# MA.912.DP.2.4

Fit a linear function to bivariate numerical data that suggests a linear association and interpret the slope and y-intercept of the model. Use the model to solve real-world problems in terms of the context of the data.

### Clarifications

Clarification 1: Instruction includes fitting a linear function both informally and formally with the use of technology.

Clarification 2: Problems include making a prediction or extrapolation, inside and outside the range of the data, based on the equation of the line of fit.

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Data Analysis and Probability
Date Adopted or Revised: 08/20
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Numerical Data
• Line of Fit
• Scatter Plot

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 8, students first worked with scatter plots and lines of fit. In Algebra I, students relate the slope and $y$-intercept of a line of fit to association in bivariate numerical data and interpret these features in real-world contexts. In later courses, students use the correlation coefficient to measure how well a line fits the data in a scatter plot, and they also work with scatter plots that suggest quadratic and exponential models.
• This is an extension of MA.912.DP.1.1, where students are working with numerical bivariate data (scatter plots and line graphs). It is good to review with students that a scatter plot is a display of numerical data sets between two variables.
• They are good for showing a relationship or association between two variables.
• They can reveal trends, shape of trend or strength of relationship trend.
• They are useful for highlighting outliers and understanding the distribution of data.
• One variable could be the progression of time, like in a line graph.
• In this benchmark, students are fitting a linear function to numerical bivariate data, interpreting the slope and $y$-intercept based on the context and using that linear function to make predictions about values that correspond to parts of the graph that lie beyond or within the scatter plot.
• Instruction includes the use of technology for students to understand the difference between a line of fit and a line of best fit. Additionally, instruction of this benchmark should be combined with MA.912.DP.2.6 and MA.912.DP.2.5, as these are extensions of this benchmark.
• During instruction is important to distinguish the difference between a “line of fit” and the “line of best fit.”
• A “line of fit” is used when students are visually investigating numerical bivariate data that appears to have a linear relationship and can sketch a line (using a writing instrument and straightedge) that appears to “fit” the data. Using this “line of fit” students can estimate its slope and $y$-intercept and use that information to interpret the context of the data.
• The “line of best fit” (also referred to as a “trend line”) is used when the data is further analyzed using linear regression calculations (the process of minimizing the squared distances from the individual data values to the line), often done with the assistance of technology.

### Common Misconceptions or Errors

• Students may not know how to sketch a line of fit.
• For example, they may always go through the first and last points of data.
• Students may be confuse the two variables when interpreting the data as related to the context.
• Students may not know the difference between interpolation (predictions within a data set) and extrapolation (predictions beyond a data set).

### Strategies to Support Tiered Instruction

• Teacher provides sketched lines of fit and has students identify the one that best models the data.
• Teacher provides a sentence frame for interpreting the data in the context of the problem using two different colored highlighters to highlight the same variable in the sentence frame and table or graph.
• Example:

• Teacher models creating a scatter plot on a piece of graph paper, then has students place a piece of spaghetti on the scatter plot to model the line of best fit. Students could also use a coordinate plane peg board to plot each point creating a scatter plot and then use a rubber band to model the line of best fit.
• Instruction includes vocabulary development by co-creating a graphic organizer for interpolation and extrapolation.

Instructional Task 1 (MTR.4.1, MTR.5.1, MTR.6.1
• Crickets are one of nature’s more interesting insects, partly because of their musical ability. In England, the chirping or singing of a cricket was once considered to be a sign of good luck. Crickets will not chirp if the temperature is below 40 degrees Fahrenheit (°F) or above 100 degrees Fahrenheit (°F). A table is given with some data collected.
• Part A. Create a line of fit based on the data. Compare your line of fit with a partner.
• Part B. What is the estimated slope and $y$-intercept of the line?
• Part C. What does the slope mean in terms of the context?
• Part D. What does the $y$-intercept mean in terms of the context?
• Part E. Using technology, determine the line of best fit. Compare this to the line of fit determine from Part A. What is the difference?
• Part F. Based on this line, predict the temperature to be if you recorded 250 chirps per minute?
• Part G. Based on this line, estimate the number of chirps per minute at exactly 50°F.

