published by
the National Council of Teachers of Mathematics

This is a two-week instructional unit for Grades 9-12 on data analysis. Students use real-life data to interpret the slope and y-intercept of least squares regression lines in the context of everyday situations. Lessons include travel distances, bathtub water levels, and automobile age vs. mileage. In each activity, students interpret the meaning of slope and y-intercept, calculate correlation coefficients, and fit the data by estimating parameters. Individual lessons can be easily parsed out to create a shorter unit.

This resource is part of a larger collection of lessons, labs, and activities developed by the National Council of Teachers of Mathematics (NCTM).

Editor's Note:Authors offer guidance to help students prepare verbal presentations of findings and to facilitate discussion that elicits critical response. Resource meets numerous national standards in mathematics and language arts.

6-8: 9B/M3. Graphs can show a variety of possible relationships between two variables. As one variable increases uniformly, the other may do one of the following: increase or decrease steadily, increase or decrease faster and faster, get closer and closer to some limiting value, reach some intermediate maximum or minimum, alternately increase and decrease, increase or decrease in steps, or do something different from any of these.

9-12: 9B/H4. Tables, graphs, and symbols are alternative ways of representing data and relationships that can be translated from one to another.

9C. Shapes

6-8: 9C/M4. The graphic display of numbers may help to show patterns such as trends, varying rates of change, gaps, or clusters that are useful when making predictions about the phenomena being graphed.

11. Common Themes

11C. Constancy and Change

9-12: 11C/H9. It is not always easy to recognize meaningful patterns of change in a set of data. Data that appear to be completely irregular may be shown by statistical analysis to have underlying trends or cycles. On the other hand, trends or cycles that appear in data may sometimes be shown by statistical analysis to be easily explainable as being attributable only to randomness or coincidence.

12. Habits of Mind

12D. Communication Skills

9-12: 12D/H7. Use tables, charts, and graphs in making arguments and claims in oral, written, and visual presentations.

12E. Critical-Response Skills

9-12: 12E/H1. Notice and criticize claims based on the faulty, incomplete, or misleading use of numbers, such as in instances when (1) average results are reported but not the amount of variation around the average, (2) a percentage or fraction is given but not the total sample size, (3) absolute and proportional quantities are mixed, or (4) results are reported with overstated precision.

Common Core State Standards for Mathematics Alignments

Standards for Mathematical Practice (K-12)

MP.2 Reason abstractly and quantitatively.

High School — Functions (9-12)

Interpreting Functions (9-12)

F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

F-IF.7.a Graph linear and quadratic functions and show intercepts, maxima, and minima.

High School — Statistics and Probability^{?} (9-12)

Interpreting Categorical and Quantitative Data (9-12)

S-ID.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 different data sets.

S-ID.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

S-ID.6.a Fit a function to the data; use functions fitted to data to solve problems in the context of the data.

S-ID.6.b Informally assess the fit of a function by plotting and analyzing residuals.

S-ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

S-ID.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.

Making Inferences and Justifying Conclusions (9-12)

S-IC.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

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.

Using Probability to Make Decisions (9-12)

S-MD.1 (+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.

S-MD.4 (+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.

S-MD.7 (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

Common Core State Writing Standards for Literacy in History/Social Studies, Science, and Technical Subjects 6—12

Text Types and Purposes (6-12)

1. Write arguments focused on discipline-specific content. (WHST.9-10.1)

2. Write informative/explanatory texts, including the narration of historical events, scientific procedures/ experiments, or technical processes. (WHST.9-10.2)

This resource is part of a Physics Front Topical Unit.

Topic: Kinematics: The Physics of Motion Unit Title: Graphing

High school students can often record data and "plug & chug", but have more difficulty in fitting or interpreting data. This exemplary two-week unit on data analysis introduces students to the statistical method known as least squares regression. Using an online tool to plot data, students then calculate regression lines and fit the data to estimated parameters.

<a href="http://www.thephysicsfront.org/items/detail.cfm?ID=8293">National Council of Teachers of Mathematics. Illuminations: Least Squares Regression. Reston: National Council of Teachers of Mathematics, February 14, 2008.</a>

Illuminations: Least Squares Regression, (National Council of Teachers of Mathematics, Reston, 2008), <http://illuminations.nctm.org/LessonDetail.aspx?ID=U117>.

Illuminations: Least Squares Regression. (2008, February 14). Retrieved November 25, 2014, from National Council of Teachers of Mathematics: http://illuminations.nctm.org/LessonDetail.aspx?ID=U117

National Council of Teachers of Mathematics. Illuminations: Least Squares Regression. Reston: National Council of Teachers of Mathematics, February 14, 2008. http://illuminations.nctm.org/LessonDetail.aspx?ID=U117 (accessed 25 November 2014).

Illuminations: Least Squares Regression. Reston: National Council of Teachers of Mathematics, 2008. 14 Feb. 2008. 25 Nov. 2014 <http://illuminations.nctm.org/LessonDetail.aspx?ID=U117>.

@misc{
Title = {Illuminations: Least Squares Regression},
Publisher = {National Council of Teachers of Mathematics},
Volume = {2014},
Number = {25 November 2014},
Month = {February 14, 2008},
Year = {2008}
}

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