Activity: SimCalc
Activity Links
SimCalc
Materials Needed
Access to website (http://www.kaputcenter.umassd.edu/products/software/), computer lab, student handout (see attached file), 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!
Ask them to imagine that a fish is swimming at 4m/s. How far do they think that fish would go in 8 seconds? How do they think they might model the fish’s activity? What would a graph that modeled speed or distance for this fishy look like? These questions and more will be explored in today’s activity!
Tutorial Video: https://www.youtube.com/watch?v=qfAAS8aA5GE
Description: Provide groups a worksheet with a series of questions. Each question is associated with a simulation on Simcalc. Once a group feels satisfied with one question, they move onto the next. The group should also attempt to represent each question as a function using GSP, where the y-axis is the speed of the farmer. The students start out by programming a fish to swim 4 meter per second, and then observe the simulation.The students considers the following questions. They will use the Simcalc program to explore the situations that the questions involve, including making speed graphs of each. If a group finishers early, they can start working on distance graphs.
Post-Activity: Come together as a class and discuss the solutions that the groups had to the questions, and the graphs that they developed. You might ask the students if there seems to be a pattern emerging in the graphs as they change. Note that there seems to be a pattern of the speed graphs having more and more steps, until the last graph is just a slope. You might ask what it would look like if we just kept increasing the number of 'steps' that the graph had while maintaining the time length as 8 seconds. The intended conclusion being, that it would look more and more like the last graph. Explain the interpretation of the pairs of graphs in the class, and talk about why it makes sense that the area underneath the speed graph is the distance graph. Note that the initial two questions on the Group discussion worksheet are meant to provide the initial intuitive understanding of this, "If a farmer travels 4 meters per second for one second, how far does he go?"Also note to the class that we made these graphs that seemed to divided up into pieces! Can we identify how to represent each piece. Talk with the kids about how they might try to describe the whole graph in terms of its pieces. End with the canonical way of writing the speed graphs as piece-wise functions. Discussing what the speed graph looks like with a constant increase in speed is optional. It can be left to the end of the class time or to be recapped in the video.
Link to the video: https://www.youtube.com/watch?v=F6X8fnmf-mc
Common Misconceptions and Difficulties
Watch the video for discussion of misconceptions and difficulties.
Resources
SimCalc
Materials Needed
Access to website (http://www.kaputcenter.umassd.edu/products/software/), computer lab, student handout (see attached file), 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!
Ask them to imagine that a fish is swimming at 4m/s. How far do they think that fish would go in 8 seconds? How do they think they might model the fish’s activity? What would a graph that modeled speed or distance for this fishy look like? These questions and more will be explored in today’s activity!
Tutorial Video: https://www.youtube.com/watch?v=qfAAS8aA5GE
Description: Provide groups a worksheet with a series of questions. Each question is associated with a simulation on Simcalc. Once a group feels satisfied with one question, they move onto the next. The group should also attempt to represent each question as a function using GSP, where the y-axis is the speed of the farmer. The students start out by programming a fish to swim 4 meter per second, and then observe the simulation.The students considers the following questions. They will use the Simcalc program to explore the situations that the questions involve, including making speed graphs of each. If a group finishers early, they can start working on distance graphs.
Post-Activity: Come together as a class and discuss the solutions that the groups had to the questions, and the graphs that they developed. You might ask the students if there seems to be a pattern emerging in the graphs as they change. Note that there seems to be a pattern of the speed graphs having more and more steps, until the last graph is just a slope. You might ask what it would look like if we just kept increasing the number of 'steps' that the graph had while maintaining the time length as 8 seconds. The intended conclusion being, that it would look more and more like the last graph. Explain the interpretation of the pairs of graphs in the class, and talk about why it makes sense that the area underneath the speed graph is the distance graph. Note that the initial two questions on the Group discussion worksheet are meant to provide the initial intuitive understanding of this, "If a farmer travels 4 meters per second for one second, how far does he go?"Also note to the class that we made these graphs that seemed to divided up into pieces! Can we identify how to represent each piece. Talk with the kids about how they might try to describe the whole graph in terms of its pieces. End with the canonical way of writing the speed graphs as piece-wise functions. Discussing what the speed graph looks like with a constant increase in speed is optional. It can be left to the end of the class time or to be recapped in the video.
Link to the video: https://www.youtube.com/watch?v=F6X8fnmf-mc
Common Misconceptions and Difficulties
Watch the video for discussion of misconceptions and difficulties.
Resources
Worksheet
Answer Key (Distance Traveled and Piece-wise Relations)
Question 1.) 32 meters
Question 2.) 32 meters
Question 3.) 32 meters
Question 4.) 32 meters
Question 5.) y=8x
Question 6.) Piecewise function:y =2x for x in [0,4]
=6x-16 for x in (0,8]
Question 7.) Piecewise function:y =1/2 for x in [0,1]
=3/2x-1 for x in (2,3]
=5/2x-3 for x in (3,4]
=7/2x-6 for x in (4,5]
=9/2x-10 for x in (5,6]
=11/2x-15 for x in (6,7]
=13/2x-21 for x in (7,8]
=3/2x-28 for x in (1,2]
Question 8.) y=(1/2)(x^2)
Question 1.) 32 meters
Question 2.) 32 meters
Question 3.) 32 meters
Question 4.) 32 meters
Question 5.) y=8x
Question 6.) Piecewise function:y =2x for x in [0,4]
=6x-16 for x in (0,8]
Question 7.) Piecewise function:y =1/2 for x in [0,1]
=3/2x-1 for x in (2,3]
=5/2x-3 for x in (3,4]
=7/2x-6 for x in (4,5]
=9/2x-10 for x in (5,6]
=11/2x-15 for x in (6,7]
=13/2x-21 for x in (7,8]
=3/2x-28 for x in (1,2]
Question 8.) y=(1/2)(x^2)
Answer Key (Graph Questions)