. Develop intuitive understanding of the expected variation in data
5. Distinguish possible errors in measurement
6. Identify possible bias in survey questions
7. Understand what is meant by simple random sampling
8. Recognize the variability in subjective and random sampling techniques and
understand how to measure it
9. Recognize how bias can occur in the sampling process
10. Draw conclusions about the effect of sample size on statistical information
11. Explore the sampling distributions of stratified random samples and clustered
samples
Activities in Unit (1)
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ALGEBRA 1 SUMMER INSTITUTE 2014 OVERVIEW
Name
The Question
Formulation
Data measurement
and variation
Data Measurement
and Variation
Uncertainty
The Survey
Predilection
The Sampling
Proclivity
Description
This investigation provides a framework for constructing questions
that can be addressed through the collection and analysis of data.
Participants will be encouraged to think about the population to be
studied, the variable to be measured, and the variation that may
occur in the measurement of that characteristic. Participants will
practice with statistical questions at levels A, B, and C according to
the GAISE report.
Participants will make measurements using two different
measurement devices. Variation, or differences in measured data,
occurs for a number of reasons. Participants will conjecture on the
possible reasons for variation
Participants will make measurements using different measurement
devices. Variation, or differences in measured data, occurs for a
number of reasons. Participants will conjecture on the possible
reasons for variation
Participants will discuss the need to be aware of the various sources
and causes of measurement error in surveys and the possible bias
implicated in the formulation of questions
How we select a sample is extremely important. Improper or biased
sample selection can produce misleading conclusions. Sample
selection is biased if it systematically favors certain outcomes.
Unit (2)
Graphical Description of Data:
The participants will learn ways to graphically represent and summarize data.
Essential Questions:
1. How can data be represented graphically?
2. What are the differences and similarities between boxplots, histograms, and
dotplots?
3. When is more appropriate to use each one of them?
Associated State Standards:
MAFS.6.SP.1: Develop understanding of statistical variability
1.2: Understand that a set of data collected to answer a statistical question has a
distribution which can be described by its center, spread, and overall shape.
MAFS.6.SP.2: Summarize and describe distributions
2.4: Display numerical data in plots on a number line, including dot plots, histograms,
and box plots.
2.5: Summarize numerical data sets in relation to their context, such as by:
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ALGEBRA 1 SUMMER INSTITUTE 2014 OVERVIEW
a. Reporting the number of observations
c. Giving quantitative measures of center (median and/or mean) and variability
(interquartile range and/or mean absolute deviation), as well as describing any
overall pattern and any striking deviations from the overall pattern with
reference to the context in which the data were gathered.
MAFS.912.S-ID.1: Summarize, represent, and interpret data on a single count or
measurement variable
1.1: Represent data with plots on the real number line (dot plots, histograms, and box
plots).
1.3: Interpret differences in shape, center, and spread in the context of the data sets,
accounting for possible effects of extreme points (outliers).
Learning Objectives:
Participants will:
1. Create histograms from data and interpret them in terms of distribution shape,
center and spread.
2. Create dotplots from data and interpret them in terms of distribution shape, center
and spread.
3. Compare bar graphs, frequency tables, histograms, and dotplots for a single set of
data.
4. Organize data in frequency tables and frequency cumulative tables.
5. Determine relative frequencies and use it to create bar graphs.
6. Make conjectures and predictions based on the data
7. Justify their reasoning and decisions
Activities in Unit (2)
Name
The Categorical
Predilection
The Grapevine
Fabrication
The Minute
Paradigm
The Old Faithful
Geyser
Prognostication
Rationale
Graphs and tables are useful for displaying distributions of
categorical data. Ideas of center and variability will be discussed in
the context of the example.
Participants will explore different graphical and tabular
representations and their usefulness in helping analyze discrete data
Participants will investigate some approaches to grouping data in
graphs and tables, and you will examine different types of statistical
answers to questions based on these grouped representations.
Participants encounter samples of measurement data that occurred
in sequence over time. Participants construct their own visual
displays of the data and then make some estimates and projections
based on the data.. The importance of attending to variability in the
data and the need to track variation in samples of the data over time
are major themes in the tasks in this activity.
