T936 Mathematics Success Grade 8 [OBJECTIVE] The student will find the line of best fit for a scatter plot, interpret the equation and y-intercept of the linear representation, and make predictions based on the line of best fit. [PREREQUISITE SKILLS] Scatter Plots (Lesson 34), plotting points, slope [MATERIALS] Student pages S459 S471 Rulers, Graphing calculators (optional) [ESSENTIAL QUESTIONS] 1. How do you find the equation for the line of best fit without a graphing calculator? Justify your answer. 2. When using the linear representation of a scatter plot in the form of y = mx + b, what does the b represent? Defend your thinking. 3. Explain how the linear model of a scatter plot can be used to solve real world problems. [WORDS FOR WORD WALL] line of best fit, y-intercept, y = mx + b, slope [GROUPING] Cooperative Pairs (CP), Whole Group (WG), Individual (I) [LEVELS OF TEACHER SUPPORT] Modeling (M), Guided Practice (GP), Independent Practice (IP) [MULTIPLE REPRESENTATIONS] SOLVE, Pictorial Representation, Verbal Description, Algebraic Formula, Table, Graph, Graphic Organizer [WARM-UP] (IP, I, WG) S459 (Answers on T945.) Have students turn to S459 in their books to begin the Warm-Up. Students will determine the type of association for each scatter plot and explain their thinking. Monitor students to see if any of them need help during the Warm-Up. Have students complete the problems and then review the answers as a class. {Algebraic Formula, Pictorial Representation, Graph} [HOMEWORK] Take time to go over the homework from the previous night. [LESSON] [2 3 Days (1 day = 80 minutes) M, GP, WG, CP, IP] SOLVE Problem (GP, WG) S460 (Answers on T946.) Have students turn to S460 in their books. The first problem is a SOLVE problem. You are only going to complete the S step with students at this point. Tell students that today they will learn about the line of best fit for scatter plots, and they will use this knowledge to complete this SOLVE problem at the end of the lesson. {SOLVE, Graphic Organizer, Verbal Description, Table}
Mathematics Success Grade 8 T937 Line of Best Fit (M, GP, CP, WG) S460, S461 (Answers on T946, T947.) M, GP, CP, WG Students will work with the scatter plot on S460 as they explore creating a linear representation of the scatter plot also known as the line of best fit. Assign the roles of Partner A and Partner B to students. {Pictorial Representation, Graph, Algebraic Formula, Verbal Description} MODELING Line of Best Fit Step 1: Have student pairs look at the scatter plot on S460 and be prepared to describe the elements of the graph. Partner A, what is the title of the scatter plot? (Hotel Floors vs. Hotel Rooms) Partner B, what is the label of the x-axis? (Number of Floors) Partner A, what is the scale of the x-axis? Explain your answer. (1 Although the numbers on the scale are by twos, there is a line between each value which means that the scale is actually 1.) Partner B, what is the label for the y-axis? (Number of Hotel Rooms) Partner A, what is the scale for the y-axis? Explain your answer. (10 When you look at the values on the y-axis, they are numbered by 20, but there is a line between each value which means that the scale is 10.) Partner B, describe any unusual features you may observe in the graph. (The y-axis does not start at 0.There is a zig-zag line to indicate that there are values not shown on the scale between 0 and the lowest value.) Step 2: Have students turn to S461. They will need to refer back to the scatter plot on S460 to answer the questions and explore the concept of linear representation and the line of best fit. Partner A, what type of association is there between the number of floors in a hotel and the number of hotel rooms? Explain your thinking. (There is a positive association. The more floors there are, the more rooms there are.) Record. Partner B, if we want to write an equation to represent a linear model of the data, what do we need to do first? (Draw a line of best fit.) Record. Have student pairs discuss how they could draw a line through the data on the scatter plot so that it would best represent the data. Partner A, how can we draw a line through the data? Explain your thinking. (The line should follow the data and if it passes through the middle of the data it would split the data in half.) Partner B, explain how to draw a line of best fit based on your discussion. (The line should pass through the middle of the data points with an equal number grouped on each side of the line.) Record.
