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Pre-AP: Strategies in Mathematics -- Analyzing and
Describing Data
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Data is a key component to all of the mathematics taught in school today. From the primary counting of objects to multivariable analyses, data forms the basis for most of the problem solving that students see.
This one-day workshop enriches the data analysis topics included in the middle and secondary grades by providing examples of activities where students collect data, use graphs and numerical summaries to get information from data, and communicate that information. Teachers assume the role of students as they discuss data collection and experimental design issues, work through exercises, and share observations and conclusions. Teachers collaborate and share ideas in the workshop just as students will do in class.
Five student-ready, multiple-day activities are given and modeled in the workshop: two dealing with univariate data and three with bivariate data. All build on topics that are already included in middle school and high school curricula and can be used as ancillaries to enrich and extend material that is routinely taught.
General Themes
- Analysis of real data
- Using summary statistics and graphs to investigate and gain information from data
- Elementary principles of experimental design
- Interpretation and communication of results
The workshop conforms to:
The workshop provides:
- Substantial content background for teachers
- Activities for students across grade levels
- Activities meant to illustrate good pedagogy (various instructional approaches including cooperative learning)
- Activities that require communication to support observations and conclusions
- Activities that utilize technology
- Opportunities for reflection
- Connections between activities
- Optional online follow-up
Agenda
Activity 1: Is Michael's Penny Unusually Old?
This activity is designed for middle school students to collect and analyze data with the goal of determining whether or not a penny dated 1980 is unusually old. Students will make a physical dotplot using their sample of pennies, observe a very strongly skewed distribution, and discover that sometimes mean and median differ in value and provide different information. Extensions of this activity include histograms and their sensitivity to class intervals, the definition of outliers, and boxplots.
Break
Activity 2: Do Short- and Long-Rotor Helicopters Differ in Drop Time?
This activity is designed for secondary students to collect and analyze data with the goal of determining whether two different models of paper helicopters differ in flight time when dropped from the same height. A pattern is provided so that students can construct their own helicopters to use in this activity. Heavy emphasis is placed on ensuring that students think carefully about how to design the experiment so that they are analyzing good data. Questions are provided to introduce students to the basic principles of experimental design. Flight times are compared using summary statistics, parallel histograms, and parallel boxplots. Standard deviation is explained and defined in this activity.
Lunch
Activity 3: Grocery Store Reach/Using Volume to Predict Mass
This activity is designed for middle school students to collect and analyze data to determine the relationship between the height and the reach height of an adult and, ultimately, to determine how high to place the highest shelves in a grocery store. Students will use a scatterplot to get general information about the relationship. Then they will draw a line on their scatterplot to summarize the relationship and use it for predicting reach height when a person's height is known. Students will analyze the distance of data points from their line to get an estimate of errors that might result when predictions are made using the line.
Activity 4: Using Volume to Predict Mass
In this activity, students are provided with a data set and asked to create a linear model they can use to predict the mass of a certain type of wood from its volume. To complete all parts of the activity, students need to know how to write equations of lines when they know two points on the line. Students are first asked to draw a freehand line to model the data, find the equation of this line, and use it to make predictions. As they compare predictions, they recognize the need for a standard, more objective line. The median-median line algorithm is provided, first graphically and then algebraically. Residuals are defined in this activity, and students are asked to interpret the information provided by the slope and intercept of their model in the context of the problem.
Break
Activity 5: The Least Square Lines
This activity, designed for secondary students, consists of three parts. The first and second parts use data collected by students to develop understanding of the least squares criterion. The final part provides students with two data sets with a specific purpose. The first data set contains influential points and is used to demonstrate the sensitivity of the least squares line to points that deviate from the general pattern. The second data set exhibits subtle curvature. It is used to demonstrate that a residual plot can provide important information about whether or not a model is a good one and also to demonstrate that a high correlation value does not guarantee a good model.
Author
Gloria Barrett has taught statistics to high school students for 20 years, 19 of them at the North Carolina School of Science and Mathematics. She presently teaches AP Statistics and other courses at Deerfield Academy. She has written a book of statistical activities for graphing calculators and served on the writing team for the NCTM publication Navigating Through Probability in Grades 9-12. She now serves on the AP Statistics Development Committee.
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