The Oasis of Troy

This past week I spent some time in Philadelphia for the EnCoMPASS Project (more on that later). During the last day, Fawn and I were walking downtown and stopped at this fountain near city hall.

There we met a man named Troy who was selling bottled water. He sat comfortably in the shade of a tree and called out to people as they walked by, "Excuse me, would you like some water?" Although it was a warm day and plenty of people were in the park, it didn't seem to me that he was selling much. This probably had more to do with the fact that people don't like to be bothered by random vendors on the street. I'm one of those people, so I could certainly sympathize with them. Or maybe they just didn't want water.

Eventually a couple walked by, and as per usual, Troy asked if they would like some water. The man said "no". Troy followed up with, "perhaps your lady is thirsty?", and without hesitation, the man again said "no". Troy turned to us and said, "he didn't even ask her!" Fawn and I lost it and Troy joked with us about how the girl was likely to start an argument with her boyfriend for not offering her some water.

In order for Troy to increase sales, I made a suggestion. Troy needed to market his product better. People had to see his corner of the park as a refuge from the hot sun...an oasis. The Oasis of Troy!

As the next customer walked by, Troy, who had not sold any bottles for quite some time said, "Welcome to my oasis! Would you like some water?" And sure enough, the man stopped, seemed to think about it for a second, and said, "Yes."

Later, it came up that Fawn and I are math teachers, and Troy became very excited. Aside from selling water, he also sells credit card processors to businesses. He explained that his main competition in credit card processing is something called The Square.

The Square charges 2.75% per credit card swipe with no additional fees. Troy charges 1.09% per swipe with a $10 monthly charge. He explained that he wasn't having much success and that many businesses turned him away, thinking that the monthly fee was too high. But he was convinced that his processor was cheaper. He just didn't know how to express it. There was probably a formula to show this, but he didn't know what it was.

After some discussion with Fawn, we came up with the following formulas:

s = 0.0275x
t = 0.0109x + 10

x is the monthly revenue, s represents the fees charged by The Square, and t represents the fees charged by Troy.

After setting The Square's charges equal to Troy's, we found that the break-even point was about $600. This seemed to be good news for Troy, because if a business takes in more than $600 in revenue, Troy's processor is cheaper. And sure, he looks worse for a revenue less than $600, but any business that takes in less than $600 per month will certainly go out of business.

I asked Troy, what do I say to students who ask, "when would I ever use this?" He said to tell them, "No, you are never going to use this, but knowing it will allow you to do the things that you want to do." Well said.

Update:

For those of you who know Fawn, you know she solves just about every math problem with rectangles. It's uncanny. Here's her solution:

Torturing Kids with Toothpicks - Part 4

If you've been following my blog, you probably know about my adventures with toothpicks. Basically I placed a bunch of them on my classroom floor for a math problem. I only realized this morning that I never showed the final product on my blog...just on my wikispace. You can find all of that work here:

http://mrkraft.wikispaces.com/Triangle+of+Toothpicks

While I received a little bit of feedback from the twitter crowd, I'd appreciate more. Any ideas on presentation? How would you use this in your classroom? What modifications would you make? Any kind of extension ideas?

Thanks!


Incidentally, a couple of weeks ago I made a change to my Three Act page. It wasn't visually appealing (all text) and I thought I could do better. Click here for the new page.

Torturing Kids with Toothpicks - Part 3

Here is the first act of my toothpick lesson. Feedback is appreciated.



Here is part 1 and part 2 of this series.

Sometimes I Dance

This has nothing to do with math, but sometimes I dance. (I'm the one wearing the vest.)

Give Me the Wrong Answer

The other day I had the following warm-up on the board:

It takes 8 elephants 8 minutes to drink 8 pints of water. If they drink at the same rate, how many elephants would you need to drink 16 pints of water in 16 minutes?

(What is it with me and elephants?)

This isn't a difficult problem if you take the time to think about it. But as I walked around the room, I saw quite a few students had it wrong. Some students were thinking proportions and assumed that the number of elephants would also double. Some students noticed that 16 was evenly divisible by 8 and wrote 2 elephants. (As if I would give them something that easy.) And some had the correct answer of 8.

So after walking around, I asked, "How many elephants?" And not surprisingly, not many hands went up. Even the students who had 8 seemed to be a bit reserved.

For me, it's a little disappointing because I've had these kids all year and I would hope that at this point I could get a little more participation out of them. Obviously some students are still not very confident and don't like putting themselves out there in front of their classmates.

