Trying to find math inside everything else

Posts tagged ‘Education’

You All Have “A”s

So I was thinking about grading a little bit, and how grading works in my classroom. I tried to ask people about grading on Twitter, but perhaps the medium is not the best for talking about it, because only one person responded. (Thanks, @algebraniac.) I wanted to get a feel for how people out there calculated grades, before I wrote about it, but I figure, what the hell! Just write about it anyway! (Maybe channeling Hedge a little bit here.)

So, like, I’m imagining the typical first day of class that happens. The teacher tells all the students, “As of right now, you all have ‘A’s.” With the intention being, of course, encouragement, because despite how bad they might have done in that subject in the past, right now, they have an A.

But when you think about it a little more…it’s really kind of terrible, isn’t it? “Right now, you have an ‘A’…and the only way to go is down.” So then the grades don’t reward good work, they only penalize bad. Your grade tracks every mistake you make, every little fuck-up, dropping in a downward spiral. And we talk about students “slipping” and “dropping the ball” and “not doing as well as they used to.” The whole terminology is pretty terrible.

On the surface, it might seem like Standards-Based Grading can help with this, like it helps with so many others. Students have standards, and if they are low they reassess and go up. At the end of the marking period or term, that certainly seems like a good system. For each individual standard, it works, but as a collective whole? Let me ask you this:

It is halfway through the (quarter/marking period/term), so report card grades are not due for another few weeks. A student comes up to you and asks what their grade is. What do you tell them? What is it calculated from? And how will the future work they do affect that grade, if they do well? What about if they do poorly?

I’d really like to know. Drop a line in the comments and tell me. I’ll follow up with people’s responses and what I do in another post.

 

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assessment

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Trig without Trig

Over the summer I made a Donors Choose page in the hope of getting some clinometers, so that we can go out into the world and use them to calculate the heights of objects like trees and buildings. And we did!
I created the Clinometer Lab as an introduction to Trigonometry. As such, prior to the lab, they had not seen any trig at all. So I started off with this video as the homework from the night before.

In class, we talked about how, when the angle is the same, the ratio of the height and shadow is the same. And how, long ago, mathematicians made huge lists of all of these ratios.
So if they told me any angle, I could tell them the ratio, and then they could just set up the proportion and solve.

So we left the building and went to the park, armed with clinometers, measuring tape, calculators, and INBs, so do some calculations. I had the students get the angles from their eyes to the height of a tree, from two different spots, and the distances, so they could set up a proportion and discover the height of the tree. (When they finished, they did a nearby building. If they finished that too, the extension problem was to switch it up: given the height of a famous building, the Metlife Tower, that they could see from the park, and determine how far away they were.)

Clinometer Park Pic

It went really well, and I think they got the idea. The fact that they could choose which tree to measure and were given free range of the park, and could choose where to stand to look at the top, but always had to come back to me to get their ratio, worked seamlessly. I could check in on them easily and keep them on the right path. (Especially with my co-teacher there to keep them all wrangled, or the AP who observed/helped chaperone the classes without my co-teacher.)

The next class, I revealed to them that I was not using that crazy chart to give them their ratio, but rather they can just get it themselves from the calculator, which basically internalized the list. Then they got it, that is was just ratios and proportions, not some crazy function thing.

Trig as a topic did not go great in my class, but that is the fault of my follow-up lesson, where I tried to squeeze in too much and didn’t do any practice, not that fault of this lesson, which still sticks in their minds. Later in the year we were doing a project about utilizing unused space, so we picked empty lots but couldn’t get in. So in order to figure out how big they were, we used clinometers, and trig. Now that’s a real world application.

The Lab

Clinometer Lab Instructions

Clinometer Lab Sheet

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anywhere, labs, lessons, math

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The Archimedes Lab

I love this lab, but I’m sure it can be improved, so I’d love to hear feedback in the comments.

When I was student teaching at Banana Kelly, they had a combined math/science program that often had many lab activities, both math labs and science labs. But I noticed the Systems of Equations unit didn’t really have a lab, so I set out to make one. I created the Archimedes lab because of that.

