CREATING LINEAR MODELS FOR DATA

Transcription

1 Topic 12: Creating linear models for data 481 CREATING LINEAR MODELS FOR DATA Lesson 12.1 Scatterplots and trend lines 12.1 OPENER Ayana and Kayla decide to sell lemonade one hot summer afternoon. They set up their lemonade stands on opposite corners of their block. Here are graphs of each girl s profit for the four hours they sell lemonade. Use the graphs to answer these questions. 1. State one thing that is similar about the two graphs. 2. State one thing that is different between the two graphs. 3. Which graph shows a linear relationship between profit and time? 4. Which graph shows an approximately linear relationship between profit and time? 12.1 CORE ACTIVITY 1. Plot the tabular data on the grid to illustrate the relationship between height and shoe size. 2. How does the pattern in the graph compare with other linear situations you have explored previously? 3. Draw a line on your graph for question 1 showing the general relationship between height and shoe size.

2 482 Unit 4 Linear functions 4. Describe the correlation between the variables in each graph as either strong or weak; and either positive or negative. a. b. c. d. 5. Record the measurements for height and arm span, rounded to the nearest centimeter, for the students your class measured. 6. Plot the measurements. Student Height (cm) Arm span (cm) Use your class data to answer questions 7 and Is there a correlation between arm span and height? If so, how would you describe the strength of the relationship (weak, moderate, or strong)? 8. If a correlation exists, add a trend line to the graph you made in question 6. Estimate the location of the trend line by drawing a line that passes near as many of the points as possible. 9. An algebra class is investigating arm span versus height. The students collect the data and plot them. On the graph they create, x represents height and y represents arm span. The students find that the trend line that best fits the data is represented by the algebraic rule y = x. The trend line is shown on the graph. Interpret the meaning of this trend line s algebraic rule in the context of the problem situation.

3 Topic 12: Creating linear models for data CONSOLIDATION ACTIVITY Complete questions 1-7 to find an equation for the given trend line, then interpret and apply the equation. 1. Using the labels for the x- axis and y- axis on the graph, define the variables that you will use in this situation. Let x = Let y = 2. Of the five points listed, which three are on the trend line? Cross out the points that are not on the trend line. (0,40) (40,0) (50,3) (60,7) (80,14) 3. Use two of the points on the trend line to find the slope of the trend line. Calculation: Answer: Slope = 4. Find the y- intercept of the trend line. Show your work in the space provided. Work: Answer: y- intercept = 5. Write an equation of the trend line for this situation in slope- intercept form: 6. What do the slope and y- intercept of the trend line mean in the context of this situation?

4 484 Unit 4 Linear functions 7. Use the equation you wrote for the trend line to predict the shoe size for a person who is 65 inches tall. Show how you arrived at your answer. 8. In this lesson, you have learned some vocabulary terms related to data collection and data analysis. Complete the math journal that follows. State your ideas in your own words. You may want to use a situation you explored in this lesson to create the examples. Vocabulary term My understanding of the term Example that shows the meaning of the term a. Scatterplot b. Trend line c. Correlation d. Positive correlation e. Negative correlation

5 Topic 12: Creating linear models for data 485 HOMEWORK 12.1 Notes or additional instructions based on whole- class discussion of homework assignment: 1. Marcus collected data during his science lab experiment. The data are shown in the following table and graph. Time Temperature (seconds) ( C) a. Is the relationship between time and temperature a function? If so, what is the domain and range? b. Does there appear to be a correlation between the two variables? If so, classify the strength of the correlation as weak, moderate, or strong. c. What is the direction of the correlation (positive or negative)? Explain what it means for the variables to have a positive or negative correlation in the context of the problem situation. d. Add a trend line to the graph. Estimate the location of the trend line by drawing a line that passes as near to as many of the points as possible. e. Calculate the slope of the trend line. f. What is the practical meaning of the slope for this situation? 2. For each graph, add a trend line. Then describe the strength and direction of the correlation you see in the data. Under each graph, write two words: either strong or weak and either positive or negative.

6 486 Unit 4 Linear functions 3. For her math project, Jacqueline is investigating whether there is a relationship between the length of a person s signature and the number of letters in the person s name. She collects data from seven different people and then plots the data. From the graph, she sees that there is a fairly strong correlation between the two variables, so she adds a trend line. Answer the following questions to help Jacqueline find an equation for the trend line and make a prediction using the equation. a. Use the labels of the x- axis and y- axis on the graph to define the variables that will be used in this situation. Let x = Let y = b. List two points that are on the trend line. c. Use the two points you listed in part b to find the slope of the trend line. Calculation: Answer: slope = d. Explain what the slope means in the context of the situation. e. Find the y- intercept of the trend line from the graph. y- intercept = f. Explain what the y- intercept means in the context of the situation. g. Write an equation of the trend line for this situation in slope intercept form: h. Use the equation you wrote for the trend line to predict the number of letters in a person s name if the signature is 6 inches long. Show how you arrived at your answer.

