1. Integrating Technology into Mathematics (6-12) MELT 2015 Appalachian State University Kayla Chandler DAY 4

2. Agenda TimeActivity 8:30 – 10:00 Schoolopoly Simulations 10:00 – 10:15Break 10:15 – 11:45Are you a sleeping sheep or a turbo charged cheetah? 11:45 – 1:00Lunch 1:00 – 2:30City vs. Hwy MPG 2:30 – 2:45Break 2:45 – 4:15Lesson Planning Work Session 4:15 – 4:30Wrap up

3. SESSION 1

4. Probability Probabilistic reasoning is difficult Intuitions Randomness – removes cause and effect Simulation activities Grounded in real world contexts Help students develop better intuition for probability

5. Schoolopoly Suppose your school is planning to create a board game modeled on the classic game of Monopoly. The game is to be called Schoolopoly and, like Monopoly, will be played with dice. Because many copies of the game expect to be sold, companies are competing for the contract to supply dice for Schoolopoly. Some companies have been accused of making poor quality dice and these are to be avoided since players must believe the dice they are using are actually “fair.” Each company below has provided dice for analysis. Investigate whether the dice sent from each company is fair or biased. Calibrated Cubes Dazzling Dice Delta’s Dice Dice Depot Dice R’ Us High Rollers, Inc.

6. Questions to Consider Do you believe the dice from each company are fair or biased? Which company would you recommend purchasing dice from? What compelling evidence do you have that the dice you of each company are fair or unfair? Use your data to estimate the probability of each outcome, 1-6, of the dice for each company.

7. Simulating Randomness in Technology Tools Pseudo-random number generators Difficult concept for students

8. Setting the Seed Value on a Graphing Calculator Type in a value (something different for each calculator!) Press the “STO  ” key (store key). Press the “MATH” key. Choose “PRB.” Select 1:rand. Press enter and the seed value will be displayed.

9. Simulating a Die Toss on the Calculator randInt command allows you to choose an integer between two other integers for a given number of times randInt(1,6,5) will choose a number between 1 and 6 five times. The outcome is based on an equiprobable die.

10. Questions to Consider Would you prefer to discuss the importance of a seed value with students or to set all of the graphing calculators with different seed values yourself and not discuss the issues with students? Explain your choice. How could you sue the fact that many calculators and computers generate the same list of pseudo- random numbers given the same initial seed value to generate discussions with students about randomness in general and the use of computers to simulate probability experiments?

11. Using Data to Estimate Probabilities & Design Simulations The rates for each school describe the actual percent of freshman from an incoming class of full-time students (population) that enrolled in courses for a second year.

12. Questions to Consider Use the retention rates at Chowan College, NC Central and NC State University to estimate the probability that a randomly chosen freshman will continue into their second year at each school. Describe how you would use a coin to simulate the experiment of deciding whether or not any given freshman will continue on the next year at Chowan College. What other objects could you use to conduct a simulation?

13. Questions to Consider If you conduct the simulation you described above with 30 trials to represent the number of freshman, what is a reasonable amount of these 30 freshman you could expect to return the following year? Why? If you repeat a simulation of 30 trials several times (decide how many times), what similarities or differences do you expect across the results from the different samples of 30 trials?

14. Questions to Consider Use a coin to conduct a simulation of several samples of 30 trials and record your results. How do the results compare with what you anticipated? What are some of the potential benefits and drawbacks of using a real world context like the freshman retention rate for introducing probability to students, especially terminology such as outcomes, sample space, experiment, trial, event, sample, and sample size?

15. Simulating Binomial Events with a Graphing Calculator Retention rate situation of Chowan College, recall our sample space has two possible outcomes: student returns and student does not return. Use the randInt function assigning two consecutive integers to represent the two outcomes.

16. Questions to Consider Suppose the freshman class at Chowan College has 500 students. What command would you use on the graphing calculator to conduct a simulation of whether or not each of the 500 freshmen stays in school with a 50% estimate for the probability for retention? Explain each part of the command in terms of the context of the situation.

