Grade 8: Statistics & Bivariate Data

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.

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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 (graphs that show pairs of related numbers as points) for bivariate data — data that involves two different measurements taken from the same source — to look for patterns between two quantities. Describe patterns such as clustering (points grouped together), outliers (points that don't fit the pattern), positive or negative association, linear association (a straight-line pattern), and nonlinear association (a pattern that isn't a straight line). (8.SP.1)

• Understand that straight lines are often used to model, or represent, the relationship between two quantities. When a scatter plot suggests a linear relationship, you can informally draw a straight line to fit the data, then judge how well the line fits by seeing how close the points are to it. (8.SP.2)

• Use the equation of a linear model to solve real-world problems using bivariate data, and be able to explain what the slope and intercept mean in that situation. For example, in a science experiment about plant growth, a slope of 1.5 cm/hr would mean that for every extra hour of sunlight a plant gets each day, it grows an additional 1.5 cm. (8.SP.3)

• Understand that patterns of association can also show up in bivariate categorical data — data made of two categories, rather than numbers — by organizing frequencies (counts) and relative frequencies (percentages or fractions) into a two-way table. Construct and interpret a two-way table that summarizes data on two categories collected from the same group of people. Use the relative frequencies in the rows or columns to describe any possible connection between the two categories. For example, you might collect data from your classmates about whether they have a curfew on school nights and whether they have chores at home, then ask: Is there evidence that students with a curfew are also more likely to have chores? (8.SP.4)

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 neatly along a straight line. But in the real world, data is rarely perfect. Even so, real data often follows patterns that we can describe using math. In this chapter, students will look for patterns of association in bivariate data. They will do this by:

• building and interpreting scatter plots,
• fitting a straight line to scatter plots that show a linear pattern, and
• using that line's equation (called a prediction function) to solve real-world problems and make predictions.

Students will also explore bivariate categorical data — data grouped into categories instead of numbers — by building and interpreting two-way frequency tables.

Connections to Content

Prior Knowledge: Before 8th grade, students mainly studied univariate data — data involving just one type of measurement. They learned to create and analyze displays of this kind of data, describe its features, and calculate numbers that show its center (like the average) and spread. In 8th grade, students will use what they already know about the coordinate plane and linear functions to analyze bivariate data and build linear models for data sets that show a straight-line pattern.

Future Knowledge: Later on, students will use technology to more precisely fit straight lines — and other types of curves — to bivariate data. They will also learn to calculate a number called the correlation coefficient, which measures how strong a linear relationship is. Students will use residual plots (a tool that shows how far off a line's predictions are) to judge how well a linear model fits the data. They will also continue studying two-way frequency tables.

Mathematical Practice Standards

Make sense of problems and persevere in solving them.

Emina loves to eat tomatoes from her garden in Salt Lake City. She asked her friend Renzo, "Don't you just love tomatoes?" Renzo crinkled his nose and replied, "Ew, tomatoes gross me out! When I see them in the grocery store, I just keep on walking." Renzo's response made Emina think: "I don't buy tomatoes at the grocery store either, because I grow them in my garden. The tomatoes from my garden are delicious, but grocery store tomatoes don't look as appealing 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 a connection between enjoying tomatoes and having a home garden. They organize the data they collect into a two-way frequency table and then study it closely. Along the way, they must think carefully about how to organize the data and what it's telling them.

Reason abstractly and quantitatively.

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

Students plot this data on a graph, placing the number of oil changes on the horizontal axis. They choose their own scale for the graph.

Then, they write a prediction function in slope-intercept form (the y = mx + b format) that could be used to predict the cost of repairs, y, based on any number of oil changes, x. They compare their equation with a partner's.

Finally, they use their prediction function to estimate how much someone would spend on repairs after 8 oil changes, and again compare results with a partner.

Throughout this chapter, students study displays of numerical data in tables and graphs. When the data suggests a straight-line pattern, they build a linear function to represent it. These functions are an abstract way of describing the patterns suggested by real data.

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 of field goals made in a game. They describe any trends or patterns they notice in the data.

Throughout the chapter, students create scatter plots from given data sets and study them to see whether two variables are related. They look for trends and patterns, including clusters and outliers, and explain what might be causing these patterns based on the real-world context. In doing so, students build arguments about the data and are expected to back up their claims using evidence and careful thinking about the situation — including any limits on what the data can tell them.

Model with mathematics.

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

The tongue twisters used are:

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 peopl—" (passage ends here)