1. This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. Click to go to website: www.njctl.org New Jersey Center for Teaching and Learning Progressive Mathematics Initiative

2. 8th Grade Data www.njctl.org 2013-07-09

3. Use Normal View for the Interactive Elements To use the interactive elements in this presentation, do not select the Slide Show view. Instead, select Normal view and follow these steps to set the view as large as possible: On the View menu, select Normal. Close the Slides tab on the left. In the upper right corner next to the Help button, click the ^ to minimize the ribbon at the top of the screen. On the View menu, confirm that Ruler is deselected. On the View tab, click Fit to Window. Use Slide Show View to Administer Assessment Items To administer the numbered assessment items in this presentation, use the Slide Show view. (See Slide 13 for an example.) Setting the PowerPoint View

4. Table of Contents Two Variable Data Determining the Prediction Equation Two-Way Table Line of Best Fit Click on a topic to go to that section

5. Two Variable Data Return to Table of Contents

6. Two Variable Data is also called Bivariate Data With bivariate data there are two sets of related data that you want to compare.

7. Example 1: An ice cream shop keeps track of how much ice cream they sell versus the temperature on that day. Temperature degrees F Ice Cream Sales $ 57.5215 61.5325 53185 60332 65406 72522 67412 77614 74541 64.5421 This table shows 10 days of data. The two variables are: Temperature and Ice Cream Sales. We can create a scatter plot by plotting the points. Temperature is the x variable Sales is the y variable.

8. Ten Days of Ice Cream Shop Sales Scatter Plot

9. What did the scatter plot show us? Using the Scatter Plot it is easy to see that: warmer weather leads to more sales. Click to reveal

10. Scatter Plots are either: Linear Non-linear

11. These scatter plots are also non-linear.

12. If a scatter plot is linear it can be described 3 ways: Negative Association Positive Association No Association

13. 1What type of scatter plot is shown from the Ice Cream Shop example 1? Anon-linear Blinear, positive association Clinear, negative association Dlinear, no association

14. Example 2: Data for 10 students math and science grades are shown in the table. Plot the points to create the scatter plot. Math Grade Science Grade 5662 9693 8581 8482 6360 10098 7881 8991 4648 75

15. 2What type of scatter plot is shown for the math and science grades from example 2? Anon-linear Blinear, positive association Clinear, negative association Dlinear, no association Click to reveal solved graph.

16. 3What kind of association is shown in the graph? Anon-linear Blinear, positive association Clinear, negative association Dlinear, no association

17. 4What kind of association is shown in the graph? Anon-linear Blinear, positive association Clinear, negative association Dlinear, no association

18. 5What kind of association is shown in the graph? Anon-linear Blinear, positive association Clinear, negative association Dlinear, no association

19. 6Which of the following scenarios would produce a linear scatter plot with a positive correlation? AMiles driven and money spent on gas BNumber of pets and how many shoes you own CWork experience and income DTime spent studying and number of bad grades

20. 7Which of the following would have no association if plotted on a scatter plot? ANumber of toys and calories consumed in a day BNumber of books read and reading scores CLength of hair and amount of shampoo used DPerson’s weight and calories consumed in a day

21. What kind of predictions can you make from looking at the graph?

22. Number of Hours Resting Heart Rate 1261 678 1070 090 1665 285 475 1462 378 187 869 A student wanted to find out if there was a relationship between the number of hours a person exercised in one week and their resting heart rate. 15 people were surveyed and the table at the right shows the results.

23. Number of Hours Resting Heart Rate 1261 678 1070 090 1665 285 475 1462 378 187 869 Plot the results of the survey on a scatter plot.

24. Is there a linear relationship? Is there a positive or negative association? According to your scatter plot, does a person who exercise generally have a lower resting heart rate than a person that doesn't exercise?

25. Hours Math Grade 296 775 486 194.597 870 290 387 1068 194 675 488 Sandy wanted to find out if there was a relationship between the number of hours a student spent browsing the Internet in each day and their math grades for the marking period. She surveyed several students and the results are shown in the table at the right.

26. Hours Math Grade 296 775 486 194.597 870 290 387 1068 194 675 488 Plot the points on the graph. Make sure to include your labels and a title.

27. Look at your results. Is the scatter plot linear or non-linear? Is there a positive or negative association? What can you say about the math scores as more hours are spent browsing the Internet?

