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onds, how far were you from the motion detector? Graph these two positions.

line touch es the ball and says “ stop. ” Once all of the class members have joined the game and the data has been

x 1 (People)

y 1 (Seconds)

3

3.02

Activity 3: Using an Equation of a Line to Predict Ball - Passing Time

6

6.65

9

10.48

In this activity the class uses linear re gression to predict how long it will take to pass the spiky - light - up ball down a line that includes all of the students in the school. The students are spaced three feet apart.

12

13.48

15

15.97

18

19.52

Figure 5: Spikey - light - up ball

21

22.38

24

26.30

gathered, each stu dent uses a table in Desmos to enter the data. The number of players in each turn

Supplies

27

29.17

1. Per student :

30

32.02

a) Post - it - note b) Pencil c) computer with Desmos graphing calcu lator a) Spikey light - up ball (see Figure 4) b) Stopwatch c) Board space

goes in x1, and the amount of time elapsed during each turn in y1. Table 1 provides an example of proper data entry for one class. The students then perform the linear regression by typing y 1 ~ ax 1 + b and generate a line of best fit. Our school serves 1891 students, thus x=1891. The line of best fit for the data in Table 1 is y = 1.06665x + 0.299333. Given this equation and our student body, the passing time is expected to be 2017.3280 seconds which is 33.62 minutes, much quicker than most students predicted. To conclude this activity, the students first use their computers to graph the line, the parent function, and then they use Desmos to respond to the follow ing challenges. Graph a line that is steeper. Graph a line that is less steep. What could we do to our po sitions in the room to make our line more or less steep? Having the students stand farther apart would make the slope steeper, and standing closer together would result in a flatter line. The emphasis here is on the line of best fit, as opposed to a typi cal function, which is illustrated by the inclusion of the point (0.299, 0). Theoretically, this point should be (0,0). This activity should conclude with a brief discussion about why extrapolating from our class room data is an oversimplification. Could we really maintain this pace as we passed the ball to each student in the school (Bazak, 2016)? One common difficulty that students have with this activity is thinking logically and systematically be fore they predict the amount of time it will take to pass the ball, down the line, to each student in the school. Students showing difficulty with this esti mate will often predict one or two days. It is help ful to ask students to divide their prediction by the Common Difficulties and Challenge

1. Per class:

Before the activity begins, each student submits a prediction of how long it will take to pass the ball to each member of the student body if the students are in a line and each student was spaced three feet apart. The class then plays a game to systematically determine how long it takes their class to pass the ball down the lines of varying lengths. Phase One : The game begins with three players each standing in a line about a yard apart. The first student passes the ball to the second student. This is timed and is recorded. The class then uses skip counting to determine how long it would take to pass the ball to each member of the class given the seconds/person rate they found earlier. For exam ple, if the pass took 2 seconds, they would count 2, 4, 6…60 seconds for a thirty - member class. This is repeated using the rate from the second to third person. Phase Two : The first three students are timed as they pass the ball down the line. After each turn, three new players are added to the line until the en tire class is playing. Players always stand three feet apart. In many classrooms, students are three tiles apart. An appointed timekeeper operates a stopwatch and records on the board both the num ber of players and the amount of time elapsed dur ing each turn. Once the players are in place, the turn begins as the timekeeper says “ go, ” and the first person in line gently tosses the ball to the next player in line. The turn ends as the last player in

Virginia Mathematics Teacher vol. 46, no. 2

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