BATTLESHIP! - Graphing Equations of Circles

I've been dying to incorporate more PrBL tasks into my classroom. For the past couple of years, our math team spent a huge deal of time and energy on a complete overhaul of our four-year math curriculum in order to more strongly align it with ACT College Readiness Standards. It was certainly a worthwhile endeavor; I'm very proud of what our awesome math team has accomplished, and I think our students will greatly benefit from what we've done so far.

At the same time, this pretty much meant I had zero time to work on any PrBL stuff, especially with moving from teaching Geometry to teaching Pre-Calculus at the same time. However, our project was finally completed this past spring, so I have been happily spending the summer working on PrBL-related curriculum mapping for my Pre-Calculus and Advanced Pre-Calculus classes.

(Yes, I just said "happily" and "curriculum mapping" in the same sentence.)

Below is one PrBL task that I've been working on for a graphing unit this school year. I think (and hope) the students will have fun with it; it's not particularly all that "real-worldy," and it definitely needs refinement, but I gotta start somewhere. Of course, as with anything I haven't tried in class yet, it's a work in progress.

This task involves understanding and graphing equations of circles. I call it: BATTLESHIP!

(Although the task is not quite the same as the classic board game.)

The Scenario: You are the commander of a mighty naval fleet in the middle of international waters. The enemy has developed a new type of submarine known as a Hyperbolic Invisibility/Deep Dive ENgine, or a H.I.D.D.EN. submarine.

The enemy's H.I.D.D.EN. submarines are capable of avoiding nearly all types of radar detection. In fact, you are only able to determine the distance a H.I.D.D.EN. submarine is from any of your naval stations.

Your task is to devise a way to pinpoint the exact location of a H.I.D.D.EN. submarine. Succeed, and your forces will be able to destroy the enemy fleet. Fail, and you're doomed. DOOMED!

(If you couldn't tell, I have an affinity for silly acronyms.)


The Entry Event: Before things really kick off, I'll give the students a few warm-up problems to assess and activate their prior knowledge. Students will need to know the parts of a circle (particularly the radius and the center), and will also need to be able to re-write a two-variable equation (i.e. solve for y in terms of x). The latter will be important for graphing circles on most graphing utilities.

To introduce the problem, I'll present the following situation to students on Activeprompt:

"You are the commander of a naval station, shown here on the grid. An enemy submarine is approaching.

The submarine has a cloaking device that hides its exact location from your radar system. However, you are still able determine how far away the submarine is from the station.

The submarine is 5 miles away from the station. Where is it?" 

(I could make things more interesting by removing the axes and labels, but I want to steer the students in a certain direction here.)

I posted this prompt on Twitter, and a couple of my friends immediately pointed out that they couldn't answer the question because they didn't know which direction the submarine was from the station. This is true, and in many ways is actually the point of this prompt; I suppose I should be more clear that I want students to guess where the submarine might be, and that I'm not necessarily looking for "the correct answer" at this stage.

Still, I had several responses to the prompt and ended up with something I would hope to see in class:

 (Interesting, isn't it?)

Hopefully, students will take one look at this picture and notice the pattern: there appears to be a circle forming around the station. At this point, students can take some time to think about further questions: Why is there a circle? What does this circle mean? What can we figure out about this circle? What does this circle have to do with finding the submarine?

After discussion, the hope is that students would come to the following conclusions:

• The circle represents all of the possible locations of the submarine, based on the information we have.
• We have no way to determine the exact location of the submarine with our current information.

That second statement is critical. The key to solving the problem lies in the realization that more information is needed.


Need-to-Knows & Scaffolding: While I'm sure that my students will surprise me (students have a habit of doing that), the need-to-know that should be immediately apparent is: "How do we locate the submarine?" In fact, we begin the process of answering this question with the entry event.

Again, one of the key realizations from the entry event is that all of the possible locations of the submarine are represented by a circle, radius 5, with the naval station as its center:

A good follow-up question would be, "How do we narrow down the number of possible locations?" The answer may or may not be readily apparent. I'd encourage students to think outside the box -- or perhaps, more appropriately, "think outside the circle."

Because we could narrow down the number of possible locations if we had a second naval station. Say, at coordinates (7, -8). And it detects the submarine at a distance of 7 miles.

 

Aha! Just like that, we've narrowed our possible locations down to two; namely, the two points (1 and 2) where the circles intersect each other. (It certainly wouldn't hurt to have the students explain why these are the only two places the submarine could be.)

From here, it probably won't be a huge leap for the students to realize that adding a third naval station will narrow our choices down to just one. We'll get back to that in a moment.

A critical issue arises from this new picture: while Point 2 clearly appears to be located at the coordinates (7, -1), it's much less clear what the coordinates of Point 1 are. This should lead to another question: "How do we accurately determine the coordinates of the point(s) where the circles intersect?"

Now, this part of the task is a bit murky for me. It's not all that difficult to come up with a good estimate of Point 1's coordinates using Geometer's Sketchpad, but the point of the task is for students to work with and understand equations of circles. To this end, I want students to be working with a graphing utility (e.g. TI-83/84) as we address this question. So, yeah... if anyone has a good suggestion for how to make sure it steers in that direction, I'm all ears!

In any case, turning to our graphing calculators should bring up the question: "How do we graph circles?" The best way to do this with our graphing calculators (or an online tool like Desmos) would be to input an equation. That, of course, leads to: "What's the equation for a circle?"

At this point, appropriate scaffolding activities and workshops could be used to help students understand how to determine the equation of a circle, given the center and the radius. I'd probably also give students a few practice problems to give them some exercise in this skill. When using a graphing calculator like a TI-83 or TI-84, students would also need to know how to re-write their circle equations for y in terms of x so they can actually enter them. (This would be one advantage of using Desmos over a graphing calculator; such a conversion isn't necessary. On the other hand, re-writing equations would also be a great chance to talk about issues such as positive and negative roots, for instance.)

Since I don't have the proper software readily available for getting some clear TI-83 screenshots, here are the two circles graphed on Desmos:

On a TI graphing calculator, students could use a combination of ZOOM and TRACE to estimate the coordinates of Point 1. CALC -> INTERSECT would also be a good option. On Desmos, we can just click on the intersection point to get an estimate of the coordinates:

If we want greater accuracy, we can zoom in really close:

Using CALC -> INTERSECT on my TI-83 yielded an estimate of (3.4461538, -1.969231), so very similar results. If we rounded to the nearest hundredth, we can pretty solidly estimate the coordinates of Point 1 to be (3.45, -1.97). (It might be interesting to have students estimate the coordinates of Point 1 prior to using their graphing utilities to see how close they came by just "eyeballing" it.)

Of course, we said much earlier that we need three stations to determine where the submarine is. We could introduce the third station much earlier in the problem, or we could hold off until now to introduce it.

So, let's say the third station is located at (-5, 4) and detects the submarine at a range of 13 miles. Students determine the equation of the circle with this center and radius, enter it into their graphing utility, and voila:

So our enemy submarine is located at coordinates (7, -1). Huzzah!


Applying the Learning: Now, I wouldn't have gone through the whole business of figuring out how to estimate coordinates using a graphing utility if the solution was always going to be as simple as (7, -1). For something more challenging that definitely requires the assistance of a graphing utility, let's say we have the following information:

• Naval Station A is located at (-16.47, -3.53). It detects an enemy submarine at a distance of 12.31 miles.
• Naval Station B is located at (5.68, -3.74). It detects an enemy submarine at a distance of 13.97 miles.
• Naval Station C is located at (5.43, 5.68). It detects an enemy submarine at a distance of 11.96 miles.

This takes a bit more work, and the answers will probably vary slightly. This might also be a good opportunity for students to debate how to get the "best" answer to this problem, since we want to be as accurate as possible when tracking down the enemy submarine.


Would You Like to Play a Game?: For something really fun at the end of this problem, we would turn our scenario into a war game. I would break the students up into teams of two or three; each team gets one H.I.D.D.EN. submarine and three naval stations. Teams get to place their submarine and naval stations at whatever coordinates they choose (within certain borders, of course).

