Archive for Education

The Math Behind Why The House Always Wins

As a part of the course I took this past semester on teaching math in secondary schools, I (with a small group of my classmates) taught the class a unit on data analysis and probability. I also had to write a few short papers about certain things we covered in our lesson. What follows will be a merged summary of two of my papers.

Before I begin I’ll provide some vocabulary for those of you that never learned these terms or learned them and have since forgotten them.

Expected Gain: How much money we should expect to gain if we continue to gamble on one game for a long time.

Fair Game: A game in which the expected value is equal to zero. In other words if we played a fair game for a long time we should break even.

Unfair Game: A game in which the expected value is positive (we will profit from the game) or negative (we will lose money by playing the game).

Now, let’s consider a game where you win $4 if you roll a fair six sided die and it lands on “1” but lose $1 if it lands on any other number. Would you be tempted to play this game because winning $4 is more than loosing $1? The way we calculate the expected gain of this game is to multiply the chance of winning (1/6) in one play by the amount we would win ($4) in one play and add that to the the chance of loosing (5/6) in one play multiplied by how much we gain for loosing (-$1). The equation looks like this:

(1/6)($4) + (5/6)(-$1) = -$0.16666….

Our expected gain is equal to about negative 17 cents. Basically it means that if we continue to play this game for a long time, we should expect to lose 17 cents on average. If we won $5 for winning instead of $4 the expected gain would be 0 and this game would be fair.

Now you might be thinking that my example was stupid because no casinos have a lame game like the one I mentioned. If you are, I have two points to make (1) idiots will bet on anything, and (2) they might not have this game but they have others like it. Consider the game of roulette.

There are plenty of casino games I could discuss, but roulette is the simpliest to understand. There are many ways you can bet on the roulette wheel, evens and odds, red and black, a third, a single number, etc. I’ll discuss the expected gain for when we place our bet on evens or odds. The payout is 1:1 (if we bet $1, we win $1 and if we loose we loose the $1 we bet).  Do you think our chance of winning is 50:50?

If you think that half of the wheel is even and half of the wheel is odd you’re wrong because you’ve failed to noticed the two green “0” slots. If you think that more of the numbers are even because in mathematics zero is, in fact, an even number you’re also wrong because it’s not that way in the casinos. Did you notice that our chances of winning are not 50:50, yet our payout is 1:1? Our chances of winning are 17:36. The expected gain in this case is:

(17/36)(1) + (19/36)(-1) = (-323/630) =  -0.05555…

This means that on average we can expect to lose about 5.6% of what we’re betting.

The reason that the house always wins is because *every* game they have is unfair in their favor. The only exception is Blackjack (21), and it’s only an exception if you can count cards. They either rig the odds or payouts of every game so that they have some sort of advantage over you. Even if we win big, it doesn’t hurt the casino because our win is made up for by all of the other player’s losses. So there you have it, the math behind why the house always wins.

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Some Statistics

Time for another math related post because I haven’t done one in a while!

Currently I’ve been really busy working on this huge arse project for my math education class (it counts for a ton of my grade).  I’m in a group with five other people and we have to teach our class a unit on probability, and fill a 5 inch binder with lesson plans and other educational goodness. We already taught our first two lessons and have three to go.

During our first lesson we talked about new ways of visually representing data, aside from xkcd’s awesome maps of the internet (2007) and (2010) we also showed them word clouds. Word clouds are clouds of words, where the most frequently used words are the biggest and less frequently used words are smaller. You can make them at wordle.net . Here are some that I’ve made of my blog and the blogs of two of my friends.

Here are links to their blogs if you’re interested: http://kristinamirasol.net/, http://chubbyriceball.wordpress.com/

Another interesting thing that I recently rediscovered is some of the other things wolfram alpha can do besides act as a super calculator.

If you put in your gender, age and height it will tell you what you should weigh. And what your appropriate lung capacity should be and how much blood should be in you and other interesting tidbits of that source.  (I don’t know where they got 131lbs for a 5’5″ woman though. I had been told that healthy weight was 100 lbs for the first 5 feet and 5 more pounds for each additional inch, give or take 5 pounds, was a healthy weight from my cousin who is a nutritionist. That would make a healthy weight be between 120 and 130 pounds.  I don’t think I’ll ever weigh 130 pounds. I’ll be lucky if I can get back up to 117.)

It will also tell you how many days and weeks you’ve been alive if you put your birthday in it. It’s also a dictionary! And for those of you that didn’t grip the meaning of “super calculator,” it will do integrals for you *and* show you the steps! It will factor huge expressions and expand them among so many other awesome things!

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Educator.com

As a BzzAgent, I’m given the opportunity to try out new products. Most recently, I got to try out the website Educator.com for a month. Educator.com has tutoring videos in a decent range of high school and college level math and science courses. Since I’m finished with all of my undergrad math classes and science requirements, I can’t really use this website to help me study. But since I’m also studying to become a math teacher I thought I would take a look at some of videos for high school level math.

After I typed “logaritms” into the website’s search bar and hit enter, I was directed to links for many videos concerning logarithms. I clicked on the first one, and learned what the format of their videos was like. Click the screen capture to enlarge.

