Summary
In this lesson, students will explore the properties of linear and nonlinear relationships by examining tables, graphs, and descriptions. They will then calculate the linear regression model for a set of data and evaluate the reliability of the model by interpreting the correlation coefficient.
Essential Question(s)
How can we find and describe the shape of data?
Learning Objectives
Create a linear regression model that describes the given data.
Use the correlation coefficient to determine the fit of the model.
Determine whether the best model for the data is linear or nonlinear.
Snapshot
Engage
Students think through potential payment options and select which one they would prefer.
Explore
Students create graphs from given data and use their calculators to find the linear regression equation, assessing for fit.
Explain
Students develop their own meaning of the variables from the linear regression model and use patterns to identify linear and nonlinear trends through guided exploratory notes.
Extend
Students complete two experiments and then analyze their data to find the line of best fit and interpret the correlation coefficient.
Evaluate
Students reflect on their original answers of the potential payment options from the beginning of the lesson and modify or justify as needed.
Materials
Lesson Slides (attached)
Pay Day handout (attached; one-half page per student; print one-sided)
Data Exploration handout (attached; one per student; print two-sided)
Calculator Guide handout (attached; one per student; print one-sided)
Guided Notes Correlation Coefficient handout (attached; for half the class; print two-sided)
Guided Notes Correlation Coefficient (Model Notes) document (attached)
Guided Notes Trends handout (attached; for the other half of class; print two-sided)
Guided Notes Trends (Model Notes) document (attached)
Experimental Drop handout (attached; one per student; print two-sided)
Meter sticks (one per group)
Golf balls (one per group)
Candy (ten pieces per group)
TI-84 Plus CE graphing calculator (one per student)
Desmos Studio calculator or other graphing calculator (optional)
Preparation
During the Extend phase of the lesson, students will be working in pairs to conduct experiments and gather data. Half of the class will be dropping golf balls onto the floor and measuring the bounce using meter sticks, while the other half of the class will be dropping candy onto a table/desk and recording how it lands. Think about how to best set up your classroom space for these experiments to run smoothly.
The candy needed for this experiment needs to have a top and a bottom for students to define as “heads” or “tails.” Candies that would work well for this experiment would be candies like M&M’S or Starburst.
Engage
5 Minute(s)
Introduce the lesson using the attached Lesson Slides. Briefly introduce the essential question on slide 3. Move to slide 4 to identify the lesson's learning objectives. Review each of these with students to the extent you feel necessary.
Introduce the class to the Preflections strategy. Give each student a half-page of the attached Pay Day handout and display slide 5. Read the prompt from their handout aloud as they follow along:
“You got a new short-term job! Your employer gives you two options for how you will receive payment. In the first option, you will receive $10,000 every day for a month (30 days). With the second option, you will receive $0.01 on the first day, but each day, the amount will double (the first five days you would get $0.01, $0.02, $0.04, $0.08, $0.16).”
Have students circle the payment option they would prefer in the Preflection section of the table on their handout and then write their reasoning.
Once students are done, collect these responses with the plan to redistribute them during the Evaluate phase of this lesson.
Explore
35 Minute(s)
Have students get into groups of 2–3. Distribute the attached Data Exploration and Calculator Guide handouts to each student. Instruct students to get a graphing calculator. The guide will walk students through the steps of how to find the line of best fit using the linear regression functionality of their calculator.
Display slide 6 and preview the activity. Explain to students that for each of the two given data sets, they need to:
Plot the given data on their handout.
Input the data into their graphing calculator.
Use their calculator to find the line of best fit.
Use that information to sketch the line of best fit on their handout.
Remind students that even though they are working in small groups, they are required to independently record their work and reasoning. Give students approximately 20–25 minutes to complete this task for the two datasets. Use the hidden slide 7 to check a, b, and r-values for the given datasets.
Have students return to their seats and display slide 8. Facilitate a discussion using the questions on the slide:
What trends did you notice in the data?