The story of Archimedes and the Crown of Syracuse says that he was able to determine that the crown was not pure gold by using mass and volume. We start by watching a short video about the story. (I had assigned the video as homework, but no one watched it. While my video homework watching rates aren’t stellar, they’ve never been this bad. I blame the fact that it’s been a while: the last video was in December.)

Then I tell them that we can do Archimedes one better. It’s been over 2000 years since he lived and we have algebra and precise tools on our side: we can figure out exactly how much gold was stolen, not just if some was.

CAM00027

To demonstrate this, we built several Mystery Blocks out of two different materials. Back at BK, we had lots of different materials in the science closet to use. Because the students would know the type of material (brass, aluminum, wood, etc) but not the size and shape, they would have to use density to solve the problem. (But I had seeded the story throughout the whole unit.) The density kit I requested last year never arrived, though, so I had to scrounge around for something, and I found the perfect thing: Cuisinaire Rods.

I noticed that, despite what I expected, the longer Cuisinaire rods are much less dense that the small single cubes. So I used some amount of cubes and some amount of rods to make the mystery blocks. Then I simply give the kids one cube, one rod, the block, a scale, and a ruler, and ask them to figure out how many if each kind are in the block.

CAM00024

This lab was much more procedural when I first made it, but I’ve embraced the problem-solving classroom since then. And while my students were unsure of what to do, I feel like we’ve built up to this kind of unstructured problem, and most of them took to it well.

When they thought they had a solution, I would press them on it. “Why do you think it’s 5 yellow blocks and 5 cubes?” If they responded that that matches up in size to the block, I would point out any others do also, like 4 yellow and 10 cubes. I would implore them to use the tools available to them. (We also talked about mass, volume, and density in the warm-up.) If they told me some combination gave them the right mass, I would point out that their volume would be off. If they told me no combination gave them the right volume AND right mass, I would ask why. Most would identify the tape/foil/wrapping paper as the culprit.

Most importantly, if they gave me a solution that was well supported, even if incorrect, I told them there was only one way to find out if they were right: unwrap the mystery block and see. The excitement at each table when they got to that point was palpable. One student said it felt like Christmas. If they were right, they were ecstatic. If they were not, they tried to figure out where they went wrong.

CAM00025

When I first made this lab, I put it at the end of the Systems unit, thinking they would need those tools. Now I have the confidence to put it at the beginning, and let them figure it out. No one used equations explicitly to solve the problem, but they definitely grappled with the idea that two separate constraints need to both be satisfied in order to solve a problem, which is an important understanding for systems.

This post is getting a little long, so I’ll probably split off into another posts with problems I had with the lesson, things that went well, and a call for suggestions. I just wanted to get this out there. It’s been a long time since I’ve blogged, because I’ve had a lot going on in my life. So I want to get back to it.

The Lab Sheet

The Archimedes Lab

Accompanying Graph

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labs, lessons, math

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Dragonbox in the Classroom

Last week, my students spent 2 double periods playing Dragonbox, the iPad (and computer) game designed to teach solving linear equations, which I think it does quite well. (I agree with many of Max Ray’s opinions when he writes about it here. Which makes sense, as Max first showed me the game this past summer.)

While one of my goals was teaching solving equations, it was not my only one, which is what I wanted to talk about here. (I’ll probably review the game itself later.) I told the students that I had forgotten to make a lesson, so we were just going to play a game on the iPad today. What I did want, though, was for them to home their ability to figure out how something works. To me, this is an even more important lesson to get than just solving equations.

To this end, I talked about how websites like GameFAQs has walkthroughs for all sorts of games, but one walkthroughs were all written by regular players, who sat down with a game right when they bought it, took notes on what they did, figured things out, and shared with others. So we were going to take that role. In their Interactive Notebooks, I told them to write down every thing they could do in the game. Whenever they came across a new rule, some new ability, or a new solution to a tough puzzle, write it down. Example: “Tap the green swirl to make it disappear.”

The surprising part was, they really did it, and quite well. Hey even discovered a lot of things about the game that I didn’t know, because I always played it “perfectly,” since I knew the rules of algebra. (Example: if you have a denominator under a green swirl (aka 0) and tap it, the while thing disappears. Or a green swirl won’t disappear if it is the only thing left on its side, which was fun to talk about later.)