After all submarines and stations are placed, I provide each team with information about how far away each enemy submarine is located from their stations. (This adds a layer of complexity to the original problem scenario, as teams now have information about multiple submarines and they have to mix & match circles in order to pinpoint them all.) The teams then race against each other to try and be the first to locate and destroy the other submarines. Winning team gets riches and glory. Well, just glory. Not much glory.


Final Thoughts:  In the end, I thought this task seemed like a fun way for students to learn about how to graph equations of circles and then apply that skill.

Hopefully, when the students share out what they learned as a result of this problem, they'll be able to articulate a deep understanding of the relationship between circles, their equations, and their graphs. It'd also be cool if some of them see the connections between equations of circles and the Pythagorean Theorem or the Distance Formula. I certainly hope they end up finding the whole thing to be a worthwhile experience.

It's definitely not perfect, but I'm looking forward to trying it out and seeing how it goes.

Should I Even Bother Reviewing For Final Exams?

Happy New Year, everyone!

It's been a while since I last released one of my incoherent ramblings into the wild jungles of cyberspace, and I have much to talk about, so expect to see a few more posts in the coming days. (And if you don't see said posts materialize, please nag me until I get them done.)

(As a side note, as of this writing, my blog has about 25,000 views accumulated since my first post in July. 20,000 of those views are attributed to a post I wrote in August about the ninja board. Apparently Google likes ninjas.)

My school resumed classes today, and 1st semester final exams are coming up in a week and a half. That means the time has come to start reviewing for finals.

I've been wondering about this lately, the idea of spending a week and a half of class time reviewing for final exams.  I'm not completely sure I ever do it the right way. Actually, I often wonder if there even is a right way.  

Does it even do any good to review for final exams?

Every semester, I take the last week and a half or so before final exams to review with my students everything that we learned over the prior 16-17 weeks, tell them what kinds of questions to expect on the final exam and how many, give them time to work on review packets/assignments/flaming obstacle courses, etc. and so forth.

I've tried various ways of helping my students to take stock of what they learned (or were supposed to learn) over the semester. We've done the "review for finals process" as a project (with a rubric and everything) where students had to develop and publish their own study guides. We've done the classic "Jeopardy!"-style review game. We've done notecards that students were allowed to use on the final. We've done review assignments with the final exam questions literally lifted from the exam itself, with the numbers changed.

And what bothers me is this: Not once, that I can recall, in the four years I've spent teaching so far, have I been able to discern whether or not these methods of reviewing have done any good to any of my students.

What appears to happen is that the students who more or less have been "getting it" (or have been perpetually on the cusp of "getting it") all along are best equipped to understand and solve the problems set before them on the review assignments. Students who have been struggling all semester -- for whatever reason -- also struggle to find success on review assignments. It strikes me as a situation where the students who benefit the most from reviewing for finals are also the ones who need it the least, and the ones who benefit the least are the ones who need it the most.

I don't know why this appears to happen. (Or, if I'm really being honest, if it actually is what happens.) Maybe I haven't been making enough of an effort to find out. Maybe it's some bizarre phenomenon that can't be explained, like Honey Boo Boo. Maybe I suck at teaching. (Okay, maybe not.)

I was discussing this matter with my lovely wife the other night, and she asked me, "well, how do you know whether or not it's helping your students?" I thought about it, couldn't come up with a great answer, got childishly frustrated then stammered something like, "it's just based on what I've observed in class, I don't know how to explain it!" Then I pouted and decided to go do something else, because I'm so mature.

The bottom line is, I've never really been confident in my approach to reviewing for finals. I haven't made it easy for myself to tell whether or not my approach has a positive (or negative) effect. Maybe that's what makes me wonder if reviewing does any actual good.

Perhaps in the naivety of being a young teacher, I've been thinking of it the wrong way.  I think the best way to describe how I've approached reviewing for finals is that I've seen it as an eleventh-hour scaffolding activity, intended to give students one last hope at having a mathematical epiphany, a lifeboat that will float them safely through the perilous, shark-infested tides of the final exam.

It never seems to really work that way. No lifeboats. Sharks with happy tummies.