Unit (3):
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ALGEBRA 1 SUMMER INSTITUTE 2014 OVERVIEW
Numerical Descriptions of Data:
Data sets can be summarized by condensing the information down into just a few numeric
statistics. The participants will learn ways to numerically represent and summarize data.
Essential Questions:
1. How are the central tendencies of mean and median alike? When is more
appropriate to use each one?
2. How are the interquartile ranges useful when describing data sets?
3. What is a normal distribution?
4. What is the probability that a random variable fall in a given interval in a normal
distribution?
Associated State Standards:
MAFS.6.SP.1: Develop understanding of statistical variability
1.2: Understand that a set of data collected to answer a statistical question has a
distribution which can be described by its center, spread, and overall shape.
1.3: Recognize that a measure of center for a numerical set of data summarizes all of
its values with a single number, while a measure of variation describe how its
values vary with a single number.
MAFS.6.SP.2: Summarize and describe distributions
2.5: Summarize numerical data sets in relation to their context, such as by:
a. Reporting the number of observations
b. Describing the nature of the attribute under investigation, including how it was
measured and its units of measurement.
c. Giving quantitative measures of center (median and/or mean) and variability
(interquartile range and/or mean absolute deviation), as well as describing any
overall pattern and any striking deviations from the overall pattern with
reference to the context in which the data were gathered.
MAFS.912.S-ID.1: Summarize, represent, and interpret data on a single count or
measurement variable
1.2: Use statistics appropriate to the shape of the data distribution to compare center
(median, mean) and spread (interquartile range, standard deviation) of two or more
data sets.
1.3: Interpret differences in shape, center, and spread in the context of the data sets,
accounting for possible effects of extreme points (outliers).
Learning Objectives:
Participants will:
1. Compute and compare the central tendency measures of mean and median on data
sets.
2. Compute and compare interquartile ranges and mean / median absolute deviations
on data sets.
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ALGEBRA 1 SUMMER INSTITUTE 2014 OVERVIEW
3. Understand the median as the value that divides the ordered data into two groups,
with the same number of data values (approximately one-half) in each group
4. Understand quartiles as the values that divide the ordered data into four groups,
with the same number of data values (approximately one-fourth) in each group
5. Investigate the concept of quartiles using several different representations
6. Summarize an entire data set with a Five-Number Summary
7. Summarize an entire data set with a box plot
8. Understand the sample mean as an indicator of fair allocation
9. Explore deviations of data values from the sample mean
10. Understand the mean as the "balancing point" of a data set
11. Learn how to measure variation about the mean
12. Understand the effect of outliers in the mean and median
13. Understand when the values of mean and median are more appropriate
14. Demonstrate skill in constructing absolute and relative-frequency graphs,
histograms, and boxplots
15. Identify similarities and differences between the types of graphs and use them to
make decisions
16. Compare data sets with unequal number of elements
17. Investigate comparative studies, which can be experimental or observational
18. Learn how to analyze and interpret results from comparative studies
19. Learn how to design a comparative experiment
20. Explore paired and unpaired comparative studies
Activities in Unit (3)
Name
Teach Statistics
Before Calculus
The Noodle
conundrum
The Fair
Allocation
Paradigm
The Mean and
Median
Fluctuation
The Migraine
Reaction
Rationale
Video. Someone always asks the math teacher, "Am I going to use
calculus in real life?" And for most of us, says Arthur Benjamin, the
answer is no. He offers a bold proposal on how to make math
education relevant in the digital age.
When working with a large collection of data, it can be difficult to
keep an accurate picture of your data in mind. One way to make it
easier to work with large data sets is to reduce the entire data set to
just a few summary measures (or numbers that describe significant
characteristics of the data). In this activity, participants will learn how
to determine the 5-number summary and its graphical representation.
Participants will explore and interpret the mean and learn several
ways to describe the degree of variation in data, based in how much
the data values vary from the mean
Using interactive software, participants can compare and contrast
properties of measures of central tendency, specifically the influence
of changes in data values on the mean and median. As participants
change the data values, the interactive figure immediately displays the
mean and median of the new data set.
This activity focuses on ways to compare data sets, from a
comparative study, when the numbers of elements in the data sets are
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ALGEBRA 1 SUMMER INSTITUTE 2014 OVERVIEW
The Words
Memorization
Saturation
not the same (unequal Ns). When comparing some data sets with
different Ns relative frequencies and boxplots are more useful than
absolute frequencies.