Basically, you see a video of the teacher on the right, see how they work out the problem in the center, and see what sub topics the video is divided into on the left. What I like about this is that you can easily skip to a certain topic in the video by clicking on it on the left part of the screen. If you feel like the teacher went too fast or didn’t catch something, you can pause the screen and let the math sink in, or you can rewind the video and listen to his or her explanation again. Many of the videos also have a nice summary of what was covered underneath them.

I also took a look at some of videos for multivariable calculus. I wish I knew about this website when I was taking multivariable calculus. My professor was horrible and I had difficulty finding tutoring videos. I used Paul’s Online Notes and Cramster for the most part, but I wished I had more. This website would have made my life a lot easier when I was taking that course because it gives you a thorough review of the subject in addition to solved practice problems.

Since a friend of mine is currently taking (and having difficulty in) Physics, I shared the code BUZZFA626 with him so he could also take part in the free trial.  He said that the physics videos were great because the professor is very relaxed and doesn’t go over things too fast. Since the professor is so relaxed, it helped him feel calm and relaxed, which helped him gain more from the online lectures. I also watched a physics video and found that this was true. I think the physics professor could possibly be the best for this reason.

Pros:

They have a decent range of topics. Their search function is very useful. You don’t need to look through anything, just type in the topic you want to learn about. You also have the ability of asking the teachers questions about their videos if you wish. I haven’t taken advantage of that aspect of the website, but I can only assume its a very good thing.

Cons:

Their prices. I don’t think any of them are really targeted at college students. A normal semester is about 4 months long, and from what I’ve seen, an “average” student wont start looking for online resources until about a month into the class when they realize they need to learn whats going on before midterms. If they had a three month plan that cost around $60 I could more easily imagine students going for this. Also, the only programming language they have that I’ve seen offered in colleges is Java. The website would have more appeal if they also offered Pearl, Python and C++ or C#

Overall:

In the end, I would say that Educator.com is a good resource for a student that needs a through review of a subject without spending hours pouring over small details. The professors are good and are able to give you a general idea and some examples in less than the time of an actual lecture. However, because I’m not too crazy about their prices, I would be more likely to buy a month subscription the month before finals to help me review, rather than pay for six months and only use the website for four months at most.

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Special Ed. Math

Yesterday I got a chance to observe a special education math class. It was a small class with only 12 students and it wasn’t what I expected. None of the kids had anything like downs syndrome or severe autism, I think they all just had various learning disabilities. I got the chance to go over two problems with the class and I really enjoyed it. The first problem was a unit conversion and the second one was a word problem about an inequality.

I saw that the way he taught the class was very similar to how I’ve helped my sister with math. After I would do one step of the problem, the teacher asked every student individually if they understood what just happened and if they had any questions. When I asked the class if they had ideas for what the next step in the problem would be, the teacher would relate it to other things they knew or concepts that were easier for them to understand. That’s exactly how I’d help my sister. I’d go over the problem with her, ask her if she understood every step (because they don’t always realize they don’t get it, or they don’t speak up about not understanding, you *must* ask them), and many times, break down a more complex problem into an easier one.

I could see a bunch of the kids getting frustrated when they didn’t know what was going on, but they also light up more than mainstream kids when they finally understand something. I also found out that there’s a unique certification for special education math. I think that might be my calling in life. Teaching math is the most fun when the kid is either very smart, or struggling. When the kid is smart you can challenge what they know with more advanced concepts and when the kid is struggling you have to figure out multiple ways of explaining one concept. But teachers for honors classes aren’t in demand. Special ed. teacher’s are. Now all I need to do is find out what the certification requirements are 😀

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Spring!

7:21 pm has past and it’s officially spring! Luckily this year the equinox occurred at a time I’m normally awake at, so I stood up some eggs!

My mom always told us you could do this during the spring equinox because something happens with gravity. For some reason this was the year I finally decided to research the topic. Every website I cam across said that this phenomenon is a myth and that if you are patient enough you can stand up eggs any day of the year. It didn’t take much effort to stand up these eggs. And I remember one year when the equinox was early in the morning I stood some up with my sister outside. It was very easy for a few minutes to stand most of the eggs up, but within a few minutes we couldn’t stand up the eggs anymore and a few fell over. So I guess this will forever remain a mystery, but at least it’s a fun mystery 😀

In other news, here’s an awesome interactive diagram I made to demonstrate Bhaskara’s First Proof of the Pythagorean Theorem.

note the squares of the sides

lift the flaps to see that the square of the hypotenuse can be rearranged into the sum of the squares of the legs!

It’s for the education class I’m taking that’s focused on incorporating ESL teaching methods into mainstream subject classes. Every so often I’ll learn something interesting, but for the most part it’s really boring.

Happy Spring!!!!!!!!

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ESL Math

So this past Friday I had to observe more classes so I’ll have 25 hours by the end of the semester. I finally got to observe an ESL class, and it was not what I expected at all.