Maybe I should be looking at reviewing for the final exam as part of the cumulative assessment itself. Reviewing should really be more of a time for reflection and fine-tuning, not making a last-ditch effort for comprehending something for the first time. That's not to say there won't be a few students that do get that benefit from reviewing, but that shouldn't be the point. The point should be to look back at all of the work we've done all semester, take stock of what we've learned and what we still have questions about, address areas that still need addressed, and perhaps even celebrate.

My angst aside, here's what I'm trying this time around. The other day, I remembered something I read on David Coffey's blog about giving students the answers to the problems and having them explain how to get that answer. In my case, I'm going to provide students with a set of problems that are similar to what's on the final exam, give them all of the answers, and require them to explain how to get each answer. This way, they focus on how to solve the problem as opposed to focusing on getting the right answer.

I don't really know if this will be any better or any worse than what I've tried in the past. But, I think it will at least alleviate some of the anxiety and second-guessing that comes with reviewing for final exams. We have a week and a half, which should be plenty of time to address any questions or concerns that arise as students work through their review assignments, particularly since I am putting the focus on articulating their mathematical thought processes.

Will it do any good? Your guess is as good as mine.

It's School Again! Huzzah! (Part 2 of 2)

The freshmen had their first day of school with us last Tuesday; on Wednesday, the rest of our students returned and I got to see my seniors for the first time since I last had them all as sophomores.

This year, one of my goals for my math class is to get my students writing more and to practice digital citizenship by focusing on communicating with peers in an online environment. To that end, one of our opening activities was for students to respond to a discussion board prompt on our echo course page.

The questions were pretty simple:

Many of the responses were encouraging to read; a lot of students stated they were planning to go to college after high school and that they were excited for graduation. Several students said they were excited to have me as their teacher again because they enjoyed how I teach (which I'm not particularly sure how to feel about since I think I was probably doing many things wrong two years ago).

Some students had their priorities straight:

While other students took a rather avant-garde approach:

Still, I learned a great deal about my students. A few of them said they wanted to go into graphic design; one wants to be a zoologist; one is thinking about cinematography or film; a few are considering getting business degrees; one wants to be a mechanic; one has aspirations of joining the FBI; some are planning to go into the military; and many, many more. There is a lengthy, eclectic list of careers my students want to pursue after high school, which is awesome.

Other students are unsure of what they want to do after they finish high school, which is also okay. I'm hoping that during this year I can connect with these students and help them figure out plans and goals for themselves for a post-high school existence.

At any rate, this discussion board activity served two important purposes. First, as I just detailed above, I learned a lot about my students. I know more about their post-graduation plans and their interests, which will help me a great deal in tailoring our class to incorporate their interests. Second, the activity established a baseline for their ability to communicate and interact with each other in a supervised online environment.

I saw some good things. The students were able to follow directions well for the most part, did an "okay" job of using polite language (save for one student who jokingly said her mom would "beat her ass" if she didn't get a good grade in math), and responded to each other's posts while making an effort to comport professionally.

I also saw some things that need a lot of work. The vast majority of the students re-posted each question and answered them in a list format. I would like to see them get away from re-posting questions and answering in a paragraph form. (Not that answering in a list format is necessarily a bad thing, but I would like to see them practice putting their thoughts together in a coherent, flowing format.) Spelling, grammar, and punctuation remains an issue; I realize that I'm a math teacher, but that doesn't mean I can't give them feedback on these things. (Actually, I minored in English at Michigan State, and am certified to teach the subject in the state of Michigan.) The replies that students wrote to each original post were also, for the most part, superficial. I saw a lot of "I agree with you"-type posts that had little depth and weren't suited to continuing the conversation. Again, not necessarily a bad thing; plus, I wasn't expecting most of the students to be able to do this on the first go. We were simply establishing a baseline to help us identify what to work on. By the end of the year, I'm hoping to see well-crafted, thoughtful responses and replies that result in deeper conversation. (To be fair, this probably requires a deeper topic than what I gave them to start with.)

Outside of the discussion post activity, the majority of the time was spent administering a math benchmark test to establish where the students are in terms of content mastery. This benchmark will help me determine what the students already know, what they still need to master, and thus where we should focus our efforts as far as mastering content is concerned.