This investigation focuses on the pattern of chance variation that
arises from randomly assigning subjects to treatment groups. After
considering the amount of chance variation, participants can probe the
experimental results for evidence of a genuine signal – a treatment
effect that could not have plausibly arisen from the random
assignment process alone
Unit (4)
Bivariate Data:
The participants will use graphical and numerical statistical methods to understand the
relationship between two sets of data.
Essential Questions:
1.
2.
3.
4.
5.
6.
7.
8.
How can distributions from two different sample sets be compared numerically?
How can distributions from two different sample sets be compared graphically?
How can bivariate data be represented in a frequency table?
What does it mean for two data sets to be correlated?
How can the correlation of two data sets be determined numerically?
How can the correlation of two data sets be estimated from a scatterplot?
What is causation?
What is the difference between correlation and causation?
Associated State Standards
MAFS.7.SP.2: Draw informal comparative inferences about two populations.
2.3: Informally assess the degree of visual overlap of two numerical data distributions
with similar variabilities, measuring the difference between the centers by
expressing it as a multiple of a measure of variability. For example, the mean
height of players on a basketball team is 10 cm greater than the mean height of
players on the soccer team, about twice the variability (mean absolute deviation)
on either team; on a dot plot, the separation between the two distributions of
height is noticeable.
2.4: Use measures of center and measures of variability for numerical data from
random samples to draw informal comparative inferences about two populations.
For example, decide whether the words in a chapter of a seventh-grade science
book are generally longer than the words in a fourth-grade science book.
MAFS.8.SP.1: Investigate patterns of association in bivariate data
1.1: Construct and interpret scatter plots for bivariate measurement data to investigate
patterns of association between two quantities. Describe patterns such as
clustering, outliers, positive or negative association, linear association, and
nonlinear association.
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ALGEBRA 1 SUMMER INSTITUTE 2014 OVERVIEW
1.4: Understand that patterns of association can also be seen in bivariate categorical
data by displaying frequencies and relative frequencies in a two-way table.
Construct and interpret a two-way table summarizing data on two categorical
variables collected from the same subjects. Use relative frequencies calculated for
rows or columns to describe possible association between the two variables. For
example, collect data from students in your class on whether or not they have a
curfew on school nights and whether or not they have assigned chores at home. Is
there evidence that those who have a curfew also tend to have chores.
MAFS.912.S-ID.2: Summarize, represent, and interpret data on two categorical and
quantitative variables
2.5: Summarize categorical data for two categories in two-way frequency table.
Interpret relative frequencies in the context of the data (including joint, marginal,
and conditional relative frequencies). Recognize possible associations and trends
in the data.
MAFS.912.S-ID.3: Interpret linear models
3.9: Distinguish between correlation and causation.
Learning Objectives:
Participants will:
1. Numerically compare distributions obtained from different samples.
2. Graphically compare distributions obtained from different samples.
3. Represent categorical bivariate data in a frequency table.
4. Calculate numerically the correlation of two data sets.
5. Estimate the correlation coefficient from a scatterplot.
6. Distinguish examples in which correlation does not imply causation from
examples in which it does.
Activities in Unit (4)
Name
Hans Rosling
video
The height and
arm span
juxtaposition
The Olympic
Quandary
Rationale
You've never seen data presented like this. With the drama and
urgency of a sportscaster, statistics guru Hans Rosling debunks
myths about the so-called "developing world." In Hans Rosling’s
hands, data sings. Global trends in health and economics come to
vivid life. And the big picture of global development—with some
surprisingly good news—snaps into sharp focus
This activity focuses on statistical problems that collect and analyze
data on two numerical variables. This kind of data analysis, known as
bivariate analysis, explores the concept of association between two
variables. Participants will work with residuals in order to determine
which of the given lines best fit the data. Participants will explore
with the use of technology the line of best fit.
The discussions in this activity focus on interpreting various
equations of least-squares regression lines. Participants will think and
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ALGEBRA 1 SUMMER INSTITUTE 2014 OVERVIEW
The Outlier
Deviation
The Correlation
vs. Causation
Turmoil
The Table
Categorization
The Weather
Turbulence
The Patterns
Observation
reason about regression lines as summary representations of data that
can be used to make conjectures about data sets collected over time.