At the end of my first double, with about 20 minutes left, I compiled all the notes they took using Novel Ideas Only (where all students stand and share things they have written, only sitting once everything they have written down is said, either by themselves or someone else), creating a master list of actions they could refer to next time.

The next class, they came in and immediately started playing. I must say, the entire time I used it, the kids were really into it, and most of them were really persistent. Some occasionally requested help, but my intervention was minimal. This time, I had this answer several questions after they had played some more, which really dove into the meat of the game. What does this card or action in the game represent in math? Why does a certain rule in the game happen that way?

One thing I really loved is how solid the game got them on how dividing something by itself won’t make it go away. It was a tactic many of them tried in several levels and it always got them stuck. I focused on the difference between “zeroing out” and “oneing out.”

We had one major downside, technology-wise, though. Each game had four save files, which worked out, because I had four sections. So one file per student. But there is nothing to stop a student in one class from playing on, or, even worse, DELETING, another student’s file. I e-mailed the company, and they said a solution would happen in a future update.

Today was the follow-up quiz, and they mostly did well. The things they stuck on was something that wasn’t well covered in the game: the distributive property. But we’ll work on that.

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games, lessons

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Steepest Stairs Redux

Last year I made a lesson about determining the steepest stairs, using pictures my co-teacher and I took and based on an idea from Dan Meyer. It took about a period, and was mostly teacher-led. But after arguments and deep thinking about slope, I wanted to go into the lesson deeper, so I turned it into a lab.

I started the same way, throwing up the (new and improved) opening slide and asking which they thought was the steepest and which was the shallowest.

Screen shot 2012-11-26 at 1.55.16 PM

I really like this new improved one because I took a picture of the toy staircase from the board game 13 Dead End Drive (middle left). Last time, there were overall agreement on the shallowest (the Holiday Market) while there was disagreement on steepest. This time, because the toy was tiny (if not shallow), we had some disagreement there, which really let us tease out some definitions of “steeper” and “shallower.”

Once we had definitions of steeper (which usually came out to something like “closer to vertical” or “at a bigger angle”), I handed out the pictures on a sheet of paper and asked them to develop a method for determine which was steeper, or the steepest. I mentioned coming up with some sort of “steepness grade” (because I thought it would be amusing to throw the word “grade” in there).

So I let them struggle, and come up with what information they had to ask me for, which I would then provide. If I had to do it again, I would also have pictures of the width of each stair, as a distracter, because some kids asked for it. Interestingly, some also asked for the angle, because of our prior experience in the year with the clinometers. I told them I didn’t have the clinometer with me at the time. One kid called me on it, because she knew I had a clinometer app on my iPad. So I told her (truthfully) some of the pictures were taken last year, before I had it.

So I had them come up with their own measures. If they tried to base it off of only height or only depth, I deflected with examples of really tall, really shallow stairs, or really short, really steep stairs. TallShallow

By the end of the classes, students usually came up with one of three different measures: slope, the inverse of slope (depth over height), and grade (that is, slope as a percentage).

IMG-20121127-00127IMG-20121127-00125

So they had to then reason as to why they might prefer height/depth to depth/over. (Their logic: it seems more natural to have bigger numbers be steeper stairs, rather than the other way around.) And so it was that point that I told them this “steepness” grade that they developed was often called “slope” by mathematicians.

At which point, I got a big “Ohhhhhhhhhhh.” Which always makes it worthwhile.

The Materials

Stairs – Portrait

Stairs – Landscape

Steepest Stairs Lab

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labs, lessons, math

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Today’s Roles: IT Department, Programmer, Lecturer, Assessor, Tutor, Co-ordinator….

5:30 – Alarm goes off. That is not happening.

6:20-7:00 – Wake up, shower, pack up, go. I decide to take the subway to school today, instead of biking, because I have too much to do to lose those 40 minutes.

7:00-7:07 – Walk to subway. Catch up on Twitter while walking.

7:07-7:40 – Subway to work. My train still isn’t running to my job because of Sandy, so I have to transfer. While I’m riding, I grade math labs. (Despite grading for several hours over the weekend, I didn’t finish.) I don’t finish by the time we arrive.

