## Unit 4 - Day 2

##### All Units
###### ​Learning Objectives​
• Calculate rates of change in the context of straight-line motion.

###### ​Success Criteria
• I can use first and second derivatives to find the velocity and acceleration of an object, given its position.

• I can determine when an object is at rest, speeding up, or slowing down.

###### Overview

A greater amount of time than usual should be spent on the Activity, Debrief, and Important Ideas sections of Topic 4.2.  Representing the components of rectilinear motion (position, velocity or acceleration) on a rectangular grid is a necessary part of this lesson.  The concepts of speeding up/slowing down and moving toward/moving away from the origin must be approached using multiple representations:  graphical, analytical, and numerical. The Activity and Important Ideas sections of Topic 4.2 address these representations. Additionally, students will have to interpret and explain their solutions as they apply to a given context.

###### Teaching Tips

Use different colors to denote the various components of motion when drawing position, velocity, and/or acceleration on the same grid. Continually point out when the values of a particular function are positive or negative and when the slope of that same function is positive or negative. Require precise language for the many functions students will see: disallow “the function,” or “the slope,” or “the graph,” when more than one function, slope or graph is present. Require students to use specific names for any function. Saying or writing “It’s increasing because it’s positive” is ambiguous and will not earn points on the AP Test.

Unfortunately, the terms “value,” “slope,” “increasing,” “positive,” and “derivative,” are often confused and appear interchangeably, almost, in student writing and speech. We can reduce some of this by using precise language ourselves and encouraging correct language from our students.

A great method for illustrating why a particle moving on a horizontal axis will slow down or speed up is to represent the particle yourself with positive (or negative) velocity. If you are moving to the right (or left) in front of your students, acceleration can easily be represented by a student acting as a force that is positive (pushing you in the same direction you are traveling) or negative (pushing against your current direction).
Let’s assume you are moving with positive velocity (to the right):

*If the student pushes you to the right (to represent positive acceleration), you will speed up: positive velocity combined with positive acceleration will speed up an object. We justify this by writing “v(t) > 0 and a(t) > 0 means the object is speeding up.” The same effect will appear if you move to the left and are pushed to the left.

*If the student pushes you to the left (to represent negative acceleration), you will slow down: positive velocity combined with negative acceleration will slow down an object. We justify this by writing “v(t) > 0 and a(t) < 0 means the object is slowing down.” The same effect will appear if you move to the left but are pushed right.

There are four non-zero combinations of v(t) and a(t). A visual example of each scenario may or may not be needed, but students must know all four outcomes and the justification for each.

Similarly, there are four non-zero combinations of s(t) and v(t) to represent motion toward or away from the origin: s(t) > 0 and v(t) < 0 means the particle is moving toward the origin (as does s(t) < 0 and v(t) > 0); if s(t) > 0 and v(t) > 0, the particle is moving away from the origin (as does s(t) < 0 and v(t) < 0).

###### Exam Insights

With 5 or 6 multiple‐choice questions and one free‐response question, motion problems usually make up about 14% of the exam. The stem of the question may introduce information graphically, analytically, or numerically (table format), so students should be familiar with all types of presentations.

See the following FRQs for excellent practice:
2011 AB 1 (calculator active; equations given)
2006 AB 4 (tabular)
2009 AB 1(graphical)
2008 AB 4 (graphical)

Multiple choice questions cover the same content as the FRQs, but in smaller pieces. See the following:
2008 MC #7, 21, 82, 86, 87
2012 MC #6, 16, 28, 79, 83, 89

###### Student Misconceptions

Although motion concepts are easy to visualize (imagine a squirrel running back and forth along a telephone wire with numerous starts and stops and changes of direction), many students will be challenged by representing this horizontal motion on a rectangular coordinate grid. For them, the y-axis has represented the height of an object for many, many years.

Now, we are using the vertical axis to measure position, velocity, and acceleration values --- sometimes with all three graphs on the same grid! And, later, the graphical interpretation of integral functions will add another wrinkle to the process.

Quite often, the words “value,” “slope,” “increasing,” “positive,” “derivative,” --- and many other terms! --- are conflated and confused and appear to be interchangeable in student writing and speech. We can reduce some of these errors by using precise language ourselves and patiently encouraging the same from our students. Lots of practice with Topics 4.1 and 4.2 is the prescription for good results on the AP Test.