Lesson 4: Geometric Exploration with Linear Functions
Rationale
This activity gives the students a chance to practice really making sense of a problem with well defined constraints and abstract ideas embedded within it. The challenging problems and scaffolding available make this a great activity for developing perseverance. Therefore, these activities definitely hit the first and second Abstract Algebra process standard: ‘Make sense of problems and persevere in solving them’ and ‘Reason abstractly and quantitatively.’
In this activity we extensively compared and contrasted linear and quadratic models, thereby hitting the MCC9-12.F.LE standard. This activity also provides great in depth practice for graphing both linear and quadratic functions, in addition to providing intuitive scaffolding for when and why there are intercepts, maxima and minima (when the speed is zero), therefore contributing to the MCC9-12.F.IF.5 standard. What I like most about this exercise, is how the student is exclusively using material they are familiar with, ie, polygons and linear equations, to develop an application for and idea they may not be familiar with, quadratic functions. In addition, this lesson provides some small measure of scaffolding for when students are introduced to finding instantaneous change (such as speed) and the area underneath a curve in a future calculus class, while only utilizing ideas from geometry and algebra. I believe there are so many opportunities for teachers to introduce seeds of ideas that are found in calculus and thereby creating a fascinating classroom activity, and secretly prep them for a future calculus course. Much of the challenge of calculus is that you are given mere weeks to digest a series of very heavy concepts. Perhaps and activity such as this could spread out some of this processing and still generate positive algebra learning outcomes.
Standards
The activity covers Common Core Georgia Performance Standards (CCGPS) for 9-12th Grade. The detail of standards could be found here.
Formative Question
In the initial activity you’ll want to check up on all the groups, and make sure they are able to correctly find the areas of the shapes they have created. The following are ways you might help students with difficulty in finding the area:
In the next activity, the students might have some difficulty generalizing the formulas for the bounded regions. Not all students have to go at the same rates! The bonus question is there for the students who to go more quickly. The teacher should go around the classroom and assess how the students are doing. If they are having trouble generalizing formulas. Offer scaffolding or a related exploration. For example, you may ask the to come up with the area at specific values of ‘a’, instead of for ‘a’ in general. Then ask them to look at how they solved each value, and what the solving processes seemed to have in common. You might also introduce the idea that all the sides of the polygon are increasing proportioanlly. You might ask them if they could figure out what the general formula would be if the height and length of their square or triangle were the same. How would such a polygon be represented with an equation and vertical line? Notice that these questions are meant to foster exploration and offer scaffolding for learning objectives. Much of the classroom formative assessment will be a natural consequence of student responses in discussion, as they work out their ideas of how to apply the mathematics.
This activity gives the students a chance to practice really making sense of a problem with well defined constraints and abstract ideas embedded within it. The challenging problems and scaffolding available make this a great activity for developing perseverance. Therefore, these activities definitely hit the first and second Abstract Algebra process standard: ‘Make sense of problems and persevere in solving them’ and ‘Reason abstractly and quantitatively.’
In this activity we extensively compared and contrasted linear and quadratic models, thereby hitting the MCC9-12.F.LE standard. This activity also provides great in depth practice for graphing both linear and quadratic functions, in addition to providing intuitive scaffolding for when and why there are intercepts, maxima and minima (when the speed is zero), therefore contributing to the MCC9-12.F.IF.5 standard. What I like most about this exercise, is how the student is exclusively using material they are familiar with, ie, polygons and linear equations, to develop an application for and idea they may not be familiar with, quadratic functions. In addition, this lesson provides some small measure of scaffolding for when students are introduced to finding instantaneous change (such as speed) and the area underneath a curve in a future calculus class, while only utilizing ideas from geometry and algebra. I believe there are so many opportunities for teachers to introduce seeds of ideas that are found in calculus and thereby creating a fascinating classroom activity, and secretly prep them for a future calculus course. Much of the challenge of calculus is that you are given mere weeks to digest a series of very heavy concepts. Perhaps and activity such as this could spread out some of this processing and still generate positive algebra learning outcomes.
