##
* *Reflection:
What Goes Up, Day 2 of 3 - Section 1: Determining Average Velocity

Two students came to the board today. The first explained – pretty clearly – how he was able to find the average velocity of the stone on a given interval, and why the calculations he was doing made sense. When he was through, it seemed that almost the entire class was on board; they saw both how 42 feet per second was calculated and why this number must be the average velocity on the first 1/2-second interval.

Then a second student raised his hand. This student claimed that “since the equation began with -16t^2, it made sense that the initial velocity would have to be 50. Oh, also because the next term is 50t.” Everyone understood *what* he was claiming, but I don’t think he won anyone over by his ‘argument’. When I asked if he could clarify his thinking, he replied that he was basing his claims on outside information he had from physics. It seems that no one else was privy to that information, and it was clear that he was unable to make it accessible by any further justification. I thanked him and made a point of differentiating between his *claim* and the first student’s *argument*. I also made sure to point out that these two students weren't disagreeing; the first number was an *average* velocity, while the second was an *instantaneous* one. Everyone understood that the average velocity was an estimate, but at least they understood it!

This little episode illustrated for me once again that I have to be ready for anything, and that I should never be afraid of what a student may propose. The important thing is that the classroom be an open environment for ideas, that all ideas be given the respect they deserve, but that in the end our discourse is about *argument* and arguments require *evidence*.

*Student Explanations*

*Student Explanations*

# What Goes Up, Day 2 of 3

Lesson 8 of 15

## Objective: SWBAT interpret the average rate of change of a quadratic function in terms of the velocity and acceleration of a projectile. SWBAT make use of the structure of the quadratic equation to compare projectile motion under a variety of conditions.

## Big Idea: Projectile motion provides context for average rates of change in the context of velocity and acceleration. What goes up, must come down!

*45 minutes*

#### Group Time

*5 min*

During this brief collaborative session, the students have just one task:

*Compute average velocities for each 1/2-second interval from 0 - 3.5 seconds and record them in a table.*

See the solutions document in the resources for an example of what this table might look like. Video: BL What Goes Up Day 2, the big idea

Watch out! Many students may simply take the change in the height as the average velocity. Remind them that this change occurred over a *half*-second. What would be the change over a *full* second? That's the velocity.

As I find the first groups completing the task, I make a point of guiding those students to the conclusion that the *average velocity* on the interval *t* = 0 to *t* = 0.5 is an estimate of the *actual velocity* at *t* = 0.25 (the midpoint of the interval). This will be crucial for understanding the average acceleration correctly! It might be helpful to suggest that they try graphing the velocity data, if they do, they will see that the velocity is a linear function. A graph will help students make sense of their calculations and also begin to see that the velocity itself is changing at a constant rate - the force of gravity!

Of course, the point of all of this is to lead students to see the meaning behind the coefficients in the displacement function. Ultimately, they will be able to identify the constant term with displacement, the linear term with velocity, and the quadratic term with acceleration.

#### Resources

*expand content*

#### What is Acceleration?

*5 min*

In this section, the students and I have a brief conversation about acceleration to prepare them to answer the remaining questions. The dialogue might go something like this:

What is *acceleration*?

[Students should recall that acceleration is the rate of change of velocity. You can calculate average acceleration on a given interval by comparing the change in the velocity to the change in time.]

Is the velocity of the stone changing over time?

[Yes, it's decreasing.]

Okay. Let's use the average velocities we've computed to find some average rates of acceleration.

[Ok, well the velocity changes by a constant amount. I guess that means the acceleration is constant?]

How much does the velocity change each second?

[It goes down by 8. No, by 16. No, by 32. Huh?]

Wait a minute! Think carefully about how much time it's taking to change. What is the *per second* rate of change?

[Well, since the displacement is given every 0.5 sec, the average velocities are given per 0.5 sec., so the average accelerations are given every 0.25 sec.! The velocity changes by -32fps every second!] *See the Solutions document for more details on this!*

Great. What does that mean?

[Um.]

Well, Isaac Newton says that the stone would just keep moving in a straight line, and at a constant velocity, unless something acted on it. So, is something acting on the stone?

[Yes, gravity! The acceleration is the force of gravity! And *that* explains why it's constant.]

Good. From here you should be able to answer the final question. When you think you're done with it, let me know. If your answers are correct, I'll give you the next part of the assignment.

(Optionally, you might consider graphing the average velocities to see that they lie on a straight line with a constant slope. It's a nice visual confirmation of what the students are seeing numerically. It also forces you to assign each average velocity to a specific time, allowing you to see more clearly the actual interval over which the velocity is changing.)

#### Resources

*expand content*

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- UNIT 1: Modeling with Algebra
- UNIT 2: The Complex Number System
- UNIT 3: Cubic Functions
- UNIT 4: Higher-Degree Polynomials
- UNIT 5: Quarter 1 Review & Exam
- UNIT 6: Exponents & Logarithms
- UNIT 7: Rational Functions
- UNIT 8: Radical Functions - It's a sideways Parabola!
- UNIT 9: Trigonometric Functions
- UNIT 10: End of the Year

- LESSON 1: What is Algebra?
- LESSON 2: The Music Shop Model, Day 1 of 2
- LESSON 3: The Music Shop Model, Day 2 of 2
- LESSON 4: Letters & Postcards, Day 1 of 2
- LESSON 5: Letters & Postcards, Day 2 of 2
- LESSON 6: Choose Your Own Adventure
- LESSON 7: What Goes Up, Day 1 of 3
- LESSON 8: What Goes Up, Day 2 of 3
- LESSON 9: What Goes Up, Day 3 of 3
- LESSON 10: The Constant Area Model, Day 1 of 3
- LESSON 11: The Constant Area Model, Day 2 of 3
- LESSON 12: The Constant Area Model, Day 3 of 3
- LESSON 13: Practice & Review, Day 1 of 2
- LESSON 14: Practice & Review, Day 2 of 2
- LESSON 15: Unit Test