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__.

**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:

**In fact, we begin the process of answering this question with the entry event.**

*"How do we locate the submarine?"*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,

*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."*

**"How do we narrow down the number of possible locations?"**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.

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:

*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:*

**"How do we graph circles?"**

**"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.

**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.