28. Year Temperature in F 2,00030.4 2,00130.1 2,00237.3 2,00326.7 2,00424.8 2,00530.3 2,00638.9 2,00737.1 2,00834.5 2,00927.3 2,01031.4 The table shows average temperatures for the month of January in New Jersey from 2000 to 2009. Is it linear? Is there a positive association, negative association, or neither?

29. Month Temperature in F 135.4 238.8 349.8 452.8 565.3 670.2 778.2 875 967 1057 1149 1240.8 The table shows average temperature by month for New Jersey. Month 1 = January, Month 2 = February, etc. Make a scatter plot using the data from the table. Is the graph linear? Is there an association?

30. Shoe Size Girl's Height in Inches 555 5.554 864 7.565 970 652 7.563 866 Shoe Size v. Girl’s Height Height in Inches Shoe Size

31. Girl's Height in Inches 555 5.554 864 7.565 970 652 7.563 866 Shoe Size v. Girl's Height Shoe Size Height in Inches 8 What association is shown in this graph? Anon-linear Blinear, positive association Clinear, negative association Dlinear, no association

32. Poll 10 girls and 10 boys from your class on their heights and shoe size. Make a scatter plot for your observations. Girl’s Height (in inches) Shoe Size Boy's Height (in inches) Shoe Size

33. Survey your classmates and to find out what time they wake up on a school day and how long it takes them to get ready. Make a scatter plot of your results. Wake Up Time How Long to Get Ready Is there an association with the time a student wakes up and how long it takes them to get ready?

34. Line of Best Fit Return to Table of Contents

35. Bivariate data plotted on a scatter plot shows us negative or positive association (correlation). A line of best fit, or trend line, can help us predict outcomes using the data that you already have. It is drawn on a scatter plot that best fits the data points.

36. Notice that the points form a linear like pattern. To draw a line of best fit, use two points so that the line is as close as possible to the data points. Our line is drawn so that it fits as close as possible to the data points. This line was drawn through (35,82) and (50,90).

37. Time spent studying Test Score Predict the test score of someone who spends 52 minutes studying Predict the test score of someone who spends 75 minutes studying

38. Predict the height of a person who wears a size 8 shoe Predict the shoe size of a person who is 50 inches tall Draw a line of best fit, or trend line, on this graph.

39. 9Consider the scatter graph to answer the following: Which 2 points would give the best line of fit? XY 39 4.58 57 65 84 93 101 B and C A B C D A and D C and D there is no pattern

40. 10Consider the scatter graph to answer the following: Which 2 points would give the best line of fit? B and C A B C D A and D C and D there is no pattern

41. XY 296 775 486 194 0.597 870 290 387 1068 194 675 488 11Which 2 points would you pick to draw the line of best fit? B and C A B C D A and D C and D there is no pattern

42. Shoe Size Girl's Height in Inches 555 5.554 864 7.565 970 652 7.563 866 12Which two points would you use to draw the line of best fit? AA and D BC and D CB and D Shoe Size v. Girl's Height Shoe Size Height in Inches

43. Using the scatter plot you created for shoe size v. girls' heights and shoe size v. boys' heights, determine line of best fit that goes through each of these scatter plots.

44. Determining the Prediction Equation Return to Table of Contents

45. The points form a linear like pattern, so use two of the points to draw a line of best fit. Our line is drawn so that it fits as close as possible to the data points. This line was drawn through (35,82) and (50,90).

46. Use the two points that formed the line to write an equation for the line. This equation is called the Prediction Equation. Find m Find b Where S is the score for t minutes of studying. The slope also shows that a student's score will increase by 8 for every 15 minutes of studying they do.

47. Prediction Equations can be used to predict other related values. If a person studies 15 minutes, what would be the predicted score? This is an extrapolation, because the time was outside the range of the original times.

48. If a person studies 42 minutes, what would be the predicted score? This is an interpolation, because the time was inside the range of the original times.

49. Interpolations are more accurate because they are within the set. The farther points are away from the data set the less reliable the prediction. Using the same prediction equation, consider: If a person studies 120 minutes, what will be their score? What is wrong with this prediction?

50. If a student got an 80 on the test, What would be the predicted length of their study time? The student studied about 31 minutes.