17. Questions to Consider Given a 50% estimate for the probability for retention, out of 500 freshmen, what is a reasonable interval for the proportion of freshmen you would expect to return the following year? Defend your expectation.

18. Model Using Calculator randInt(0, 1, 500)  L1. This generates a list of 500 random integers between 0 and 1. A result of 0 will represent a student does not return while a result of 1 represents the case when a student returns. Now we want to see a histogram of this to see the frequency of 0s and 1s. Activate Plot1 on the StatPlot menu. Select the Histogram icon for the Type of plot and make sure the Xlist displays L1. Set the window to have range from 0 to 2 for x (scale of 1) and -50 to 350 for y (scale of 50) with x resolution 1. Now if you press the graph key, you should see the histogram.

19. Questions to Consider Explain why the values used in the Window setting create the appropriate graphical display of two bars with the frequency of the 0s in the first and the frequency of the 1s in the second. In your response, explain why a y max of 350 is appropriate. Determine the proportion of freshmen who will return to Chowan College next year.

20. Questions to Consider For our problem we are interested in how much the proportion of freshmen returning to Chowan College will vary from the expected 50%. Use the skills you have learned to repeat the simulation of deciding the retention of 500 freshman many times. Record the proportion of freshmen that continue on next year for each of the samples of 500 trials. How many of the sample proportions fall within the interval you predicted in question 2? Discuss why the results may or may not have been consistent with your earlier prediction.

21. Questions to Consider Recall that in our simulation, each freshman had a 50% chance of returning to school. Did you get exactly 50% of the freshmen returning in the simulation? Discuss how and why the empirical proportions from the simulation may have varied from 50%. If we reduced the number of trials to 200 freshmen, what do you anticipate would happen to the interval of proportions from the empirical data around the theoretical probability of 50%? Why? Conduct a few samples with 200 trials and compare your results with what you anticipated.

22. Questions to Consider If we increased the number of trials to 999 freshmen, what do you anticipate would happen to the interval of proportions from the empirical data around the theoretical probability of 50%? Why? Conduct a few samples with 999 trials and compare your results with what you anticipated. Based on your experience with the simulations and the empirical data you collected, what would be a reasonable interval for the proportion of freshmen that administrators can expect to return to Chowan College for freshmen class sizes of 200, 500, and 999? Explain your predictions.

23. Questions to Consider Discuss why it might be beneficial to have students simulate the freshman retention problem for several samples of sample size 500, as well as sample sizes of 200 and 999.

24. Questions to Consider An important aspect of conducting a simulation of a repeated random event is to be sure that students have a conceptual understanding of how the simulation and the commands used in the computing tool represents the context of the problem. Consider the following context. Explain how you would help students use the graphing calculator to simulate this context. Explicitly describe what the commands represent and how the students should interpret the results. Suppose a university gives away token spirit gifts to all incoming freshmen. As they check in to pick up their class schedule, they get to randomly choose three cards. Each card displays a different gift: keychain, window decal for a car, t-shirt with college logo. Most freshmen would prefer the t-shirt.

25. Questions to Consider Suppose that after conducting the above simulations, a student gets a result that a t-shirt was chosen 20% of the time. The student is surprised and claims that this proportion is too low from the expected 33.3%. What might this student be misunderstanding? What are some questions you could pose and further simulations you could suggest that might help this student?

26. SESSION 2

27. Are you a sleeping sheep or a turbo-charged cheetah? Activity about reaction times and measuring reaction times Possible opening questions to use with students?

28. Review & Brainstorm Experiment Read through the activity – but do NOT start it yet! What should we consider before collecting data?

29. Data Collection & Analysis You will now use the applet to collect data as we discussed. Record yours and your partner’s data and input it into TinkerPlots. Who has the faster reaction time? Be prepared to defend your argument!

30. Analyzing the Task What did you like about the task? Not like? What was challenging? What did you think about the other approaches presented? Would you modify this task for your classroom? If so, how? If not, why? What would your students like about this task? Dislike? Is the task engaging enough for students? Explain. What are some problems that may arise during the implementation of this task? What are some misconceptions students might have during this task? What could you do to alleviate some of these problems/misconceptions without taking away from the discovery? What sort of things could you reflect on as a teacher for this activity? As a student? What are possible extensions of this task?

