C. A. R. E. Curriculum Assessment Remediation Enrichment Algebra 1 Mathematics CARE Package #6 Modeling with Linear Functions

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1 C. A. R. E. Curriculum Assessment Remediation Enrichment Algebra 1 Mathematics CARE Package #6 Modeling with Linear Functions Domain Statistics and Interpreting Data Standards MAFS.912.S-ID.2.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, and exponential models. b. Informally assess the fit of a function by plotting and analyzing residuals. c. Fit a linear function for a scatter plot that suggests a linear association. MAFS.912.S-ID.3.8 Compute (using technology) and interpret the correlation coefficient of a linear fit. MAFS.912.S-ID.3.9 Distinguish between correlation and causation. Clarifications Students will represent data on a scatter plot. Students will find a linear function, a quadratic function, or an exponential function using regression. Students will use a regression equation to solve problems in the context of the data. Students will calculate residuals. Students will create a residual plot and determine whether a function is an appropriate fit for the data. Students will determine the fit of a function by analyzing the correlation coefficient. Students will distinguish between situations where correlation does not imply causation. Students will distinguish variables that are correlated because one is the cause of another. STUDENT PROBLEM-SOLVING This links to the Student Problem-Solving Algebra I site that provides interactive activities for students. This wiki is a work in progress and will be updated regularly. PERFORMANCE TASK Considerations This task makes use of Rubrics (C1) and Graphs (I6). Students can use digital tools and Software (I4) for graphing functions. DIFFERENTIATION Some students may benefit from working with a partner to complete the task. Encourage students who completed the performance task early to share their rationale and how they arrived at the solutions with a partner.

2 CURRICULUM Performance Task - Laptop Battery Charge Jerry forgot to plug in his laptop before he went to bed. He wants to take the laptop to his friend's house with a full battery. The pictures below show screenshots of the battery charge indicator after he plugs in the computer. a. Let t represent time in minutes since the laptop was plugged in. Let b represent the charge of the battery. Graph the data and find a line of best fit. b. When can Jerry expect that his laptop battery is fully charged? c. At 9:27 AM Jerry makes a quick calculation: The battery seems to be charging at a rate of 1 percentage point per minute. So the battery should be fully charged at 10:11 AM. Explain Jerry's calculation. Is his estimate most likely an under- or over-estimate? How does it compare to your prediction? d. Compare the average rate of change of the battery charging function on the first given time interval and on the last given time interval. What does this tell you about how the battery is charging? e. How long would it take for the battery to charge if it started out completely empty?

3 f. Calculate the residuals for your line of best fit equation. Make a residual plot. t b b Residual actual predicted g. Evaluate the suitability of a linear fit and the goodness of the fit. Commentary This task uses a situation that is familiar to students to solve a problem they probably have all encountered before: How long will it take until an electronic device has a fully charged battery? A linear model can be used to solve this problem. The task combines ideas from statistics, functions and modeling. It is a nice combination of ideas in different domains in the high school curriculum. Focus in high school means finding connections between the different topics that are covered. Lines of best fit are a perfect example of this idea since you are using linear functions to analyze data. Students are not told how to come up with a model for the situation. Part of their job is to turn the screenshots into data that can be visualized with a scatter plot and analyzed with a linear function. The task can be used for instruction with group work and whole class discussion. The last two parts of the task lead towards the question if a line is the best model for this situation. Close inspection shows that there seems to be some downward curvature to the data, which may agree with students' experiences when waiting for batteries to charge. On the other hand, while a line might not be the most accurate model, it is the simplest model that gives a result that is perfectly adequate for the situation, which is an important aspect of mathematical modeling. The task provides good opportunities for students to engage in SMP 2 - Reason Abstractly and Quantitatively as well as SMP 4 - Model with Mathematics. Solution: 1 a. In this situation we are looking at two variables: time, t and battery charge, b. There are several ways we can choose units. A reasonable choice is "time in minutes since the laptop was plugged in" and "battery charge in percentage of full, %" The laptop started charging at 9:11 a.m. and it was initially 41% charged. If we let t be time since the laptop was plugged in, this information corresponds to the point with

4 coordinates (0,41). Similarly, we can translate the other screenshots into coordinate points:(16,56), (25,64), (37,74), (44,79), (57,86), (66,91). We can now make a scatter plot of the data to get an idea how the two variables are related. It looks like there is a linear relationship between the variables. Using technology such as a graphing calculator or WolframAlpha we find that b= t. We can extend the line of best fit and see at what time it will reach an output value of 100%. This happens about 75 minutes after the computer was plugged in. We can also use the equation to solve b= t=100 for t to obtain t 74.5 minutes.