51. 13Consider the scatter graph to answer the following: What is the slope of the line of best fit going through A and D? XY 39 57 65 84 93 101 A B C D

52. 14Consider the scatter graph to answer the following: What is the y-intercept of the line of best fit going through A and D? XY 39 4.58 57 65 84 93 101 A B C D 9 11 12

53. 15 Consider the scatter graph to answer the following: The equation for our line is y = -1x + 12. What would the prediction be if x = 7? Is this an interpolation or extrapolation? XY 39 4.58 57 65 84 93 101 5, extrapolation A B C D 5, interpolation 6, interpolation 6, extrapolation

54. XY 39 4.58 57 65 84 93 101 -4, extrapolation A B C D -4, interpolation -2, interpolation -2, extrapolation 16 Consider the scatter graph to answer the following: The equation for our line is y = -1x + 12. What would the prediction be if x = 14? Is this an interpolation or extrapolation?

55. XY 39 4.58 57 65 84 93 101 1, extrapolation A B C D 1, interpolation 2, interpolation 2, extrapolation 17 Consider the scatter graph to answer the following: The equation for our line is y = -1x + 12. What would the prediction be if x = 11? Is this an interpolation or extrapolation?

56. 18In the previous questions, we began by using the table at the right. Which of the predicted values: (7,5) or (14, -2) will be more accurate and why? XY 39 4.58 57 65 84 93 101 (7, 5); there already is a 5 and a 7 in the table A B C D (7, 5); it is an interpolation (14, -2); it is an extrapolation (14, -2); the line is going down and will become negative

57. 19What is the slope of this best fit line that goes through A and C? XY 36 25 59 48 13 610 712 914 C A A B C D

58. 20What is the y-intercept of the line of best fit that goes through A and C? A B C D XY 36 25 59 48 13 610 712 914 C A

59. 21The equation for the line of best fit is. What would the prediction be if y = 4.5? Is this an interpolation or extrapolation? XY 36 25 59 48 13 610 712 914 1, extrapolation A B C D 1, interpolation 2, interpolation 2, extrapolation

60. 22The equation for the line of best fit is. What would the prediction be if y = 8? Is this an interpolation or extrapolation? A B C D interpolation extrapolation interpolation extrapolation XY 36 25 59 48 13 610 712 914

61. Shoe Size Girl's Height in Inches 555 5.554 864 7.565 970 652 7.563 866 Shoe Size v. Girl's Height Shoe Size Height in Inches Calculate the prediction equation using the two labeled points.

62. Shoe Size Girl's Hei ght in Inc hes 555 5.554 864 7.565 970 652 7.563 866 Shoe Size v. Girl's Height Shoe Size Height in Inches 23What is the slope of the prediction equation for this graph?

63. Shoe Size Girl's Height in Inches 555 5.554 864 7.565 970 652 7.563 866 Shoe Size v. Girl's Height Shoe Size Height in Inches 24What is the y-intercept for the prediction equation for this graph?

64. 25A girl with a size 7 shoe and height of 56 inches will be an interpolation. A True B False Shoe Size Girl's Height in Inches 555 5.554 864 7.565 970 652 7.563 866 Shoe Size v. Girl's Height Shoe Size Height in Inches

65. 26A girl with a size 4 shoe and height of 51 inches will be an interpolation. A True B False Shoe Size Girl's Height in Inches 555 5.554 864 7.565 970 652 7.563 866 Shoe Size v. Girl's Height Shoe Size Height in Inches

66. 27What will the height be of a girl with a size 8.5? Shoe Size Girl's Height in Inches 555 5.554 864 7.565 970 652 7.563 866 Shoe Size v. Girl's Height Shoe Size Height in Inches

67. 28A girl with a size 10 shoe and height of 70 inches will be an extrapolation. A True B False Shoe Size Girl's Height in Inches 555 5.554 864 7.565 970 652 7.563 866 Shoe Size v. Girl's Height Shoe Size Height in Inches

68. 29 Using the prediction equation, what will the height be of a girl who has a size 10 shoe? Shoe Size Girl's Height in Inches 555 5.554 864 7.565 970 652 7.563 866 Shoe Size v. Girl's Height Shoe Size Height in Inches

69. Using the scatter plot you created for the shoe size v. girls' heights and shoe size v. boys' heights from your class, determine the prediction equation for each graph. Using the equation, how tall is a girl that wears a 9.5 size shoe? How tall is a boy that wears a 6.5 shoe?

