First published in 2013 by the University of Utah in association with the Utah State Office of Education.

Copyright © 2013, Utah State Office of Education. Some rights reserved. This work is published under the Creative Commons Attribution License ("CC BY"), available online at http://creativecommons.org/licenses/by/3.0/. This license allows others to reuse, revise, remix, and redistribute the work. This is an open educational resource (OER).

Table of Contents

CHAPTER 6: STATISTICS — INVESTIGATE PATTERNS OF ASSOCIATION IN BIVARIATE DATA (2 WEEKS)

• 6.0 Anchor Problem: Tongue Twisters
• Section 6.1: Construct and Interpret Scatter Plots for Bivariate Data
• 6.1a Class Activity: Read and Interpret a Scatter Plot
• 6.1a Homework: Read and Interpret a Scatter Plot
• 6.1b Class Activity: Create and Analyze a Scatter Plot
• 6.1b Homework: Create and Analyze a Scatter Plot
• 6.1c Classwork: Patterns of Association
• 6.1c Homework: Patterns of Association
• 6.1d Self-Assessment: Section 6.1
• Section 6.2: Construct a Linear Model to Solve Problems
• 6.2a Classwork: Lines of Best Fit
• 6.2a Homework: Lines of Best Fit
• 6.2b Class Activity: Fit a Linear Model to Bivariate Data
• 6.2b Homework: Fit a Linear Model to Bivariate Data
• 6.2c Self-Assessment: Section 6.2
• Section 6.3: Construct and Interpret Two-Way Frequency Tables to Analyze Categorical Data
• 6.3a Class Activity: Construct Two-Way Frequency Tables Using Categorical Data
• 6.3a Homework: Construct a Two-Way Frequency Table
• 6.3b Class Activity: Interpret Two-Way Frequency Tables
• 6.3b Homework: Interpret Two-Way Frequency Tables
• 6.3c Class Activity: Conduct a Survey
• 6.3d Self-Assessment: Section 6.3

Chapter 6: Statistics — Investigate Patterns of Association in Bivariate Data (2 weeks)

Utah Core Standards:

• Construct and interpret scatter plots for bivariate data (data made up of two related measurements for each subject) to look for patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and non-linear association. (8.SP.1)
• Recognize that straight lines are commonly used to model the relationship between two quantitative (numerical) variables. When a scatter plot suggests a linear relationship, informally fit a straight line to the data, and judge how well that line fits by looking at how close the data points are to it. (8.SP.2)
• Use the equation of a linear model to solve problems involving bivariate data, and interpret what the slope and the y-intercept mean in context. For example, in a linear model for a biology experiment, a slope of 1.5 cm/hr means that each additional hour of sunlight per day is associated with an extra 1.5 cm in a plant's mature height. (8.SP.3)
• Understand that patterns of association can also show up in bivariate categorical data (data grouped into categories rather than numbers) by displaying frequencies (counts) and relative frequencies (proportions) in a two-way table. Construct and interpret a two-way table that summarizes data on two categorical variables collected from the same group of people. Use the frequencies calculated for rows or columns to describe a possible association between the two variables. For example, you might collect data from your classmates on whether they have a curfew on school nights and whether they have assigned chores at home, then ask: Is there evidence that students with a curfew also tend to have chores?

Academic Vocabulary: experiment, outcomes, sample space, random variables, realizations, quantitative (numerical) variables, categorical variables, univariate data, bivariate data, scatter plot, association, positive association, negative association, no apparent association, linear association, non-linear association, weak association, strong association, perfect association, cluster, outlier, line of best fit, linear model, prediction function, two-way frequency table, marginal frequencies, relative frequencies.

Chapter Overview:

So far, students have studied data that lines up perfectly along a straight line. In the real world, though, data is rarely perfect. Even so, data often follows patterns that can be described mathematically. In this chapter, students will explore patterns of association in bivariate data by building and interpreting scatter plots, fitting a linear function to scatter plots that suggest a linear relationship, and using that function to solve real-world problems and make predictions. Students will also examine categorical bivariate data by constructing and interpreting two-way frequency tables.

Connections to Content:

Prior Knowledge: Before 8th grade, the study of statistics has focused on univariate data — data involving just one variable. Students have created and analyzed displays of univariate data, described its features, and calculated numerical measures of center (like mean and median) and spread (like range). In 8th grade, students apply what they already know about the coordinate plane and linear functions to analyze bivariate data and to build linear models for data sets that show a linear relationship.

Future Knowledge: Later, students will use technology to more formally fit linear functions — and other types of functions — to bivariate data. They will calculate correlation coefficients, a number that measures how strong a linear relationship is. They will also use residual plots to judge how well a linear model fits the data, and they will continue studying two-way frequency tables.

Mathematical Practice Standards

Make sense of problems and persevere in solving them.

Emina loves eating the tomatoes from her garden in Salt Lake City. She asked her friend Renzo, "Don't you just love tomatoes?" Renzo wrinkled his nose and said, "Ew, tomatoes gross me out. When I see them at the grocery store, I just keep walking." Renzo's answer made Emina think: "I don't buy tomatoes at the grocery store either — I grow my own. The tomatoes from my garden taste great, but grocery store tomatoes just don't appeal to me. I wonder if there's a connection between liking tomatoes and having a garden at home?"

In this problem, students help Emina figure out whether there's an association between liking tomatoes and having a home garden. They organize the data they collect into a two-way frequency table and then analyze it. Along the way, they have to make decisions about how to organize the data and figure out what story it tells.

Reason abstractly and quantitatively.

A table (not shown here) gives data comparing the number of oil changes a car gets over two years to the cost of its repairs.

Students plot this data on a graph, placing the number of oil changes on the horizontal axis and choosing their own scale. They then write a prediction function in slope-intercept form (y = mx + b) that could be used to predict the cost of repairs, y, for any number of oil changes, x, and compare their function with a partner's. Finally, they use the prediction function to estimate repair costs for someone who gets 8 oil changes, again comparing results with a partner.

Throughout the chapter, students study numeric data sets shown in tables and graphs. When a data set suggests a linear relationship, students build a linear function to model it. These functions are an abstract way of representing the relationships the data suggests.

Construct viable arguments and critique the reasoning of others.

Using a scatter plot, students determine whether there's a relationship between the number of field goals attempted and the number made, describing any trends or patterns they notice.

Throughout the chapter, students create scatter plots from given data sets and analyze them to determine whether an association exists between two variables. They look for trends and patterns, including clusters and outliers, and explain — using the context of the data — why those associations, trends, and patterns might occur. In doing so, students build arguments about the data and are expected to back up those arguments with evidence, while also thinking critically about the context and limitations of the data.

Model with mathematics.

Students take turns saying a chosen tongue twister aloud, one student at a time. In the first round, only the first student says it. In the second round, the first and second students both say it. Each round adds one more student to the chain, and the total time it takes is recorded.

Tongue twisters used:

A. Work will win when wishy-washy wishing won't.
B. Three witches wished three wishes, but which witch wished which wish.
C. Peter Piper picked a peck of pickled peppers.
D. Picky people...