##
* *Reflection:
What Goes Up, Day 1 of 3 - Section 2: Individual Time

Many students assumed that the highest data point actually represented the maximum height of the stone. When two students pointed out that they had tested some other *t*-values and found greater heights, I asked the class to calculate the *exact* maximum. No one knew what to do. After a bit of discussion, it was decided that we could find the other time at which *h* = 5, then “go to the middle” and that would be the time of maximum height.

Interestingly, one student pointed out that this question was a lot like the ones we’ve recently been answering since it was about finding a maximum or minimum of some sort. I thought this was a keen insight, but it sent one other student on a wild goose chase to somehow use a system of inequalities to locate the vertex. Oh, brother!

One student used an interesting method. He reduced the equation to 16*x*^2 = 50*x* and then divided by *x* to obtain 16*x* = 50, or *x* = 25/8. I applauded him for this solution (since no else had one at all) but then pointed out that we had only found one solution. We already knew there was a second solution (*x* =0), but for some reason it didn’t show up. Why? No one could say.

After pointing out that we had eliminated the second solution when we divided by *x* (we'll return to this concept in great depth later), I reminded the class of the Quadratic Formula, and they were all able to use it to solve the equation - thank goodness!

*Working with Quadratic Equations*

*Working with Quadratic Equations*

# What Goes Up, Day 1 of 3

Lesson 7 of 15

## Objective: SWBAT model projectile motion using polynomial functions. SWBAT answer questions about velocity and acceleration using quadratic function models.

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

*45 minutes*

#### Individual Time

*10 min*

Hand out What Goes Up and ask students to begin working individually for about 10 minutes.

In that time, they are expected to do three things:

1. Understand the situation.

[Some students may need help understanding that the equation does not model the actual *path* of the stone, but the relationship between its *height* and *time*. The dynamic GeoGebra application may help with this.]

2. Complete the table and graph.

[Encourage students to use *at least* 1/2-second intervals for their table.]

3. Find maximum height & flight time.

[This is a good test of how well students have retained their skills from Algebra 1. The problem is open to interpretation: does the stone hit the "ground" at *h *= 5 or at *h* = 0? Depending on the class, it might be worth coming to a consensus before moving ahead.]

Please see my **Strategy Video on Individual Time** for more details.

*expand content*

#### Group Time

*5 min*

Now, announce to the class that they may begin working with their groups (pre-assigned, three or four students max.) for five minutes.

Tell that class that you expect them to:

1. Compare & comfirm their answers to parts 1 and 2

[If all goes well, this should take very little time, but this is the main purpose of the group time.]

3. Begin describing *velocity* in qualitative terms (increasing/decreasing, upward/downward, etc.)

[I expect students to be a bit confused by this question, but I want them to begin thinking about it and discussing it in preparation for the class discussion that's coming next.]

For more details, please see my **Strategy Video on Group Time**.

*expand content*

#### Checking In

*10 min*

In a 10-minute class discussion, I hope to briefly summarize and clarify the students' solutions to parts 1 and 2 of the problem, and then get them ready to investigate the average velocity of the projectile.

I plan to ask four questions to make this happen:

1. Are we all looking at the same graph?

[I will use the **GeoGebra **resource here to make the model more intuitive. If there is confusion about the actual path of the stone, this resource can help.]

2. What features are important?

[We will point out the mathematical features: symmetry, vertex, domain, intercepts. Then we will interpret these in the context of the situation. (**MP 2**)]

3. What is *velocity*?

[Students should be familiar enough with this concept, but I want to emphasize the concept of **rate of change** and also describe it qualitatively as "how quickly an object is moving in a particular direction". Of course, we will also distinguish speed from velocity.]

4. How do you calculate it?

[It is good to remind students of the general velocity formula, as well as to discuss the unit of measure for velocity. What I do NOT want to do here is to tell students how they are to compute the average velocity in this situation. (**MP 1**)]

Once we've reached the point of recalling *generally* how to calculate average velocity, I'll drop the conversation and announce: *Good. Back to work!*

*expand content*

#### Group Time

*15 min*

My expectations during this final 15-minute collaborative work time are:

1. Qualitatively describe the change in the velocity over time. As you circulate, encourage students to be attentive to precision in their writing. Saying something clearly is not an easy thing to do! (**MP 6**)

2. Estimate actual velocity at three specific times.

(For this, I expect students to use several small time intervals and consider the average rate of change of the height over that interval. They have some freedom here, but they might be encouraged to ask how the interval they choose affects the estimate. The best estimates in this case come from small intervals that are centered on the time in question, but it isn't necessary for all groups to do this. In fact, it makes for a more interesting class if different groups come up with different intervals for different (but good) reasons!)

If students get as far as considering acceleration, that's great! Encourage them to find a way to compute *average acceleration *by first finding *average velocities* on a series of equal time intervals (1/2-seconds, for instance). See the solutions resource for an example of what this might look like.

Tomorrow, we will discuss the conclusions they've drawn about the velocity of the stone, and then we'll work to answer the remaining questions.

*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