1. Metro RESA...Building leaders of teaching and learning Digging Deeper Into Mathematics Clayton County Summer Math Academy Algebra 1 – Curves of Best Fit Sarah Ledford June 15, 2015 1

2. Metro RESA...Building leaders of teaching and learning Falling Dominoes What mathematical questions can you ask about falling dominoes? How could you design experiments to answer your questions? Do you think your experiment would answer your question? Can you use your experiment to make predictions & test them? 2

3. Metro RESA...Building leaders of teaching and learning Falling Dominoes Our experiment will investigate the relationship between the number of dominoes spaced one inch apart in a straight line and the time it takes for all the dominoes to fall down. We will need timers!! 3

4. Metro RESA...Building leaders of teaching and learning Falling Dominoes We’ll begin by sticking a piece of masking tape (about 3 feet) to the floor or the table, and marking it off every inch. 4

5. Metro RESA...Building leaders of teaching and learning Falling Dominoes Next, line up dominoes at every mark to match each row of the data collection table. 5

6. Metro RESA...Building leaders of teaching and learning Falling Dominoes Complete the table in groups. Why is there a column for “mean time?” 6

7. Metro RESA...Building leaders of teaching and learning Falling Dominoes Graph the data 7

8. Metro RESA...Building leaders of teaching and learning Falling Dominoes Should your graph go through the origin? Why or why not? Use a piece of raw spaghetti to place on your graph. Try to determine the line of best fit. This line should go through as many of your data points as possible, and should have roughly the same number of points below the line as above the line. Use a ruler to draw your line of best fit. 8

9. Metro RESA...Building leaders of teaching and learning Spaghetti Regression Why do we say that the line in the top graph fits the points better than the line in the bottom graph? Can we say that some other line might fit them better still? Usually we think of a close fit as a good fit. But, what do we mean by close? 9

10. Metro RESA...Building leaders of teaching and learning Falling Dominoes Determine the equation of your line of best fit. Use the equation of your line of best fit to predict how long it would take to knock down a line of 60 dominoes. Test your prediction by building a line of 60 dominoes and timing how long it takes for all of them to knock down. How good was your prediction? 10

11. Metro RESA...Building leaders of teaching and learning Calculator Let’s find the line of best fit using our calculators. We are going to enter the data on our scatterplot so we need the ordered pairs. 11

12. Metro RESA...Building leaders of teaching and learning Calculator To enter data in lists: “stat” “edit” enter data in L1 (inputs) & L2 (outputs) To show data on scatterplot: “y=“  highlight “Plot1” 12

13. Metro RESA...Building leaders of teaching and learning Calculator To do regression on data: “stat”  “calc”  Choose the kind of regression you want to do. We want linear so “Lin Reg(ax + b)” is a good option for us. Let’s talk about what we see on our screens. 13

14. Metro RESA...Building leaders of teaching and learning Calculator In order to be able to graph equation on the scatterplot as an example, we want to type “LinReg L1, L2, Y1” L1 can be typed with “shift” “1”; L2 is “shift” “2”; etc. Comma is beside x 2 button Y1 is found in “vars”  ”y-vars”  “function”  ”Y1” 14

15. Metro RESA...Building leaders of teaching and learning Calculator To get the correlation coefficient: go to “catalog” scroll down to “diagnostic on,” hit enter make sure your calculator says “done” The closer to 1, the better the fit! 15

16. Metro RESA...Building leaders of teaching and learning Hmmm… I surveyed my students by asking them how long they studied for an exam. Then I made a scatter plot showing the relationship between minutes spent studying for an exam and the exam results. What do you think this scatter plot looked like? Good questions for math teaching: Why ask them and what to ask grades 5–8. Math Solutions Publications: Sausalita, CA. Schuster, L. and Andeson, N.C. (2005). Pg. 127. 16

17. Metro RESA...Building leaders of teaching and learning Texting and Grades II https://www.illustrativemathematics.org/illustrations/1028 Medhavi suspects that there is a relationship between the number of text messages high school students send and their academic achievement. To explore this, she asks a random sample of 52 students at her school how many text messages they sent yesterday and what their grade point average (GPA) was during the most recent marking period. Her data are summarized in the scatter plot on the next slide. The least squares regression line is also shown. 17 REFERENCE THE source (GaDOE

18. Metro RESA...Building leaders of teaching and learning Texting and Grades II 18 REFERENCE THE source (GaDOE

19. Metro RESA...Building leaders of teaching and learning Texting and Grades II The equation of the least squares regression is GPA = 3.8 – 0.005(Texts sent). Interpret the quantities -0.005 and 3.8 in the context of these data. 19 REFERENCE THE source (GaDOE

20. Metro RESA...Building leaders of teaching and learning Task Review What GSE (content) was addressed? – Domain, cluster, and standards What SMPs were addressed? 20

21. Metro RESA...Building leaders of teaching and learning Algebra I GSE S.ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. S.ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. S.ID.8 Compute (using technology) and interpret the correlation coefficient “r” of a linear fit. 21