### Instructional Items

Instructional Item 1
• Below is data from a variety of fast food chains.
• Part A. Create a scatter plot based on the data above and estimate an equation for a line of fit.
• Part B. What do the slope and $y$-intercept tell us about the relationship of total fat and total calories in these fast food items?
• Part C. If a fast food item has 10 grams of fat, estimate the total calories of that item.

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
1200310: Algebra 1 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200320: Algebra 1 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200370: Algebra 1-A (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1210300: Probability and Statistics Honors (Specifically in versions: 2014 - 2015, 2015 - 2019, 2019 - 2022, 2022 and beyond (current))
7912070: Access Mathematics for Liberal Arts (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 - 2023, 2023 and beyond (current))
7912080: Access Algebra 1A (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))
1200315: Algebra 1 for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200375: Algebra 1-A for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912075: Access Algebra 1 (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))
1210305: Mathematics for College Statistics (Specifically in versions: 2022 and beyond (current))
1207350: Mathematics for College Liberal Arts (Specifically in versions: 2022 and beyond (current))
1200388: Mathematics for Data and Financial Literacy Honors (Specifically in versions: 2022 and beyond (current))
1200384: Mathematics for Data and Financial Literacy (Specifically in versions: 2022 and beyond (current))
7912120: Access Mathematics for Data and Financial Literacy (Specifically in versions: 2022 - 2023, 2023 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.912.DP.2.AP.4: Fit a linear function to bivariate numerical data that suggest a linear association and interpret the slope and y-intercept of the model.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Formative Assessments

Tuition:

Students are asked to use a linear model to make a prediction about the value of one of the variables.

Type: Formative Assessment

Foot Length:

Students are asked to interpret the line of best fit, slope, and y-intercept of a linear model.

Type: Formative Assessment

Swimming Predictions:

Students are asked to use a linear model to make and interpret predictions in the context of the data.

Type: Formative Assessment

House Prices:

Students are asked to informally fit a line to model the relationship between two quantitative variables in a scatterplot, write the equation of the line, and use it to make a prediction.

Type: Formative Assessment

Slope for Human Foot Length Model:

Students are asked to interpret the meaning of the slope of the graph of a linear model.

Type: Formative Assessment

Slope for Life Expectancy:

Students are asked to interpret the meaning of the slope of the graph of a linear model.

Type: Formative Assessment

Intercept for Life Expectancy:

Students are asked to interpret the intercept of a linear model of life expectancy data.

Type: Formative Assessment

Bungee Cord Model:

Students are asked to interpret the meaning of the constant term in a linear model.

Type: Formative Assessment

## Lesson Plans

Spreading the Vote Part 3:

Students will explore voter turnout data for three gubernatorial elections before and after the passage of the 19th amendment. They will fit linear functions to the data and compute predicted values for raw and percentage of voter turnout. Students will draw some conclusions concerning the relationship between eligible voters and voter turnout, including possible causes behind the fluctuation in voter participation in this integrated lesson

Type: Lesson Plan

Compacting Cardboard:

Students investigate the amount of space that could be saved by flattening cardboard boxes. The analysis includes linear graphs and regression analysis along with discussions of slope and a direct variation phenomenon.

Type: Lesson Plan

A Day at the Park:

In this activity, students investigate a set of bivariate data to determine if there is a relationship between concession sales in the park and temperature. Students will construct a scatter plot, model the relationship with a linear function, write the equation of the function, and use it to make predictions about values of variables.

Type: Lesson Plan

You Can Plot it! Bivariate Data:

Students create scatter plots, calculate a regression equation using technology, and interpret the slope and y-intercept of the equation in the context of the data. This review lesson relates graphical and algebraic representations of bivariate data.

Type: Lesson Plan

How Hot Is It?:

This lesson allows the students to connect the science of cricket chirps to mathematics. In this lesson, students will collect real data using the CD "Myths and Science of Cricket Chirps" (or use supplied data), display the data in a graph, and then find and use the mathematical model that fits their data.

Type: Lesson Plan

Basketball - it's a tall man's sport - or is it?:

The students will use NBA player data to determine if there is a correlation between the height of a basketball player and his free throw percentage. The students will use technology to create scatter plots, find the regression line and calculate the correlation coefficient.

Basketball is a tall man's sport in most regards. Shooting, rebounding, blocking shots - the taller player seems to have the advantage. But is that still true when shooting free throws?

Type: Lesson Plan

What happens to available energy as it moves through an ecosystem?:

This activity is a lab exercise where students look at the passing of water in cups and compare it to the loss of available energy as it moves through an ecosystem. Students will collect data, calculate efficiency, graph the data and respond to reflection questions to connect the data to what happens in an ecosystem. The end of the activity includes a connection to the 10% rule where only 10% of energy from one trophic level is available at the next level.

Type: Lesson Plan

Heart Rate and Exercise: Is there a correlation?:

Students will use supplied heart rate data to determine if heart rate and the amount of time spent exercising each week are correlated. Students will use GeoGebra to create scatter plots and lines of fit for the data and examine the correlation. Students will gather evidence to support or refute statistical statements made about correlation. The lesson provides easy to follow steps for using GeoGebra, a free online application, to generate a correlation coefficient for two given variables.

Type: Lesson Plan

Span the Distance Glider - Correlation Coefficient:

This lesson will provide students with an opportunity to collect and analyze bivariate data and use technology to create scatter plots, lines of best fit, and determine the correlation strength of the data being compared. Students will have a hands on inquire based lesson that allows them to create gliders to analyze data. This lesson is an application of skills acquired in a bivariate unit of study.

Type: Lesson Plan

Scatter Plots:

This lesson is an introduction to scatterplots and how to use a trend line to make predictions. Students should have some knowledge of graphing bivariate data prior to this lesson.

Type: Lesson Plan

Study of Crowd Ratings at Disney:

In this lesson, students develop a strong use of the vocabulary of correlation by investigating crowd ratings at Disney. Students will determine weekly crowd rating regression lines and correlations and discuss what this means for a Disney visit.

Type: Lesson Plan

Hand Me Your Data:

Students will gather and use data to calculate a line of fit and the correlation coefficient with their classmates' height and hand size. They will use their line of fit to make approximations.

Type: Lesson Plan

What Will I Pay?:

Who doesn't want to save money? In this lesson, students will learn how a better credit score will save them money. They will use a scatter plot to see the relationship between credit scores and car loan interest rates. They will determine a line of fit equation and interpret the slope and y-intercept to make conclusions about interest and credit scores.

Type: Lesson Plan

What does it mean?:

This lesson provides the students with scatter plots, lines of best fit and the linear equations to practice interpreting the slope and y-intercept in the context of the problem.

Type: Lesson Plan

Why do I have to have a bedtime?:

This predict, observe, explain lesson that allows students to make predictions based on prior knowledge, observations, discussions, and calculations. Students will receive the opportunity to express themselves and their ideas while explaining what they learned. Students will make a prediction, collect data, and construct a scatter plot. Next, students will calculate the correlation coefficient and use it to describe the strength and magnitude of a relationship.

Type: Lesson Plan

Is My Model Working?:

Students will enjoy this project lesson that allows them to choose and collect their own data. They will create a scatter plot and find the line of fit. Next they write interpretations of their slope and y-intercept. Their final challenge is to calculate residuals and conclude whether or not their data is consistent with their linear model.

Type: Lesson Plan

Students will make a scatter plot and then create a line of fit for the data. From their graph, students will make predictions and describe relationships between the variables. Students will make predictions, inquire, and formulate ideas from observations and discussions.

Type: Lesson Plan

Mass Mole Relationships: A Statistical Approach To Accuracy and Precision:

The lesson is a laboratory-based activity involving measurement, accuracy and precision, stoichiometry and a basic statistical analysis of data using a scatter plot, linear equation, and linear regression (line of best fit). The lesson includes teacher-led discussions with student participation and laboratory-based group activities.

