Unit 8: Bivariate Data
Students combine their knowledge of linear functions with knowledge of data representations and analysis to make the jump from univariate data in one variable to bivariate data in two variables.
In Unit 8, eighth-grade students make the jump from univariate data in one variable to bivariate data in two variables. They re-engage in the major work of the grade, analyzing scatterplots for positive or negative linear trends (MP.7), using lines to represent relationships between the variables, writing linear equations, interpreting these equations in context (MP.2), and using the equations to make predictions beyond the scope of the data (MP.4). Throughout the unit, students analyze scatterplots and two-way tables for trends in the data, asking themselves, Is there evidence in this graph or table to suggest an association between the variables? (MP.2)
Prior to eighth grade, students explored how and why data is collected—by thinking about statistical questions, samples, populations, and various ways to analyze data representations. Students worked with line plots, histograms, and box plots, and they considered what the shape, center, and spread of these data sets said about the data itself.
In high school, students’ understanding of statistics is formalized. They analyze bivariate data using functions, design and carry out experiments, and make predictions about outcomes based on probabilities. Students use their knowledge of association between variables as a basis for correlation. They develop nonlinear models for data and formally analyze how closely the model fits the data.
Pacing: 12 instructional days (9 lessons, 2 flex days, 1 assessment day)
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The following assessments accompany Unit 8.
Have students complete the Pre-Unit Assessment and Pre-Unit Student Self-Assessment before starting the unit. Use the Pre-Unit Assessment Analysis Guide to identify gaps in foundational understanding and map out a plan for learning acceleration throughout the unit.
Pre-Unit Student Self-Assessment
Have students complete the Mid-Unit Assessment after lesson 5.
Use the resources below to assess student mastery of the unit content and action plan for future units.
Post-Unit Assessment Answer Key
Post-Unit Student Self-Assessment
Use student data to drive your planning with an expanded suite of unit assessments to help gauge students’ facility with foundational skills and concepts, as well as their progress with unit content.
Suggestions for how to prepare to teach this unit
The central mathematical concepts that students will come to understand in this unit
Terms and notation that students learn or use in the unit
double bar graph
line fit to data
segmented bar graph
To see all the vocabulary for Unit 8, view our 8th Grade Vocabulary Glossary.
The materials, representations, and tools teachers and students will need for this unit
To see all the materials needed for this course, view our 8th Grade Course Material Overview.
Topic A: Associations in Bivariate Numerical Data
Define bivariate data. Analyze data in scatter plots.
Create scatter plots for data sets and make observations about the data.
Identify and describe associations in scatter plots including linear/nonlinear associations, positive/negative associations, clusters, and outliers.
Informally fit a line to data. Judge the fit of the line and make predictions about the data based on the line.
Write equations to represent lines fit to data and make predictions based on the line.
Interpret the slope and $$y$$-intercept of a fitted line in context.
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Topic B: Associations in Bivariate Categorical Data
Create and analyze two-way tables representing bivariate categorical data.
Calculate relative frequencies in two-way tables to investigate associations in data.
Complete two-way tables and identify associations in the data.
The content standards covered in this unit
— Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
— Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
— Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.
For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.
— Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.
For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?
Standards covered in previous units or grades that are important background for the current unit
— Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.
For example, the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
— Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
— Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
— Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
— Use proportional relationships to solve multistep ratio and percent problems.
Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
— Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.
— Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
— Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.
— Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.
For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.
— Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
Standards in future grades or units that connect to the content in this unit
— Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
— Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
— Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
— Compute (using technology) and interpret the correlation coefficient of a linear fit.
— Distinguish between correlation and causation.
— Make sense of problems and persevere in solving them.
— Reason abstractly and quantitatively.
— Construct viable arguments and critique the reasoning of others.
— Model with mathematics.
— Use appropriate tools strategically.
— Attend to precision.
— Look for and make use of structure.
— Look for and express regularity in repeated reasoning.
Pythagorean Theorem and Volume