Modeling Projectiles

My Functions for Modeling classes are ending on Tuesday. These are the introductory math classes at Baxter Academy. This course is paired with Modeling in Science, which has a focus on science inquiry and the physics of motion – kinematics. The final assessment is a ballistics lab, where the student groups have to measure the launch velocity of their projectile launcher and, along with some other measurements and a few guesses, identify the best launch angle and launch point to fire the projectile at a vertical target. The students are not allowed to have test shots or simulated shots. They are expected to gather the necessary data and complete all of their calculations before testing their theory with one, single shot. The target is fairly forgiving, but students are still amazed when their predictions result in a projectile going through the target.

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How do you teach Project Management?

One of the founding principles of Baxter Academy is the idea that working on big, long-term projects is worthwhile. These are projects that students design, sometimes in conjunction with a faculty member, sometimes on their own. While the process worked pretty well last year, the student feedback indicated that they wanted to be held more accountable for their project proposal, for the work they did on the project, and what they were able to accomplish. So this year we ramped up the proposal process. Project teams have to write a proposal using this template and submit it to a review panel consisting of a teacher, administrator, and student senator. Almost every proposal fails to be approved on the first try, but that’s okay. The review process has made the proposals more focused and much stronger.

Since we devote every Friday to this project work, it’s important that the project teams have a clear idea of what they want to accomplish and what they need to do each week. Which begs the question, “How do you teach project management?” Gantt charts are nice, but require good guesses at tasks and completion times. I have experience teaching the critical path method, which I think it pretty cool, but again, you need to have an idea of what the tasks are and how long each task will take. And the “projects” were all hypothetical, from a textbook. Nobody was actually trying to do them. Our engineering teacher is really partial to Scrum boards. I kind of get how they work, but have a difficult time conveying their use and importance to students.

Do you just dive in and make guesses at times and adjust along the way?

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It’s all in how you ask the question

I teach half of an integrated math & science modeling class. On the math side, we focus on functions and a little bit of right triangle trigonometry. The science side is all about motion, one dimensional and two dimensional – hence the trigonometry. We’re now entering the final few days of the trimester, and have gotten into that 2D motion part. Did I mention that this is the introductory math/science class for 9th graders at Baxter Academy?

We started with Dan Meyer‘s Will It Hit the Hoop? concept, slightly modified. Showed Act 1 video, but captured this picture for analysis.

hoop shot-1

Interesting conversation begins. Many students are convinced that the ball will fall short of the hoop because “it is slowing down.” What makes them think that, I wondered. Maybe because up until this point, the conversation in science has been about constant velocity motion, in one dimension. Showed Act 3 of course and those who were sure the ball would go in were vindicated. But their comments still nagged at me. Maybe they just need more experiences – this was, after all, just the first day of 2D motion.

We watched part of an episode of Mythbusters, the one where they fire a bullet and drop a bullet and have them land in the same spot at the same time. It’s really a good episode. It really helps to drive home the fact that the forward motion of the bullet has nothing to do with how much time it takes to fall to the ground. It means that horizontal and vertical motion can be thought of, and modeled, independently of each other. On the science side of things they had developed the kinematics model: z(t)=\frac{1}{2}at^2+v_0t+z_0. So then we adapted that model for horizontal and vertical motion. We went back to the basketball shot. Analyzing the photo against the graph, we estimated the the initial position of the ball is at (1, 8) and the final position of the ball would be (19.75, 10). We also figured that the ball was in the air for about 1.8 seconds. From this information my students calculated the initial horizontal and initial vertical velocities to be 10.4 ft/s and 30 ft/s, respectively.

hoop shot

 

 

 

 

 

 

But Dan did not throw the ball only horizontally or only vertically. He threw it at an angle – so that it could reach the hoop, presumably. So I asked the question: “What was the launch velocity of the basketball?” and accompanied the question with this image:

hoop shot 2

 

 

 

 

Class was over, so I left them to work that out for homework.