Type: Lesson Plan

Scatter Plots and Correlations:

Students create scatter plots, and lines of fit, and then calculate the correlation coefficient. Students analyze the results and make predictions. This lesson includes step-by-step directions for calculating the correlation coefficient using Excel, GeoGebra, and a TI-84 Plus graphing calculator. Students will make predictions for the number of views of a video for any given number of weeks on the charts.

Type: Lesson Plan

Cat Got Your Tongue?:

This lesson uses real-world examples to practice interpreting the slope and y-intercept of a linear model in the context of data. Students will collect data, graph a scatter plot, and use spaghetti to identify a line of fit. A PowerPoint is included for guidance throughout the lesson and guided notes are also provided for students.

Type: Lesson Plan

If the line fits, where's it?:

In this lesson students learn how to informally determine a "best fit" line for a scatter plot by considering the idea of closeness.

Type: Lesson Plan

Doggie Data: It's a Dog's Life:

Students use real-world data to construct and interpret scatter plots using technology. Students will create a scatter plot with a line of fit and a function. They describe the relationship of bivariate data. They recognize and interpret the slope and y-intercept of the line of fit within the context of the data.

Type: Lesson Plan

Scrambled Coefficient:

Students will learn how the correlation coefficient is used to determine the strength of relationships among real data. Students use card sorting to order situations from negative to positive correlations. Students will create a scatter plot and use technology to calculate the line of fit and the correlation coefficient. Students will make a prediction and then use the line of fit and the correlation coefficient to confirm or deny their prediction.

Students will learn how to use the Linear Regression feature of a graphing calculator to determine the line of fit and the correlation coefficient.

The lesson includes the guided card sorting task, a formative assessment, and a summative assessment.

Type: Lesson Plan

Spaghetti Trend:

This lesson consists of using data to make scatter plots, identify the line of fit, write its equation, and then interpret the slope and the y-intercept in context. Students will also use the line of fit to make predictions.

Type: Lesson Plan

Correlation or Causation: That is the question:

Students will learn how to analyze whether two events/properties demonstrate a correlation or causation or both. They will learn what factors are involved when evaluating whether correlated events demonstrate causation. If two events are claimed to be causal when they are not, they will be able to determine why, and which (if any) causal fallacies are present. At the close of the lesson students will be given situational data and develop a newscast that assumes causation when in fact there is no causal link. Students who are observing will analyze each presentation and determine which (if any) causal fallacy was used (or explain why the newscast is correct in their assumption of causality).

Type: Lesson Plan

How technology can make my life easier when graphing:

Students will use GeoGebra software to explore the concept of correlation coefficient in graphical images of scatter plots. They will also learn about numerical and qualitative aspects of the correlation coefficient, and then do a matching activity to connect all these representations of the correlation coefficient. They will use an interactive program file in GeoGebra to manipulate the points to create a certain correlation coefficient. Step-by-step instructions are included to create the graph in GeoGebra and calculate the r correlation coefficient.

Type: Lesson Plan

Smarter than a Statistician: Correlations and Causation in the Real World!:

Students will learn to distinguish between correlation and causation. They will build their skills by playing two interactive digital games that are included in the lesson. The lesson culminates with a research project that requires students to find and explain the correlation between two real world events.

Type: Lesson Plan

Linear Statistical Models:

In this lesson, students will learn how to analyze data and find the equation of the line of best fit. Students will then find the slope and intercept of the best fit line and interpret the meaning in the context of the data.

Type: Lesson Plan

Slope and y-Intercept of a Statistical Model:

Students will sketch and interpret the line of fit and then describe the correlation of the data. Students will determine if there’s a correlation between foot size and height by collecting data.

Type: Lesson Plan

Line of Fit:

Students will graph scatterplots and draw a line of fit. Next, students will write an equation for the line and use it to interpret the slope and y-intercept in context. Students will also use the graph and the equation to make predictions.