7:40-7:50 – Walk to work plus breakfast.

7:50-8:55 – Enter my classroom to discover 1) it’s a sauna, and 2) that the wi-fi is down in my classroom (and only my room). This is awesome, because I have a computer based lesson today. Also, the person in charge of the laptop cart doesn’t get in until later. Luckily, I am technologically proficient, so I spent this time creating an ad-hoc network and setting it up on the ancient Dell laptops (after tracking down the AP to unlock the tech room) so the students could get and give files. I also spent some of this time inputting the grading I did into the gradebook.

8:55-9:46 – Start of contracted time. Embassy class, which is our special version of advisory. We finally have a curriculum to follow, so I need to modify for my students.

9:46-11:30 – Math class, students brought in survey results to analyze. So I pass out laptops and walk them through the analysis excel file I made yesterday. Minor tech problems, so most of my time is spent fixing those and running around teaching the quirks of excel while the students do data entry and create conclusions. I explain the requirements for their project, stop a student from hacking into one of the computers, and general maintenance. We also have problems getting the files back to me, because they don’t follow directions.

11:30-11:50 – Putting away the laptops and making sure all files are saved.

11:50-12:20 – Run and get lunch, while planning the next lesson with my co-teacher. When we get back, we meet with a third teacher about two students who need resource room (since I’m the programmer, and can change it.)

12:20-1:20 – Student lunch period, so some kids come up to my classroom to work on their projects. I continue setting up laptops (since my afternoon class is larger). At 12:45, the wi-fi returns, so I switch the computers back to that instead of the ad-hoc network.

1:20-2:00 – First opportunity to use the bathroom. I run to the programming office to change a schedule. Then I go back to grading, or, more accurately, data entry.

2:05-3:45 – Another math class, this one with 4 languages spoken and no ESL support. The tech problems seem even worse at first, but balance out in the end. Unlike my morning class, which is very industrious, several pairs in this class did not come prepared and needed to do alternate work/catch up work. End of contracted time.

3:45-3:55 – I bring some of my students to the Teacher Work Room to make copies of their surveys for them, so they can catch up on their project. I get a cookie from a co-worker.

3:55-4:15 – Break and decompress, including short chats with coworkers in the hall.

4:15-4:45 – Back on the grading grind.

4:45-5:30 – I plan with my co-teacher on Thursday’s lesson, which we won’t have time to do tomorrow because of other meetings. (I’ll have a Math for America meeting in the evening.) We adjusted my Lying with Statistics Stations because they were confusing and ill-timed last year, opting this year for a looser flow. I finish grading while I do this.

5:30-5:35 – I write an e-mail to a parent because her son is way behind on the project.

5:45-6:20 – Time to head home. I grab a hot dog on the way. True to my pledge to not bring work home (even though I broke it over the weekend), I play my 3DS on the subway ride home. I almost fall asleep on the train, and my game freezes, losing all progress.

6:20-6:35 – I stop at the supermarket on the way home, to get some stuff for dinner and breakfast.

6:45-7 – I forgot the shallot. So I change my plans, because I don’t want to go back out. It’s a tough decision, I seriously thought about it for 5 minutes because I wanted the shallot but was so tired. I make more bachelor-y food.

7-7:45 – Watch Daily Show/Colbert while catching up on tweets/blogs.

8-9 – Leisure Time

9-9:20 – I try to go into the Global Math Department meeting about homework, but the audio is too messed up, so I duck out early. Now I’m going to lay down and read the news/play with my DS probably until around 11, when I’ll hit the sack.

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schooling

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How Many Representatives Should We Have

Back in 1788, James Madison wrote up 20 proposed articles to amend to the constitution. 12 of those were approved by Congress. The latter 10 were ratified by the states and became the Bill of Rights. The second was ratified over 200 years later and became the 27th Amendment. But Article the First was never ratified. Here’s what it said (corrected):

After the first enumeration required by the first article of the Constitution, there shall be one Representative for every thirty thousand, until the number shall amount to one hundred, after which the proportion shall be so regulated by Congress, that there shall be not less than one hundred Representatives, nor less than one Representative for every forty thousand persons, until the number of Representatives shall amount to two hundred; after which the proportion shall be so regulated by Congress, that there shall not be less than two hundred Representatives, nor less than one Representative for every fifty thousand persons.