7 Topic 12: Creating linear models for data 487 STAYING SHARP What are the slopes of line A and line B in the graph? Practicing algebra skills & concepts 2. Write equations for lines A and B. The pictures of these balances give you information about the relationships among the five weights. Preparing for upcoming lessons Use the pictures to complete each statement with equal to, less than, or greater than. Explain your reasoning. 3. The weight of C is the weight of D. How I know: 4. The total weight of A and E is the total weight of C and D. How I know: 5. Write an expression for the perimeter of the rectangle. Write the expression in the form ax + b, where a and b are numbers. 6. Write an expression for the area of the rectangle. Write the expression in the form ax + b, where a and b are numbers. Focus skill: Simplifying expressions

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9 Topic 12: Creating linear models for data 489 Lesson 12.2 Finding equations for trend lines 12.2 OPENER For each set of variables, tell whether you would expect a relationship to exist between the variables. If you feel a relationship exists, predict the direction of the relationship (positive or negative) and the strength of the relationship (strong, weak, or moderate). Variables 1. Independent: Outside temperature Dependent: Popsicle sales 2. Independent: Number of absences Dependent: Grade point average 3. Independent: Distance of flight Dependent: Airfare 4. Independent: A person s age Dependent: Time spent on cell phone Do you expect a relationship to exist between the variables? If a relationship exists, predict the direction. If a relationship exists, predict the strength. Yes No Positive Negative Strong Moderate Weak Yes No Positive Negative Strong Moderate Weak Yes No Positive Negative Strong Moderate Weak Yes No Positive Negative Strong Moderate Weak 12.2 CORE ACTIVITY 1. Complete the table with the data from the experiment. 2. Make a scatterplot of the data. Be sure to label the axes. Number of weights added 0 Distance from metal arm to bottom of weight hanger (cm)

10 490 Unit 4 Linear functions 3. In question 2, you made a scatterplot of the data from the experiment. The variables are (number of weights added, distance from metal arm to bottom of weight hanger). a. Based on your scatterplot, what conclusions can you make about the relationship between the variables? b. How would you classify the direction and the strength of the correlation between the variables? c. Consider the slope between any two points on the graph that shows a relationship between two variables. What must be true if the data are truly linear? 4. You have used a graph to analyze the data from the experiment. Now, use the table to analyze the data. Answer the following questions to write an equation for a trend line of these data. a. Find the rate of change each time one weight is added. Number of weights added Distance from metal arm to bottom of weight hanger(cm) b. Is the rate of change constant each time one weight is added? Is it approximately constant? c. What rate of change will you use for the slope of a trend line? Provide a justification for the rate that you use. d. What is the initial distance of the bottom of the weight hanger from the metal arm (when the number of weights added = 0)? e. Use the information in the previous parts of this question to write an equation of a trend line for these data. f. What would be an appropriate domain and range for the trend line that models the data? (Consider only the model and not the mathematical function.)

11 Topic 12: Creating linear models for data Use the table of data from the experiment and the answer choices provided to complete each statement below a. The y- intercept is. This value represents the distance from the metal arm to the bottom of the weight hanger when there are weights on the weight hanger. b. The average rate of change is cm/weight. c. An equation for the trend line modeling these data could be y = x Refer to the data from this experiment, as shown in the table in question 4, to answer the following questions, a. How far did the spring stretch from its initial length when 6 weights were added? b. Suppose 6 more weights were added to the same spring (for a total of 12 weights). According to Hooke s Law, how far would you expect the spring to stretch from its initial length?? 7. Suppose you saw this scatterplot in a newspaper. The plot shows a positive correlation between the number of ice cream cones sold and the number of shark attacks. a. Based on this scatterplot, could you make a reasonable prediction of the number of shark attacks if you knew the number of ice cream cones sold?. b. Does this mean that eating ice cream causes sharks to attack? 8. Consider another example. These data show gasoline consumption and curb weight (the weight of the car with no cargo or occupants) for a sample of different car models. What conclusions might you draw from these data? Is there a causal relationship between weight and fuel economy? 9. Now consider these two situations that you explored previously. In both cases, there is a correlation between the variables. Is there a causal relationship in either of these situations?

