Twosday Things: Hearts, Stars, Messy Numbers

Time again for Twosday Things!

Taking a cue from last Tuesday's post, I'll discuss two teaching-related things (however big or small) that happened over the past week. I'm trying to post about two things every Tuesday throughout the school year (hence the title, "Twosday Things"). This makes the second week in a row; so far, so good.

Thing #1:
Something I've noticed that happens A LOT in my class:

• Student is working through a (typically algebraic) problem.
• Student gets a non-integer answer (i.e. a "decimal answer").
• Student immediately assumes they must be wrong. Often accompanied by asking the teacher, "am I supposed to get a decimal for my answer?"

This is a near-daily occurrence in my class, despite my frequent insistence that "decimals are numbers, too!" ("Fractions are numbers, too!" is similarly used often.) I cannot even count the number of times this happens in a school year.

How does this happen? How do our students reach the point where they automatically assume that "decimal answers" must be wrong? How do we let them get to high school with this assumption cemented into their mathematical psyche?

Yesterday, I took this question to my Twitter feed:

Some super-awesome math-types from the Twittersphere chimed in with their thoughts on the topic:

"Give them messiness." I love that. I feel like our students need more practice and earlier exposure to "messy numbers," because real-world math is messy and complex. Students need to learn that decimals, fractions, irrationals, etc. are all numbers, too.

At the same time, I don't think it's inherently bad that students question their answers every time they get something "messy." Sometimes (often, in fact), their answer actually is the result of a mathematical mistake, and they need to be able to figure out where the mistake was made.

I can see some potentially good habits here: stopping to think about whether the answer makes sense in the context of the problem; double-checking work for mathematical mistakes; and so forth. I just don't think that "getting a messy answer" should be the sole reason a student thinks they did something wrong. If anything, students should be trained to question "messy" answers and "clean" answers. Students should be in the habit of doubling back and re-checking their work to make sure their reasoning makes sense.

Maybe the mistrust in "messy" numbers can be a good thing; but if it is, it needs to be applied to all numbers. Equal opportunity, darn it!


Thing #2:
Today in class, I had a few students who asked for help with the following problem:

We discussed the fact that the problem mentioned "two numbers." We had no idea what those two numbers were, offhand. But, we had enough information to be able to set up a couple of equations. We just needed to pick two variables to represent the numbers first.

"We can call these two numbers anything we want," I said. "We can call them x and y. We can call them a and b, or c and d. We could even call them stuff like, 'dollar sign' and smiley face.' What do you want to call these two numbers?"

One of my students said, "heart and star."

Math, learning, and hilarity ensued:

I had a terrible time keeping a straight face, especially when I said things like, "so what expression do we plug in for heart?" or "yep, we have to simplify by combining our star terms, so star plus eight equals twenty-four," or "there we go, star equals sixteen and heart equals forty."

It was a fun little way to talk about the concept of representing unknown values with variables. Why settle for boring old x and y when you can have a bit of fun?

Multiple Solutions (A follow-up to "When Is the Right Answer the Right Answer?")

A couple of weeks ago, I wrote this post about how I wanted my students to determine equations of lines, given certain information. The broader point, I think, was realizing that my students had more than one option for determining answers to the problems they were working on, and being okay with that. (Why wouldn't I be?)

I had another "when is the right answer the right answer?" moment in class yesterday that I thought was really super-cool.

Two students were working together on the same problem. They came up with what they thought were different answers, so they were wondering who was correct. Their work is shown below:

So both students used point-slope form for their equations, and came up with two answers that looked different. This peculiarity made them wonder who was right and who was wrong. (Which, in turn, makes me realize that I still have a lot of work to do with teaching them about making sense versus being right.) They called me over to ask me who had the correct equation.

I must have been really busy at that moment and not really thinking, because I looked at their answers and said, "actually, you're both right." Not that I was wrong in saying so; but I regret that I didn't recognize the teachable moment that had presented itself. This would have been a great opportunity to ask each of them what they thought about their equations, how they came up with them, why they thought their answers made sense, why the other person got something different, and whether or not it made a difference which point they used for point-slope form. Still, it was a really cool moment: two students have a spirited debate over who had the "right" equation, when really they were both right. It was my favorite moment of class from yesterday.

