12th Grade Math : Unit #9 - Full Course Review via Motion : Lesson #10

Calculus and My Car's Dashboard

Objective: SWBAT make and implement a plan to compute displacement given velocity data from a car’s dashboard. SWBAT identify and correct common mistakes on the spring break AP question set.
Standards: MP1 MP4 MP5
Subject(s): Math
60 minutes
1 Video Narrative - Lesson Overview - 0 minutes
Calculus and My Car's Dashboard, Video Narrative, Lesson Overview.mov
2 Warm-Up + Homework Review - 15 minutes
Warm-Up + Homework Review.docx
3 Investigation: Car Dashboard Video Project - 35 minutes

To begin our investigation of the Car Dashboard Video Project, I play the Car Dashboard Video (make sure it’s the hidden version!) through the video until the end.  Then ask students what question comes to mind – inevitably students will ask “How far did the car travel?” 

Generally, I prefer letting students formulate their questions as much as possible – this approach increases engagement and interest with the problem, and students cannot just depend on the teacher to always do the question formulation for them.  Depending on my students’ ability to use calculus for modeling motion (SMP #4), it may be fruitful to ask preliminary questions that get students thinking about ways to connect their prior calculus knowledge with the authentic velocity data in this video. 

Once students have collected the time and velocity data from the video, the most common approach to this project is using regression to find several functions that collectively model the velocity function over partitions of the full driving time, then summing each of the integrals of the regression equations on the applicable intervals.  I expect my students to use either Right or Left Rectangular Approximation Methods (RRAM or LRAM) or the Trapezoidal Approximation Method (TRAP). Though I will not stop students from approaching the problem this way because it is a mathematically acceptable approach, I will try to have a conversation with these students after they complete the project about more efficient approaches. 

I expect that some of my students may still be unsure how to approach this problem. Rather than discussing regression or RAM techniques with them, I will ask them to recall a project we completed earlier this year that might be applicable to this project.  If students recognize and connect the technique they used for gathering measurements from a bottle and then estimating its volume, they will be more likely to recognize the need to gather measurements from my car’s dashboard and then use calculus to estimate its change in position.

DIFFERENTIATION:  Scaffolds that I may use for the Car Dashboard Video Project might include:

  • Asking questions that relate multiple derivative relationships of f, f ’, and f ” to motion relationships of position, velocity, and acceleration.  
  • Asking questions that use the behavior of f ’ to answer questions about f, such as “When is my car’s odometer increasing the fastest?  What evidence from the video supports your answer?”

The assignment I will give my studentsA written report consisting of 1-2 paragraphs explaining your approach and solution to this problem is due in 1 week.  Make sure to summarize your approach to solving this problem and interpret the calculus computations in the context of the car’s motion (SMP #6)

Once students have submitted their reports, I will play the video file that reveals the odometer’s reading (Car Dashboard Video - Reveal - no song).  I tell students that if they are absent on the due date then they must e-mail their written report to me before class begins on that date in order to receive credit for the project, since the answer will be displayed in class.  This is noted on the Car Dashboard Video Project - Description, but it is always good to discuss these kinds of policies directly with students.


4 Investigation: Review of AP Practice Set - 10 minutes

When displaying the compiled data to students to identify their most frequently missed questions on the 1997 AP Multiple Choice Problems, I use students’ preliminary submissions, not the final submissions.  The preliminary data provides a more accurate picture of my students’ current understandings of these questions. I have given my students feedback from me over the break about the correctness of their preliminary answers. 

As I review the AP question set today and identify mistakes that led students to choose common distractors, I ask students to maintain their own log of mistakes. I believe that keeping a "preparation log" helps students recognize their errors and avoid repeating them in the future.  The log is effectively a customized reference sheet and it will be VERY useful as students scale up their preparation for the AP test. 

The 1997 MC Calculus AB - Student Guidance for Most Missed Questions identifies the content of the most frequently missed questions, and, it lists a citation to a helpful chapter or section in the textbook. 

Teacher's Note: The specific questions that your students frequently miss will likely vary somewhat from mine, and the textbook citations will have to be customized to your specific textbook, so be sure to make these modifications before photocopying and distributing this sheet to students. 

Rather than rushing through and solving all of the most frequently missed questions myself in class today, I prefer to select a few of the most “important” questions that are worth spending class time discussing at length, and then holding a comprehensive discussion about those questions, common student errors, and trying to discover the errors represented by each of the distracters.  Students are much more likely to retain their learning with this level of discussion and analysis, rather than ending our work with each question once we get the answer.



5 Closure + Homework - 2 minutes
Closure + Homework.docx