Probably the coolest thing of the week that happened was Friday. Many students still needed to finish their benchmark from Wednesday/Thursday, while others were already finished and weren't going to have much else to do. This seemed like a great opportunity to preview the election-themed project that we're going to be launching when we come back from Labor Day weekend.

I decided to get together the students who were already finished with their benchmark in each class for a Critical Friends session. The students had seen the Critical Friends protocol in their sophomore English class and were somewhat familiar with the procedures, so I gave each class a quick refresher before starting.

It went alright in my 1st period class, but the students more frequently got off-task in 2nd period. I realized that I needed to designate a few students to be responsible for steering the "I Like/I Wonder/Next Steps" portion of the session and keeping everyone focused on the task at hand. So, in my 3rd period, I asked the group if anyone was comfortable leading the discussion. Three students immediately spoke up, so I told them they were responsible for keeping everyone on task. I presented the project idea and sat back to let the students discuss it.

I hadn't expected what transpired next.

One of the students I designated to lead the discussion immediately chose a student to read the entry document out loud. The other students all listened intently as she read through the entry doc. After she was done, one of the other student leaders grabbed a dry erase marker to start writing "I Likes," "I Wonders," and "Next Steps" on the board while the other two called on students for their feedback.

I was amazed. I was very proud and excited. I thought to myself, "SOMEONE HAS TO SEE THIS!!"

So I shot a quick Skype message to our assistant principal, who came down a few minutes later as the session got in full swing. We were enraptured by how well the students had taken over the conversation, listing several "I Likes," "I Wonders," and "Next Steps" while conducting themselves in an orderly fashion:

I was very impressed with what the students were able to do on their own. I had actually intended to listen to their conversation and write down all of their feedback myself (as I had done for the first two periods), but they completely took care of that for me! The only thing I'd done was to assign a few students to lead the discussion, and they took it from there! It was really awesome to watch.

I did the same thing with my 4th period class and got similar results. Our principal stopped by my room during that period and was very proud of the students for what they were doing -- she even joined in and gave some Critical Friends feedback herself!

As I said, the students had previous experience with the Critical Friends process in their sophomore English class, so I made sure to track her down and let her know what had transpired in my class. When I told her they not only remembered Critical Friends, but successfully ran a session on their own, she did a happy dance.  I imagine the news must have been incredibly satisfying -- it showed that these students had actually been listening to her two years ago.

So my week ended on a high note. The students gave me some great feedback for our project, and most of them seemed to be interested in the idea.

I would love to expound more on Week 1 (and I did give an update on the Ninja Board), but there's still much I need to do for Week 2! A teacher's work is never done. Until next time!

Keepin' It Real

I'm going into my fourth year of teaching math at New Tech High @ Zion-Benton East. I love my job. I've discovered that I'm actually, perhaps, maybe, starting to get halfway decent at it. At least there's a chance that I am. There's perpetual room for improvement, and today I wanted to talk briefly about one area I hope to improve this year.

In my first few years of attempting to teach a PBL/PrBL math class, I've come up with some projects that are pretty good at simulating authentic real-world tasks: Creating a different type of Oreo cookie package for Nabisco; creating a scale drawing depicting how furniture should be laid out in a dorm room; and designing a hole for a miniature golf course.

These projects are certainly useful ways to help students see how math can be used for creating and improving products in a real-world context. On the other side of the coin, they fall short when it comes to real-world results. We didn't actually create a real Oreo package for Nabisco; we didn't actually have a real room with real furniture for our scale drawings; we didn't actually build a real mini-golf course.

They were real-world projects without real-world results.

As Dennis Littky probably would put it, I had the students doing "fake real work" instead of "real real work." Something like that.

I want to change that this year. As I continue on my journey of teaching PBL/PrBL math, I believe one of my next steps is to move my students away from the "fake real work" and into the "real real work." I want students to use math to actually create things; to innovate; to predict; to think critically; to affect their community in a positive way.

How do I do this? I haven't completely figured that out. I think I have a good start with the election-themed project idea I blogged about last time. I'm hoping my students can use their experience with this project to learn more about important issues, about making informed decisions based on available data/information, and about making defensible predictions.

A few of my students will even be voting this year; this might really help them learn about being informed voters.

