I have the opportunity to be a long term substitute for eighth grade in my local middle school. Part of the curriculum calls for "academic enrichment," a 40 minute period to study something fun (but still educational). I want to spend my time doing math that isn't part of the curriculum and isn't something the students are likely to see in high school.
Yesterday I introduced the students to the Euler Characteristic of a plane by having them draw random graphs and counting the number of vertices, edges and areas for each. After about 30 minutes, half of the class discovered that V-E+F = 1 for all graphs.
Next week I plan on having the students build the Platonic Solids out of drinking straws and yarn. I also hope to work with some number sequences.
Any ideas that I might add to my curriculum? The idea is to get the students actively engaged in doing mathematics, rather than just learning mathematics.
There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.
i just did something with my class they thought was interesting,
the monty hall problem...
It touches on theoretical/experimental probability, outcomes, events, and can provide a very valuable lesson on assuming things based on what you see.
I started off telling a story about Marilyn Vos Savant and how the monty hall problem came to be in her life. The fact is that she answered the question correctly yet so many mathematicians, doctors, and professors wrote her and told her to correct her mistake because they thought she was wrong!
I tell the kids that they are going to solve the problem today and be able to understand something that all of those doctors, mathematicians, and professors could not!
The problem is like this:
The teacher is the host, a student comes up to play the game.
There are 3 doors in front of the classroom. Behind 2 of the doors are goats, and behind one of the doors is a car. The student picks one of the doors. (I used playing cards, where (2) 2's were the goats and an A was the car)
The host then opens one of the doors that has a goat behind it since he knows what is behind all of the doors.(they picked one card, i looked at the other two and showed a 2)
Then the host gives the student a choice: stick with the same door you orginally picked, or switch to the other one that hasn't been opened.
The question here is: Should you stick, or should you switch, or does it not matter?
Before reading any farther, if you have not heard this problem before, think of how you would answer this question. Everyone who I have talked with this about has thought the same thing initially.... See if you do too!
The same thing that everyone thinks is: it doesn't matter if you stick with your original choice or switch to the other card since there are only two cards left and it is a 50 50 chance you win. Most people will stick with their card when given the choice to change(this might be an interesting psychological question!)
The fact is,
this is the wrong choice, since when you change you choice you have a 2/3 chance of winning the car!
At this point, I go around the room and have everyone play the game and we tally the results based on our classroom playing the game( I try to have 20-30 trials).
A great way to do this is online with a monty hall simulator:
but you can do it the long way too. Just keep track of wins and losses for keeping your choice and changing your choice. (good for block scheduling)
On the above website you can also run 1000's of simulations with keeping choice and changing chioce as well.
This would be considered the experimental probability of this situation. You can find it to be 67% chance of winning when you change your choice by running a couple thousand simulations, same for keeping your choice comes out at 33%. The law of large numbers can also be explained here, since in the beginning of the experiment when the number of trials was low, the percentages of wins and losses were not what they theoretically should be.
The mistake most people make when assuming that it is just a 50 50 chance at the end is they forget that there was 3 original choices you could have made. If you keep your choice, then you will only have a 1/3 chance of winning, because you are sticking with your original 1 of the 3 cases all the way. When you change it becomes different.
Think about it, each time the host has to show a goat. You have a 2 out of 3 chance of picking one of those goats in the beginning. So if you pick one of those goats initially, and the host shows you the other one, then when you change you will win the car. The only time you lose when changing is when you pick the car initially, which is a 1 out of 3 chance. So you win 2 out of 3 when changing and you win 1 out of 3 when keeping your original choice.
I have them fill out a diagram like this to do the theoretical probability of this situation.
The three original choices you could make are:
Car Goat 1 Goat 2
Stick Change Stick Change Stick Change
Win Car Win Goat Win Goat Win Car Win Goat Win Car
There are 6 possible outcomes, 3 you stick, 3 you change. Of the 3 when you stick you win a car 1/3 times. Of the 3 when you chage, you win a car 2/3 times.
All in all, it is the better choice to change since you have a 2/3 chance of winning when you do so. You can talk for a while about these types of things all while having them engaged.
Its called the monty hall problem because of its similarity to the lets make a deal game show.
I like the Doomsday Algorithm. In case you don't know it, it allows you to compute the day of the week for any date in history. Understanding it requires a little modular arithmetic, but nothing too complicated. Kids love it - they will go around telling their friends the day that they were born afterwards. A quick internet search will provide all the notes you need.
If you're going to be taking the SAT, check out my book: