Lesson 7
Confident Models
These materials, when encountered before Algebra 1, Unit 3, Lesson 7 support success in that lesson.
7.1: Math Talk: Ordering Decimals (5 minutes)
Warm-up
The purpose of this Math Talk is to elicit strategies and understandings students have for ordering positive and negatve decimals. These understandings help students develop fluency and prepare students to interpret correlation coefficients.
Launch
Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.
Student Facing
Mentally order the numbers from least to greatest.
20.2, 18.2, 19.2
-14.6, -16.7, -15.1
-0.43, -0.87, -0.66
0.50, -0.52, 0.05
Student Response
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Activity Synthesis
Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:
- “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
- “Did anyone have the same strategy but would explain it differently?”
- “Did anyone solve the problem in a different way?”
- “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
- “Do you agree or disagree? Why?”
7.2: Ranking Models (15 minutes)
Activity
The purpose of this activity is for students to look at different scatter plots and rank them in order from the linear model being a very bad fit to the data to the linear model being a very good fit to the data.
Launch
Allow students to work individually or with a partner. Allow students about 10 minutes for the first part of the task statement.
Student Facing
- Here are scatter plots that represent various situations. Order the scatter plots from “A linear model is not a good fit for the data” to “A linear model is an excellent fit for the data.”
A
B
C
D
E
- Here are two scatter plots including a linear model. For each model, determine the \(y\) when \(x\) is 15. Which model prediction do you think is closer to the real data? Explain your reasoning.
Graph F. \(y = 150-5x\)
Graph G. \(y = 100-1.3x\)
Student Response
Teachers with a valid work email address can click here to register or sign in for free access to Student Response.
Activity Synthesis
The goal of this discussion is for students to understand that they can rank the goodness of fit of a linear model to data in a scatter plot based on the linear model’s ability to predict outcomes in the data. Discuss how they ranked the scatter plots and how to use this to interpret data. Here are sample questions to promote class discussion:
- “How did you determine which scatter plot would be very badly fit by a linear model? Fit very well?” (I looked for the scatter plot that looks least like a straight line. Then, I looked for the scatter plot that is closest to forming a straight line.)
- “Some of the scatter plots show a negative association and some show a positive association. How does that affect your ranking?” (Whether the data has a positive or negative association is not important when determining how good of a fit a linear model is to the data. The main focus is on how close the data are to being in a straight line.)
- “How did you determine which of the two linear models is most likely to predict accurate outcomes in the data?” (When I found the \(y\) for each model when \(x\) is 15, I looked at which predicted \(y\) is closest to the actual \(y\) given from the data set. Whichever predicted \(y\) was closer to the actual \(y\), I decided that model is closer to the real data. However, it is important to consider all data points before making a decision about a linear model being an accurate predictor of the outcomes.)
- “When a linear model is used to describe a relationship between two variables, how does a linear model’s goodness of fit to a scatter plot help to interpret the data?” (The closer the model is to the data, the stronger the relationship between the variables. The more I can use a model to predict accurate outcomes, then the more dependable that model is and I can feel more confident in using it to describe the relationship.)
7.3: Predicting Value (20 minutes)
Activity
The purpose of this activity is for students to synthesize data displayed in a scatter plot using aspects of a linear model. Students practice using the linear models to interpret the data.
Student Facing
Here are situations represented with graphs and lines of fit. Use the information given to complete the missing fields for each situation.
- The model predicts how much money, in dollars, the coach will make based on how many athletes sign up for one-on-one training. The model is represented with the equation \(y=200 + 25x\).
- The slope of the model is \(\underline{\hspace{.5in}}\) (positive or negative).
- What does the model predict would be the amount the coach makes when there are 10 athletes present?
- Using the data points and the model as a reference, what is a reasonable range of money the coach will make when there are 10 athletes present?
- This model is a \(\underline{\hspace{.5in}}\) (great, good, okay, or bad) fit for the data.
- Using numbers between 0 and 1, rate your confidence in the model where 0 is no confidence and 1 is total confidence.
- The model predicts the annual salary of a worker in a certain government position based on years of experience. The model is represented with the equation \(y=1.5x + 35\).
- The slope of the model is \(\underline{\hspace{.5in}}\) (positive or negative).
- What does the model predict would be the employee’s salary when the employee has 10 years of experience?
- Using the data points and the model as a reference, what is a reasonable range for the salary of a worker based on 10 years of experience?
- This model is a \(\underline{\hspace{.5in}}\) (great, good, okay, or bad) fit for the data.
- Using numbers between 0 and 1, rate your confidence in the model where 0 is no confidence and 1 is total confidence.
- The model predicts the number of absences a school will have based on the number of incentives given per month. The model is represented with the equation \(y= \text-2.18x + 54.78\).
- The slope of the model is \(\underline{\hspace{.5in}}\) (positive or negative).
- What does the model predict would be the number of absences when 10 incentives are given for the month?
- Using the data points and the model as a reference, what is a reasonable number of absences when there are 10 incentives given?
- This model is a \(\underline{\hspace{.5in}}\) (great, good, okay, or bad) fit for the data.
- Using numbers between 0 and 1, rate your confidence in the model where 0 is no confidence and 1 is total confidence.
Student Response
Teachers with a valid work email address can click here to register or sign in for free access to Student Response.
Activity Synthesis
The purpose of this discussion is for students to make a connection between scatter plots and how to use a linear model to interpret the relationship between the variables presented in the scatter plot. Discuss how students made sense of the problems and ranked their confidence in the models. Here are sample questions to promote class discussion:
- “How did you determine the ‘reasonable ranges' of values?” (I looked at how close the points are to the line and used that to determine a range on either side of the line for the given \(x\)-value.)
- “How did you determine what type of fit the linear model is to the data presented in the scatter plot?” (I look at how close the points are to the line. If the points are very close to the line, it is a good fit. If the points are farther from the linear model, then the fit is not very good.)
- “What things did you consider when determining how confident you are in the linear models?” (I considered if the linear model was a good fit to the data or not, how close the predicted \(y\) values were to the actual \(y\) values, and the strength of my confidence in the model.)