And, because I want my students to produce real-world results, I need them to have a real-world audience. That's why students will be publishing their findings on our class blog (link coming soon) for the community and the rest of the interwebs to see, as well as sharing them with the Obama and Romney campaigns (fingies crossed that they'll actually take a look).

I think that's a good start in my goal to move away from "fake real work" and giving my students the chance to do "real real work."

But I need more. It probably sounds overly ambitious to the point of absurdity, but I want my students to always be using math to become better citizens and to benefit their community. I think the key to this is "real real work." I would love to have 100% of the school year consist of "real real work." (At this point, I'd be thrilled to even get 25% of the school year that way.)

So that's one of my goals this year. I want my students doing "real real work" that has a positive impact beyond the classroom. I'll certainly be scouring and engaging the blogosphere, Twittersphere, and meatspace for ways to accomplish this.

Project Idea: Math, Social Studies, and 'MURRICA!

I've had a half-baked idea for a project tossing around in my head for the past few weeks that I've been meaning to share. It's nowhere near perfect or ready to go, but I think it has some really cool potential. So, here we go:

It's an idea for a math and social studies project centered around the 2012 election.

(Math and social studies! I know, right?)

The idea is simple: Students work to answer the driving question, "What are the keys to winning the 2012 presidential election?"

Anyone who has been paying attention to the news (or who haven't been living under a rock at any point since 2008) probably have an idea of what the hot-button issues are, or which swing states will be most crucial to securing the presidency. For the math end of this project, however, numbers will tell the story.

As part of the process to answer the driving question, students will examine various sources of polling data. Gallup, for instance, has a daily tracking poll and plenty of polling data broken down by demographics. RealClearPolitics gathers and averages polling data from battleground states. Various electoral maps, such as this one on CNN's website, are available as well. Rasmussen Reports has polling data showing what issues are most important to Americans today. In short, lots of data to examine and interpret.

Students will gather and examine polling data to determine a few key points, including which states the candidates should focus most of their resources on and which issues the candidates should focus on. Their data analysis will be used to justify why they identified particular states and issues as being the most important to focus on.

For the final product in the math portion of this project, students will create a multimedia presentation to deliver their findings and make recommendations to both the campaigns of President Obama and Governor Romney as to how they should focus their campaigns in the final weeks leading up to the election. These presentations are to be posted to our class blog (which I have yet to set up -- I'd better get going on that) and will also be forwarded to both campaigns. (Hopefully, they'll even take time to look at them!)

I've been talking with the social studies teacher on my grade-level team about this project. It sounds like he and his English co-facilitator are planning to run a debate project at the start of the year that this could actually fit into. I think having the students use data to identify what issues are most important to Americans would then lead them to investigate why those issues are important, which would lend itself well to research for a debate. The math can inform their approach to debating various issues.

So that's my half-baked project idea to this point. There's certainly much more that needs to be thought about as I develop this into something workable.

For instance, I talked about students "using data analysis," but haven't gotten very far on how students will actually learn what it is and how to apply the skill. I think I could especially use some help there.

Also, I'm wondering if there's a place for linear modeling in here with the polling data (particularly since the first unit of the year is supposed to be linear equations/inequalities).

Other ideas I've had to far include: utilizing social media to talk directly to people in battleground states and survey them on what issues are important to them; convincing someone from Gallup or another polling agency to Skype with the class and talk about how they conduct their polls; convincing someone from either the Obama or Romney campaigns to Skype with the class about how they use polling data or other statistics to drive decisions about how they conduct their campaigns.

(Also, it would be really cool to come up with a way to make this work with #MYParty12.)

Anyway, that's it. As I said, I think there's lots of potential here, but I can definitely use as much help as I can get. If even one or two of you out there have thoughts or "I wonders" on this, please share! Otherwise, thanks for reading!

Twenty-Six-Point-DOOM: The Marathon Man Task

Okay, sometimes I enjoy a bit of over-the-top drama when I title stuff. Something you should probably know about me going forward.