12 492 Unit 4 Linear functions 12.2 CONSOLIDATION ACTIVITY 1. Complete the following five steps to create a scatterplot and trend line for the data from the Hooke s Law experiment. Step Step 1: Enter the data from the spring experiment into the list editor. Pictures and notes to help me remember how to do this step Distance from metal arm Number of weights to bottom of weight added hanger(cm) Step 2: Set an appropriate window for the graph. Record the window that you used. Step 3: Create a scatterplot of the data. Draw a sketch of your plot on the axes given. Minimum x- value: Maximum x- value: Increment for x- axis: Minimum y- value: Maximum y- value: Increment for y- axis: Step 4: In 12.2 Core Activity, question 4e, you found an equation to model the trend line for the data. Enter the equation into your calculator. Record the equation you entered. Step 5. Use your calculator to graph your trend line. Draw a sketch of the trend line over the scatterplot you sketched in Step 3. Now that you have made your scatterplot and added a trend line, you can answer some questions related to the graph. 2. You made a scatterplot and added a trend line. Complete these sentences to describe the graph you created. a. The sign of the slope for the trend line is because the distances are as more weights are added. b. The units for the slope of the trend line are per. 3. You can also use the equation of your trend line, along with the capabilities of your graphing calculator, to make predictions for the distance from the metal arm to the bottom of the weight hanger, in centimeters, for different numbers of weights. a. Based on the trend line you found, what distance would you predict if 4.5 weights were added to the hanger? Show how you arrived at your answer. b. Based on the trend line you found, what distance would you predict if 7 weights were added to the hanger? Show how you arrived at your answer.

13 Topic 12: Creating linear models for data Here are data describing handwarmer sales during the days of the Village Winter Festival. Create a scatterplot and trend line for the data on your graphing calculator. (Re- reading the steps in question 1 may help you.) Daily low temperature (degrees F) Number of handwarmers sold that day Record the window that you used to graph your data. Minimum x- value: Maximum x- value: Increment for x- axis: Minimum y- value: Maximum y- value: Increment for y- axis: 5. Record the equation you found to model the trend line for the data:. 6. Complete these sentences describing the relationship: a. The sign of the slope for the trend line is because the number of hand warmers sold as the temperature increases. b. The units for the slope of the trend line are per.

14 494 Unit 4 Linear functions HOMEWORK 12.2 Notes or additional instructions based on whole- class discussion of homework assignment: 1. The table below shows the results of a new spring experiment. a. Calculate the differences for distance for adjacent rows in the table. Write your answers in the spaces provided. b. Calculate the differences for number of weights added for adjacent rows in the table. Write your answers in the spaces provided. c. Use your work from questions 1a and 1b to calculate a rate of change for adjacent rows in the table. You should calculate five separate rates of change. Show your calculations. d. Do the results you found in question 1c indicate that the relationship between the variables is approximately linear? Explain. e. Using your results from question 1c, calculate an average rate of change for the data in the table. f. What value can you use for the y- intercept of the linear equation that models these data? Explain your answer. g. Using your results from questions 1e and 1f, write an equation for the trend line that can be used to model these data. h. Based on your trend line, how far would you predict the spring to stretch if 5 weights were added to the hanger? i. Based on your trend line, how far would you predict the spring to stretch if 15 weights were added to the hanger?

15 Topic 12: Creating linear models for data For each set of data, sketch a trend line on the graph. Use the table and/or the scatterplot to write an equation for your trend line. Show how you arrived at the equation for your trend line. a. x y b. x y

16 496 Unit 4 Linear functions STAYING SHARP 12.2 Practicing algebra skills & concepts 1. Describe a situation that could be represented by the graph. 2. Aiesha leaves her house and rides her bike to the mall at a steady speed. She hangs out with some friends for a while, and then realizes she has stayed too long and must hurry home. She rides home at a steady speed, but faster than the speed at which she rode to the mall. Sketch a graph that represents this situation. The pictures of the scales give you information about the relationships among the weights of the different fruits. Preparing for upcoming lessons 3. Which weighs more, a pear or a pair of cherries? How do you know? 4. According to the three scales, a banana and a plum have the same weight. Justify that this is true. 5. Write an expression for the perimeter of the rectangle. Write the expression in the form ax + b, where a and b are numbers. 6. Write an expression for the area of the rectangle. Write the expression in the form ax + b, where a and b are numbers. Focus skill: Simplifying expressions

17 Topic 12: Creating linear models for data 497 Lesson 12.3 Transforming linear functions 12.3 OPENER In the arm span versus height activity, you considered the line with equation y = x. (In that context, the line represented people whose arm span and height were equal.) The solid line is modeled by the equation y = x. Imagine placing a two- sided mirror along the x- axis. The dotted line shows the reflected image of the line with equation y = x. What is the equation of this new (reflected) line? 12.3 CORE ACTIVITY 1. What is the parent function for linear functions? 2. How do other linear functions compare to the parent function? 3. One way to transform the graph of the line is to modify its slope. a. Explain how slope affects the graph of a line. b. Explain how positive and negative values for slope affect the graph of a line. c. Recall the equation for the trend line for the shoe size versus height situation: y = 0.35x 14. How does the slope of 0.35 in the equation affect the graph of the line when compared to the graph of its parent? 4. Another way to transform the graph of the line is to modify its y- intercept. a. Explain how the y- intercept, in general, affects the graph of a line when compared with the graph of its parent. b. How does the y- intercept of - 14 in the equation y = 0.35x 14 affect the line compared to its parent?

