Trip Planning Activity

For an (imaginary) locally developed math class, we did an interesting open-ended activity. We were given a choice between going bowling or going to a movie, and had to predict how much money we'd need for the evening. This included transportation costs, food, admission, and anything else that may apply. It was a very real application of math. There is sometimes a tension between the exactness of math and the practicality of estimation. I often think estimation and guesswork is crammed into problems where it doesn't belong, so it was nice to see a perfectly appropriate and realistic use of estimation. As the students, we were required to figure out bus fare or approximate the cost of gas. We needed to research the cost of a movie ticket and and try to anticipate what food we'd want and how much it would cost. All in all it was a very useful, dare I say fun, activity. The natural danger of course is that students will not use their phones or computers for the task at hand, and some classroom management will be needed to meet that challenge. However, if you can keep your students on task, I think if would be a very rewarding lesson for them.

"Deal or No Deal" Probability Game

A very fun game was played in the math class during a sample lesson on probability. We played "Deal or No Deal", based on the game show. We were divided into groups of two. In each pair, one person was the banker and the other was the contestant. We had a chart covered in cards, each of which having a dollar value on the underside of the card. The contestant picked a card at random, and the banker and contestant then haggle over how much the banker will pay for the card. During each round, the values on the remaining cards are gradually revealed, giving the contestant and banker information with which they can try and figure out how much the contestant's card might be worth. The trick is that the banker wants to buy the card for less than it's worth and the contestant wants to sell it for more than its worth. To take an extreme case, suppose all the cards have been revealed and we know that the contestant's card is either worth $1 or $500,000. If the banker offers $250,000, each player has a 1/2 chance of making or losing $250,000. Would that be a good offer from the banker? Perhaps. The interesting part of this game is that while the probability is important, there is also a lot of room for judgement. If I was the contestant, I would take the $250,000. If I was the banker, I would probably offer a lot less than $250,000 since I would rather earn $500,000 or nothing than make or lose $250,000. However, since the contestant only stands to gain, there may be a "bird in the hand" type of reasoning that would make the guarenteed $250,000 more attractive than the $500,000 or nothing. In short, there's a lot of higher-level thinking that can go into strategy for this game, and could have interesting applications in financial math and economics as well. If I ever get to teach data, it's definitely a game I would try to incorporate.

Student-Made Review Booklets

A very interesting idea was presented last week in our Grade 11 presentations. We were shown how to make little review booklets. These can serve as a quick reference material for the student, a study guide, perhaps even as a formula sheet for a test. I liked that this approach requires the students to make their own (as opposed to using a pre-made reference sheet). In grade 8, my history teacher had us make a table of contents in the front of our binders. We'd add to the table of contents as we took notes (and put page numbers on the top corners of our notes). This was a great lesson in organization and I see something similar in this. By this method, students have a short reference of everything they need to know, which may help their studying (since textbooks can be cumbersome sometimes). This would be a great method for physics as well, which has a lot of content and can be difficult to organize.

Generating Plots Using Motion Detectors

This week, an interesting application of TI graphing calculators was demonstrated. A calculator was connected to a motion detector which allowed a person walking in front of the detector to create a position-time graph. The exercise we were given was to recreate a given set of plots using the motion detector. It was a great way to connect slope with rate of change. The obvious physics applications didn't escape me. Kinematics, the unit that traditionally begins 11 Physics, requires developing an intuitive understanding of the position-time graphs. Students often have a difficult time relating what they see on the plot to a mental picture of actual motion. Curvature, as it relates to acceleration is particularly tricky. There is a difference between acceleration and "getting faster". Being able to create their own motion and seeing it plotted would be a very useful experience. Another difficulty students have is understanding that at turning points of a position-time curve, the speed is instantaneous zero, even though the acceleration is always non-zero. This would also be a useful exercise in calculus, when the topic of second derivatives come up.

 

Introducing the Concept of Proof to Students

This week we saw four more interesting miniature lessons. By a good coincidence, three of them dealt with mathematical proof to differing extents. I want to speak about one lesson in particular, which covered the following geometry problem:

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The object is to determine the sum of the three angles indicated in the figure. I believe this puzzle originated with Martin Gardner (an excellent source for clever math problems and riddles) but I am not positive. We were given Geoboards and elastic bands as tools to help solve the problem. In addition, we we cut up a copy of the figure in order to see that the three angles should sum to a right angle. Although it's a very good problem, this was a rather difficult exercise (perhaps due the fact that we weren't expecting it). It made me wonder how it would be received by grade 10 students. However, if they had been working up to this exercise over the term by doing simpler geometric proofs, perhaps it would not catch them as off-guard. It is probably easier in the beginning to learn proof by proving the obvious, so whether this problem is at-level would depend on how many similar exercises the class had already seen. It is definitely the type of thing that needs to be seen by students more often, and early. If I had a choice, I would have done an algebraic proof, but being forced to do a geometric proof showed me how bad I really am at that kind of thing.

My first real encounter with proof was a second year university algebra course and it completely revolutionized my impressions of mathematics. It was not difficult, it taught me to think, and I really wished that I had seen something like that earlier. Another benefit of proof, particularly of the geometric kind, is that is does not use numbers. It is a purely logical/visual expression of math which I imagine many students may find refreshing, particularly those whose numeracy may be lagging behind their peers. Does the prevalence of calculation, data, word problems etc. perhaps create an implicit bias in the students' minds with respect to what mathematics actually is? Even geometry, when it is taught, is less about structure and more about numerical quantities of area, volume and so on. Which is more fundamental after all, shape or quantity? In schools, quantity seems to be hands-down winner of that debate (whether the debate ever took place I do not know). Perhaps it may be that the math class could do with fewer numbers.

Story Based Math Problems and Proportional Reasoning

Last week in our math course, we saw presentations of three condensed lessons. The topics covered were finance, graph transformations, and calculating volumes. Although all were good, I was to focus on the “volumes” lesson for this post. Nam (the presenter) based his lesson on the film adaption of the book “Holes”, which is a very common presence in the junior school English class. For those who haven’t read it, Holes is a young adult novel about a boy named Stanley who is wrongly convicted of theft and sent to a sort of juvenile disciplinary camp where the inmates are forced to dig a hole every day out in the desert. The depth of the hole and the diameter must be equal to the length of the inmate’s shovel. Nam started the class with a clip from the movie in which Stanley accidently gets in trouble with another inmate for grabbing a shovel slightly shorter than the rest of them. The question Nam posed to the class is, “does it really matter if the shovel is a bit shorter”. He then had us try and figure out how much less dirt we would have to dig over the course of a year if we used the shorter shovel (it turned out to be about a month’s worth of holes). I liked this set up for a lesson because it starts with familiar territory. Practically everyone in the class had read Holes so when we worked on this problem, we already had a context in mind for the problem. I also liked that I was surprised by the answer, I didn’t expect the difference in volumes would be so significant. All in all, it showed me the effectiveness of using stories to engage students with a problem (as opposed to the more generic types of word problems we often encounter). If I had one thing to improve upon, it would be the organization of the problem. The problem was posed in feet and inches and we were made to convert everything to metric units. This made the problem a bit numbers heavy. Additionally, it was a bit ambiguous what was meant by “how fewer holes …” since we had two different volumes. This is nitpicky of course, and easy to fix. An alternate approach could be to do away with inches and metres altogether and use the standard shovel length as your unit length. For example, if the shorter shovel was 7/8 of the standard shovel, the smaller hole has (7/8)³ the volume of a standard hole. This greatly simplifies the calculation and may be a useful way of incorporating proportional reasoning.