T938 Mathematics Success Grade 8 Hotel Floors vs. Hotel Rooms Number of Hotel Rooms 200 180 160 140 120 2 4 6 8 10 12 Number of Floors This line is not correct, because none of the points are above the line. Hotel Floors vs. Hotel Rooms Number of Hotel Rooms 200 180 160 140 120 2 4 6 8 10 12 Number of Floors This is not a good line of best fit, because none of the points are below the line. Hotel Floors vs. Hotel Rooms Number of Hotel Rooms 200 180 160 140 120 This line is a good example. 2 4 6 8 10 12 Number of Floors
Mathematics Success Grade 8 T939 Step 3: Use a ruler to model how to draw a line of best fit. Explain to the students that you want the line to be in the middle of the data. A good rule to follow is that about half of the data points are above the line, and half of the data points are below the data line. Place your ruler in different places. Ask the students if they think the line follows that rule, and if the line would be the best fit for the data. (Refer to the scatter plots on T938.) Step 4: Partner A, why is this line of best fit a good example? (About half of the data is above the line, and half the data is below the line.) Model how to draw the line as students draw the line. Step 5: Have student pairs discuss what they notice about the line on the scatter plot. (It passes through three points.) Partner A, how many points are needed to write the slope of a line? (2) Have students list two points on the line that was drawn. [(4, 127) and (10, 187)] Record for Question 5. Step 6: Partner B, when we write the equation of a line when we know two points, what do we need to do? (Find the slope.) Review the formula for determining the slope using two points. (change in y over change in x) m = y 2 y 1 x 2 x 1 = 187 127 10 4 = 60 6 = 10 Step 7: Partner A, once we find the slope, how can we use that information to find the y-intercept. Explain your thinking. (Substitute in the slope for m and then choose one of the points to substitute in the values for x and y. Solve the problem for the variable of b which will be the y-intercept.) Model the steps for finding the y-intercept with the students, and then write the equation in slope-intercept form. (y = mx + b) y = mx + b y = mx + b 127 = 10(4) + b y = 10x + 87 127 = 40 + b 40 40 87 = b Step 8: Have students look at Question 7. Partner B, explain how we can use the equation of the line of best fit to estimate how many rooms the new hotel will have. (Write the equation of the line and then substitute in 15 for the value of x, which is the number of floors.) y = 10x + 87; so if x = 15, then y = 10(15) + 87; y = 150 + 87; y = 237, so the hotel will have 237 rooms.
T940 Mathematics Success Grade 8 Line of Best Fit and Data Prediction (GP, IP, WG, CP, M) S462 (Answers on T948.) M, GP, CP, WG Students will work with scatter plots to draw a line of best fit, find the equation for the line of best fit and make predictions based on the linear represesntation. {Graph, Algebraic Formula, Verbal Description, Graphic Organizer} MODELING Line of Best Fit and Data Prediction Step 1: Have students look at Scatter Plot A on S462. Partner A, what is the title of the scatter plot? (Snakes) Partner B, what is the label of the x-axis? (Length in feet) Partner A, what is the scale of the x-axis? (1) Partner B, what is the label for the y-axis? (Weight in pounds) Partner A, what is the scale for the y-axis? (1) Step 2: Have students draw a line of best fit for the scatter plot. Partner B, what do we need to remember when drawing the line of best fit. (About half the data should be above the line and half the data should be below the line.) Partner A, is there anything else we want to keep in mind? (If we can draw the line so that it passes through two points, we can use those two points to determine the equation of the line.) Step 3: Let s choose the two points of (2, 2) and (4, 3) to determine the equation for the line of best fit. Partner A, what is the first step to find our equation of the line? (Substitute in the values of the two points into the formula for slope.) m = y 2 y 1 x 2 x = 3 2 1 4 2 = 1 2 Partner B, what is the slope of our line? ( 1 or 0.5) 2 Partner A, what is the next step. (We substitute in the value of the slope for m and then choose the x-and y-coordinates from one of our points and substitute those values into our equation.) y = mx + b 2 = 0.5(2) + b 2 = 1 + b 1 = b
Mathematics Success Grade 8 T941 Partner B, what is the slope of our line? ( 1 ) Explain what this means. 2 (For every unit we move up, we move to the right two units.) Partner A, what is the value of the y-intercept? (1) Explain the meaning of this value. (This is where the line will meet the y-axis.) Partner B, what is the equation of the line? (y = 0.5x + 1) Step 4: Have student pairs look at the last column. Partner A, what will we be doing in the last column? (Using the equation to make predictions about the scatter plot.) Partner B, what is this problem asking us to find? (the weight, if the length is 12 feet) Partner A, explain how we can use the linear equation to find this information. (We can write the equation and then substitute in the value of x, which is the length of the snake, to find the value of y, which is the weight of the snake at that length.) Step 5: Partner B, what is the equation of the line? (y = 0.5x + 1) Record. Partner A, explain our next step. (Substitute in the known value of the snake length, 12 feet, for the x-value in the equation.) Record. Partner B, what is the final step? (Complete the computation and find the value of y, which is the weight of the snake.) Record. Partner A, what would be the weight of a snake that is 12 feet in length? (7 pounds) Record. y = 0.5x + 1 y = 0.5(12) + 1 y = 6 + 1 y = 7 The snake would weigh 7 pounds. IP, CP, WG: Have students work with partners to complete the questions about Scatter Plots B and C on S462. Students will draw the line of best fit, select two points, find the equation of the line and then use the equation to make predictions about data that is not plotted on the scatter plot. When students are done with the graphic organizer, review the answers as a whole group. {Verbal Description, Graphic Organizer, Graph, Algebraic Formula}
T942 Mathematics Success Grade 8 Scatter Plots and Real World Situations (GP, IP, WG, CP, M), S463, S464, S465, S466 (Answers on T949, T950, T951, T952.) M, GP, CP, WG Students will be given a real world problem and a table of data. They will have to create a scatter plot and then identify the association, draw a linear model, identify the equation of the line and y-intercept. Be sure that students know their designation as Partner A or Partner B. {Graph, Algebraic Formula, Verbal Description, Algebraic Formula} MODELING Scatter Plots and Real World Situations Step 1: Have students look at Problem 1 on S463 and read it with their partners. Partner A, what information are we going to plot? (the minutes studied and the math scores) Partner A, what would be a good title for the scatter plot? (possible answer: Math Exam) Record. Partner B, what is the label of the x-axis? (Minutes Studied) Record. Partner A, what would be an appropriate scale for the x-axis? (suggested answer: 30) Defend your thinking. (I found the maximum amount, which is 150, and the minimum, which is 30. I noticed that the times were in half hours, so I chose 30 for the scale) Record the scale on the graph. Partner B, what would be a good label for the y-axis? (Scores) Record. Partner A, what would be an appropriate scale for the y-axis? (5) Defend your thinking. (I found that the lowest score is 60 and the highest score is 98. I decided to start the graph at 50 and made the mark to indicate a broken scale. I started with the value of 50 and used a scale of 5.) Record the scale on the y-axis. Step 2: Have student pairs plot the points on the coordinate graph. *Teacher Note: You can plot the points on a graph on the wall or using technology so that students can compare their graphs to the model. Step 3: Partner B, what type of association is there between the time spent studying and the student scores? Explain your thinking. (There is a positive association. The more time spent studying, the better grade a student will earn.) Record. Step 4: Have students draw a line of best fit for the data. Partner A, what is one thing that we want to keep in mind when drawing the line of best fit? (Approximately half of the data should be above the line and half of the data should be below the line.)
Mathematics Success Grade 8 T943 Partner B, what is another thing to keep in mind? (We want to try and have the line pass through two points in order to make it easier to find the equation for the line of best fit.) Step 5: Have students turn to S464. Let s choose the two points of (60, 72) and (150, 98) to determine the equation for the line of best fit. Partner A, what is the first step to find our equation of the line? (Substitute in the values of the two points into the equation.) m = y 2 y 1 98 72 x 2 x = 1 150 60 = 26 90 = 13 45 Partner B, what is the slope of our line? ( 13) 45 Partner A, what is the next step? (We substitute in the value of the slope for m and then choose the x- and y-coordinates from one of our points and substitute those values into our equation.) y = mx + b 72 = 13 45 (60) + b 72 = 52 3 + b 54 2 3 = b y = 13 45 x + 54 2 3 Step 6: Partner B, what does the y-intercept represent in the context of the problem? (the score a student would expect to get if he/she did not study at all) Record. Partner A, what is the slope of the line? ( 13 ) Record. Explain the 45 meaning of the slope in the context of the problem. (For every 45 minutes studied, the score was increased by 13 points.) Record. Step 7: Partner B, what score would you expect from a student who studied 105 minutes? Let s determine the answer by looking at the graph and then by substituting in the value of 105 into the equation of our line. Partner A, what is the score for 105 minutes studied if you look at the graph? (85) Record. Partner B, model how to determine the answer by substituting in the value of 105 into the equation we found. y = 13 45 (105) + 54 2 3 y = 30 1 3 + 54 2 3 = 85 Partner A, were the two answers in Problem 7 the same? (Yes) Record. Do you imagine that will always be the case? (No, because you may not always be able to pick the exact point by looking at the scatter plot.) Record.