Posts
9 years ago
Friday, October 31, 2008
Thursday, October 30, 2008
The Snickers Challenge
Some of my fondest memories of middle and high school math were the puzzles. In between memorizing algorithms and vocabulary in class, there was an occasional puzzle thrown in which I invariably found more interesting than whatever I was supposed to be learning on the lesson plan.
This is still true today; an example that comes to mind is last Thursday night (early Friday morning, really), as I lay in bed after preparing my little talk for class on Friday. I couldn't help mulling over some of the sequences of numbers I had prepared for class (this is a recurring theme for me in algebra class; I come in with 5 numbers that correspond to a pattern, and I ask the students to try to deduce the sixth). This particular day, I had the first five numbers of several sequences: the triangle numbers (1,3,6,10,15,...), the perfect squares, the cubes, the primes, and a special sequence called the "highly composite numbers" (look it up!).
I always like to try to think of different patterns the kids might see in these sequences, because there's often more than one way to get to the right answer, and I try to avoid calling any answer wrong, instead just pointing out where (usually) their perceived pattern breaks down. In this case, I knew that the triangle numbers can also be seen as the recursive sequence a_n=a_(n-1)+n, or in simpler terms you add 2, then 3, then 4, etc, running up through the positive integers. Similarly, for the squares you successively add odd integers +3, +5, +7, etc. I wondered if there was a pattern for the cubes? Well, it was 2 am but I still spent a few minutes pondering in that half-awake state trying to see what the pattern was, and I realized that
1^3 + 3 + 4 = 2^3
2^3 + 9 + 10 = 3^3
3^3 + 18 + 19 = 4^3
4^3 + 30 + 31 = 5^3
5^3 + 45 + 46 = 6^3
There's a couple of patterns here -- you could say that the first number added is 3 times the triangle numbers we just saw above, or equivalently that the first of the two sum numbers forms a pattern of +6, +9, +12, ... A deeper pattern is that the difference is 3*n^2+3*n+1, which is mathematically the difference between two successive cubes, but that's kind of cheating. There are actually more, and I think just before I fell asleep I had an even better one, but.. where was I going again? Oh, right.
The point of this tangent is that puzzles are often way more fun than what you're supposed to be learning. Especially if there is some sort of prize or at least some bragging rights for solving them. With that in mind, I created the Snickers Challenge, a collection of math puzzles for the algebra and stats classes.
Each time I go into class, I also try to create a new puzzle for the students. Sometimes, stats and algebra get the same puzzle, other times I give separate puzzles owing to the different skill sets of the classes. I usually try to present the puzzles in class, but since kids often forget the details between classes or just plain aren't paying attention, I thought a website would be a good companion to the in-class mentions, and also perhaps be a way of integrating technology into the classroom.
The website URL is mathisnofun.blogspot.com; this is purposely a tongue-in-cheek domain because I have secret ambitions of using reverse psychology on my students. The first post lays out the ground rules, but basically the first person to present a solution to a challenge gets a Snickers bar (or other candy of their choice; I had several students who were not Snickers fans). I am not above bribing students to think about math with 79 cent candybars -- so sue me.
Note to other TFs and higher-ups: I purposely created a new blogspot username for this other blog, and have not linked it in any way to the TF blogs. This is on purpose, since I would prefer the TF community of blogs to remain as private as such things can be on the internet. Since we often discuss our students, even under aliases, and since I feel it would be difficult to speak frankly about our experiences knowing that our students will read our words, I also ask that no one become a "follower" this blog, as this would also provide a link back to the TF blog community. I may link to the Snickers Challenge from this blog, but only once I make sure that leaves no trace on the receiving end of the link.
This is still true today; an example that comes to mind is last Thursday night (early Friday morning, really), as I lay in bed after preparing my little talk for class on Friday. I couldn't help mulling over some of the sequences of numbers I had prepared for class (this is a recurring theme for me in algebra class; I come in with 5 numbers that correspond to a pattern, and I ask the students to try to deduce the sixth). This particular day, I had the first five numbers of several sequences: the triangle numbers (1,3,6,10,15,...), the perfect squares, the cubes, the primes, and a special sequence called the "highly composite numbers" (look it up!).
I always like to try to think of different patterns the kids might see in these sequences, because there's often more than one way to get to the right answer, and I try to avoid calling any answer wrong, instead just pointing out where (usually) their perceived pattern breaks down. In this case, I knew that the triangle numbers can also be seen as the recursive sequence a_n=a_(n-1)+n, or in simpler terms you add 2, then 3, then 4, etc, running up through the positive integers. Similarly, for the squares you successively add odd integers +3, +5, +7, etc. I wondered if there was a pattern for the cubes? Well, it was 2 am but I still spent a few minutes pondering in that half-awake state trying to see what the pattern was, and I realized that
1^3 + 3 + 4 = 2^3
2^3 + 9 + 10 = 3^3
3^3 + 18 + 19 = 4^3
4^3 + 30 + 31 = 5^3
5^3 + 45 + 46 = 6^3
There's a couple of patterns here -- you could say that the first number added is 3 times the triangle numbers we just saw above, or equivalently that the first of the two sum numbers forms a pattern of +6, +9, +12, ... A deeper pattern is that the difference is 3*n^2+3*n+1, which is mathematically the difference between two successive cubes, but that's kind of cheating. There are actually more, and I think just before I fell asleep I had an even better one, but.. where was I going again? Oh, right.