Fortunately, the same thing happened today, on the same problem, with the same work as shown above, between a different pair of students. Grateful for a second chance, I was able to stop and facilitate an awesome math discussion between the two of them.

One student was adamant that the "first" point, (-4, 3), had to be plugged in for point-slope form instead of the "second" point, "because they're X₁ and Y₁," she reasoned. She said this because she had labeled the coordinates as such when using the slope formula to determine the slope:

And point-slope form was written on the board as Y - Y₁ = (X - X₁). So I could see where she was coming from.

I asked her, "so, how would you label these points if the order was swapped?" In other words, what if the problem listed the points "(6, 1) and (-4, 3)" instead of the order they were given? She responded that she would have labeled (6, 1) as (X₁, Y₁) and (-4, 3) as (X₂, Y₂).

My next question was, "So would that change things? Would you get a different slope, for instance?" The student initially thought that yes, she would get a different slope. The other student, who was working with her, said that the slope should be the same. I had both of them determine the slope of the line with the different designations for the coordinates; naturally, the slopes turned out to be the same as in their original work.

I asked, "how did changing the order of the points affect the slope?" The student replied that the order of the points didn't change the slope at all. "Cool," I said. "So what about the two different equations you guys came up with? What difference does choosing one point over the other [when plugging a point into point-slope form] make?" The first student still wasn't quite convinced that it didn't matter what point she chose; her partner said it didn't matter what point was chosen for the point-slope form of the equation.

We decided to have each of them solve their equations for y, so they'd both be in slope-intercept form. When they did so, they came up with the same equation, and the first student was finally convinced that it didn't matter which of the two points she chose. Both students were convinced that they'd both determined correct equations for the line described in the problem. "Why doesn't it matter which point you choose?" I asked. The first student wasn't quite sure. The second student guessed, "because both points are on the same line?" I replied, "that sounds like it makes sense."

I love when students find different (yet equally valid) solutions to problems like this. It makes for some great discussion. I need to keep myself aware that it's more important to ask my students to make sense of their work instead of telling them that they're right; I missed out on having a great conversation with two students yesterday, but I'm glad I had another chance at it today.

When Is the Right Answer the Right Answer?

This week, my students have been working on determining equations of a line based on properties of parallel and perpendicular lines (GRE 604 from the ACT College Readiness Standards for Mathematics), which involves problems like this one:

Several concepts popped up throughout the week while working on this skill: determining slope, slope-intercept form, point-slope form, and the relationships of slopes between lines that are either parallel or perpendicular to each other.

Throughout the week, I have been insisting that my students give their solutions to these problems in slope-intercept form, as shown in this student's work:

(There are some other things going on here that would also be interesting to talk about, but that will have to wait for another day.) 

Perfectly reasonable solution method, isn't it? Put the original line equation in slope-intercept form, determine the slope, use point-slope form to get the equation of the parallel line, and then solve for y to put that equation in slope-intercept form.

This morning, I found myself wondering why I was insisting on having my students put their answer in slope-intercept form.

Is it really necessary? I mean, couldn't the student have just stopped at point-slope form and still been correct? I mean, plug a few things into Desmos and it's hard to argue otherwise:

I've been thinking about this and struggling with this all morning. The focus of this particular ACT skill isn't necessarily for students to determine the equation of line and put it in slope-intercept form; the skill is just to determine the equation of a line based on properties of parallel and perpendicular lines.

In the problem above, the student is given the equation of a line and a point on another line that is parallel. The student knew to look for the slope of the original line, knowing that the parallel line they were looking for would have the same slope. After determining the slope, the student created the equation of the parallel line using point-slope form.