In this investigation, participants will use GeoGebra to investigate
the effect of outliers in the regression line.
The main objective of this presentation is to point out the difference
between correlation and causation.
Methods for analyzing categorical data are developed in this lesson.
Participants also work with a random sample and build on their
understanding of a random sample.
Participants will work and reason with bivariate data that show a
linear relationship. They will determine how strong is the
relationship calculating a correlation coefficient. They will also
calculate the coefficient of determination and interpret its result
based on the context of the problem. Finally they will do the
calculations to determine the equation of the regression line.
Participants will work and reason with bivariate data that show linear
and non-linear relationship.
Unit (5):
Estimating Probabilities:
The participants will study and engage in the estimation and application of probability and
random sampling.
Big Ideas:
1. The difference between theoretical and experimental probability
2. Discrete vs. continuous probability distributions
Essential Questions:
1. How can one generate a random sample via simulation?
2. What do the results of a computer random number generator mean?
3. What does “the probability of tossing a head equals one-half” mean?
Associated State Standards:
MAFS. 7.SP.3: Investigate chance processes and develop, use, and evaluate probability
models
3.5: Understand that the probability of a chance event is a number between 0 and 1
that expresses the likelihood of the event occurring. Larger numbers indicate
greater likelihood. A probability near 0 indicates an unlikely event, a probability
around ½ indicates an event that is neither unlikely nor likely, and a probability
near 1 indicates a likely event.
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ALGEBRA 1 SUMMER INSTITUTE 2014 OVERVIEW
3.6: Approximate the probability of a chance event by collecting data on the chance
process that produces it and observing its long-run relative frequency, and predict
the approximate relative frequency given the probability. For example, when
rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200
times, but probably not exactly 200 times.
3.7: Develop a probability model and use it to find probabilities of events. Compare
probabilities from a model to observed frequencies; if the agreement is not good,
explain possible sources of the discrepancy.
a. Develop a uniform probability model by assigning equal probability to all
outcomes, and use the model to determine probabilities of events. For example,
if a student is selected a random from a class, find the probability that Jane
will be selected and the probability that a girl will be selected.
b. Develop a probability model (which may not be uniform) by observing
frequencies in data generated from a chance process. For example, find the
approximate probability that a spinning penny will land heads up or that a
tossed paper clip will land open end down. Do the outcomes for the spinning
penny appear to be equally likely based on the observed frequencies.
3.8: Find the probabilities of compound events using organized lists, tables, trees, and
simulation.
a. Understand that, just as with simple events, the probability of a compound
event is the fraction of outcomes in the sample space for which the compound
event occurs.
b. Represent sample spaces for the compound events using methods such as
organized lists, tables and tree diagrams. For an event described in everyday
language (e.g.,”rolling double sixes”), identify the outcomes in the sample
space which compose the event.
c. Design and use a simulation to generate frequencies for compound events. For
example, use random digits as a simulation tool to approximate the answer to
the question: If 40% of donors have type A blood, what is the probability that it
will take at least 4 donors to find one with type A blood?
MAFS.912.S-IC.1: Understand and evaluate random processes underlying statistical
experiments
1.1: Understand statistics as a process for making inferences about population
parameters based on a random sample from that population.
1.2: Decide if a specified model is consistent with results from a given data-generating
process, e.g., using simulation. For example, a model says that 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?
MAFS.912.S-ID.2: Summarize, represent, and interpret data on two categorical and
quantitative variables
2.5: Summarize categorical data for two categories in two-way frequency table.
Interpret relative frequencies in the context of the data (including joint, marginal,
and conditional relative frequencies). Recognize possible associations and trends
in the data.
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ALGEBRA 1 SUMMER INSTITUTE 2014 OVERVIEW
Learning Objectives:
Participants will:
1. Explain whether a given sampling technique is random
2. Use random sampling to draw an inference or answer a question
3. Explain the probability of events from a discrete probability distribution
4. Explain the probability of events from a continuous probability distribution
5. Determine the probability distribution in a model context
6. Generate an estimate for the probability distribution using technology.
7. Identify the sample space for a given context.
8. Design a simulation for compound events.
Activities for Unit (5):
Name
The ESP
Verification
The Fair Unfair
Polarization
The Poker
Manipulation
The Birthday
Paradox
The Stick, Flip,
Switch
Corollary
Rationale
In this activity, participants will explore some basic ideas about
probability, and some of the relationships between probability and
statistics.