The point of this tangent is that puzzles are often way more fun than what you're supposed to be learning. Especially if there is some sort of prize or at least some bragging rights for solving them. With that in mind, I created the Snickers Challenge, a collection of math puzzles for the algebra and stats classes.
Each time I go into class, I also try to create a new puzzle for the students. Sometimes, stats and algebra get the same puzzle, other times I give separate puzzles owing to the different skill sets of the classes. I usually try to present the puzzles in class, but since kids often forget the details between classes or just plain aren't paying attention, I thought a website would be a good companion to the in-class mentions, and also perhaps be a way of integrating technology into the classroom.
The website URL is mathisnofun.blogspot.com; this is purposely a tongue-in-cheek domain because I have secret ambitions of using reverse psychology on my students. The first post lays out the ground rules, but basically the first person to present a solution to a challenge gets a Snickers bar (or other candy of their choice; I had several students who were not Snickers fans). I am not above bribing students to think about math with 79 cent candybars -- so sue me.
Note to other TFs and higher-ups: I purposely created a new blogspot username for this other blog, and have not linked it in any way to the TF blogs. This is on purpose, since I would prefer the TF community of blogs to remain as private as such things can be on the internet. Since we often discuss our students, even under aliases, and since I feel it would be difficult to speak frankly about our experiences knowing that our students will read our words, I also ask that no one become a "follower" this blog, as this would also provide a link back to the TF blog community. I may link to the Snickers Challenge from this blog, but only once I make sure that leaves no trace on the receiving end of the link.
Wednesday, October 29, 2008
Three Goals
This is one of those things I should have published long ago. After talking with George, I've come up with three goals for me during my teaching fellowship:
1) Make sure students are actively involved in at least some small part of every talk. A great example of this was when I had the stats classes all flip coins for our normal distributions discussion. I also recently had students in the algebra class list their ideal careers on the blackboard and vote on which career we'd discuss in detail that day (being sure to fit in math along the way). Weaker examples of class involvement would be just asking students questions and getting them to answer, but I prefer active involvement with some sort of motion or task to be completed.
2) Get feedback from George after each class to see where he thought I was losing attention, needed to speed up / slow down, or if I missed any opportunities to work in class material.
3) Follow along in each class' textbook between visits to look for ways to relate discussions to class concepts. My best example of this was when I had the students flip coins and build bell curves just before the class talked about normal distributions.
1) Make sure students are actively involved in at least some small part of every talk. A great example of this was when I had the stats classes all flip coins for our normal distributions discussion. I also recently had students in the algebra class list their ideal careers on the blackboard and vote on which career we'd discuss in detail that day (being sure to fit in math along the way). Weaker examples of class involvement would be just asking students questions and getting them to answer, but I prefer active involvement with some sort of motion or task to be completed.
2) Get feedback from George after each class to see where he thought I was losing attention, needed to speed up / slow down, or if I missed any opportunities to work in class material.
3) Follow along in each class' textbook between visits to look for ways to relate discussions to class concepts. My best example of this was when I had the students flip coins and build bell curves just before the class talked about normal distributions.
Update / Status Overview
Apologies for the very long delay between postings. It has been over a month, which is completely unacceptable! Between running my first set of solo experiments as a grad student, trying to write my first conference paper and taking a long-planned week vacation to DC, I find I really need to catch up on my blog posts.
A brief overview first, before I go into detail on my sessions. I have just about memorized all my students' names, which makes it much more comfortable to start conversations with them, and they in turn have become much more comfortable with asking me for help with their classwork. I have given several talks to both classes by now, and have pulled out some lessons along the way which I'll share in upcoming posts. George and I have reached a good arrangement where when I talk, I take 15-20 minutes or so either at the beginning or end of the hour, with a primary goal of keeping the students engaged and inquisitive in some technical discussion, and a secondary goal of relating the discussion back to current material when possible.
Coming up next, in no particular order: my talks about primes, football & predicting the future, what we can learn from starlight, the snickers challenge, and more.
A brief overview first, before I go into detail on my sessions. I have just about memorized all my students' names, which makes it much more comfortable to start conversations with them, and they in turn have become much more comfortable with asking me for help with their classwork. I have given several talks to both classes by now, and have pulled out some lessons along the way which I'll share in upcoming posts. George and I have reached a good arrangement where when I talk, I take 15-20 minutes or so either at the beginning or end of the hour, with a primary goal of keeping the students engaged and inquisitive in some technical discussion, and a secondary goal of relating the discussion back to current material when possible.
Coming up next, in no particular order: my talks about primes, football & predicting the future, what we can learn from starlight, the snickers challenge, and more.
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