18 498 Unit 4 Linear functions 5. How would you transform the graph of y = x to produce the graph of y = 3x + 1? a. Explain in words. b. Use your answer to question 5a to graph the line for the equation y = 3x How would you transform the graph of y = x to produce the graph of y = x 3? a. Explain in words. b. Use your answer to question 6a to graph the line for the equation y = x How would you transform the graph of y = x to produce the graph of y = 3x + 2? a. Explain in words. b. Use your answer to question 7a to graph the line for the equation y = 3x Explain how you would transform the graph of y = x to produce the graph of y = ax + c, where a and c are any real numbers.

19 Topic 12: Creating linear models for data CONSOLIDATION ACTIVITY For each of the following problems, describe how the graph was transformed from the parent function, y = x. Then write an equation of the function for the transformed graph. An example is provided. Example: Graph How the graph was transformed from the parent function, y = x Equation of the function of the transformed graph Reflect the line y = x across the x- axis. Draw a new line that is 3 times as steep as the reflected line. Shift the resulting line up 1 unit. y = 3x

20 500 Unit 4 Linear functions 4. Graph How the graph was transformed from the parent function, y = x Equation of the function of the transformed graph Now, create a mystery graph for your partner to analyze. Then analyze the mystery graph your partner created. a. Graph a line on this coordinate plane. b. Write the equation corresponding to your line on your whiteboard or a sheet of paper. (Do not show your partner this equation!) Trade Student Activity Books with your partner. c. Find the equation of your partner s graph. d. Check: i. Did your partner find a correct equation for your line? ii. Did you find a correct equation for your partner s line?

21 Topic 12: Creating linear models for data 501 HOMEWORK 12.3 Notes or additional instructions based on whole- class discussion of homework assignment: 1. For each transformation described, sketch the graph that results on the grid provided. (Sketch all three on the same grid.) Label each graph with the appropriate letter. Then, write the function rule that represents the transformation. a. Shift the graph of y = x down 2 units. b. Shift the graph from part a up 5 units. Function rule: Function rule: c. Create a graph with the same y- intercept as the graph in part b, but that is 4 times as steep. Function rule: 2. How would you transform the graph of y = x to produce the graph of y = x + 4? Explain in words. 3. How would you transform the graph of y = x to produce the graph of y = 2x + 1? Explain in words. 4. How would you transform the graph of y = x to produce the graph of y = 2x + 4? Explain in words.

22 502 Unit 4 Linear functions 5. For each of the following problems, describe how the graph was transformed from the parent function, y = x. Then write an equation for the transformed graph. a. Graph How the graph was transformed from the parent function, y = x Equation of the function of the transformed graph b. c. d. e.

23 Topic 12: Creating linear models for data 503 STAYING SHARP Find the slopes of line A and line B in the graph. Practicing algebra skills & concepts 2. Write the equations for lines A and B. The pictures of the scales give you information about the weights of tennis balls, golf balls, and softballs. Preparing for upcoming lessons 3. How many tennis and golf balls balance 2 softballs? (Hint: What can you do to both sides of the first scale?) 4. Use your answer to Question 3 and the second scale to find the number of tennis balls that will balance 1 softball. Focus skill: Simplifying expressions 5. Rewrite each expression in the form ax + b, where a and b are numbers. a. 4x x + 5 b. 4x 7 + 2x 5 c. 4x 7 2x 5 6. Each equation is given in point- slope form, (y y 0 ) = m(x x 0 ). Rewrite them in slope- intercept form, y = mx + b. a. (y 9) = 2(x 1) b. (y + 7) = 4(x 6)

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25 Topic 12: Creating linear models for data 505 Lesson 12.4 Matching equations and graphs of lines 12.4 OPENER Huey, Louie, and Dewey were given this graph by their teacher. Their teacher asked them to find the rate of change of the linear relationship shown in the graph. 1. Describe the error in each student s work. Work Huey Louie Dewey Error 2. Calculate the correct rate of change from the graph CORE ACTIVITY Objective: Create sets of cards that represent the same relationship. Each set will have an information card, an equation card, and a graph card. Materials: There are eight information cards, eight equation cards, and eight graph cards. Instructions: Start with an information card, then take turns with your partner matching cards. When you find a match, explain to your partner how you know the cards match. Your partner should either agree with your explanation, or question it if it is not correct or clear. When you both agree on a set of matching cards, tape together the cards that form the set. To more easily check your answers, tape each set with the information card on the left, the equation card in the middle, and the graph card on the right, as shown here. Information card Equation card Graph card