5 b. At 9:27 a.m. Jerry has two data points: (0,41) and (16,56). So the battery charged 15 percentage points in 16 minutes. That is almost 1 percentage point per minute. At 9:27 a.m. the battery still needs to charge 44 more percentage points, which will take approximately 44 minutes. So 44 minutes after 9:27 a.m. puts us at 10:11 a.m. Jerry's estimate will probably be an underestimate for the time needed to fully charge the battery. He is overestimating the rate at which the battery is charging. Therefore, his calculation comes up with a shorter amount of time than is actually needed. We can visualize Jerry's method with a line of slope 1 through the point (16,56). c. The average rate of change of the battery charging function during the first time interval is =1516= percentage points per minute. During the last time interval the average rate of change is = percentage points per minute. This means that as the battery gets charged, the rate at which it charges is going down. So it will take longer to go from 90% to 100% than it took to get from 40% to 50%. d. We can use the line of best fit to find an estimate for the time it would take for the battery to charge from 0 to 100%. If we extend the line to the left, we see that it crosses the horizontal axis at about 60. So we can estimate that it would have taken 60 minutes for the battery to go from 0% to 41% and we already estimated that it would take about 75 minutes to go from 41% to 100%. So it would take about 135 minutes to charge the battery all the way from 0 to 100%. This is probably not a bad estimate. Our average rate of change calculations show that we are probably underestimating the time it takes to charge the last 20 percentage points. But on the other hand, we probably overestimated the time it takes for the battery to go from 0 to 40%, so we should be close overall. For parts f. and g. examine student work. The residual data and graphs

6 will be specific to the equation that a student chose for line of best fit. Have students share graphs and compare residual plots to determine which lines are a better best fit. ASSESSMENT The Mini-MAFS includes standards MAFS.912.S-ID.2.6a, MAFS.912.S-ID.2.6b, MAFS.912.S-ID.2.6c, MAFS.912.S-ID.3.8/3.9. Utilize the table to assist in remediation efforts. Questions Standards Algebra 1 ACE Lessons 1-3 MAFS.912.S-ID.2.6a Section MAFS.912.S-ID.2.6b Section MAFS.912.S-ID.2.6c Section MAFS.912.S-ID.3.8/3.9 Section 8.1

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12 Key Vocabulary- Building Functions Causation Correlation coefficient Linear Regression Line of Best Fit Residual Plot REMEDIATION / RETEACH Correlation Interpolation Least Squares Regression Line Residual Scatter Plot Remediation/Reteaching Resources Building Functions Resource Links Multi-lingual glossary Calculating residuals Correlation Coefficient Line of Best Fit Reteach Problem solving - Line of Best Fit Best-Fit Geogebra Doggie Data Fitting a Line to data Description ELL students may utilize this link to access a multi-lingual glossary. Video instruction and examples Reteaching, ELL instruction, and exercises for reaching all learners. Geogebra Applet allows students to manipulate points in the coordinate plane and examine the effect on the line of best fit equation. Students can examine the correlation coefficient to explore strong and weak correlations in data. This lesson allows students to use real-world data to construct and interpret scatter plots using technology. Students will create a scatter plot with a line of best fit and a function. They describe the relationship of bi-variate data. They recognize and interpret the slope and y-intercept of the line of best fit within the context of the data. Khan Academy tutorial video that uses Excel spreadsheet and actual income data to predict annual income and expresses why lines and models are useful and interesting.

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15 Student Resources Performance Task- Laptop Battery Charge Name: Class: Date: Jerry forgot to plug in his laptop before he went to bed. He wants to take the laptop to his friend's house with a full battery. The pictures below show screenshots of the battery charge indicator after he plugs in the computer. a. Let t represent time in minutes since the laptop was plugged in. Let b represent the charge of the battery. Graph the data and find a line of best fit. b. When can Jerry expect that his laptop battery is fully charged?

16 c. At 9:27 AM Jerry makes a quick calculation: The battery seems to be charging at a rate of 1 percentage point per minute. So the battery should be fully charged at 10:11 AM. Explain Jerry's calculation. Is his estimate most likely an under- or over-estimate? How does it compare to your prediction? d. Compare the average rate of change of the battery charging function on the first given time interval and on the last given time interval. What does this tell you about how the battery is charging? e. How long would it take for the battery to charge if it started out completely empty? f. Calculate the residuals for your line of best fit equation. Make a residual plot. t b b Residual actual predicted g. Evaluate the suitability of a linear fit and the goodness of the fit.

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