Closing the Teach For America Blogging Gap
Feb 17 2010

Related Rates Wrap Up

What do explosions, Calculus, cookies, Tetris, raps songs, ballet, and 16 excited teenagers have in common? My classroom. All in one day. I came home thinking that it was the most exciting day of teaching yet. Mainly because of the dry ice and water explosion. As far as I can remember no one has even worn safety goggles in my classroom before (though years ago my dad did offer to send down a box of goggles with a paintball gun to help me control my freshmen boys.) Wow. I’ve come a long ways from that.

Now I’m left with the curriculum design question: How do I make every day project day? How do I let them see how much math has to do with the world while still covering important procedures? The students projects were filled with negative exponents, tricky constants, ugly fractions, long algebraic manipulation, real world interpretations, unit analysis and more. It was all of the things you teach them in little chunks coming together into one awesome problem. But it took a week of class time. One obvious solution is to have them write up problems and not make posters and presentations and cookies to go with them. Less fun but certainly more efficient. Part of what drove the creation of these tough problems was the drive to find out how dancing or their cars worked.
One of the more epic events was the rap written about Related Rates. I’ll need to post it here because it really accurately described what the problem was asking. What happens to the forward motion of a car when a wheel is inflated at a certain rate while the axle is rotating at a fixed rate. The theory is that the radius of the “torus” will expand and then the car will go faster. The end of it concluded with a line about using Calculus to break the speed limit “with the help of Ms. B” It was adorable.

A group of girls brought in a lazy susan type device borrowed from the physics teacher and demonstrated the effects of moving your hands closer to your body while spinning. The physical demonstration was incredibly helpful and great fun to watch. Erica and Rachelle are two adorable, sweet girls who are not always extremely confidant in their mathematical abilities. Their project required them to read ahead in the physics book and use equations that I had never even studied. After learning something new and challenging they did an incredible job illustrating the situation by drawing dancers and labeling the appropriate parts with dv/dt and dr/dt to indicate the spin of the dancer and changes in the radius of her arms. The entire poster was so clear and interesting that I’m sure they could make a good career designing instructional materials. Of course, I’m probably the only nerd in the room who is trying to do that! I hope(and based on talking to them I think it worked) that they took away just how hard of a problem they could solve with their own abilities. I think that they were quite impressed with themselves. Of course two days later I was trying to get students to figure out the area under curves using Riemann Sum approximations and Rachelle seemed to think that she couldn’t possibly make sense of a table of values from a speedometer. With a hint I got her going again but she still seems resistant to the idea of figuring out math without my prior example.

Scott and Richard presented their results about relating the surface area of a ball to it’s volume while inflating it. While the equations got rather hairy in the middle of the problem most everything canceled out and they were left with a simple formula relating the rate of change of surface area to volume. They were pleased at the elegance and seemed so excited to have actually conceived of a problem, done a bunch of messy algebra and had the result come out neatly.

The best part of their project(aside from the elegant result) was that they wrote the entire problem on a basketball. It immediately captures everyone’s attention and shows them that the math problem is connected to the real world. In fact it’s connected to something you can pick up, throw around and inflate at a constant rate. You can touch, see and comprehend what is going on and then the awestruck middle schoolers who were playing with it today at least have the sense that some very complicated looking math is describing something quite simple.

While visiting Berkeley I sat in on Alan Schoenfeld’s research group “Functions” and listed to a presentation of a pre-service teacher who was trying to understand why students didn’t like word problems.

Despite her inexperience with teacher she thought up the great warm up to linear equations and was trying to form research questions around the data. She gave students the linear equation 3x+8=17 and asked them to write a story problem related to that equation. The results seemed rather like a mad-libs game and revealed a few important things.
If Sally buys a greeting card for 8 dollars and buys x pens for 3 dollars each and pays 17 dollars in total how many pens did she buy. Alan pointed out that 8 dollars was an absurd amount for a greeting card and that kids were playing the game of story problems. Their books are filled with nonsensical word problems where reason about the real world is abandoned. The game is to translate between a certain class of word problems and they symbols that they represent. If students use real world thinking and point out that greeting cards are not actually 8 dollars and you might want to check the price tag they are asked to suspend their reason about the situation. I wondered if the new teacher had told the students not to make the problem like an example and hadn’t given them numbers if the results would have been any different. I’m not sure if the success I saw with related rates would extend to Algebra or not. I certainly saw my kids coming up with “sensible answers to sensible questions” in that class and not trying to mad lib some new words and situations into the same old ladder falling or sphere expanding problems. Of course sometimes the real world situation got so complicated that simplification to the point of absurdity was required.

The group trying to model the rate of licking of a tootsie pop decide to make it a log function thinking that people would slow down licking as they got sick of it. Of course no one licks food like a log function and only those truly committed to finding how many licks it takes to get to the center of a tootsie pop(google says 419) are going to have the willpower not to bite it towards the end.

The cookies we backed were ellipsoids and the ramps we analyzed were frictionless. Real world problems either seem impossible simple(Three friends are spitting a 20 dollar bill and want to tip 18 percent. How much does each owe? Give the combination of ones and fives they have how can they all pay evenly?) or way too hard. When you take air pressure, friction, air resistance, the vagaries of human behavior and everything else into account no one, not even me knew how to solve the problems my kids were coming up with.

I’m not convinced that problems must be related to the real world to be good. The truncated chessboard problem is one of my favorites and intrinsically interesting even though no one really cares about covering chessboards. (Actually, maybe it has to do with city planning or some other design, who knows. Need to go to school to give me the time to find out!) Real world sure seems to motivate though.

2 Responses

  1. Roxanne

    Congrats on your acceptance to Berkeley! I myself was recently admitted to their law school. Unfortunately, I don’t have the cash on hand to visit the campus, how was it?

  2. “The group trying to model the rate of licking of a tootsie pop decide to make it a log function thinking that people would slow down licking as they got sick of it. Of course no one licks food like a log function and only those truly committed to finding how many licks it takes to get to the center of a tootsie pop(google says 419) are going to have the willpower not to bite it towards the end.”

    This cracked me up!! I love reading your blog; your energy and passion for the subject is so infectious, and I’m sure it affects your students, too.

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Learning more about life than math…

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