26 506 Unit 4 Linear functions 12.4 CONSOLIDATION ACTIVITY 1. Earlier in the course, you learned some useful strategies for thinking about your thinking. Reflect on how you applied metacognitive strategies when working on the matching activity. The questions below are provided to guide your reflection. a. When and how did you make a plan for how to approach the matching activity? How did this help? b. When and how did you monitor and evaluate your progress in the matching activity? How did this help? c. Did you loop back to try another plan during the matching activity? d. Did you use the Mathematical Problem- Solving Routine from Unit 1? If so, how? 2. Use the Co- construction Routine to identify ways you can find a linear equation using various types of information. Discuss these questions with your partner and write at least two responses to each question on your whiteboards: a. From what information can you find the equation of a line? b. How can you determine the slope given an equation, a graph, or other information? c. How can you determine the y- intercept given an equation, a graph, or other information? Co- construction Routine 1. Work with your partner. Write features and methods on your whiteboard. 2. Discuss your observations with the class. Decide, as a class, which conclusions can accurately be drawn. 3. Record the conclusions in your Student Activity Book. Record the co- constructed conclusions here. Different information about a line from which you could find its equation: Ways to determine the slope of a line from an equation, a graph, or other information: Ways to determine the y- intercept of a line from an equation, a graph, or other information:

27 Topic 12: Creating linear models for data 507 HOMEWORK 12.4 Notes or additional instructions based on whole- class discussion of homework assignment: 1. Find an equation of the linear function corresponding to each situation. Write your function rule in slope- intercept form. a. The graph has a slope of 2 and passes through the point (3,10). b. The graph passes through the points in this table. x y slope = y- intercept = Equation of line in slope- intercept form: c. The graph is a line that passes through the points (0,1) and (2,4). d. The graph has a slope of 0 and passes through the point (11,5). e. The graph is shown here. f. The graph is a line that passes through the points (1,3) and (7,4). 2. A line passes through the points (7, 3) and (7,11). a. What is the equation for the line? b. Does the equation you wrote in part a represent a function? Explain why or why not.

28 508 Unit 4 Linear functions 3. Describe how each graph could be produced by transforming the parent function, y = x. a. Function Description of how to transform graph from parent function, y = x y = x 6 b. y = 2x + 1 c. y = - x 9 d. e. f.

29 Topic 12: Creating linear models for data 509 STAYING SHARP 12.4 Use the graph to answer questions 1 and Determine the speed of Car B. Practicing algebra skills & concepts 2. Which car left first, A or B? How much earlier did it leave? The pictures of the scales give you information about the weights of strawberries, limes, and oranges. Preparing for upcoming lessons 3. Here are five statements describing relationships among the fruits. Circle the one true statement. 1 orange = 2 limes 1 lime = 2 strawberries 1 orange = 4 strawberries 1 lime = 3 strawberries 1 orange = 1 lime + 1 strawberry 4. Justify your choice with an explanation or work. Focus skill: Simplifying expressions 5. For each expression write an equivalent expression without parentheses. a. 4(3x + 5) b. 4(3x 5) c. 4(3x + 5) d. 4(3x 5) 6. Show that the expressions are equivalent. Expression 1: 4x x x + 2 Expression 2: 6x x x + 4

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31 Topic 12: Creating linear models for data 511 Lesson 12.5 More on transforming linear functions 12.5 OPENER Complete the following puzzle by matching each transformation with its verbal description. Write the letter of the graph in the space beside its description. You may use some answers more than once. Graph descriptions: 1. The graph of y = 3x 2 shifted up 4 units 2. A graph that is as steep as the graph of y = x and is shifted down 1 unit 3. The linear parent function reflected across the x- axis and shifted up 2 units 4. A graph that is 3 times as steep as the graph y = x and shifted up 2 units 5. The graph of y = x reflected across the x- axis and shifted down 1 unit 12.5 CORE ACTIVITY 1. Use equations to reproduce the X pattern of the flag shown on the left on your graphing calculator. When you are done, your graphing calculator screen should look something like the image on the right. Use what you have learned about transforming linear equations to help you.

