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!

Mario: Life or Death? (A Math Task)

I love math. I also love video games. So when I find a way to combine the two, I'm like, super-mega-happy.

We're in the earlier stages of a unit on graphing, and I want my students to get used to the idea that they'll need to interpret & analyze figures, diagrams, etc. and make predictions or generalizations based on solid mathematical reasoning. This is a skill that I believe is crucial for a student's success in Pre-Calculus and Calculus (and beyond).

This week, my Advanced Pre-Calc students did a one-period math task I developed called, "Mario: Life or Death?" It wasn't anything terribly complex or flashy, but we did get at a few important ideas about graphing, symmetry, and making predictions from what we know conceptually rather than numerically.

(Also, after I came up with the idea for a Mario-based problem, I found out about this Mario problem from Nora Oswald that's way, way better than what I did. I was actually inspired to follow a similar format after finding it.)

I was taking (or trying to take) a 3-Act Math approach with this math task. It definitely still needs some tweaking all around, but considering I'd never tried anything of the kind before, I was pretty pleased. (I don't know how truly 3-Act-ish this task actually is, but, eh.)


The Opening
I played this video for the students:

The students also had the first two pages of this worksheet (I purposefully withheld pages 3 and 4 until later).

As students watched the video, I had them think about what questions came to mind. They came up with some pretty good ones:

• Will Mario make it to the other side?
• When should Mario jump to make it to the other platform?
• What is Mario doing wrong that's not letting him jump across the cliff?
• When does Mario reach the apex of his jump?
•  How can we calculate what height he needs to jump to make the distance?

And some pretty good non-math questions, too:

• Why doesn't he use the chain to make it across?
• Did the player hold down the A button long enough?

So right away, the students came up with some important concepts, including distance, height, and particularly "reaching the apex" of the jump. These were important things to notice, and were very closely related to the problem I posed (which a few of them nailed):

"Will Mario make it across on the 3rd jump?"

Working On the Problem
Next, the students were asked to come up with a few ideas. What information did they think would be needed to solve the problem? I gave them a little time to think of a few ideas on their own, and then they shared with each other. A couple of examples from their work:

With some group discussion, we pulled together some important concepts to think about: the distance between platforms, the height of Mario's jump, and in a few cases, the ideas of "arcs," "parabolas," and "symmetry."

I had the students explore these ideas further by having them create sketches of what they thought the path of Mario's jump looked like:

We talked about the shape of Mario's jump. The words "semicircle" and "arc" came up a lot. One or two students already knew the word "parabola."

I asked the students to think about what they knew about arcs; what is their structure? How do they behave? Particularly, I asked them if they've ever seen any "lopsided" arcs. The students said no; arcs are made up of "mirror images," or are "symmetrical." I asked them what they meant by that. They replied that an "arc" can be cut into two halves that are mirror images of each other; particularly, the arc is cut at the "highest point" into two congruent halves.

The students were asked to draw the path of Mario's jump again, this time on a grid. They were also asked to take the observations from their discussion into account:

After looking at the gridded drawings, the students had a few moments to think about how their observations about Mario's jump could be useful in predicting the outcome. They wrote their thoughts down:

After giving them time to think about how to use their observations to predict whether or not Mario would make the jump, I gave them pages 3 and 4 of the worksheet. Using the graphs of Mario's jumps, I asked the students to make their predictions and to give their mathematical reasoning:

Everyone shared their predictions and gave their reasoning. The consensus was that Mario would make his jump, but just barely.


The Thrilling Conclusion
After everyone made their predictions, the moment of truth arrived. I played the solution video:

So the students had predicted correctly! They seemed a bit amazed at how accurate their predictions were -- not only had they figured Mario would make his jump, but many of them said specifically that Mario would barely make his jump, which is what happened. They had made a mathematical prediction using conceptual observations about the path of Mario's jump, and very minimal use of numbers or calculations.

To wrap things up, we tied our discussion back to the idea of symmetry. The type of symmetry used in this problem was just one type of symmetry (even symmetry, and arguably symmetry about the y-axis). We talked about other types of symmetry besides the type used in this problem. The students took down some notes for their problem sets they were working on from our textbook.

I hope that the students came away from the lesson with a bit of a deeper understanding of symmetry and how to use it to make predictions. In the larger picture, I hope students started to see the usefulness of applying what they know and observe in order to gleam new information. As I've said before, there's so much more to math than just calculations and number-crunching.

