Chapter 14 Understanding Change: Relationships between Time, Distance, and Rate

1. Reconceptualizing Mathematics for Elementary School Teachers Second Edition Chapter 14Understanding Change: Relationships between Time, Distance, and Rate Judith Sowder, Larry Sowder, Susan Nickerson Lecture Presentation

2. Chapter 14 Concepts • Distance-Time and Position-Time Graphs • Using Motion Detectors • Graphs of Speed Against Time • Interpreting Graphs • Issues for Learning: Common Graphing Errors

3. 14.1 Distance-Time and Position-Time Graphs Activity 1

4. Example 1 The following graph, sometimes called a qualitative graph, captures the relationship between distance and time, without being concerned about specific numerical relationships: First, it is important to note here that distance refers to the total distance traveled by Wile E. Coyote, and not how far he is from the cave. Do you see what the segments A, B, and C represent? Do you see what the slope of those segments represents?

6. Activity 2 Now create a qualitative graph of the relationship between Wile E. Coyote’s position (his distance from the cave) and time. Be sure to label each axis with a quantity.

7. Take-Away Message… • Even without numbers, a qualitative graph can give information about a situation. • Drawn with care, both distance-time graphs and position-time graphs can reflect information about speed. • Distance-time graphs show the total distance run. • Position-time graphs tell how far, over time, the object is from the starting point.

8. 14.2Using Motion Detectors This section involves the use of motion detectors. The goal of working with motion detectors is to create graphs using real motion. The motion detector is a physical device that measures how far you are from the detector over a given time interval. It works by emitting ultrasonic pulses and then recording the length of time it takes for those pulses to return. It calculates a distance approximately 20 times per second. The range of the motion detector is between .45 and 6 meters. At this time it would be helpful for the instructor to demo the motion detector to the class.

9. Activity 3 continued…

10. Activity 3 (continued)

11. Activity 3 (continued)

12. Activity 3 (continued)

13. Take-Away Message… • Real-time movement graphs can help us visualize the relationship between position and time and can help us notice errors in graphing.

14. 14.3 Graphs of Speed Against Time

15. Activity 4

16. Activity 4 (continued)

17. Take-Away Message… • Because speed is a quantity that can change over time, a graph showing speed versus time makes sense. • When a speed involves a decrease in distance, it is reasonable to consider that a negative speed, but for many purposes doing so is not essential.

18. 14.4Interpreting Graphs Activity 5

19. Activity 6 Consider the following: “A child climbs up a slide at a constant speed, stops at the top, and then slides down.” Examine the graphs on the next slide, and then for each graph answer the following questions: 1. Which graph best models the situation? Explain your reasoning. 2. Which graph looks most like a playground slide? Could this graph represent the situation? Why? 3. For each of the graphs that don’t model the situation, explain why they do not.

20. Activity 6 (continued)

21. Activity 7

22. Take-Away Message… • Interpreting speed-time graphs requires careful thought, with a focus on how the speed (not the distance) is changing. • Part of a teacher’s job is to explain. Giving good explanations takes practice and, in part, careful attention to the language used.

23. 14.5Issues for Learning: Common Graphing Errors Algebra has all too often focused on symbolic representations alone. Visual representations are frequently ignored, and even when graphs are used, sometimes they are treated in a way that is void of true meaning for the students. We have already emphasized that being able to construct an accurate picture or graph goes a long way to not only understanding the problem, but also to come up with a meaningful and accurate solution.

24. Students often have a great deal of difficulty constructing meaningful graphs. Graphing errors often emerge when students are not aware of the conventions of drawing graphs, or they don’t fully understand the problem situation. The following conventions can help students overcome some of this:

26. continued… Given the data below and graphs on the following slides, determine the error made by each student and describe what you think the student did not understand in each case.

27. continued…

28. Now try making a correct graph of the data.

29. Discussion 2 continued… Examine the graph below followed by the student interpretations of it:

30. 1. Explain what each student might have been thinking. 2. Why does it make sense to say that both of these students had a “graph as picture” misconception? That is, they interpret the graph itself as a realistic image of the event. 3. What is the correct interpretation here?

32. 14.6 Check Yourself