32 512 Unit 4 Linear functions Here are some tips: Use a viewing window of [- 10,10] [- 10,10]. Start with the parent function y = x. You will need four functions to create the pattern. (The diamond in the center is formed by the crossing of the four lines.) Once you have created the X pattern, you may want to turn off the graph axes to more closely match the flag. Report the equations you used to create the image. Equation 1: Equation 2: Equation 3: Equation 4: 2. After reproducing the X flag pattern, see if you can reproduce these patterns as well. Note the equations you used to do so. a. b. c. d ONLINE ASSESSMENT Today you will take an online assessment.

33 Topic 12: Creating linear models for data 513 HOMEWORK 12.5 Notes or additional instructions based on whole- class discussion of homework assignment: 1. Describe how you would transform the graph of y = x to produce the graph of each of the following functions. Your descriptions should use vocabulary like shift up (or down), reflect across the x- axis, and make three times as steep. Function Description of how to produce graph from parent function, y = x a. y = 2x 4 b. y = 0.5x + 2 c. y = - x Review your graphing skills using the intercepts method. For each of the following equations, find the intercepts and use them to create a graph. Then describe how the graph you create compares to the graph of the parent function y = x. Finally, write the equation of the line in slope- intercept form, either by rewriting the equation, or using the graph. (A graph of the parent function y = x is provided as a dotted line on each coordinate grid to help you.) Function, intercepts, and description of how graph compares to y = x a. 2x + 3y = 6 x- intercept: y- intercept: Graph Description of how graph compares to parent function, y = x: Slope- intercept form:

34 514 Unit 4 Linear functions b. x + y = 3 x- intercept: y- intercept: Description of how graph compares to parent function, y = x: Slope- intercept form: c. x y = 4 x- intercept: y- intercept: Description of how graph compares to parent function, y = x: Slope- intercept form: d. 4x + 5y = 20 x- intercept: y- intercept: Description of how graph compares to parent function, y = x: Slope- intercept form: e. x + 2y = 2 x- intercept: y- intercept: Description of how graph compares to parent function, y = x: Slope- intercept form:

35 Topic 12: Creating linear models for data 515 STAYING SHARP 12.5 Practicing algebra skills & concepts 1. Graph the line that contains the points (10,2) and (4,8) and determine its slope. 2. Write an equation for the line with y- intercept 5 that is perpendicular to the line with equation y = x + 2. Preparing for upcoming lessons 3. Bob and Ted sit on one side of a seesaw. When Adam, who weighs 110 pounds, sits on the other side, the seesaw tilts down on Adam s side. a. What can you conclude about Bob and Ted s combined weight? b. What can you conclude about Bob and Ted s individual weights? 4. Suppose a package of pasta and a jar of sauce exactly balance a loaf of bread and a container of jelly on a scale. If the pasta weighs 1.5 pounds, the sauce weighs 1.2 pounds, and the bread weighs 2.4 pounds, how much must the jelly weigh? 5. Rewrite each expression as simply as possible. 6. Are the two expressions equivalent? Focus skill: Simplifying expressions a x x b. 9 3x + 6 4x c x + 5 8x Expression 1: 4(2x 5) Expression 2: 8x 5 Answer with supporting work:

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37 Topic 12: Creating linear models for data 517 Lesson 12.6 Modeling data with linear functions 12.6 OPENER Janelle and Danielle are working together on an experiment. They drop a tennis ball from various heights, then record both the drop height and the bounce height. After they graph their data, they add a trend line to the scatterplot. When they compare their graphs, they notice that the data points in both graphs are the same, but they have different trend lines. Here is part of their discussion: Janelle: Danielle: Janelle: Why do our trend lines look so different? How did you make yours? Well, I saw that there were six data points. So I made sure that three were above the line and three were below the line. That way, my trend line goes through the middle of the data. How did you make your trend line? I just looked at the graph and thought about a line that came as close as possible to all of the points. I don t think a trend line has to have half the points above it and half below it. Janelle s Graph Danielle s Graph Consider the graphs, along with the explanations each student gave for how they found their trend lines. 1. Which student has found a trend line that more accurately represents the data? Explain why. 2. Do you agree with Danielle s statement that half of the data points have to be above the trend line and half below it? Explain why or why not.

38 518 Unit 4 Linear functions 12.6 CORE ACTIVITY 1. Here are two sample trend lines modeling the relationship between height and shoe size. Which one do you think fits the data better? Why do you think so? 2. Create a scatterplot on your calculator and use the linear regression feature to find the line of best fit. Record the information on your calculator screen, then use that information to write the regression equation. Round to the nearest hundredths place if necessary. 3. Write your equation in context, replacing the variables with the labels. 4. Describe the correlation between the height and shoe size. This is related to the r- value shown on your calculator screen. 5. Your calculator showed an r- value that is close to 1. What does an r- value close to 1 indicate? What does an r- value close to - 1 indicate? What does an r- value close to 0 indicate?

