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.
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.
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!
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.