Should it stop there? After all, the student correctly applied the properties of parallel lines and determined a correct equation. That's what the skill is all about, right? Why was I insisting that the student put their answer in slope-intercept form? I'm not sure it's necessary, and I think it also creates a situation where the student can make a simple algebra mistake and come up with an equation that is no longer "correct." On the other hand, expecting students to be able to put the equation in slope-intercept form isn't all that unreasonable, is it? After all, the student did just that with the equation of the original line in the problem, in order to determine the slope of the parallel line. Is that a good enough reason to insist on it, though?

This is just one specific case. I know this isn't the only instance in mathematics where something like this happens. When is the right answer the right answer?

Coffee Spills! Sales Sheets! Math!

I like to think I have good taste in music. When I was a kid, I played a lot of video games. Video games are super fun. The best part about video games, arguably, is the music. I will always hold the opinion that the Super Nintendo era gave us some of the best video game tunes in the history of ever. EVER.

So, these days I listen to a lot of video game music (VGM) cover bands. One of my current favorites is a recently-formed act, The Returners. They're based in Austin, TX and they totally rock.

But they don't totally rock just because of their music. They totally rock because the band's founder, Lauren Liebowitz, recently helped me out with putting together a math task that involved coffee spills and band shirts.


Background
Over the past couple of years, the math team at my school worked together to put a four-year curriculum in place that's closely aligned to the ACT college readiness standards in mathematics. The skill that my students are currently working on is XEI 602:

"Write expressions, equations, and inequalities for common algebra settings."

We wrote a ton of problems related to each skill. For this particular skill, we wrote problems such as the following:

(And actually, that should be 46 cakes, not 44. Typo. Oops.)

To supplement this skill, I thought of a different way to present this type of problem. Instead of spelling out the necessary mathematical information in a word problem, I wanted to present a more realistic situation and have the students work a little bit more to dig up the mathematics of what was happening.

So I thought of the following scenario: Suppose you were selling a few different items and keeping track of your sales on a sheet, such as this:

And then, suppose you accidentally spilled coffee all over it:

Some of the information is lost! How could we figure out the information that was ruined by the coffee spill? (Obviously, there isn't enough mathematical information in the above example, which is purely for show. But given the right info, this becomes a challenging math task. Also, as it turns out, it's pretty challenging to simulate a coffee spill. And ink is pretty resilient these days.)


The Task
I spent some time thinking about what product(s) to include on the sales sheet that I was going to spill coffee on. One night, I was folding laundry and I came across my official "The Returners" t-shirt. My brain was like, "BAM. T-shirts!"

I messaged Lauren to pretty much say, "Hey, I'm using your band in a math problem!" And she basically replied, "Cool! Can I do anything to help?" And actually involving her hadn't really occurred to me, but she was totally willing to assist and I couldn't turn that chance down.

I put together a "sales sheet" listing three different types of Returners shirts, with information about prices and inventory. Then I spilled coffee on it:

Lauren included the following e-mail, and also sent along a picture of a few of her band's shirts (rolled up neatly into little shirt burritos):

Armed with the above information, my students were set to the task of helping Lauren figure out how many shirts were sold at her most recent show.


How The Task Went
I put the students in groups of three for this task. They were each given a copy of the e-mail, the ruined sales sheet, and the photo of the remaining shirts. I also gave them a worksheet with a few questions that each group member needed to contribute to.

My first period jumped into this task right away. There were a lot of good conversations going on at the start: they were looking through the documents, figuring out what information was important, and talking about how they were going to represent each type of shirt as variables in equations.

There was debate over how to represent "blue" versus "black," since both started with the same letter. Some students decided to use b for one and bl for another, but soon found that there was still no distinction (since both colors start with the letters bl). One student finally suggested using k for black, which certainly helped things.

Students were able to figure out important bits and pieces of information needed to set up an equation to represent the total sales: from the sales sheet, they found the total money made and the cost of each shirt. From the picture, they were able to determine that there were 5 blue shirts and 3 gray shirts that went unsold from the original inventory.