In this activity, participants will determine mathematical and
experimental probabilities and compare them to determine if a game is
fair or unfair.
During this activity, participants will have the opportunity to study the
applications of probabilities in the poker game. Different examples
involving the counting principle, permutations, and combinations will
be studied but the focus will be on investigating and interpreting
probabilities for different types of 5-card poker hands: a pair, two
pairs, full house, four cards of a kind, royal flush, and straight flush.
This activity demonstrates the Birthday Paradox. Participants use
Excel to run a Monte Carlo simulation with the birthday paradox and
perform a graphical analysis of the birthday-problem function.
This lesson plan presents a classic game-show scenario. A student
picks one of three doors in the hopes of winning the prize. The host,
who knows the door behind which the prize is hidden, opens one of the
two remaining doors. When no prize is revealed, the host asks if the
student wishes to "stick or switch." Which choice gives you the best
chance to win? The approach in this activity runs from guesses to
experiments to computer simulations to theoretical models. This lesson
was adapted from an article written by J. Michael Shaughnessy and
Thomas Dick, which appeared in the April 1991 issue of the
Mathematics Teacher.
Unit (6)
Sampling and Inference:
Random sampling is useful for making generalizations from a sample to the entire
population. This unit describes the properties of random samples, how to generate them,
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ALGEBRA 1 SUMMER INSTITUTE 2014 OVERVIEW
and using simple numeric estimates from samples to infer information about the general
population.
Essential Questions:
1. What is a random sample and what do they represent?
2. How can random samples be generated?
3. How can random samples be used to predict information regarding a population?
How accurate are those predictions?
Associated State Standards
MAFS.7.SP.1: Use random sampling to draw inferences about a population.
1.1: Understand that statistics can be used to gain information about a population by
examining a sample of the population; generalizations about a population from a
sample are valid only if the sample is representative of that population. Understand
that random sampling tends to produce representative samples and support valid
inferences.
1.2: Use data from a random sample to draw inferences about a population with an
unknown characteristic of interest. Generate multiple samples (or simulated
samples) of the same size to gauge the variation in estimates or predictions. For
example, estimate the mean word length in a book by randomly sampling words
from the book; predict the winner of a school election based upon randomly
sampled survey data. Gauge how far off the estimate or prediction might be.
Learning Objectives:
Participants will:
1. Generate a random sample.
2. Predict information regarding a population based on statistics from a random
sample.
3. Estimate the accuracy of predictions based on statistics.
4. Practice summarizing data in two-way tables.
5. Create a simulation to model the study
6. Compare and analyze the shape, center, ad spread of the simulated sampling
distributions based on three sets of simulations
7. Use the simulated sampling distributions to consider how infrequent an event must
be for it to be regarded as “rare”
8. Examine a data analysis problem from different perspectives
9. Formulate appropriate questions that can be answered by data analysis
10. Use appropriate graphical displays to make informed decisions in the presence of
variability
11. Compare different analyses of a single scenario
12. Infer whether the results occurred by chance variation or as a result of other
factors.
13. Test a data set for normality
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ALGEBRA 1 SUMMER INSTITUTE 2014 OVERVIEW
14. Construct an associated confidence interval about the mean.
15. Learn that assumptions about the underlying population characteristics are
important to consider when generalizing from a sample and that not all data sets
can be modeled as a normal distribution.
16. Discuss what generalizations can be inferred from a sample.
Activities in Unit (6):
Name
The
Discrimination
Fragmentation
The Cholesterol
Diet Emanation
The Distribution
Normality
The Walking
Instability
Rationale
The activities in this session invite students to explore how to
approach a problem or question of interest that has no clear cut
answer, and how to arrive to a conclusion though a decision-making
process that involves some degree of uncertainty. Though simulation,
participants will explore these points as they examine an experiment
that involves categorical variables.