39 Topic 12: Creating linear models for data Match the r- values with their graphs. Possible r- values: r = 0.33; r = 0.87; r = Use the trend line equation y = 0.35x 14 to answer the following questions. a. Find a prediction for the shoe size of a boy who is 67 tall. b. There actually was a boy 67 tall. Did the equation predict the correct shoe size for that boy? If not, was the prediction an overestimate or an underestimate? 8. Now use the equation to find a prediction for someone your height. Did the equation correctly predict your shoe size? If not, was the prediction an overestimate or an underestimate? 9. Locate the point on the graph that represents the 67 tall boy. Draw a vertical line from the point to the trend line. Do the same for the remaining points. What do you notice about the number and location of overestimates and underestimates?

40 520 Unit 4 Linear functions 10. Here are three trend lines of the (height, shoe size) data. Based on the total error illustrated by the area of the error- squares, which of these is the best model for the data? 11. Look at the regression equation you found in question 3. Use this equation to determine the predicted shoe size for each student. Then find the residual (the overestimate or underestimate) for each point. Height in inches Actual Shoe Size Predicted Shoe Size Residual = Actual Predicted 12. Complete the residual plot for these data by plotting the residual value for each corresponding student height.

41 Topic 12: Creating linear models for data REVIEW ONLINE ASSESSMENT You will work with your class to review the online assessment questions. Problems we did well on: Skills and/or concepts that are addressed in these problems: Problems we did not do well on: Skills and/or concepts that are addressed in these problems: Addressing areas of incomplete understanding Use this page and notebook paper to take notes and re- work particular online assessment problems that your class identifies. Problem # Work for problem: Problem # Work for problem: Problem # Work for problem:

42 522 Unit 4 Linear functions HOMEWORK 12.6 Notes or additional instructions based on whole- class discussion of homework assignment: Next class period, you will take an end- of- unit assessment. One good study skill to prepare for tests is to review the important skills and ideas you have learned. Use this list to help you review these skills and concepts, especially by reviewing related course materials. Important skills and ideas you have learned in the unit Linear functions: Use the connection between constant rate of change and slope to analyze and graph linear functions Use common, or first differences, to determine if a relationship is linear or approximately linear Write the equation of a line in different forms (slope- intercept, standard, and point- slope forms) Identify the strength and direction of correlation for approximately linear data Fit a trend line to approximately linear data and write an equation for a trend line Understand the relationship between the slope and y- intercept of the graph of a linear model and the situation being modeled Understand the effects of changing m or b on the graph of y = mx + b and transform the parent function y = x to create other linear functions Use slope to classify lines as parallel, perpendicular, or neither Understand that maintaining motivation while engaged in learning tasks can result in more effective effort Recognize that setting goals can help maintain motivation Understanding characteristics of effective goal setting Practice collecting like terms Practice applying the distributive property Homework Assignment Part I: Study for the end- of- unit assessment by reviewing the key ideas listed above. Part II: Complete the online More practice for this topic. Note the skills and ideas for which you need more review, and refer back to related activities and animations to help you study. Part III: Complete Staying Sharp 12.6.

43 Topic 12: Creating linear models for data 523 STAYING SHARP 12.6 Practicing algebra skills & concepts Use the graph to answer questions 1 and Does the graph represent a proportional or a non- proportional relationship? Explain. 2. Write an equation to model the situation represented by the graph. Preparing for upcoming lessons 3. When Ian and Leo sit opposite from Adam on a seesaw, it is perfectly balanced. When Ian and Leo sit opposite Howard and his dog, it is again perfectly balanced. What will happen if Adam sits opposite Howard and his dog? Answer with explanation: 4. If a = b and c d, which symbol makes the following statement true:,,, >, <, or =? Justification: a + c b + d Focus skill: Simplifying expressions 5. Rewrite each expression without parentheses. a. 4(3x + 5y) b. 7(2x 5 + 6y) c. 3(7x + 9 4y) 6. Rewrite the following expression as simply as possible, as the sum of two terms (without parentheses). 3(2x + 4) 2x

44 524 Unit 4 Linear functions

45 Topic 12: Creating linear models for data 525 Lesson 12.7 Checking for understanding 12.7 OPENER 1. Reflect on your work in the unit Linear functions. List two ideas that you understand well. 2. What are some ways to relax during a test- taking situation? 12.7 END-OF-UNIT ASSESSMENT Today you will take an end- of- unit assessment CONSOLIDATION ACTIVITY 1. The table contains data collected about Old Faithful. The table shows the relationship between duration of an eruption and the time until the next eruption. Find a line that fits the data in the table using the methods described below. o Method 1: Plot the data on the grid provided. Place a trend line on the graph. Write an equation for the trend line. o Method 2: Plot the data on a graphing calculator with an appropriate viewing window. Then, find the line of best fit. Record the equation of this line along with the associated r- value. Duration of eruption (minutes) Time until next eruption (minutes)