First I saw a class that was supposed to be mixed ESL and IEP kids (kids with learning disabilities), but it was just the IEP kids because the ESL kids were at some activity thing. The IEP kids were way more motivated to learn about stuff than the mainstream kids I had observed in weeks prior. When they found out I was a college student they were asking me what college was like, if I dormed and how long they would need to go to college to do various types of professions. They had TONS of questions about everything. They also wanted to know if a tsunami could happen here and they were hanging on my every word when I was explaining how tsunamis happen and why tsunamis and earthquakes can happen together. Then I started talking to them about Pangaea and how India is a sub continent and they thought it was cool. They said I should teach social studies.. then I told them this was Earth Science, not social studies and that I learned almost none of what I told them in school. I learned most of it from watching Discovery Channel, Science Channel and Thirteen.

Now I’m left wondering why the IEP kids are more motivated to learn than the mainstream kids… I honestly have no idea.

The next class I got to see was an ESL math class. Most of the kids were from South American and Arab/Middle Eastern countries.  These kids were also very motivated to learn. 95% of the class would raise their hands when the teacher asked a question. And many of them were very polite and respectful to me. When the teacher gave them work to do, they all did it and worked together. Many of them worked with other kids that spoke their language, but there were children that spoke different languages working together in English.

When I went around helping the kids I could easily see which ones were new and which ones had been here for a while. All of the lessons are in English, but the teacher makes sure to always give them vocabulary, visual examples and structured ways of doing things so it’s easier for them. In fact a lot of what she did would make learning for *everyone* easier in a mainstream classroom. I have no idea how anyone could teach math without first giving the class vocabulary. I think if more students knew the definition of “variable” and if that definition was stressed more often, they’d more easily understand basic algebraic concepts.

Also, since it was a seventh grade class many of them still looked very young. And their accents were adorable!

Next week I’m observing a math class that’s a mix of IEP and mainstream kids.

 

In other news I filled out my student teaching forms. I am so excited for next year!!!!!

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The Concept of Integer Addition

7th graders don’t get it. It’s pretty sad. Now I did not say addition and subtraction because they are the same thing. Subtraction is just the addition of a negative.

Anyway, onto a recap of the classes I observed this Friday. The teacher who’s classes I’m observing had to go to a workshop, so I got to see how the substitute (a retired math teacher) handled things. The normal teacher left his classes a few review sheets that involved addition and subtraction of positive and negative numbers… also known as integers. The sub went over how to do addition and subtraction and went over the problems in sets of 5-10. The first sheet was on addition, the second on subtraction. This is what they were told to do for addition:

If the signs of the numbers are the same, add the numbers and keep the signs

If the signs of the numbers are different, subtract and keep the sign of the larger number

This is what you’re supposed to do, none of the 8th graders had problems but some of the 7th graders were confused. I tried to relate negative numbers to the concept of owing someone money when helping out a few students, but they totally didn’t get the concept of owing money. It wasn’t until later that I realized they were too young to really get that when people lend you money you need to pay them back, and that I should have explained this concept to them with the use of a number line.

Instead of explaining -5 +3 by saying, if you owe someone five dollars, and then you get three dollars and give that back to them, how much do you still owe them, I should have drawn a number line. I would have said that first we move 5 away from 0 in the negative direction, then we move 3 forward. Or I could have said we dig a 5ft deep hole, then we fill it in 3ft, how deep is it now? The concept of owing money isn’t easy to visualize, but the concept of digging a hole is. Why couldn’t I have thought of that sooner.

I ended up caving in and just teaching them how to use the rules properly. Then when it came to the subtraction sheet they were told something we’ve all heard:

Keep, change, change

Meaning if we have, -5 – -2 it can be rewritten as -5 + 2, or if we have 4 – 6 it can be rewritten as 4 + – 6. Basically they’re changing it into addition, but none of the kids can see that it’s the same thing. I also would have been able to similarly explained how this worked using a number line or the idea of digging a hole, but I came up with the idea too late. I will remember to use that idea when I student teach/teach though. Once again I just caved in to teaching them how to use the rules.

It’s pretty sad that they couldn’t see that -5 + 10 and 10 – 5 were the same thing. I was happy that I got one girl to see it. It pratically blew her mind.

I think the solution to this problem would be elementary school teachers teaching addition with a number line and 3d cubes, instead of just 3d cubes. You can’t have negative 3d objects…. they should also be teaching negative numbers in elementary school if they aren’t. If you just give kids rules you’re only giving them shit to memorize, not teaching them how to understand anything. If you give a kid an understanding of a concept, they’ll always know how to do the problem.

In other news I saw two boys get into a fight! The sub told me in this school the kids are pretty good in that the class will break up the fight rather than join in on the fight like they do in crap schools. She also told me if you can break up two boys they’ll chill out, but you can never turn your back on two girls fighting or you’ll end up with a chair in your back. That was some pretty useful advice that makes a lot of sense.

My apologies for those of you who didn’t understand this post… though if you don’t it’s pretty sad. Assuming you’ve graduated high school. I’d be more than happy to explain these concepts to you in a variety of ways if you’re confused!!!

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