Next day, I had them check in with each other and then asked, “How did you think about this problem?” Overwhelmingly, they agreed that the launch velocity must be the average of the horizontal and vertical velocities. This happened with both groups.  I asked them why they thought it should be the average. I asked if they thought the launch velocity should be greater than 30 ft/s, between 10.4 ft/s and 30 ft/s, or less than 10.4 ft/s. They were convinced that the launch velocity should be somewhere between 10.4 ft/s and 30 ft/s. Some thought that it should be closer to 30 ft/s since the ball is “going more up than over,” but that it would still be less than 30 ft/s.

A student in one of the classes convinced that group that it couldn’t be the average with the following reasoning: Suppose that the ball was thrown straight up. That means that the vertical velocity is 30 ft/s and the horizontal velocity is 0 ft/s. If the launch velocity is the average, then that would be 15 ft/s, but we know that the launch velocity is 30 ft/s. So it can’t be the average! So, of course I asked, “Then what could it be?” And they went with the idea that it must be the sum of the two velocities. But that would give us a launch velocity greater than 30 ft/s. We talked about this for a few minutes. They weren’t sure.

Then I showed them this picture:

hoop shot 3

 

 

 

 

Only after seeing this picture did they make any connection to a right triangle, or Pythagorean Theorem, or trigonometry. It had taken the better part of an hour to arrive at this conclusion, and then it took only 5 minutes to find the solution.

What would have happened if I had jumped directly to the right triangle representation? They would have had a quick solution, but they wouldn’t have had the opportunity to think about whether or not the launch velocity is the average of the horizontal and vertical. Maybe you think it was a waste of class time to allow my students to engage in such discussion. Maybe it was, but I don’t think so. My students had to take some time to construct meaning. They had to confront their misconception and convince themselves and each other that taking the average didn’t make sense. Sure, I could have told them, it would have been more efficient, but would that really have helped their understanding?

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Problem Solving with Algebra

That’s the name of one of the classes I’m teaching this term. We have trimesters. So each term is 12 weeks long and we have a week of “intersession” in between the terms. Except that this first term is not quite 12 weeks. The expansion area of the building wasn’t quite finished when we started school, so we had some alternative programming called “Baxter Foundations.” It included stuff like my Intro to Spreadsheets workshop. Classes started this week. And one of my classes is called Problem Solving with Algebra. I came up with that name, and I honestly don’t know exactly what it means. I have a rough idea, but it could go in a lot of different directions. Mostly, I want my students (and all the students taking this course) to think and puzzle and use algebra and solve problems.

Then I got an email that Jo Boaler has published a short paper called The Mathematics of Hope. In it she discusses the capacity of the human brain to change, rewire, and grow in a really short time based on challenging learning experiences. We’re not talking about learning experiences that are so challenging that they’re not attainable, but productive struggle. Challenging learning experiences that produce some struggle, but are achievable. The ones that make you feel really good when you solve them. You know the ones I mean.

So I decided to start this class with a bunch of patterns from Fawn Ngyuen‘s website visualpatterns.org. The kids are amazing. They jumped right in. Okay, so I taught most of them last year and they know me and what to expect from me, but seriously. Come up with some kind of formula to represent this pattern. Kinda vague, don’t you think? And I’m pushing them to come up with as many different formulas as they can, and connect those formulas to the visual representation. For example, an observation that each stage adds two cubes to the previous stage would result in a recursive formula like: C(n) = C(n-1) + 2 when C(1) = 1 (which is a recursive formula for pattern #1).

On Tuesday, different groups of students were assigned different patterns. Wednesday, each group presented what they were able to figure out. Some had really great explicit formulas, while others had really great recursive formulas. A few had both. Most were stumped at creating an explicit formula for pattern #5, pattern #7, and pattern #8.

Tuesday night, I received this email from Sam, a student:

“After staring at the problem for 2 hours, (5:45 to 7:45) and scribbling across the paper as well as two of my notebook pages, I am still unable to find a explicit equation. Then, reading the directions, I realized that the way they are worded allows the possibility of no explicit equation, as well as the fact that I only had to come up with equations as I can find. So after 2 hours, several google searches, lots of experimentation and angry muttering, I decided I have all that I can muster, and must ask you in the morning.”