70. Two-Way Tables Return to Table of Contents

71. We can also organize data gathered in a two-way table. Two-way tables display information as it pertains to two different categories. Here is an example of a two-way table: Take a Bicycle to School Do Not Take a Bicycle to School Total Take the Bus to School 5712 Do Not Take the Bus to School 61218 Total111930

72. What does the two-way table show us? The table below shows information gathered from 30 students. They were asked if they took a bus or a bicycle to school. Take a Bicycle to School Do Not Take a Bicycle to School Total Take the Bus to School 5712 Do Not Take the Bus to School 61218 Total111930

73. As you can see from the table, some students take the bus, other students ride their bicycles, take the bus or ride a bicycle to school. Several students do not take a bus nor ride their bicycles to school. Let's answer some questions using the data from the table. Take a Bicycle to School Do Not Take a Bicycle to School Total Take the Bus to School 5712 Do Not Take the Bus to School 61218 Total111930

74. From this table, how many students take the bus or ride their bicycle to school? Take a Bicycle to School Do Not Take a Bicycle to School Total Take the Bus to School 5712 Do Not Take the Bus to School 61218 Total111930

75. 31 Take a Bicycle to School Do Not Take a Bicycle to School Total Take the Bus to School 5712 Do Not Take the Bus to School 61218 Total111930 How many students take the bus, but do not ride their bicycles to school?

76. 32 Take a Bicycle to School Do Not Take a Bicycle to School Total Take the Bus to School 5712 Do Not Take the Bus to School 61218 Total111930 How many students do not take the bus to school?

77. 33 Take a Bicycle to School Do Not Take a Bicycle to School Total Take the Bus to School 5712 Do Not Take the Bus to School 61218 Total111930 How many students ride their bicycles to school, but do not take the bus?

78. Henry surveyed students from several classes to find out if they did chores and received an allowance. 65 students did chores. Of those 65 students, 49 received an allowance. There were 26 students that did not do chores and did not receive an allowance. 10 students that did not do chores, but received an allowance.

79. Set up your table, and label the categories. AllowanceNo AllowanceTotal Chores No Chores Total

80. AllowanceNo AllowanceTotal Chores65 No Chores Total 65 students did chores. Where would you write that number? Notice that the "Chores" and "No Chores" categories are in the rows, and the "Allowance" and "No Allowance" categories are in the columns.

81. Of those 65 students, 49 received an allowance. Where would you write the 49? Look at the "Chores" category, then "Allowance" since the 49 students who did chores received an allowance. AllowanceNo AllowanceTotal Chores4965 No Chores Total

82. AllowanceNo AllowanceTotal Chores4965 No Chores26 Total There were 26 students that did not do chores and did not receive an allowance. Look at the "No Chores" category and "No Allowance" category.

83. 10 students that did not do chores, but received an allowance. Look for the "No Chores" category then "Allowance" category. AllowanceNo AllowanceTotal Chores4965 No Chores1026 Total

84. AllowanceNo AllowanceTotal Chores4965 - 49 = 1665 No Chores102610 + 26 = 36 Total49 + 10 = 5916 + 26 = 42 65 + 36 = 101 or 59 + 42= 101 This is the table filled using the information that was given. Although some of the cells are not filled, you can easily find the rest of the information with simple math. If you did your math correctly, the total row and column should be the same.

85. Here is the final table. Now you can answer some questions using the data. AllowanceNo AllowanceTotal Chores491665 No Chores102636 Total5942101

86. 34How many students took this survey? AllowanceNo AllowanceTotal Chores491665 No Chores102636 Total5942101

87. 35How many students do chores, but do not receive an allowance? AllowanceNo AllowanceTotal Chores491665 No Chores102636 Total5942101

88. 36How many students do not do chores, but still receive an allowance? AllowanceNo AllowanceTotal Chores491665 No Chores102636 Total5942101

89. Survey your class to find out if each student has a laptop computer and/or desktop computer at home. Make a two-way table showing your results. Laptop ComputerNo Laptop ComputerTotal Desktop Computer No Desktop Computer Total

90. Using two-way tables, we can calculate relative frequencies. Relative frequencies are ratios that compares the value of a certain category to the subtotal in that category. As you have previously learned, the frequency is the quantity of just how many of a certain event occurs. Relative frequency is how many compared to the subtotal. The relative frequency is written as a fraction or decimal.