22. Metro RESA...Building leaders of teaching and learning 6-8 GSE 6.SP.4 Display numerical data in plots on a number line, including dot plots (line plots), histograms, and box plots. 8.SP.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. ADD THESE!! 22

23. Metro RESA...Building leaders of teaching and learning 6-8 GSE 8.SP.2 Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. 8.SP.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. ADD THESE!! 23

24. Metro RESA...Building leaders of teaching and learning Price vs. Quantity A manufacturer wants to examine the relationship between the price of one of its products (x) and the quantity demanded (y). The scatterplot here shows the results and the line of best fit y = -5.8x + 105.7. 24

25. Metro RESA...Building leaders of teaching and learning Price vs. Quantity Using the equation for this line of best fit, what is the number of units demanded at a price of $15? y = -5.8x + 105.7 A.25,000 B. 20,000 C. 15,000 D. 30,000 25

26. Metro RESA...Building leaders of teaching and learning Price vs. Quantity Which equation best describes this data? A.y = 7.2x – 14.4 B.y = 7.2x + 1.6 C.y = 0.12x – 8.16 D.y = 0.12x + 0.28 26

27. Metro RESA...Building leaders of teaching and learning EOC like question A positive correlation always exists on a scatter plot when A. y remains unchanged as x increases. B. y changes randomly as x increases. C. y decreases as x increases. D. y increases as x increases. 27

28. Metro RESA...Building leaders of teaching and learning Non-linear data? What if our data isn’t linear? We can use regression to find several different kinds of equations & then determine which we think is best based on the correlation coefficient, r. Using our calculators we can do exponential, quadratic, cubic, trigonometric, logarithmic, etc., regression. 28

29. Metro RESA...Building leaders of teaching and learning Water Fountain We can find an equation that could tell us the trajectory of the water. What does the shape of the water remind you of? 29

30. Metro RESA...Building leaders of teaching and learning Water Fountain We need to find a few ordered pairs that lie on the stream of water, put them in a table, & do regression… 30

31. Metro RESA...Building leaders of teaching and learning Model with Mathematics… 31

32. Metro RESA...Building leaders of teaching and learning Model with Mathematics… 32

33. Metro RESA...Building leaders of teaching and learning Model with Mathematics… 33

34. Metro RESA...Building leaders of teaching and learning Model with Mathematics… 34

35. Metro RESA...Building leaders of teaching and learning Model with Mathematics… 35

36. Metro RESA...Building leaders of teaching and learning Model with Mathematics… 36

37. Metro RESA...Building leaders of teaching and learning Model with Mathematics… 37

38. Metro RESA...Building leaders of teaching and learning Olympic Pole Vault Let’s look at the data. What kind of model do you think will best fit the data? This is REAL LIFE MODELING!! https://www.nsa.gov/academia/_files/collected_learning/high_school/algebr a/beginning_modeling.pdf 38

39. Metro RESA...Building leaders of teaching and learning Olympic Pole Vault Let’s look at the data. What kind of model do you think will best fit the data? This is REAL LIFE MODELING!! https://www.nsa.gov/academia/_files/collected_learning/high_school/ 39

40. Metro RESA...Building leaders of teaching and learning Spaghetti Regression From GaDOE CA Unit 4 Framework We need to discuss “good fit.” Handout & 3-5 pieces of spaghetti. 40

41. Metro RESA...Building leaders of teaching and learning Spaghetti Regression On your own: On the scatter plot, tape a piece of spaghetti to represent the line of best fit. In your group: Decide who in your group has the best line of best fit. 41

42. Metro RESA...Building leaders of teaching and learning Spaghetti Regression On your own: Using another piece of spaghetti, measure the distances between each data point and your line. Break off the spaghetti to represent each distance. In your group: Decide how your group will measure this distance. 42

43. Metro RESA...Building leaders of teaching and learning Spaghetti Regression There are at least three ways to measure the space between a point and the line: vertically in the y direction, horizontally in the x direction, and the shortest distance from a point to the line (on a perpendicular to the line.) 43

44. Metro RESA...Building leaders of teaching and learning Spaghetti Regression Jfd In regression, we usually choose to measure the space vertically. These distances are known as residuals. 44

45. Metro RESA...Building leaders of teaching and learning Why would you want to measure this way? What do the residuals represent in relation to our function? Consider the purpose of the line and the following diagram. 45

46. Metro RESA...Building leaders of teaching and learning The vertical distance represents how far off the prediction is from the actual data poin.t Residual = actual – predicted = y i −f(x i ) 46

47. Metro RESA...Building leaders of teaching and learning Spaghetti Regression On your own: Measure the total length of your spaghetti pieces that represent the residuals. In your group: Determine who actually has the closest fit. 47

48. Metro RESA...Building leaders of teaching and learning Spaghetti Regression Formative Assessment Questions: How did you decide where to put your line of best fit on the scatter plot? How did you choose measure the distance from each point to the line of best fit? Why? What do the residuals tell you about your line of best fit? 48