I left them with the challenge to find an explicit formula related to one of these patterns. Their choice. Just put some thought into it before we meet again on Monday. Wednesday night, Sam sent me this followup email:

“After another hour at work, I found the explicit formula. I realized that the equation was quadratic, not exponential, and youtubed a how-to for quadratic formulas from tables. I kid you not, the man said the word “rectangle” and from that, I solved the problem. Then I watched the video through and took quick notes for future reference.”

Then, Dan Meyer posts this: Real work vs. Real world. Makes me think – as always. What am I asking of my students? This is real work – they are engaged and they are thinking. Sam, and the others, were not going to be defeated by a visual pattern. The fact that they are working in a “fake world” doesn’t matter.

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The Power of Interesting Questions

Today I led two groups of students through an introduction to spreadsheets as part of our Baxter Foundations workshops. Our framing question was, “How much is that Starbucks habit costing you?” Many students, of course, said $0, but we widened the question to include other vices, like Monster drinks, Red Bull, going across the street to Portland Pie every day, or down the street to Five Guys for lunch. And we broadened the question to, “What if you put your money into a retirement fund instead?” To make this real for my students, my friend Tracy admitted to her Starbucks habit and offered to be our real case study.

Before we started creating anything, I asked the students to complete this quick survey to figure out what they knew and what they didn’t. Then we looked at the results as a group. Here’s what we found:

Group 1: Mostly sophomores

Group 1: Mostly sophomores

grp2-1

Group 2: All freshmen

grp1-2

Group 1: Mostly sophomores

grp2-2

Group 2: All freshmen

grp1-3

Group 1: Mostly sophomores

grp2-3

Group 2: all freshmen

Clearly, the sophomores were bringing more to the table than the freshmen. After all, they had been instructed in spreadsheets in their engineering class last year, but they were still a bit unsure of what they knew. They thought they probably knew more than they had indicated, but didn’t know what I meant by “cell reference,” for example. And remember, I teach in Maine where 7th graders are given their own digital device. It used to be a laptop, but last year many districts changed to iPads. I would have expected the 9th graders to have had much more experience with spreadsheets, but I’m seeing that the switch to iPads is having an impact on that. Very sad.

I began by explaining the situation: Tracy spends $x each day on her Grande Soy Chai at Starbucks. If we want to figure out how much she spending, and what she could be earning instead, what information do we need? And then I had them brainstorm for a couple of minutes.

Information needed: cost of the drink, how much spent each month, and interest rate for the investment.

We made a few assumptions:

  • Tracy could find a mutual fund, or other investment, that earns an average of 7% annually
  • that she is 25 years from retiring (I don’t actually know this)
  • that the price of coffee would not change over the life of the investments (we knew this was unreasonable)
  • that Tracy would invest the same monthly amount for the life of the investment (also unlikely)

But this is also part of problem solving. Take a few minutes to watch Randall Monroe’s TED Talk and you’ll understand what I mean.

So here’s the spreadsheet that we came up with.

So what did we learn?

  • Tracy spends a lot of money on her Grande Soy Chai. But, it’s possible that the drink adds some value to her life and is worth the price.
  • Investing early and for a long time really can pay off, even if the amount invested isn’t all that much each month.
  • Learning about spreadsheets can be fun if you have an interesting question to answer.

Do I think the students in this 90-minute workshop will remember everything that we discussed? Of course not – I’ve been doing this job way too long to think that. But here’s the beauty of it all – they have their own model to reference, be it Google or Excel, they all created one and can take another look at any time. I heard from another teacher that a couple of his advisory kids started talking about making their own coffee instead. A couple of my advisory students commented on the experience at the end of the day. One said, “It was interesting to see how the numbers involved in the Starbucks added up if invested in a retirement fund. The actual application was nice.” Another said, “The spreadsheet exercise this morning was fun. I think it was the funnest way to learn how to do a spreadsheet I have ever done. So thank you.”

