Lesson 3: Velocity with Linear Equation
Rationale
The discussion sections of this lesson and the formative assessment of the teacher give the students a chance to construct arguments for why their piece-wise function is appropriate for modeling their fish’s situation, thereby allowing the students to work on the third and fourth process standard for Advanced Algebra: ‘Construct viable arguments and critique the reasoning of others’ and ‘Model with mathematics.’ The activity also asks the students to see abstract ideas by observing repeated reasoning in the graphs. The basis of much of this project was the idea that increasing the speed changes systematically in the provided way changed the shape of the graph, but not the ending distance, thereby allowing students to work on the eighth Advanced Algebra process standard: ‘Look for and express regularity in repeated reasoning.’
This activity provides real motivation for why we would explore and become proficient at piecewise problems, thereby hitting the MCC9-12.F.IF.7b. This is introduced first with the constant speed graphs, giving them something simple to start with. This piecewise activity quickly increases in complexity with the distance graphs who need to have vertical graph adjustments. The connection the two graphs have to each other and to the Simcalc fish story gives something concrete the students can connect to when exploring these new ideas.
We use these equations to model and understand how the fish’s progress changes given different speed choices, thereby working on MCC9-12.A.CED.1. Distance and speed traveled is not only a very accessible introduction to modeling with linear equations, it also strengthens the idea of slope for the student to see it expressed in a separate speed graph, but also plants the calculus of idea of taking the area underneath a function to get the distance.
Standards
The activity covers Common Core Georgia Performance Standards (CCGPS) for 9-12th Grade. The detail of standards could be found here.
Formative Question
A quick assessment of student progress can be achieved by observing the shape of their graphs. This allows a quick analysis of well they understand the linear equation concepts and the activity. Asking for justification will have the multiple purposes of helping the teacher further assess the students’ understanding, help the students self correct, and help the students reflect on the ideas presented in the activity. The teacher should start with asking a justification for the shape of the students’ graph(s). A teacher shouldn’t initially give an indication of whether the distance graph is correct, but any justifications you ask for should be related to their misconceptions. Questions could include asking a question that will cause them to come to a contradiction or by asking what speed or distance the fish would be at during a certain times. If they are correct with their graph, you might ask how they knew the graphs would be of that shape, or ask how their speed graph relates to a distance graph (or vice versa) thereby developing the slope idea.
Additional Resources
The discussion sections of this lesson and the formative assessment of the teacher give the students a chance to construct arguments for why their piece-wise function is appropriate for modeling their fish’s situation, thereby allowing the students to work on the third and fourth process standard for Advanced Algebra: ‘Construct viable arguments and critique the reasoning of others’ and ‘Model with mathematics.’ The activity also asks the students to see abstract ideas by observing repeated reasoning in the graphs. The basis of much of this project was the idea that increasing the speed changes systematically in the provided way changed the shape of the graph, but not the ending distance, thereby allowing students to work on the eighth Advanced Algebra process standard: ‘Look for and express regularity in repeated reasoning.’
This activity provides real motivation for why we would explore and become proficient at piecewise problems, thereby hitting the MCC9-12.F.IF.7b. This is introduced first with the constant speed graphs, giving them something simple to start with. This piecewise activity quickly increases in complexity with the distance graphs who need to have vertical graph adjustments. The connection the two graphs have to each other and to the Simcalc fish story gives something concrete the students can connect to when exploring these new ideas.
We use these equations to model and understand how the fish’s progress changes given different speed choices, thereby working on MCC9-12.A.CED.1. Distance and speed traveled is not only a very accessible introduction to modeling with linear equations, it also strengthens the idea of slope for the student to see it expressed in a separate speed graph, but also plants the calculus of idea of taking the area underneath a function to get the distance.
Standards
The activity covers Common Core Georgia Performance Standards (CCGPS) for 9-12th Grade. The detail of standards could be found here.
Formative Question
A quick assessment of student progress can be achieved by observing the shape of their graphs. This allows a quick analysis of well they understand the linear equation concepts and the activity. Asking for justification will have the multiple purposes of helping the teacher further assess the students’ understanding, help the students self correct, and help the students reflect on the ideas presented in the activity. The teacher should start with asking a justification for the shape of the students’ graph(s). A teacher shouldn’t initially give an indication of whether the distance graph is correct, but any justifications you ask for should be related to their misconceptions. Questions could include asking a question that will cause them to come to a contradiction or by asking what speed or distance the fish would be at during a certain times. If they are correct with their graph, you might ask how they knew the graphs would be of that shape, or ask how their speed graph relates to a distance graph (or vice versa) thereby developing the slope idea.
Additional Resources
End the class with the wrap up video. Alternatively, with a reminder to watch the wrap up video once they are in the comfort of their own homes, with the company of some delicious crumpets.
Additional Resources:
Incline - http://www.shodor.org/interactivate/activities/Incline/
Graphing Stories - http://graphingstories.com/
Incline - http://www.shodor.org/interactivate/activities/Incline/
Graphing Stories - http://graphingstories.com/