Do you think the linear equation you found was an accurate representation of the data?
What do you think the r-value is for?
What might explain the relationship between:
income and obesity?
days and zombies?
Explain
40 Minute(s)
Distribute a copy of the Guided Notes Correlation Coefficient handout to half of the students and a copy of the Guided Notes Trends handout to the other half of the students. Display slide 9 and have students with the same handout form small groups of 2–3 to work on the handout together. Introduce the Jigsaw strategy to the students and explain that they are responsible for learning the information on the front side of their handout and will need to explain it to a classmate who has a different handout. Emphasize that all students within each small group need to work together to make sure their partners understand all of the information, as they are each expected to be able to individually explain what they learned.
Students with the Guided Notes Correlation Coefficient handout will be comparing tables, scatter plots, and linear regression models to determine the significance of the variables from the given linear regression model results, including the correlation coefficient, r. When groups have completed the directions and recorded their observations on the front of the handout, they should formalize their findings by filling out the top four bullet points on the back of their handout.
Students with the Guided Notes Trends handout will be analyzing the rates of change from tables of data, where some of the data form a linear trend and some of the data is nonlinear (specifically, exponential). When groups have completed the directions and recorded their observations on the front of the handout, they should formalize their findings by filling out the top four bullet points on the back of their handout.
Give students approximately 15 minutes to observe and formalize their observations, and then move to slide 10. Ask students to form groups of 3–4. Each group should contain students who worked on both the Guided Notes Correlation Coefficient handout and the Guided Notes Trends handout. Give groups around 15 minutes to explain their findings to each other. Emphasize to students the importance of explaining what they learned and helping their peers understand the reasoning behind the conclusions they drew, rather than allowing their peers to simply copy their notes.
After groups have had a chance to share and learn from each other, come together as a whole class. Transition through slides 11–12 and summarize the information, ensuring students understand the meaning of the variables in the linear regression model and how to determine the difference between the change in linear and exponential data.
Transition through slides 13–19 and have the students work through the example problem at the bottom of the handout. Depending on your students’ needs and level of understanding, this may be completed through a whole class discussion, small group work, or individually.
Extend
35 Minute(s)
Give each student a copy of the attached Experimental Drop handout. All students will be doing both experiments, but half will complete the Ball Drop experiment first, while the other half completes the Candy drop experiment first. Have students find a partner or assign partners. Then direct half of the pairs to begin the Ball Drop experiment and the other half to begin the Candy drop experiment. Display slide 20. Have students gather their materials:
For the Ball Drop experiment, each pair will need a meter stick, a golf ball, and a calculator.
For the Candy Drop experiment, each pair will need 20 pieces of candy and a calculator.
Give the students around 10 minutes to complete the experiment, record their results on their handout, and answer the questions. Here, students are asked to use their calculator to find the linear regression model and then identify and interpret the correlation coefficient.
Move to slide 21 and have pairs now conduct the other experiment.
After another 10 minutes, bring the class together and lead a discussion about their findings. Consider using the questions on slide 22 as talking points:
Which data was linear?
Which data was nonlinear?
Did anyone get an r-value of 1 for their regression models?
Why do you think different groups got different results?
Evaluate
10 Minute(s)
Return each student’s Pay Day handout from the Engage phase of the lesson and display slide 23. Ask students to reflect on what they learned during the lesson and use their knowledge to evaluate their original payment choice. Have students indicate their payment option preference on their handout, and if their mind changed (from the beginning of the lesson), have students write what made them change their mind.
Resources
K20 Center. (n.d.). Desmos Studio. Tech Tools. https://learn.k20center.ou.edu/tech-tool/2356
K20 Center. (n.d.). Jigsaw. Strategies. https://learn.k20center.ou.edu/strategy/179
K20 Center. (n.d.). Preflections. Strategies. https://learn.k20center.ou.edu/strategy/191