Back in 1911, Congress froze its size at 435 members of the House of Representatives, and so the amount of people representative by each representative has grown extraordinarily. (Note that this is before we even had all the states (only 46), so the Reps continued to spread thinner.) The average district size now is about 700,000 people, which is a lot of people and opinions to accurately represent. Of course, if we followed Article the First to the letter, we would now have about 6300 representatives, which seems like a lot.

Source: thirty-thousand.org

But what if the article is a formula, not meant to stop at districts of 50000? The way it is written, it seems like every 100 Representatives would prompt an increase in the size cap of districts by 10000. So how could we model that to determine how many reps we need?

Well, in general, the population divided by the number of people in the average district should give us the number of reps. So if P = U.S. Population, R = number of representatives, and D = max size of district, then R=\frac{P}{D}.

To represent Article the First, since 0-100 reps have 30000 each, 100-200 have 40000 each, 200-300 have 50000, etc, it seems like we could say R=\frac{D-20000}{100} to give a rough estimate. (Anyone have anything more precise?) So we can substitute, as well as plugging in 308,745,000 for P (according to the 2010 census), to get

\frac{D-20000}{100}=\frac{308745000}{D}, and solving for D gets us approximately 186000 people per district. Plug in for D to get 1660 representatives. (Exact amount varies by the precise district make-up.) That seems quite possible, not even four times as many as we have now.

Follow-up questions to consider:

  • Is 700000 people too many to represent? Is 190000? What would be an ideal amount?
  • How would representing 30000 people in 1790 be different from representing that many people now? How does technology change how effectively we can represent people?
  • How could we accommodate having 1700 representatives? What changes would need to be made?
  • What other representative systems could you come up with? How would it work?
  • How would having more representatives change our current representation?
  • How are representatives apportioned in other countries? What methods do they use for determining the size?

For that last one, I think it’s interesting to just look at the Congressional districts of New York City as an example.

I live in District 12. It’s easy to see that the district is half in Manhattan, in the affluent Upper East Side, and half in Queens, in Astoria/Sunnyside/Long Island City. I think it would be very easy to believe that the desires of the people on the UES don’t always line up with the desires of the Queens constituents. Yet we are represented by just one person. However, with a smaller district, they can be divvied up more logically. All of Astoria has 166,000 people, which is almost a full district, and it would be nice to have a district that is clearly where you live.

Since US History doesn’t usually line up with Algebra, this idea might be hard to implement in math. Though it could work fine in Algebra 2. And it might work even better as a history lesson with a bit of math, instead of a math lesson with a bit of history? I dunno. But I think it can definitely be food for thought for any class.

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history

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Is Algebra 2 Necessary?

So, of course, Andrew Hacker’s article “Is Algebra Necessary?” had caused quite the stir, and the obvious answer to that question was “Yes, algebra is necessary.” But the article makes you think if all of what we learn of algebra is necessary. And I think it isn’t, but that comes from thinking about what high school is for.

Do we expect that, when a student gets to college, they can skip the lower levels of Biology because they took bio in high school? No, of course not. (Excepting AP courses, of course.) So what is our goal for learning biology in high school? It’s to provide a general foundation of the subject, that most people should know, and it prepares you for a college level course or major in Biology.

Really, all of what we learn in high school is designed to broaden our horizons, to provide experiences and content we wouldn’t see otherwise, and to provide a baseline of knowledge that we feel everyone should have.

I remember reading from someone, though I don’t recall who, that they had struggled through Algebra 2 and Pre-Calculus, slogging along, and then when they got to Calculus a light turned on. “This was why we’ve been learning everything we’ve done in the past two years! It was all for this!” Even the wikipedia page on Pre-Calc says “…precalculus does not involve calculus, but explores topics that will be applied in calculus.” It’s putting the work before the motivating problem, again.