46 526 Unit 4 Linear functions 2. Write an equation for your trend line and the line of best fit. Explain how you found the equation for your trend line. 3. Use both equations to predict the amount of time until the next eruption, given the duration of the previous eruption. Show how you arrived at your answers. How different are the estimations? a. Previous eruption lasted 2.25 minutes b. Previous eruption lasted 4.8 minutes 4. After you write and apply your equations, discuss these questions with your partner. You will also discuss them as a whole class. a. What variable do the park rangers use to make predictions? What variable are they trying to predict? Given your answers to these questions, what are the independent and dependent variables? How do you know which variable to put on each axis of a graph representing these data? b. How did you decide where to place your trend line on your scatterplot? How did you use your trend line to find the slope and y- intercept? c. Do you expect different students to find the same equation for a trend line? d. Does the y- intercept have real meaning in this situation? What about the slope? Do the values and their signs make sense in this situation? e. If the last eruption lasted 2.25 minutes, what does your equation tell you about how long park visitors will have to wait until the next eruption? Does your equation tell you for sure? f. How would different trend line equations affect the predictions for the time until the next eruption? g. How does the trend line you wrote compare to the line of best fit computed by the calculator? How strong is the correlation? How do you know?

47 Topic 12: Creating linear models for data Answer the following questions to reflect on your performance and effort this unit. a. Summarize your thoughts on your performance and effort in math class over the course of this unit of study. Which areas were strong? Which areas need improvement? What are the reasons that you did well or did not do as well as you would have liked? b. Set a new goal for the next unit of instruction. Make your goal SMART. Description of goal: Description of enabling goals that will help you achieve your goal:

48 528 Unit 4 Linear functions HOMEWORK 12.7 Notes or additional instructions based on whole- class discussion of homework assignment: A small town has two museums. The Railroad Museum charges an admission fee of $5. The Town History Museum requests a donation of $3 for admission, but it does not require patrons to pay this fee. The town s Director of Cultural Affairs collects data on the two museums for ten different days. The data are shown in the tables. The number of people visiting each day is reported in the input column, and the money collected (in dollars) in the output column. Railroad Museum Town History Museum Number of patrons Dollars collected Number of patrons Dollars collected Plot the data for the Railroad Museum on the following axes. 2. Plot the data for the Town History Museum on the following axes.

49 Topic 12: Creating linear models for data Answer the following questions to compare the data from the two museum situations. a. Describe the pattern of the data points for the Railroad Museum graph. b. Explain why the data take this pattern for the Railroad Museum graph. c. Write an equation to represent the data for the Railroad Museum graph. d. Describe the pattern of the data points for the Town History Museum graph. e. Explain why the data take this pattern for the Town History Museum graph. f. Explain why it might be hard to write an equation to exactly represent the data for the Town History Museum graph. 4. Add a trend line to the Town History Museum graph you made in question 2. Then write an equation for the trend line. Show how you determined the equation. 5. What would be an appropriate domain and range for each of the two relationships you graphed? 6. Now enter these data into a calculator and find the correlation coefficient and line of best fit for both sets of data. a. What are the two correlation coefficients? b. What does the correlation coefficient tell you about the data in each case? c. How close were your trend lines to the lines of best fit reported for these two data sets? 7. If you did not finish it in class, complete the Consolidation activity. Then, summarize the results of that activity here. a. An overall rating of my effort in the unit Linear functions (excellent, good, okay, or need to improve): b. An overall rating of my performance in the unit Linear functions (excellent, good, okay, or need to improve): c. A goal for the next unit in the course: d. Steps that will help me achieve my goal:

50 530 Unit 4 Linear functions STAYING SHARP 12.7 Practicing algebra skills & concepts Use the graph to answer questions 1 and What are the x- intercepts and y- intercepts of line A and line B? 2. Use the intercepts you found in question 1 to calculate the slopes of line A and line B. Preparing for upcoming lessons 3. Check whether each value in the table makes the inequality 5x + 3 > 38 true when substituted for x. Value True or false 4. Hylenne is thinking of a number that is at least 9. Identify each statement as definitely true, could be true or false, or definitely false. Explain your answers. a. Hylenne s number is greater than 10. b. Hylenne s number is greater than 8. c. Hylenne s number is less than Focus skill: Simplifying expressions 5. Rewrite the following expression as simply as possible, without parentheses. 5(2y + 3) + 6y 6. Rewrite the following expression as simply as possible, without parentheses. 4(2x + 5y) + 5(3x + 4y)