Standards
The activity covers Common Core Georgia Performance Standards (CCGPS) for 9-12th Grade. The detail of standards could be found here.
Formative Question
In the initial activity you’ll want to check up on all the groups, and make sure they are able to correctly find the areas of the shapes they have created. The following are ways you might help students with difficulty in finding the area:
- If they have difficulty with a rectangle, remind them of how you found the rectangle in the previous class. If they have difficulty with a right triangles, recall that they are just rectangles that are split ‘in half’. If they have trouble with the trapezoids, just remind them that they are simply a triangle added to a rectangle.
In the next activity, the students might have some difficulty generalizing the formulas for the bounded regions. Not all students have to go at the same rates! The bonus question is there for the students who to go more quickly. The teacher should go around the classroom and assess how the students are doing. If they are having trouble generalizing formulas. Offer scaffolding or a related exploration. For example, you may ask the to come up with the area at specific values of ‘a’, instead of for ‘a’ in general. Then ask them to look at how they solved each value, and what the solving processes seemed to have in common. You might also introduce the idea that all the sides of the polygon are increasing proportioanlly. You might ask them if they could figure out what the general formula would be if the height and length of their square or triangle were the same. How would such a polygon be represented with an equation and vertical line? Notice that these questions are meant to foster exploration and offer scaffolding for learning objectives. Much of the classroom formative assessment will be a natural consequence of student responses in discussion, as they work out their ideas of how to apply the mathematics.
Class Discussion
We have shown the class how quadratics have emerged out of geometry and linear equations. This demonstrates one interpretation of quadratics. Note that this also exposes them to seeds of ideas that will be useful in a future calculus course. Start a discussion of the various uses or interpretations of quadratics. What does the class think of the shape interpretation? What would a story be like if a quadratic was a story’s distance function, ie, determined the location of a person in a story? (A concept touched on in the previous class) How would the person's speed be changing?
Optional Discussion:
Discuss with the students why a speed graph determines a graph’s instantaneous rate of change and why this makes sense. How might students determine a distance graph for a relation if the speed graph was linear? (Relates to ideas touched on in the previous class) Note that this turns out to be an application of the formula discovered at the end of class today.
You might ask the student how they would determine the speed of an object was moving given that their position was determined by a quadratic. (Also relates to ideas from the previous class) Could the students determine what speed graph would create the given quadratic distance graph? You might also explore the slopes of speed graphs and why these lead to acceleration graphs.
From here there are many extensions available to you. You may continue to work with quadratics. You may create a distance problem to determine when a person is farthest or closest to you. This allows you to bring up ideas such as minimums, maximums, and that these must occur when the corresponding speed function is zero. (Why does this make sense?)
We have shown the class how quadratics have emerged out of geometry and linear equations. This demonstrates one interpretation of quadratics. Note that this also exposes them to seeds of ideas that will be useful in a future calculus course. Start a discussion of the various uses or interpretations of quadratics. What does the class think of the shape interpretation? What would a story be like if a quadratic was a story’s distance function, ie, determined the location of a person in a story? (A concept touched on in the previous class) How would the person's speed be changing?
Optional Discussion:
Discuss with the students why a speed graph determines a graph’s instantaneous rate of change and why this makes sense. How might students determine a distance graph for a relation if the speed graph was linear? (Relates to ideas touched on in the previous class) Note that this turns out to be an application of the formula discovered at the end of class today.
You might ask the student how they would determine the speed of an object was moving given that their position was determined by a quadratic. (Also relates to ideas from the previous class) Could the students determine what speed graph would create the given quadratic distance graph? You might also explore the slopes of speed graphs and why these lead to acceleration graphs.
From here there are many extensions available to you. You may continue to work with quadratics. You may create a distance problem to determine when a person is farthest or closest to you. This allows you to bring up ideas such as minimums, maximums, and that these must occur when the corresponding speed function is zero. (Why does this make sense?)