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.

Huh? What? Where?

Yesterday morning, an e-mail found it's way to my inbox to alert me that crazedmummy left a comment on a previous entry of mine.

The comment read, "I nominated you, get back to work!" and was accompanied by a healthy list of questions about me.

At first I was like, "HUH? WHAT? WHERE?" because I had no idea what I was being "nominated" for. I thought maybe there had been a secret gathering of blogging ninjas that decided to select other bloggers among those who have been inactive for months and leave cryptic messages on their blogs, telling them to get back to work. And if a blogger didn't get back to work, then the secret blogging ninjas would break into their house/apartment/cardboard box in the middle of the night while they slept and do something horrible, like tattoo the University of Michigan logo on their forehead. I mean, that's just awful. Like, death would be preferable.

But then I did a bit of digging around, and found that I was really being nominated for a Liebster Award. It's a prestigious award that involves widespread fame and endless riches.

Or maybe not.

In reality, it's not really an "award," but rather a way for bloggers to recognize other bloggers that are either getting started or have less than 200 followers on their blog. (I'm just shy of this number at a hefty 6 followers total.)

I'm not sure where it originated from. As far as I can tell, in order to accept the nomination for the Liebster Award, I must complete these tasks:

1. Link back to the blog that nominated you
2. Nominate 5 blogs with fewer than 200 followers
3. Answer the questions posted by your nominator
4. Share 11 random facts about yourself
5. Create 11 questions for your nominees
6. Contact your nominees and let them know you nominated them

Sounds like a great opportunity for me to get off my butt and resume blogging, thus proving to the rest of the world that I am, in fact, not at all dead.

So, step one is to link back to the blog that nominated me. Did that at the top of this post, but hey, here's crazedmummy's blog once more.

Step two, nominate 5 blogs with fewer than 200 followers. I don't really know how to check for that, but I don't think the blogging ninjas are gonna come after me for this one, so here's five blogs:

The Problem Bank

The Pai Intersect

The Education of Future Math Ninjas

Prime Factors

Writing to Learn to Teach


Step three, answer the questions that were posed to me. Here goes!

1. Why did you start this blog?
Because all the cool kids were doing it! Actually, when I started this blog last summer, I was intending to use it as a place to share ideas for problem-based learning, to talk about ongoing issues in education, and to network with other educators who are active on the blogosphere. Things kinda ground to a halt as the school year got underway; I'm hoping to be more diligent in my blogging this coming school year.

2. What do you like most about your life?
I'm happily married to my best friend in the world; we have a wonderful dog, a place to live, and food to eat; and, I love my job. There's not much that isn't to like!

3. What one thing would you improve about yourself?
Well, I'm hoping to blog more and do a much better job of staying engaged with other teacher/bloggers this year than I did last year. I feel like I'm missing out on a lot!

4. What sort of vehicle do you drive to work?
A 2000 Nissan Altima. Previously, it was a 2003 Dodge Neon, but in February I hit a deer and totaled it. Ah well, life goes on. (Except for the deer.)

5. How often do you cook?
My wife and I split it pretty evenly. I actually rather enjoy cooking!

6. What one show do you watch on TV ( if any)?
I'm actually into a couple of shows right now: my wife and I have been working our way through Doctor Who (the 2005 and onward series), and this summer I've gotten into Breaking Bad.

7. What non-school talent do you have?
I sing with my wife in our church choir, and I write creatively from time to time. Those count, right?

8. What is your favorite exercise?
Running! I'm currently training for the Chicago Marathon and I'm pretty excited for it.

9. Where would you go for vacation if you had unlimited funds?
I can't pick just one place, but since I have unlimited funds I supposed I could spring for a grand tour of various destinations. Among these would definitely be Hawaii, Australia, Tahiti, England, Ireland, Italy, and France.

10. What one thing do other teachers do that you think is completely crazy (only one).
Keep coming back every fall. Come on, we're all crazy for liking this job, aren't we?

11. If you were not a teacher, what else would you do?
Oh, let's dream here. Maybe I'd go into radio or do public address announcing. Maybe I'd spend time learning how to use music software and compose soundtracks for video games. Maybe I'd write TV sitcoms. I dunno, those all sound fun. But I love teaching.

Phew, this is a lot of work, but good for getting the blogging juices flowing. Alrighty, step four:


Eleven Random Facts About Me!

1. I love many classic video game series, like Mario, Zelda, Mega Man, Final Fantasy, etc. But one series that I'm really into that you're far less likely to have heard of is Phoenix Wright: Ace Attorney. Great storylines, awesome characters, and sweet soundtracks.

2. Speaking of video games, I'm totally into video game music (VGM) cover bands. Go onto Google and check out The One-Ups, The Megas, The Protomen, Descendants of Erdrick, The World is Square, and Bit Brigade for starters.

3. A life ambition of mine is to visit all 50 states and all 7 continents with my wife. So far, we're at like, 6 states and 1 continent, or something. But we have future travel plans!

4. My favorite pizza is Domino's, hands down. However, I do miss getting Pokey stix from Gumby's in East Lansing. I lived off of those during my year of student teaching (terrible idea, by the way).

5. I once broke a window with a computer processor. Long story.

6. There's a song about me on YouTube that I got for donating to a Kickstarter.

7. When I was attending Michigan State University for my undergrad, I worked for the freshman orientation program during the summer. This included giving campus tours daily; I became adept at walking backwards while spouting all kind of random facts and figures about the university.

8. I also worked a a summer camp for a few years after I finished college; my favorite thing was to threaten the kids with liver and onions for dinner if they didn't manage to successfully shoot their archery targets. It was an excellent motivator; liver and onions were never served once.

9. I took a job working as a night receptionist while at Michigan State purely because I was having major trouble with insomnia at the time. I figured if I was going to be awake all night, I might as well be getting paid for it. I worked a few semesters, including one summer. Working night reception during the summer was the best because there was pretty much nobody around, and the only people who came to the door had their own keys anyway. I'd take my TV and PS2 and just play video games all night long, and get paid for it. Sometimes I miss that.

10. I'm deathly afraid of roller coasters. I think it stems from a trip to Disneyland when I was 3 years old. My parents took me on Space Mountain and it scared the life out of me. I've never been a fan of roller coasters and I don't think I ever will be.

11. My favorite flavor of any kind just might be pumpkin. I haven't met anything pumpkin-flavored that I haven't liked. In fact, one of my favorite pumpkin-flavored things is O'Fallon Pumpkin Beer, which sadly was very limited last fall. I'm totally going to stock our fridge with it next fall.


Almost done! Now it's on to step five, create eleven questions for the five blogs I nominated:

1. What is the one thing you love most about teaching?
2. What is the best gift you have ever received or given?
3. Outside of teaching, what is one thing you do to try and make the world a better place?
4. Who is your favorite U.S. president, and why?
5. What is your favorite "me time" activity?
6. What is your survival strategy for the imminent zombie apocalypse?
7. What "energizes" you during the school year?
8. What is your favorite holiday?
9. What one thing do you miss about your childhood?
10. Where do your lesson ideas come from?
11. What one thing do you want your students to remember you for?


BAM! Pretty much done. All that's left now is to contact each of these bloggers and let them know of the torture that awaits them.

This was a fun exercise! I certainly hope that I'll have more to blog about soon, but this is a good start.

In the meantime, I am happy to say that I have accepted this nomination; and I can rest easy, knowing that I won't have to be looking over my shoulder for any blogging ninjas... for now.