This was totally the highlight of my week. However, I know that this task is far from perfect. It could certainly go deeper and touch on quadratics, graphing equations, and so forth. I'm almost certain this task could be made into something richer than its current form. I'll definitely be taking another look at it after some time passes and I'm free enough to sit down and make revisions.

But in the meantime, I welcome any and all feedback/comments/insults from any readers out there. Don't be shy!

Week 1: Why My First Day Activity Didn't Go At All As I Had Hoped (and Why That's Awesome)

Phew, the first week has come and gone and I found myself utterly exhausted on Friday. Thank goodness for the holiday weekend; I've been able to get more sleep in the past 2-3 days than I have in quite a while.

As part of the kickoff to our school year, I had my seniors work on an "opening day" activity that I lovingly borrowed/blatantly stole from Nadji (who blogs at Physix Coolisms!) that involves grids, writing your name (a lot), and using that data to generalize a pattern.

The activity I snagged is called "What Is Math?" and is described by Nadji from 4:25 to 12:30 of this First Day of School Activities presentation from Global Math Dept. I won't re-post the entire activity here, but basically the aim of the activity is to challenge students' perceptions of what it means to do math.


The Activity
The first part of the activity has students answer the following questions:

1. What is math? What does it mean to you?
2. List 7 mathematical words or phrases that come to mind when doing math.

Often, students respond to these questions with the mindset that "doing math" means working with numbers and calculations and equations.

After answering these questions, students then fill out several square grids by writing the letters of their name over and over again.

After doing that, the students shade the first letter of their first name and then fill out a table to record the patterns that show up.

From there, students are asked to make predictions about the patterns, such as:

• Predict what pattern would appear in a 41x41 grid.
• Predict how the patterns would be affected if the second letter of each name was shaded instead.
• Predict how the patterns would be affected if students started by writing their name in the bottom right corner and filling out the grid backwards.

And so on.

At the end of the activity, students are asked the beginning two questions again; by this point, the hope is that students will start to see that math is much more than just working with numbers and calculations and equations. There is much more to mathematics: finding patterns, making generalizations, predicting unknown events, thinking critically, etc.


How Things Went:
Before I get into this, one side (yes, side, not snide) comment: I had my students fill out grids up to 10x10. If I do this activity again, I might have them go up to 12x12. I have many students with names that are 6, 7, or 8 letters long, and their patterns don't really start to become apparent until the grids get bigger. As I checked in on students and looked through the tables they were filling out, it seemed to me that the "pattern of the patterns" would be more apparent if they had more data. Something to think about for next time.

At any rate, student responses to the opening two questions went pretty much as I expected. Many students came up with responses like "math is the study of numbers," or "math is the tool of Satan," and so forth. The lists of 7 mathematical terms often included "addition, subtraction, multiplication, division, square root, equation, numbers," and the like.

I decided to collect answers to the first two questions via Socrative, so I could quickly generate a bunch of text and then dump them into a Wordle. I thought it would be cool to generate a visual snapshot of student responses from before and after the activity so I could compare.

Here is the "before" Wordle:

As you can see, there's a great deal of "number-ish, calculation-y" stuff. I expected to see this.

Based on what I was seeing from the students as they were working on the activity and the conversations they were having (with each other and with me), I expected to see a dramatically different Wordle from the post-activity responses. After all, they were noticing patterns, making predictions about how patterns would look in grids that were far larger than they had time to fill out, and working together to describe a "rule" for making such a prediction. They weren't really doing "stuff with numbers."

So, here's how the post-activity Wordle turned out:

So uh... um... not really all that different. I mean, "patterns" showed up a lot more in this one, but there was still a bunch of "number-ish, calculation-y" stuff.

I'll admit, at first I was a little bummed that I seemingly hadn't changed very many minds or shifted very many paradigms after doing this activity.

But then I thought about it. And I became okay with it.

In fact, it's actually pretty awesome that I didn't change their minds so easily, and here's why:

This becomes a new challenge for me. This allows me to set a goal for myself. I want my students, by the end of the year, to understand that there's a lot more to mathematics than just crunching numbers and solving numerical problems.

Math is recognizing patterns and trends. Math is making use of those recognitions to make predictions. Math is critical thinking.

Math is art. Math is visual, spatial, tangible.

Math is freaking everywhere and freaking awesome.

I don't get to spend just one day trying to convince my students of this. I get to spend an entire year trying to convince my students how super-cool math is. I have a lot of convincing to do, but that's okay with me. I want to earn it.

That's one thing I learned from doing this activity. That's one thing this activity has given to me: a theme for this year: Math is freaking everywhere and freaking awesome.

It's going to be a great year.