49. Metro RESA...Building leaders of teaching and learning GA DOE Implementation Resources https://www.georgiastandards.org/Common-Core/Pages/Math.aspx – Common Core Tools – Schedule of Unit webinars – Recordings: Standards for Mathematical Practices Grade level overviews Unit webinars – List Serve (subscribe to receive emails from DOE)

50. Metro RESA...Building leaders of teaching and learning GA DOE GaDOE Wiki – Grades 6-8 Please visit http://ccgpsmathematics6-8.wikispaces.com/ to provide feedback, ask questions, and share ideas and resources. http://ccgpsmathematics6-8.wikispaces.com/ Multi-Grade Resources: http://ccgpsmathematics6- 8.wikispaces.com/Multi+Grade+Resources http://ccgpsmathematics6- 8.wikispaces.com/Multi+Grade+Resources Secondary Mathematics Specialist Brooke Kline – bkline@doe.k12.ga.usbkline@doe.k12.ga.us 50

51. Metro RESA...Building leaders of teaching and learning GA DOE GaDOE Wiki – Coordinate Algebra & Analytic Geometry Please visit http://ccgpsmathematics9-10.wikispaces.com/ to provide feedback, ask questions, and share ideas and resources. http://ccgpsmathematics9-10.wikispaces.com/ Multi-Grade Resources: http://ccgpsmathematics9- 10.wikispaces.com/Multi+Grade+Resources http://ccgpsmathematics9- 10.wikispaces.com/Multi+Grade+Resources Secondary Mathematics Specialists Brooke Kline – bkline@doe.k12.ga.usbkline@doe.k12.ga.us 51

52. Metro RESA...Building leaders of teaching and learning Implementation Resources Progressions – “offer explanations for the sequence of the standards, potential cognitive difficulties, and pedagogical solutions which may be useful in teacher preparation and professional development, organizing curriculum, and writing textbooks.” http://ime.math.arizona.edu/progressions/ 52

53. Metro RESA...Building leaders of teaching and learning Implementation Resources Illustrative Mathematics Project – “illustrates the range and types of mathematical work” http://illustrativemathematics.org/ 53

54. Metro RESA...Building leaders of teaching and learning Implementation Resources The Mathematics Common Core Toolbox – “ This site is a resource designed to support districts working to meet the challenge and the opportunity of the new standards. Here you will find tools and instructional materials that help you to better understand and to implement the CCSSM. …created through a collaboration of the Charles A. Dana Center at the University of Texas at Austin and Agile Mind with partial funding from the Bill & Melinda Gates Foundation.” http://www.ccsstoolbox.org/ 54

55. Metro RESA...Building leaders of teaching and learning Implementation Resources Inside Mathematics – “a professional resource for educators passionate about improving students' mathematics learning and performance. This site features classroom examples of innovative teaching methods and insights into student learning, tools for mathematics instruction that teachers can use immediately, and video tours of the ideas and materials on the site.”classroom examplestools for mathematics instructionvideo tours http://insidemathematics.org/index.php/home 55

56. Metro RESA...Building leaders of teaching and learning Implementation Resources Engage NY – A comprehensive collection of units of instruction and assessments to support the implementation of the Common Core State Standards in New York State. https://www.engageny.org/ 56

57. Metro RESA...Building leaders of teaching and learning Implementation Resources Mathematics Assessment Project – “The project is working to design and develop well- engineered assessment tools to support US schools in implementing the Common Core State Standards for Mathematics.” http://map.mathshell.org/ 57

58. Metro RESA...Building leaders of teaching and learning Implementation Resources Math Common Core Coalition – “The Mathematics Common Core Coalition works to provide expertise and advice on issues related to the effective implementation of the Common Core State Standards for School Mathematics (CCSSM).” www.mathccc.org 58

59. Metro RESA...Building leaders of teaching and learning Implementation Resources Achieve the Core Free resources Founded by one of the contributing authors of the CCSS (Student Achievement Partners), now devoted to successful implementation 59

60. Metro RESA...Building leaders of teaching and learning Assessment Resources Initial Set of Test Items and Task Prototypes Released http://www.parcconline.org/initial-set-test-items- and-task-prototypes-released http://www.parcconline.org/initial-set-test-items- and-task-prototypes-released http://www.smarterbalanced.org/sample-items-and- performance-tasks/ http://www.smarterbalanced.org/sample-items-and- performance-tasks/ Georgia Online Formative Assessment Resource (GOFAR) 60

61. Metro RESA...Building leaders of teaching and learning Outcomes: Dig into the domains to determine the development of the content across the grade levels Work tasks that help develop conceptual understanding Discuss the integration of the Standards for Mathematical Practice (SMPs) Identify teacher actions to develop content and mathematical practices while addressing TKES 61

62. Metro RESA...Building leaders of teaching and learning Now What? What does this mean for me and my students? Follow us on Twitter @MRESAmathTwitter Please complete the evaluation. http://bit.ly/ref_eval