College and Career-Readiness Conference Summer 2014 FOR HIGH SCHOOL MATHEMATICS TEACHERS

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 standards within each grade/course.

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?  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  Introduce the concept of deviations from the mean and their effect on spread.  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  Groups will be completing parts of the Illustrative Mathematics Task Understanding the Standard Deviation.  The next 3 slides include the questions each group should answer.  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? Explain your choice.

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:

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, quadratic, and exponential models. *

Cluster C. Interpret linear 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. 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  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 y- axis

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 Don’t Jump to Conclusions!

Activity 2 Instructions  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 YearTime YearTime YearTime Sketch the scatter plot with the line of best fit (x: years since 1900). Also sketch the residual plot. Discuss correlation vs. causation.

Discussion  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: orts/olympics/the-100-meter-dash-one-race-every- medalist-ever.html?_r=1& orts/olympics/the-100-meter-dash-one-race-every- medalist-ever.html?_r=1&

What have you done that works? Best Practices

Additional Resources  Illustrative Mathematics Illustrative Mathematics  PARCC Practice Test (go to Algebra 1 Item 20) PARCC Practice Test  Engage NY Module Engage NY Module  Mathematics Vision Project (Module 8 is Data) Mathematics Vision Project

What are the muddiest points? Record any question you still have after today’s presentation on your post-it note. Please provide your name and address. 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