31. SESSION 3

32. City vs. Hwy MPG Is there a relationship between City MPG and Hwy MPG for this data?

33. Importing Data into Fathom Open the 2013CarData Excel file. Select all and copy. Open Fathom. Start new collection. Paste cases.

34. Exploring the Data Create dot plots for each attribute of interest (i.e., City, Hwy).

35. Questions to Consider What do you anticipate might be a reasonable relationship between City mpg and Hwy mpg? Explain how it might be helpful to use the linked representations of the two dot plots (or box plots) to assist students in examining the covariation between two quantitative attributes.

36. Moving Towards Two Dimensions Dot plots (and box plots) are one dimension. Scatterplots are two dimensions. Useful for examining if there is an association between quantitative attributes. Change both dot plots to box plots. Change the window sizes and orient the two box plots as if they are representing your axes with City on the x-axis and Hwy on the y-axis. Create a scatterplot with City on the x-axis and Hwy on the y-axis.

37. Questions to Consider Use form, direction, and strength to describe the relationship between City and Hwy mpg. Describe a typical City and Hwy mpg for this set of vehicles. Explain how you determined what you would consider as “typical”. How can displaying the means in a scatterplot help or hinder students’ ability to think about variation in bivariate data?

38. Relationship? Does there appear to be a relationship between the attributes? If so, what kind? Can we predict a vehicle’s Hwy MPG?

39. Computing r Using Fathom Drag down a Summary object. Drag “City” onto a row in the table and “Hwy” onto a column. The correlation coefficient is automatically calculated.

40. Questions to Consider What does the value of the correlation coefficient imply about the relationship between City and Hwy mpg? Insert a moveable line (under “Graph”) and adjust it until you feel it best models the data. Interpret the slope and y-intercept in the equation of your linear model. Is the slope of your line the same as the value of the correlation coefficient? Should they be the same? Why or why not?

41. Questions to Consider How can the ability to overlay a moveable line on a scatterplot help or hinder students’ understanding of the use of a linear equation to model a relationship between two variables? Often, students do not have confidence in their solutions in situations like these where everyone might have a different equation. How could you help students understand the differences in solutions are acceptable and expected within the context of estimating a linear model?

42. Finding the Line of Best Fit How can we determine which line best fits the data? Residual: difference between actual data value and predicted value Sum of Squares: sum of squares for each residual Least Squares Regression Line: linear model that minimizes sum of squares

43. Showing the Sum of Squares Remove the mean(City) and mean(Hwy) lines by right clicking on their formulas and clearing them. Under “Graph” click to show the sum of squares. Manipulate the movable line to explore whether it is possible to create a line that is far from several points but still has a small sum of squares. Explain.

44. Showing the Residuals A residual plot displays the ordered pairs (x- value, [y-value – predicted y-value]). View a plot of the residuals (under “Graph”). In general, if we are minimizing our residuals, what do we want to see in the residual plot?

45. Questions to Consider A student placed their moveable line in the scatterplot for the 2013 Car data that resulted in the residual plot. Estimate the location of the predicted linear model based on the residual plot above in the graph.

46. Questions to Consider Describe some conceptual difficulties students may have in interpreting and using the residual plot. How will you help them understand the residual plot and its usefulness in analyzing a linear model?

47. The Least-Squares Regression Line Find the least squares line (under “Graph”). Note that the square of the correlation coefficient, r 2, is called the coefficient of determination and can be interpreted as the proportion of variation in the response variable that can be attributed to the variation in the predictor variable by the least squares line.

48. Questions to Consider Compare the function rule for the least squares model with the function rule for your estimated linear model (your moveable line). Interpret the coefficient of determination for this least squares linear model. Do you believe the least squares line is a good model for the 2013 Car City and Hwy data? Explain.

49. SESSION 4

50. Planning Session This session will be devoted to the development of your technology lesson.