You’re welcome.

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Baxter Foundations

We are beginning this year with a week of what we’re calling Baxter Foundations. We realized at some point last year that we should have done more to bring our students together at the beginning of the year. They are coming from 30+ different towns and educational backgrounds. While I completely support and value the time spent building community last week, we could also use some time building that academic foundation and getting out into the community. These are also foundations of our philosophy. Full disclosure: Our Phase 3 renovation of the building isn’t going to be ready to begin classes on Monday, so we find ourselves with this opportunity.

Beginning tomorrow, our students will participate in such varied experiences as going on college visits, listening to an entrepreneur’s journey, seeing a planetarium show, walking the Portland Freedom Trail, watching a film at nearby SPACE Gallery, touring the waste-water treatment plant across the bridge, visiting the Wells Reserve to learn about the environment, and learning about agriculture by working on Wolfe’s Neck Farm. Those who stay in the building will participate in workshops as varied as learning about Flex Friday, understanding rubrics, plagiarism, research strategies, reading strategies, and teen dating violence. Other workshops include introductions to spreadsheets, probeware, and modeling. Some students will participate in some fitness and conditioning while others are working with local experts on creative writing or art. We have a workshop in food chemistry (Is that really food?) and other in project management. 

I’m excited about the opportunity to begin the year this way. It won’t be perfect, but we will take those pieces that worked, fine-tune them, and improve Baxter Foundations for next year. After all, that’s what the design process is all about. And that’s what we do at Baxter Academy.

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Baxter Academy: Year 2 Day 2

This Day 2 was different from last year’s Day 2 – of course. There was no furniture to build, at least not yet. We have more students now. Only 9th & 10th graders were in the building today, but there are still more of them, which is great. Today was about orientation, and reorientation. As with any project, any design challenge, reflecting and improving are important steps. We’ve done that. There are changes to what we plan to do with kids this year. Some rearranging of the schedule, a bit more time with our advisory groups, nothing huge. Our philosophy is still the same: we want our kids to be able to stand up and say what they know and believe. We want them to be fearless, and to learn from mistakes.

Today’s schedule was organized around our advisory groups. We teamed together a 9th grade group with a 10th grade group. Those groups then rotated through a series of presentations about our school: Open Campus, Flex Friday, Baxpectations & Standards Based Grading, Baxtitution, Student Government, and Student Judiciary Board, Mentoring & Advisory, and a slide show of the teachers. All of these presentations were designed by teams of 10th graders, student senators, who had minimal guidance from a faculty member. Was I nervous about what they would say? Sure, I was. Did they do an amazing job? Of course they did. That’s what impressed me the most. Not only did twelve 10th graders agree to put in extra time to create a presentation about a specific topic special about our school, they delivered that information six times and answered countless questions. And the 10th graders in the audience asked great questions, too. It was really a good morning, led by kids. Could it have been better? Sure. That’s the reflection and iteration part. Next year it will be slightly different, slightly tuned, and more effective. But I doubt we will change the overall purpose or how it is student-led. Those six topics were selected by students who spent the last day of school last June brainstorming and designing today’s orientation model.

After lunch, we got outside (another gorgeous Maine day), walked to the Old Port with our 9th grade partners (my advisory group really wanted to get to know them better), learned about next week’s “Baxter Foundations” schedule (more about that next week), and had our own design challenge for “Baxter Swag.” When you work in a school that revolves around problem solving, it’s good to get in as many design challenges as you can.

When I asked my students to sum up their first two days (10th graders do not have school tomorrow), many of them said how great it was to be back in school. Even with the changes (new students, new faculty, new building spaces, moved classrooms). One student said, “All summer, people asked me about Baxter. They know it’s a charter school and they want to know about it. The best way I could sum it up is to tell them that I am excited to come to school every day. I’m so glad to be here.” Ditto that.

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