This summer, I have been training for the Prairie State Marathon on October 6th of this year. It's my first-ever marathon, and I'm totally psyched for it. In fact, I'm so psyched about running that I've already registered for the F^3 Lake Half Marathon -- which takes place along Lake Michigan in freaking January -- and am planning to run the Wisconsin Marathon next May. Needless to say, I'm addicted to running beyond all measure of common sense (which I'm not certain is actually measurable).

My fanaticism for endurance running aside, I was feeling particularly inspired after reading this post by Nat Banting (@NatBanting) about a problem he's developing to determine how to minimize the amount of water wasted by his sprinkler. I found myself wondering what real-life situations I could use to similarly create an authentic PrBL experience for my students.

The Scenario

As I was out for a 7-mile run this morning, I realized that such an experience might lie in this:

This is a Nathan Trail Mix 4 hydration belt, which I take with me on my long-distance training runs. Each bottle has a capacity of 10 ounces, which means I can take 40 ounces of liquid with me. Most of the time, I consume Gatorade while running. With longer distances, however, I also take packets of GU energy gel with me for supplementing my glycogen stores.

(Incidentally, Chocolate Outrage is my favorite flavor. 

Also, "GU" is pronounced "goo.")

Energy gel needs to be consumed with water to dilute it enough for the body to absorb it quickly, or else cramping and sometimes vomiting (eww) can occur. (Also, taking it with Gatorade causes the gel to become thick and sticky like molasses, which doesn't sit well either.) Thus, I have to take both Gatorade and water with me on my long distance runs:

My longest training run to this point has been 16.5 miles; thus far, I have been able to ration my Gatorade and water appropriately to get me through each run. In addition, there will be hydration stations spread throughout the course stocked with water and sports drink. However, as race day approaches, I've been frequently asking myself this question: Will I be able to take enough Gatorade and water with me to get me through 26.2 miles before I run out of both?

The Task

This leads to the driving question I would put to my students for this task: What is the best plan to keep Mr. B hydrated and energized during the marathon?

I am not yet sure what the entry event will look like, but my intention is for it to include only a few pieces of information:

• The task is to create a "consumption schedule" that tells Mr. B when to drink liquid or eat a gel packet.
• A marathon is 26.2 miles long.
• Mr. B has a hydration belt for carrying water, Gatorade, and energy gel.
• Energy gel must be taken with water, and must not be taken with Gatorade.
• There are also hydration stations located throughout the course that carry water and sports drink.

Potential Student Need-To-Knows

My hope is that the limited information from the entry event leads to several student-generated need-to-knows. Here are a few that I was able to come up with on my own:

• How much liquid does each bottle hold?
• How much Gatorade and how much water should be taken?
• How often should Mr. B consume Gatorade?
• How many hydration stations are there on the course?
• Where are the hydration stations?
• How many gels will Mr. B consume?
• How fast does Mr. B run?

Some of these need-to-knows can be answered rather quickly. As I mentioned above, each bottle on my hydration belt has a capacity of 10 ounces, for a total capacity of 40 ounces. If the students ask, I can simply tell them this one.

If students inquire about the number/locations of hydration stations on the marathon course, I will be able to furnish them with this map from the race web site. The map marks all of the hydration stations throughout the course.

The frequency and amount of Gatorade consumption is where I'm sure many groups will diverge in their solution paths. Some runners consume a few ounces of Gatorade every few miles or so. My personal preference is to sip about an ounce or two or Gatorade at every mile marker, though for the purpose of this task I'm not married to that notion. Chugging an entire 10-ounce bottle at any point, however, would be inadvisable, as it would probably result in me throwing up (eww).

In any case, the Gatorade consumption can be modeled with, say, a linear inequality. Is the total amount of Gatorade consumed going to be equal or less than the amount of Gatorade available to me during the race? Each group's plan will need to address this question, and there are a number of ways to answer it.

Now, I haven't forgotten about those last two need-to-knows:

• How many gels will Mr. B consume?
• How fast does Mr. B run?

These two questions are very closed tied to each other; plus, the gel consumption will also dictate the water consumption.

Gel needs to be consumed at regular intervals throughout the race in order for me to maintain my energy stores; in fact, most packages of gel carry the advice of consuming one packet every 45 minutes. With that in mind, it becomes incredibly important to know how quickly I can be expected to finish the race. Without a sense of how fast I run, students will be unable to determine how much gel and water I will need to consume.

