Activity 2: Working with GSP
Materials Needed
Access to Geometer's Sketchpad Software, computer lab, paper, pencil.
Sequence of Lesson
Warm-up: Remind the class of all the applications of relations, and how they are used to describe the relationship between two sets of objects. Ask the class what were some objects that could be modeled using linear functions. If distance or speed with respect to time is not brought up, bring it up with the at the end of the discussion. Make sure that the basics are covered, i.e., the x-value indicates the time, and the y-value indicates the distance or speed at a given time.
End by getting them excited to model some fishies!
Description: Now the class should mostly be on the same page in terms of what their speed functions should be. This next activity involves looking at what the distance graph will be for each ‘fishy’ situation. The first distance graph is a nice warm up, particularly since it is given to the students by Simcalc. The next two will require some thought by the student. To graph the second fish question (The fish travels 2 m/s for 4 seconds and then 6m/s for 4 seconds) by themselves will require a good understanding of both piecewise functions and slope. Note that Simcalc can two-part piecewise functions by restricting the domain of the student fish to (0, 4) and the domain of the teacher fish by (4, 8). Of course, without the right y-intercept for the teacher fish, the two graphs will be discontinuous at 4. Therefore, actually writing out the piecewise functions presents an extra challenge for the students, since this forces them to consider that since fishies can’t teleport, they must determine how to make sure that graph is clear about where the fish is at the fourth second mark.
Post-Activity: Come back to a class discussion of the students' findings and what they thought of the connection between the speed graphs and the distance graphs. Discuss the reasons why the area underneath a function determines a speed graph's distance graph and why the slope of a function determines a distance graph’s speed graph. Also mention the last graph, that involved a speed graph with a slope of one. Briefly discuss the students’ ideas about this graph. Let them know that the answers to these exciting questions will be answered next time! This will give them something interesting to think about and keep their wheels turning when it comes to math. It turned out that the distance graph was a parabola. You may also discuss how the students might attempt to find the speed graph if it was not already available to them or they only had a distance graph if the graph wasn’t linear. Advanced discussion might include might include using tangent lines. Be sure to open up the discussion for any questions on the subject for students to share their points of view on the subject.
Access to Geometer's Sketchpad Software, computer lab, paper, pencil.
Sequence of Lesson
Warm-up: Remind the class of all the applications of relations, and how they are used to describe the relationship between two sets of objects. Ask the class what were some objects that could be modeled using linear functions. If distance or speed with respect to time is not brought up, bring it up with the at the end of the discussion. Make sure that the basics are covered, i.e., the x-value indicates the time, and the y-value indicates the distance or speed at a given time.
End by getting them excited to model some fishies!
Description: Now the class should mostly be on the same page in terms of what their speed functions should be. This next activity involves looking at what the distance graph will be for each ‘fishy’ situation. The first distance graph is a nice warm up, particularly since it is given to the students by Simcalc. The next two will require some thought by the student. To graph the second fish question (The fish travels 2 m/s for 4 seconds and then 6m/s for 4 seconds) by themselves will require a good understanding of both piecewise functions and slope. Note that Simcalc can two-part piecewise functions by restricting the domain of the student fish to (0, 4) and the domain of the teacher fish by (4, 8). Of course, without the right y-intercept for the teacher fish, the two graphs will be discontinuous at 4. Therefore, actually writing out the piecewise functions presents an extra challenge for the students, since this forces them to consider that since fishies can’t teleport, they must determine how to make sure that graph is clear about where the fish is at the fourth second mark.
Post-Activity: Come back to a class discussion of the students' findings and what they thought of the connection between the speed graphs and the distance graphs. Discuss the reasons why the area underneath a function determines a speed graph's distance graph and why the slope of a function determines a distance graph’s speed graph. Also mention the last graph, that involved a speed graph with a slope of one. Briefly discuss the students’ ideas about this graph. Let them know that the answers to these exciting questions will be answered next time! This will give them something interesting to think about and keep their wheels turning when it comes to math. It turned out that the distance graph was a parabola. You may also discuss how the students might attempt to find the speed graph if it was not already available to them or they only had a distance graph if the graph wasn’t linear. Advanced discussion might include might include using tangent lines. Be sure to open up the discussion for any questions on the subject for students to share their points of view on the subject.