### Math HS PP for Difficult Standards

```FOR
HIGH SCHOOL
MATHEMATICS TEACHERS
Summer 2014
Cluster A. Summarize, represent, and
interpret data on single count or
measurable variable.
Cluster B. Summarize, represent, and
interpret data on two categorical and
quantitative variables.
Cluster C. Interpret linear models.
TODAY’S OUTCOMES
Participants will:
1. Briefly review the instructional shift, COHERENCE.
2. Look at the PARCC model content framework for the
high school statistics and probability standards.
3. Take an in-depth look at the S-ID standards taught
in Algebra 1.
4. Share best practices and identify muddy points.
OUTCOME 1
Participants will:
1. Review the instructional shift of
COHERENCE.
A purposeful placement of standards to create
logical sequences of content topics that bridge
across the grades and courses, as well as across
In what grade does each standard fall?
SP.A: Investigate patterns of association in
bivariate data.
SP.B: Draw informal comparative inferences about
two populations.
SP.B: Summarize and describe distributions.
HS.S.ID.A: Summarize, represent and interpret
data on a single count of measurable variable.
HS.S.ID.B: Summarize, represent and interpret
data on two categorical and quantitative
variables.
HS.S.IC.B: Make inferences and justify conclusions
from sample surveys, experiments, and
observational studies.
OUTCOME 2
Participants will:
2. Look at the PARCC model
content framework for the
high school statistics and
probability standards.
PARCC Model Content Framework
Algebra 1
PARCC Model Content Framework
Algebra 2
OUTCOME 3
Participants will:
3. Take an in-depth look at the
S-ID standards taught in
Algebra 1
Cluster A. Summarize, represent, and
interpret data on single count or
measurable variable.
Standard 1. Represent data with plots on the
real number line (dot plots, histograms, and
box plots).
Standard 2. Use statistics appropriate to the
shape of the data distribution to compare
center (median, mean) and spread (IQR,
standard deviation) of two or more different
data sets.
Standard 3. Interpret differences in shape,
center, and spread in the context of the data
sets, accounting for possible effects of
extreme data points (outliers).
Mathematical Practices
1. Make sense of problems and
persevere in solving them.
 3. Construct viable arguments and
critique the reasoning of others.
 4. Model with mathematics.
 5. Use appropriate tools strategically.

What does Bill McCallum say?
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Students should be looking at large data sets using
technology, in which case the software will be
reporting the measures.
Students should understand what measures of center
and spread mean and how they are computed.
Students are not required to calculate standard
deviation by hand.
Students should be seeing small data sets and
calculate all measures by hand.
So what does this all mean for
teachers?
Teaching Skew
Teaching Standard Deviation

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Introduce the concept of deviations from the mean
Explain how to calculate standard deviation using
the formula. *Students should not be assessed on
calculating by hand! Show them so they understand
what the concept is.
Use technology to calculate standard deviation and
discuss the need for precision.
Activity Instructions
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

Groups will be completing parts of the Illustrative
Deviation.
The next 3 slides include the questions each group
Materials: chart paper and markers.
Part 1
Below are dot plots for three different data sets. The
standard deviations for these three data sets are 5.9,
3.3, and 2.3. Looking at the dot plots and without
calculating the standard deviations, match the data
sets to the standard deviations.
Part 2
Which of the two histograms below represents the
data distribution with the greater standard deviation?
Part 3


Write two sets of 5 different numbers that have the
same mean but different standard deviations.
Write two sets of 5 different numbers that have the
same standard deviations but different means.
Gallery Walk
Photo source:
http://amrcs.aspirail.org/features/new-gallery-walk-instruction-style-introduced/
Cluster B. Summarize, represent, and
interpret data on two quantitative and
categorical variables.
Standard 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,
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.
Cluster C. Interpret linear models.
Standard 7: Interpret the slope (rate of change) and the
intercept (constant term) of a linear model in the context
of the data.
Standard 8: Compute (using technology) and interpret
the correlation coefficient of a linear fit.
Standard 9: Distinguish between correlation and
causation.
Residuals
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Defined as the prediction error
Smaller values = better fit
Residual plots show the relationship between an x
value and the corresponding residual value
Technology should be used to create residual plots
A residual plot showing random points is linear
while a residual plot showing a curved pattern is
non-linear
A scatter plot that appears linear may not be when
looking at the residual plot with an exaggerated yaxis
Correlation vs. Causation

Correlation: There is a relationship between Event A
and Event B
Can be used to make predictions
 Can be used to design further investigations
 Should be evaluated for linking and lurking variables


Causation: Event A causes Event B
There may be outside factors that are not taken into account
 Additional research/experiment is needed to determine
causation

Activity 2 Instructions
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Premise: There is a correlation between finishing time
and the year for the Olympic Games men’s 100-meter
dash.
Small groups will use the data set for the finishing times
for the Olympic gold medalist in the men's 100-meter
dash for many previous Olympic games to calculate the
equation for the line of best fit and make a residual
plot. (Use years since 1900)
Group members should discuss the validity of the linear
model using the residual plot. Group members should
also discuss correlation vs. causation.
Data Set
Year
Time
Year
Time
Year
Time
1900
11.0
1936
10.3
1980
10.25
1904
11.0
1948
10.3
1984
9.99
1906
11.2
1952
10.79
1988
9.92
1908
10.8
1956
10.62
1992
9.96
1912
10.8
1960
10.32
1996
9.84
1920
10.8
1964
10.06
2000
9.87
1924
10.6
1968
9.95
2004
9.85
1928
10.8
1972
10.14
2008
9.69
1932
10.3
1976
10.06
2012
9.63
Sketch the scatter plot with the line of best fit (x: years since 1900).
Also sketch the residual plot. Discuss correlation vs. causation.
Discussion
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What did you find as the equation for the line of
best fit?
What did the residual plot show?
What came up in the discussion of correlation vs.
causation?
Follow-up video:
http://www.nytimes.com/interactive/2012/08/05/sp
orts/olympics/the-100-meter-dash-one-race-everymedalist-ever.html?_r=1&
Best Practices
What have you done that works?
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Illustrative Mathematics
PARCC Practice Test (go to Algebra 1 Item 20)
Engage NY Module
Mathematics Vision Project (Module 8 is Data)
What are the muddiest points?
Record any question
you still have after
today’s presentation
name and email
Stick your post-it on the door as you leave
today, and we will respond. Thank you!
Teaching the Common Core content using
the Standards for Mathematical Practice to
reach progressively higher levels of
proficiency attains mathematical rigor.
-Hull, Balka, and Harbin Miles
```