I happen to have recorded the times of each of my long-distance training runs from the past few months. Mostly this is due to the fact that I am a shameless braggart:

Since I've recorded all of my times, however, it means that instead of giving my students my own estimate of how fast I run, I can provide them with a table of data and have them estimate how fast I can run. Linear regression models, anyone?


Where To Go From Here?

I'm only about 8 hours removed from when the idea for this task first formed in my head, so naturally it's nowhere near perfect. There are many pieces of the task that I was admittedly rather vague in articulating. But, I do think I'm onto something cool here.

One thing I am not sure about is how to have my students present their solution. Live presentation? Blog post? Physical document? Scribbles on the back of a napkin? All of these options and more?

Any thoughts? This is the first PrBL idea I've really come up with on my own, so I gladly welcome feedback!

Deeper Learning: Passion + Conversation = Want-To-Knows

This past week, I participated in #PBLChat on Twitter for the first time. I'm still pretty new at this whole "chatting on Twitter" thing (I hadn't even known about TweetChat until halfway through), but the experience was awesome. I "met" a lot of great teachers in the network and had a chance to build my PLN, which I think will be invaluable as we continue our conversations in the coming months.

With the theme of NTAC 2012 being "Dive Into Deeper Learning," the question posed to the chat was, naturally, "What is 'deeper learning?'" As I read response after response, two over-arching themes grabbed my attention: passion and conversation.

It really all begins with passion. Many in the chat agreed that deeper learning requires an initial deep desire to learn. I remember when I was in high school, I used to write out the proof of the quadratic formula in my notebook whenever I was bored, because I preferred math more than any other subject. I liked doing it, so in turn I gave it more attention. It's what we like and what we want to do that we seem to become best at. Passion is probably why I can quote every line of "Anchorman," but I couldn't tell you the laws of thermodynamics without looking them up. I was never that interested in science, but dude, I can go on for days about whether Brick actually loves lamp or is just looking at the lamp and saying he loves it.

Where passion fuels deeper learning, conversation helps us make sense of our passion. In the chat, this aspect of deeper learning was commonly articulated as "talking about process" or "being able to explain or teach the concept to someone else." These are fine examples; I think, more generally, the conversation aspect of deeper learning means to engage in an exchange of interpretations or viewpoints in order to refine one's own knowledge. In a way, conversation is confrontational. It forces the learner to articulate what they believe they know about a problem or a topic, and can even lead them to realize or admit what they do not yet know. This is vital to deeper learning; the very act of identifying what we do not know gives us direction for our pursuits.

To a PBL teacher, this probably sounds suspiciously like I'm talking about "need-to-knows." In the context of deeper learning, it might be more appropriate to call them "want-to-knows," since deeper learning is driven by passion. Maybe I'm overgeneralizing, but I think there's an important difference between the two. The idea of "need-to-knows" is definitely important as an organizational and learning tool in PBL, to be sure. "Needing" to know something can sometimes, I think, imply that the learner has to know the thing for the sake of completing the project. "Wanting" to know something, on the other hand, comes from sheer curiosity and is motivated by genuine interest rather than academic requirements.

For example, one (not-really-that-great) project I ran in one of my geometry classes had students design a new type of mini-Oreo package based on different types of 3-D shapes they were learning about (spheres, prisms, cylinders, pyramids, cones). So, in order to complete the project, students "needed" to find out what these shapes were and how to calculate their volume -- not a very exciting or motivating prospect for anyone who isn't already a math geek. One student, however, "wanted to know" about other, more complex 3-D shapes. There was a desire within him to research and find out what other shapes existed. So, I told him to have at it. A few days later, he came back to me with a package prototype in the shape of a conical frustum. I was amazed! I had never even heard of this shape before; had the student not presented me with his "want-to-know," I might still have no idea what a frustum is.

This is only a small example of a student pursuing a "want-to-know," but I think part of our responsibility as educators is to give our students room for such pursuits of any magnitude. We must continue to provide our students with opportunities to find their passions and make sense of them. One of the greatest things we can do for our students is to equip them to chase down their "want-to-knows."