91. Example: There are 12 girls in a class of 20 students. The frequency of number of girls in a class is 12. The relative frequency of the number of girls in the class is or 0.60. What is the frequency of girls in your class? What is the relative frequency? What is the frequency of boys in your class? What is the relative frequency?

92. Take a Bicycle to School Do Not Take a Bicycle to School Total Take the Bus to School 5712 Do Not Take the Bus to School 61218 Total111930 Calculate the relative frequency for the two-way table from earlier by row and then by column.

93. Calculate the relative frequency for the two-way table from earlier by row and then by column. For this cell, the relative frequency of students taking a bicycle to school or the bus to school is divided by the total number of students that take the bus to school.

94. Calculate the relative frequency for the two-way table from earlier by row and then by column. For relative frequency by column, the number of students that take a bicycle to school or take a bus to school is divided by the number of students that take a bicycle to school.

95. Let's answer some questions using the relative frequencies. What is the relative frequency of students that take a bicycle to school and also take a bus to all students taking a bus to school? The relative frequency is which is approximately 0.42. Tap for Answer

96. What is the relative frequency of students that do not take a bicycle to school and do not take a bus to all students that do not take a bus to school? The relative frequency is which is approximately 0.68. Tap for Answer

97. 37 What is the relative frequency of students that take a bicycle to school but do not take a bus to the total number of students that do not take the bus?

98. 38What is the relative frequency of the students that do not take a bicycle to school, but do take the bus to the all the students that take the bus to school?

99. 39What is the relative frequency of students that take a bicycle to school and also take a bus to school, to the total number of students that take a bicycle to school? Answer questions using the relative frequency found by column.

100. 40What is the relative frequency of students that do not take a bicycle to school and do not take the school bus to the total number of students that do not take a bicycle to school?

101. 41What is the relative frequency of students that take a bicycle to school, but do not take the bus to all students that take a bicycle to school?

102. AllowanceNo AllowanceTotal Chores491665 No Chores102636 Total5942101 Use the following two-way table to calculate the relative frequencies by row and by column.

103. AllowanceNo AllowanceTotal Chores No Chores Total Calculate relative frequency by row: Allowance No Allowance Total Chores491665 No Chores102636 Total5942101

104. Why do we calculate relative frequencies? We can use relative frequencies to determine if there is an association between the two categories. For example, does there seem to be a relationship between whether or not a student receives an allowance compared to whether or not a student does chores? Approximately 0.75 or 75% of students that receive an allowance do chores, and out of those that do chores only 0.25 or 25% of students receive no allowance.

105. AllowanceNo AllowanceTotal Chores No Chores Total Calculate relative frequency by column: AllowanceNo AllowanceTotal Chores491665 No Chores102636 Total5942101 Is there a relationship between students that chores to the amount of students that receive an allowance?

106. Kelly surveyed the students and teachers from her school to see how many students had pets at home, and if they had a pet dog or pet cat. She found that 49 people had dogs in her school. Out of the 49 people, 30 people had cats. 50 people had cats in her school. 22 people had neither cats nor dogs at home.

107. Construct a two-way table using the following information. Kelly found that 49 people had dogs in her school. Out of the 49 people, 30 people had cats. 50 people had cats in her school. 22 people had neither cats nor dogs at home. CatNo CatTotal Dog No Dog Total

108. Using the two-way table, calculate the relative frequencies by column and by row. CatNo CatTotal Dog No Dog Total CatNo CatTotal Dog No Dog Total By row: By column:

109. CatNo CatTotal Dog No Dog Total 42What is the relative frequency of the people who have a cat and a dog at home to the number of people that have cats? CatNo CatTotal Dog301949 No Dog202242 Total504191

110. CatNo CatTotal Dog No Dog Total 43What is the relative frequency of the people who have a dog and a cat to the number of people that have a dog?

111. CatNo CatTotal Dog No Dog Total 44What is the relative frequency of the people who have no cat, but have a dog to the number of people that have no cats?

112. Survey your classmates to find out if they play sports and/or play an instrument. Construct a two-way table displaying the results. (Write "yes“ or "no") Then calculate the relative frequencies by row and by column. Student SportsInstrument 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Is there a relationship between the number of students that play sports v. the number of students that play an instrument?