Where we ran into trouble was figuring out how to actually set up equations representing the total sales. Groups during my first period class initially set up their equation as:

5.25k + 4.75b + 4.50g = 397.25

where k represents black shirts, b represents blue shirts, and g represents gray shirts. The above equation was close, but incorrect; students from my first period continued working through the problem using this equation and ended up getting answers that didn't make sense in the context of the problem (e.g. they got non-integer values when they solved for k, b, and g).

They were on the right track, but they didn't account for the fact that a few shirts went unsold; this was kind of the "tricky" part of the task and led to a lot of frustration among my first period students. I let them have some time to try and sort things out on their own and dropped a few hints to try and point them in the right direction. Eventually, I saw it was going to be best to stop and have a quick whole-class discussion about the unsold shirts.

We talked about thinking of k, b, and g as the number of shirts that were originally in the inventory as opposed to the number of shirts that were sold. I asked the students to look through their documents again and tell me what they could find about the number of shirts that were sold and the number of shirts that were still left. We talked about what expressions we should write to represent the number of shirts that were sold. Eventually, we came up with the following:

• The black shirts were sold out, so k black shirts were sold.
• There were 5 blue shirts remaining, so b - 5 blue shirts were sold.
• There were 3 gray shirts remaining, so g - 3 gray shirts were sold.

The students adjusted their equations to reflect an accurate model of the total sales:

After my first period class, I adjusted my lesson plan so that we talked about the correct expressions for shirts sold toward the beginning of the activity. This adjustment made things go more smoothly in my other three classes. Although, in my second period class, students were having trouble with question #2, which required them to re-write their equation from question #1 in terms of one variable. We stopped to have another conversation about what to do.

I asked students to re-read their e-mails and to look specifically for relationships between the different types of shirts and how many of each kind there were. The students noticed the following info:

• There were twice as many black shirts as either the blue or gray shirts.
• The number of blue shirts was the same as the number of gray shirts.

From here, we talked about how to use this information to write a few small equations that would help us with question #2. The students came up with the following equations:

Once we were armed with these equations, we were able to go back to our total sales equation from question #1 and use substitution to re-write it in terms of one variable:

I made another adjustment to my lesson plan for my third and fourth period classes to include a discussion about representing these relationships toward the beginning of the task as well. With this guidance, students were able to successfully determine how many shirts had been sold:

What I Would Change Next Time
This task was given to the students while we were in the middle of our mini-unit on writing equations and expressions based on information from word problems. After doing this task and reflecting on how things went, I think this task has a lot of merit as one of two things:

1. A guided task at the beginning of, or during, the unit, with appropriate scaffolding included; or
2. A performance task at the end of the unit.

My students were resilient and we were able to have a lot of good mathematical conversations during this task. I could also tell there was frustration stemming from confusion about what to do at each problem. I think this might have been the result of not yet having enough practice with the skill, and I also think I didn't provide an appropriate amount of scaffolding, originally. In my later classes, I made adjustments and discussed some important aspects of the problem at the beginning of the task; this seemed to help students successfully solve the problem.

In the future, I would probably embed further scaffolding and questions into this activity. For instance, I would probably ask the students:

• What is the problem you are being asked to solve? What information are you supposed to determine?
• Choose a variable to represent each different type of shirt. Write expressions to represent the amount of each shirt that was sold.
• From the e-mail, what information can you determine about the number of shirts that Lauren originally had? Write equations that represent these relationships.

By having the students doing this work first, the rest of the task would probably go more smoothly as is.

I might also change the prices and the types of merchandise on the sales sheet in the future; a few students looked at the prices and thought, "why didn't they just sell shirts for $5? It'd be easier to make change." In reality, the shirt prices are actually more like that. I made each shirt a different price; otherwise, the task would have been pretty easy to solve. Next time, I'd probably include other merchandise such as CDs, buttons, etc. and let the prices reflect something that would actually be charged at a show.

Overall, I was pleased with this task. I certainly learned a lot, and I hope to come up with even better tasks in the future and I continue trying to figure out how to do PrBL!

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.