When this happens, I try a different approach. After seeing that only a few hands were raised, I said, "Wait. I didn't mean to ask that. Let me try again. What do you think is a common wrong answer?"

Then, a beautiful thing happened. About half the class had their hands raised. Many of these students did not know the right answer, but at some point during the process, they figured out what wasn't right. These students now felt like they had something worthwhile to contribute and were more comfortable in becoming a part of the discussion.

A student gave one of the incorrect answers I was looking for. I asked, "Why is it wrong?" She said something about rates not being the same (1 pint per 8 minutes versus 1 pint per 16 minutes). Using that, I follow up with, "Many of you thought that it would only take 2 elephants. Can someone explain why that doesn't work either?"

More hands went up.

Finally, I asked my original question, "So how many elephants?" At this point, many more students were ready to share. The students that had it wrong fixed their mistakes. The students that were right in the first place were more confident in their answers and willing to share.

On a related note: I always hate those pictures in educational magazines that show a smiling teacher in the front of a classroom, filled with students, all of whom have their hands raised high in the air. Why do I get the feeling that these pictures are all staged with the photographer telling the kids, "OK everybody! Pretend to be really excited about class today and put your hands high in the air!" I find this photo to be especially irritating because A) there's no way those kids are old enough to be learning about quadratic equations and B) no teacher would let their students have bottles of water on their desks. Someone would be asking to go the bathroom every five minutes!

I Shall Never Play a Review Game Again!

Update 8/3/15: I've decided that you probably shouldn't use this game in class and I blogged about that here. I could just delete this post, but I figure there's always the chance that someone will think of a way to improve it.

This is the second part of my ramblings about exponents. You should probably go read the first part before you read this!

I don't think there was ever a moment in my life where I said, "Hey! That review game worked really well today. It really helped to prepare my students for that big test tomorrow. Everyone was fully engaged and asking questions about the problems they didn't understand. They had so much fun and I was really impressed by their good sportsmanship! I can't wait to do it again."

Here's why it never worked for me.... The kids who struggle don't want others to notice, so they make some excuse about how they don't want to play the game. The game is "stupid". So I have to give them a worksheet to do instead. Then somebody notices somebody cheating and demands that points be taken away. Then I have to explain that I didn't catch them, but I will certainly be watching to see if anyone is cheating from now on. Then someone will get caught, and I take points away from that team, and then that team gets mad, and then the other team says something nasty to them, and then they say something nasty back, and then I explain that it's just a game, but that doesn't help, because after a few questions they're arguing again, and then I threaten to give them all boring worksheets to do, and so on and so forth.

I don't know. Maybe I'm doing something wrong, but my kids can't seem to handle competition.

So after one horrible day, I declared, "I shall never play another review game again!" That sounds overly dramatic, but that's how it went down.

So today, after seven months of not allowing any games, I decided to play a game.

It was very spontaneous. Someone asked if we could "do something fun", which annoys me to no end, but after practicing solving systems for the last couple of days, I agreed that something fun was in order. I also ran out of Oreos from a task we did the other day, so bribing them to do work was out of the question.

I remembered reading a post a few months back by Kara Wilkins called Grudgeball. It was some sort of game where everybody gets so many X's and you want to be the last student/group with an X. Kids take turns answering questions, and if the question is right, they toss a basketball into a hoop, and if they make a shot, they can erase one of their opponent's X's. I basically did the same thing, minus the basketball. (I decided to name it "Grudge"...but there has to be a better name for it.)

I wrote everyone's names on the Smartboard and everybody got three X's. It looked something like this:

(I know what you're thinking...does he really have kids named Fawn? Hedge? Timon? Bowman??? No. They're completely made up. No one would actually name their kids that.)

I then explained the rules of the game to my students. An expression with exponents would be written on the board. Everyone would write an answer on their little whiteboards, and then anyone with a right answer would erase one X from the board. The last student with an X under their name wins.

First problem:

This isn't tough. In fact, they just learned how to expand these expressions and rewrite them yesterday. So everybody got it right. Awesome. And everybody got to remove an X. Our board looked like this:

Monya, Kishi and Andrew took Fawn out pretty quickly. Fawn took a little bit of revenge on Andrew, but it didn't do much good.

I gave the students another problem, but this time it was something a little different.

Several of my students point out that they've never seen a problem like this before. With the utmost seriousness, I said, "No. You've never seen this. But you're smart enough to figure it out. Just think about what's going on here." It was just one of those moments where you tell the kids that they're on their own and they need to believe in themselves. Maybe a little corny, but it works.