Type: Lesson Plan

How Fast Can You Go:

Students will apply skills (making a scatter plot, finding Line of Best Fit, finding an equation and predicting the y-value of a point on the line given its x-coordinate) to a fuel efficiency problem and then consider other factors such as color, style, and horsepower when designing a new coupe vehicle.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal studentsâ€™ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

Type: Lesson Plan

Don't Mope Over Slope:

This is an introductory lesson designed to help students have a better understanding of the interpretation of the slope (rate of change) of a graph.

Type: Lesson Plan

Scatter plots, spaghetti, and predicting the future:

Students will construct a scatter plot from given data. They will identify the correlation, sketch an approximate line of fit, and determine an equation for the line of fit. They will explain the meaning of the slope and y-intercept in the context of the data and use the line of fit to interpolate and extrapolate values.

Type: Lesson Plan

## Perspectives Video: Experts

Oceanography & Math:

A discussion describing ocean currents studied by a physical oceanographer and how math is involved.

Type: Perspectives Video: Expert

Assessment of Past and Present Rates of Sea Level Change:

In this video, Brad Rosenheim describes how Louisiana sediment cores are used to estimate sea level changes over the last 10,000 years. Video funded by NSF grant #: OCE-1502753.

Type: Perspectives Video: Expert

Analyzing Antarctic Ice Sheet Movement to Understand Sea Level Changes:

In this video, Eugene Domack explains how past Antarctic ice sheet movement rates allow us to understand sea level changes. Video funded by NSF grant #: OCE-1502753.

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

Determining Strengths of Shark Models based on Scatterplots and Regression:

Chip Cotton, fishery biologist, discusses his use of mathematical regression modeling and how well the data fits his models based on his deep sea shark research.

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

Slope and Deep Sea Sharks:

Shark researcher, Chip Cotton, discusses the use of regression lines, slope, and determining the strength of the models he uses in his research.

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

Linear Regression for Analysis of Sea Anemone Data:

Will Ryan describes how linear regression models contribute towards his research on sea anemones.

Type: Perspectives Video: Professional/Enthusiast

Mathematically Modeling Eddy Shedding :

COAPS oceanographer Dmitry Dukhovskoy describes the process used to mathematically model eddy shedding in the Gulf of Mexico.

Type: Perspectives Video: Professional/Enthusiast

Analyzing Wildlife Data Trends with Regression :

Dr. Bill McShea from the Smithsonian Institution discusses how regression analysis helps in his research.

This video was created in collaboration with the Okaloosa County SCIENCE Partnership, including the Smithsonian Institution and Harvard University.

Type: Perspectives Video: Professional/Enthusiast

Texting and Grades II:

The purpose of this task is to assess ability to interpret the slope and intercept of the line of fit in context.

## STEM Lessons - Model Eliciting Activity

How Fast Can You Go:

Students will apply skills (making a scatter plot, finding Line of Best Fit, finding an equation and predicting the y-value of a point on the line given its x-coordinate) to a fuel efficiency problem and then consider other factors such as color, style, and horsepower when designing a new coupe vehicle.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal studentsâ€™ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

## MFAS Formative Assessments

Bungee Cord Model:

Students are asked to interpret the meaning of the constant term in a linear model.

Foot Length:

Students are asked to interpret the line of best fit, slope, and y-intercept of a linear model.

House Prices:

Students are asked to informally fit a line to model the relationship between two quantitative variables in a scatterplot, write the equation of the line, and use it to make a prediction.

Intercept for Life Expectancy:

Students are asked to interpret the intercept of a linear model of life expectancy data.

Slope for Human Foot Length Model:

Students are asked to interpret the meaning of the slope of the graph of a linear model.

Slope for Life Expectancy:

Students are asked to interpret the meaning of the slope of the graph of a linear model.

Swimming Predictions:

Students are asked to use a linear model to make and interpret predictions in the context of the data.

Tuition:

Students are asked to use a linear model to make a prediction about the value of one of the variables.

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

Texting and Grades II:

The purpose of this task is to assess ability to interpret the slope and intercept of the line of fit in context.