The activities in this session look at numerical data. Participants will
describe the graphical representations used and interpret those
descriptions in context. Sampling distributions and simulations will
be done as well as deciding exactly what it is that one wants to know
This is a basic overview of what is a normal distribution and how to
use the tables to find probabilities.
In this activity, participants will conduct an investigation to collect
data to determine how far participants can walk blindfolded down the
length of a hallway before they deviate from a straight walking path
and go ‘out of bounds’. This data will be used to test the theory that
humans are not able to walk in a straight line without being able to
see landmarks. Participants will test the data set for normality and
build a 95% confidence interval to estimate the mean distance
participants can walk.
Unit (7)
Modeling Data:
This unit explores simple statistical methods for modeling the relation between two sets of
data. Linear and exponential models are explored.
Essential Questions:
1.
2.
3.
4.
How can the relation between two data sets be modeled?
What is a line of best fit? What does it represent?
What is the effect of rescaling in the line of best fit?
What is the degree of fit for a linear relation? How is the degree of fit related to
correlation?
Associated State Standards
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ALGEBRA 1 SUMMER INSTITUTE 2014 OVERVIEW
MAFS.8.SP.1: Investigate patterns of association in bivariate data
1.2: Know that straight lines are widely used to model relationships between two
quantitative variables. For scatter plots that suggest a linear association, informally
fit a straight line, and informally assess the model fit by judging the closeness of
the data points to the line.
1.3: Use the equation of a linear model to solve problems in the context of bivariate
measurement data, interpreting the slope and intercept. For example, in a linear
model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an
additional hour of sunlight each day is associated with an additional 1.5 cm in
mature plant height.
MAFS.912.S-ID.2: Summarize, represent, and interpret data on two categorical and
quantitative variables
2.6: 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, 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.
MAFS.912.A-SSE.1: Interpret the structure of expressions
1.1: Interpret expressions that represent a quantity in terms of its context.
a. Interpret parts of an expression, such as terms, factors, and coefficients.
b. Interpret complicated expressions by viewing one or more of their parts as a
single entity. For example, interpret P(1+r)n as the product of P and a factor
not depending on P.
MAFS.912.S-ID.3: Interpret linear models
3.7: Interpret the slope (rate of change) and the intercept (constant term) of a linear
model in the context of the data.
3.8: Compute (using technology) and interpret the correlation coefficient of a linear fit.
Learning Objectives:
Participants will:
1. Identify ways of modeling the relation between two data sets.
2. Interpret the best linear fit in terms of the context.
3. Indicate how a best fit would change if the variables are rescaled.
4. Explain the relation between correlation and degree of fit for a linear relation.
5. Formalize intuition about data collection procedures
6. Generate hypothetical distributions for the sample results
7. Judge whether the actual observed sample result is consistent with the random
sampling variability from the population.
8. Develop an appreciation for the power of statistics to allow generalizations t a
larger population from a smaller, but random sample
9. Foster the skills of a statistics critic.
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ALGEBRA 1 SUMMER INSTITUTE 2014 OVERVIEW
10. Recognize both carefully conducted, high quality studies and those that could be
fatally flawed.
Activities in Unit (7):
Name
The Starbuck
Customer
Hypothesis
The Treatment
Effect
Acquisition
The Soft Drink
and Heart
Disease
Acquisition
Rationale
The main statistical ideas of this investigation revolve around the use
of random sampling to make tentative conclusions about a larger
population. This activity lends itself to small-large discussion, a
teamwork approach to data collection, and exploration of hypothetical
models for generating the data.
This investigation focuses on the pattern of chance variation that
arises from randomly assigning subjects to treatment groups. After
considering the amount of chance variation, investigators can probe
the experimental results for evidence of a genuine signal - a treatment
effect that could not have plausibly arisen from the random
assignment process alone. This investigation exposes participants to a
very powerful simulation approach that can easily be extended to
different scenarios while also developing their intuition about whether
an outcome could have occurred by chance and what factors might
affect that assessment.
The main statistical ideas in this investigation involve questioning
aspects of data production and the procedures and design of a
statistical study. Participants should be able of identifying potential
problem areas in the study, to be able to point to its potential
limitations, and to raise clarifying questions about the statistical
claims based on it.
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