A little over half of the class got the right answer. We talked about what people did to simplify the expression, and for the most part, everyone was engaged because they wanted to make sure they were good at playing the game. And if that's the carrot that I have to dangle to get my students to think algebraically with 11 days left in the school year, so be it.

Note on game play: Although Fawn, Hedge, and Nathan were out of the game, they could play a zombie role and "attack" other players by erasing their X's. This motivated them to keep playing, even though they could no longer win.

Another problem was given that was similar to the last one. That time just about everybody got it. Then we tried dividing powers. Only two or three students figured it out. Then we did another, then half the class figured it out. You see where I'm going with this?

I kept introducing something new. Some students got it right away. For some, it took a couple tries. But eventually, everybody learned how to do it.

The game was finished when all of the X's were gone except for one. Usually the kid that wins is the one you least expect.

If you remember, I said I'd never play a review game again. And technically, I didn't. This wasn't a review. This was mostly new material. I wasn't using a game to trick my students into practicing something one last time before a test. I was using a game to encourage them to learn something new.

Caution: This game seems to work differently for different groups of kids. If there's a lot of animosity in the room, it's probably not a good idea.

Exponents Rules are Lame.

I originally wrote one long post about teaching exponents and incorporating a game. It got to be so long that I split it up into two posts. So bear with me.

Admit it. For most students, exponent rules can seem really lame. It's easy to confuse them and there's not much real-world application for these things. Textbooks try really hard to make it sound like there is...but they're a bunch of liars. The only applications I could find in my textbook (that weren't related to scientific notation) looked this like:

I don't think this is a bad problem. In fact, I like that it makes you think about area. But this isn't a real-world application because you'd never actually need to find the length of a rectangle like this. (Trust me. I was an engineer for six years. I never had to do anything like this.)

And kids pick up on that lameness. This is one of those lessons, up there with factoring polynomials and simplifying radicals, where students will at some point ask, "When will I ever use this?" This question usually comes up when a particular student is getting frustrated with the lesson and suddenly says to himself, "I don't get this. This is stupid and not worth my time. I'll probably never use this outside of school anyway."

And then we, the teachers, come up with a myriad of excuses: you need it for Algebra 2, you'll need it for the test, you'll need it when you're doing sciencey-kinds of things some day, you'll need it when you're flipping burgers at a rate of 2x^6/x^6 per minute, etc.  None of these answers are satisfying to the students.

On top of all of this lameness, we make things worse by explicitly teaching students the rules for exponents. We don't give them the chance to figure them out on their own. I think there is a huge danger in just teaching students rules (or procedures or tricks). If you don't let students develop the meaning behind those rules and procedures, they are subject to error...like when a student tries to multiply two fractions, but for some reason he or she cross-multiplies. Or when they get really good with rise over run, and when you ask them to plot (4, 3), they move 4 up from the origin, and then over 3. Without meaning, rules and procedures become confused and are applied in the wrong situations.

Now, I always thought that I did a good job with teaching exponents. I always explained everything. I showed them how to expand expressions and put them back together.

The logic was there. The kids who were carefully listening "got it". But there were a lot of kids who only got the rule...just add the exponents.

As time went on, and I explained more and more of these rules, students became confused. They didn't really understand what was going on. They didn't know when to add or multiply exponents. They forgot that powers had to have the same base to combine them. They started to think that 2 times 3 was 5!

Here's a nice little graphic that shows how I feel when I teach exponents. You can see what happens as more and more rules get piled on. Students get confused and I'm left wondering, what happened? It was all going so well.

Some teachers are using a different approach. They're allowing students to discover the rules on their own. Just read these sweet posts by Andrew Stadel and Timon Piccini. (It's worth mentioning that both of these posts were inspired by Michael Pershan, who is doing a lot of awesome stuff himself.) I believe that exponents, like many things we teach, are a lot easier to understand if we de-emphasize the rules. And when students understand, they are less likely to ask, "When are we going to use this?", and they might actually enjoy mastering this concept.

Many times, we think we have to show our students every little step before they can do it on their own. Over the last few years, I've discovered that this really isn't the case. Don't believe me? Next time you teach something new, ask yourself if any of your students can figure it out without you explaining it to them. If you believe that at least one of your students can solve it without your help, let them try it. Just shut up, and let them work. Let them come to terms with what you're asking them to do. Let them wrestle with it a bit. Even if they don't get it, at least they're invested in the problem and they have some motivation to learn about its solution.

When you allow students to find their own solutions, they're going to appreciate it much more than you telling them how to do it...if only you'd shut up for once.

Here's the sequel to this post...."I Shall Never Play a Review Game Again!"