But now thinking about the normal course sequence for a student that is not advanced: Algebra –> Geometry –> Algebra 2 –> Pre-Calculus –> Graduated from High School, so no Calc! So these students will have two whole years of math without the payoff that shows why we do it.

And as teachers we know that you need to start with the motivating factor, not have it at the end. So why don’t we have calculus first, before those two? If we consider our goal in high school is to spread ideas people might not see otherwise, I think Calculus has a lot of important ideas people should see that would improve their lives. Optimization? The very idea of it can improve how you look at all the problems in your life. Related rates, limits, the idea of changing rates and local rates, the relationships between functions, these are all good ideas to be familiar with.

Can the students learn these things without having done Algebra 2/Pre-Calc? I think so. As Bowman Dickson says, “The hardest part of calculus is algebra.” So what if we taught it in a way that didn’t rely on that? We can get the ideas across without jumping into the nitty-gritty of a lot of it. Save that for AP level classes, or for college calc. What you take in college is more in depth that high school, so it should be the same here.

Now, there would certainly be some stuff from Algebra 2/Pre-Calc that we really need first. But why not have those in Algebra 1? I accidentally taught several things from Alg 2 when I taught Alg 1 my first year, because they seemed like natural extensions of what we were doing, and I didn’t know they weren’t required until I started planning for the next year. But also, consider this. If we made Probability & Statistics one of the main courses of the math sequence, I don’t have to teach it in Algebra 1. I spent about 7 weeks on those topics last year. That’s 7 weeks of Alg 2 content I could fold in, without worrying about reviewing old stuff because we just did it.

So then the new math sequence could be Statistics –> Geometry –> Algebra –> Calculus. (And I think that might fit well with the science sequence of Biology –> Earth Science –> Chemistry –> Physics.)
Thoughts?

Category:

math, schooling

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Math Needs to Be the Spark

At Twitter Math Camp I gave the following talk. The abstract from the program said:

When planning interdisciplinary projects, math teachers need to take the lead in order to create cohesive and authentic projects, and to ensure that the project doesn’t just become psuedocontext for their math goals. Uses two major interdisciplinary projects developed at my school as examples of how to bring all the subjects together, so math isn’t left out in the cold.

Here’s the talk:

Math Needs to Be the Spark from James Cleveland on Vimeo.

After that I opened to questions. The one that I remember was asked by @JamiDanielle: “How can you get other teachers who might not be on board for these types of projects to join in?” And I think this process is actually how. If you go to a teacher with an idea and just dump on them to figure out how to connect it to their class, it’s not going to end well. It’s easier and less work to just not take part. But if you go to them with an idea already half-formed of how they can implement it, it is much easier to build off of that idea and will make teachers more willing to work together.

The Projects

High Line Field Guide v5 – This is the High Line field guide project mentioned in the video, and first mentioned in this blog post, “The Start of the New Year.”

Intersession Project Requirements – It would be difficult to post everything we did in the Intersession project, but the overview from the video and this packet of requirements for the product should be useful. Anyone interested in more can ask.

Category:

biology, conference, ela, history, math, TMC

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No Right Answer

A bit ago I got yelled at by a commenter on Kate’s blog who claimed that being always right is why we like math. The problem with that point of view is that, while yes, you can always be right while doing computation, math isn’t just computation. So the other day I was talking with a friend of mine, and that prompted me to post the following tweets:

My friend Phil (@albrecht_letao) responded to the question, and he came up with an answer of $20/hr. When I worked it out with my friend, we came up with $14.25. Does that mean one of us is wrong, since we got different numbers?

No, of course not. What happened is we approached the problems in different ways. Phil only calculated the monetary value: with his amount, my friend would earn the same amount of money she does now. He figured this was an important way to look at it, for paying bills and whatnot. Our calculation came from thinking about how her time is being compensated. Since those 16 hours are being wasted (she has to work them for free; actually, she pays to lose that time), we calculated her “real” hourly rate and used that.

There can be more answers than even these two, depending on what you think is important. But it’s a clear example of a problem, solved using math, with no one right answer. That’s what math is about. I tweeted it thinking maybe it could be a problem worth considering in class, to show that essential idea to students.

What do you think?

P.S. The right answer, of course, came from @calcdave:

 

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anywhere, math

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