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A student stands about 0.5 meters in front of the detector and is asked to walk backward at a slow, constant, steady pace. The teacher asks students to predict the shape of the graph, that shows the dis tance between the motion detector and the student. The teacher asks the students to sketch their predic tion on their lesson sheet. Next, the student is asked to walk when the motion detector is activated. The graph produced is a straight - line segment. The teacher points to the place where the graph inter cepts the y - axis and asks what part of the motion is described by that point on the graph. The teacher asks why the graph is tilted upward (Hakkenberg, 2017). The teacher asks for another volunteer and posi tions this student 0.5 meters in front of the motion detector (facing the detector). This time, the stu dent is asked to walk at a quick, steady, constant pace backwards from the detector. Students in the class are asked to draw a graph of their predictions. The teacher asks what characteristic of the graph should be the same as the previous scenario. Many students correctly predict that the tilt will still be upward because distance from the detector increas es with time. The teacher then asks if it will be the same tilt. Students often know that the tilt will be steeper because the rate of change has increased. What characteristic of the motion would produce a steeper graph? Will the graph still be a straight line? Why or why not? What words in the de scription of how the person is supposed to walk suggest it might be a straight line? While keeping the graph from the first experiment on the screen, the student is asked to walk quickly and at a steady, even pace backwards. The new graph is superim posed on the same grid as the first graph so the two can be compared and the students predictions veri fied. Students are asked for the mathematical term that represents the tilt of the lines that have been pro duced during the first two experiments. They are also asked for the mathematical term that identifies the location where the line touches or crosses the vertical axis. Sometimes the line does not rise im mediately, but has a small horizontal segment be fore it rises. Students are asked what caused that horizontal segment. Usually, a few students can explain that it was caused by the delayed reaction time between the start of the motion detector and the actual movement of the student. Finally, anoth er student is asked to stand about two meters away from the detector, facing the detector. This student is asked to walk at a slow, steady, and constant pace toward the detector. Again, students predict

the graph that will be produced and explain what will be similar and what will be different from the previous graphs and they will explain why. Next, students are shown a graph with a positive slope, followed by a zero slope, followed by a neg ative slope and asked to describe how a student would have to move to create each graph. The soft ware has some simple graphs that can be projected onto the screen so that students can take turns try ing to walk in ways that will match the graph. Their motion graphs are superimposed on the “ target ” graph that is displayed. This is quite en joyable for the students. Usually three or four stu dents, in sequence, are given opportunities to try to match the graph by walking. Homework and assessments give students descrip tions of movements and asks them to sketch graphs of the movements, or give students graphs and ask them to describe the movement represented by the graph. The main concepts reinforced in this lesson in clude: slope is related to velocity, and the distance from the motion detector is captured by the upward tilt or downward tilt of the line segment over time. The positive or negative slope has meaning with regard to the physical movement of the student, and the segment is straight because the movement

is “ steady and constant. ” The lesson also reinforces the meaning of the y intercept.

Common Difficulties and Challenges

Some student have difficul ties labeling the axis. They are not sure which should be time and which should be

Figure 4: Students stand three feet apart and pass the ball down the line.

distance. If this occurs, it provides an excellent opportunity to discuss dependent and independent variables. Some students will predict that a student walking backwards results in a negative slope. Why does the line go up? Noting the units on the graph is often helpful at this point. That is, as time increases, so does distance. Other students may have difficulty knowing how to start to make pre dictions. If this is the case, referring them back to the Green Screen activity is helpful. For example, what are the x - and y - intercepts at the beginning and end of each round? At the beginning of the Round 1, 0 seconds, how far were you from the motion detector? At the end of the Round 1, 5 sec

Virginia Mathematics Teacher vol. 46, no. 2

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