Speeding

Thanks to this post on Dan Meyer’s blog, and an ensuing conversation between Dan and Curmudgeon, I was pointed to an article that I think would make a pretty compelling problem in Math 10C or Math 10-3 measurement.

The article describes a 17 year old driver who was given a $190 ticket for going 62 miles an hour in a 45 mile an hour zone. His parents, however, had installed a GPS system in his car to track his speed and driving habits, and they claim the GPS proves their son was only going 45 miles an hour at the time the ticket was issued. It appears to have taken two years of legal wrangling, before the ticket was finally upheld, and he had to pay the fine.  I wouldn’t tell the students that yet, though.

Here’s a link to the article: Speeding Teenager

Lesson Plan

1.  Present the problem.

Give the students the following excerpt from the article:

Shaun Malone was 17 when a Petaluma police officer pulled him over on Lakeville Highway the morning of July 4, 2007, and wrote him a ticket for going 62 mph in a 45-mph zone.

Malone, now 19, was ordered to pay a $190 fine, but his parents appealed the decision, saying data from a GPS system they installed in his car to monitor his driving proved he was not speeding.

What ensued was the longest court battle over a speeding ticket in county history.

In her five-page ruling, Commissioner Carla Bonilla noted the accuracy of the GPS system was not challenged by either side in the dispute, but rather they had different interpretations of the data.

All GPS systems in vehicles calculate speed and location, but the tracking device Malone’s parents installed in his 2000 Toyota Celica GTS downloaded the information to their computer. The system sent out a data signal every 30 seconds that reported the car’s speed, location and direction. If Malone ever hit 70 mph, his parents received an e-mail alert.

Malone was on his way to Infineon Raceway when Officer Steve Johnson said he clocked Malone’s car going 62 mph about 400 feet west of South McDowell Boulevard.

The teen’s GPS, however, pegged the car at 45 mph in virtually the same location.

At issue was the distance from the stoplight at Freitas Road — site of the first GPS “ping” that showed Malone stopped — to the second ping 30 seconds later, when he was going 45 mph. Bonilla said the distance between those two points was 1,980 feet.

2.  Ask the students to discuss the article.  In the end they will come to the question we want explored.  Was young Shaun guilty of speeding?

3.  Let them answer the question.  Have them prepare a defense for Shaun, or an argument for the prosecution.

4.  Show them the Commissioner’s conclusion, based on mathematics.

Bonilla said the distance between those two points was 1,980 feet, and the GPS data confirmed the prosecution’s contention that Malone had to have exceeded the speed limit.

“The mathematics confirm this,” she wrote.

Teacher Resource

A possible solution

An extension, eventually.

I have been attempting to contact the person mentioned in this local article, but so far he hasn’t responded to me.  Similar mathematics could prove he wasn’t driving as excessively fast as the red light camera claimed, but I would need to get a copy of his ticket to show that.

Red Light Camera

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Fishing Story

Here’s a WCYDWT for Alberta’s Math 10-3 on Ratios and Rates.

This fish, a 60 pound sturgeon, was recently caught and released in a small self-contained pond on a golf course near the Red Deer River.  When local biologists heard about the catch, they decided to investigate.  They surmise he (or she) came into the pond when the river flooded five years ago, and then spent the next five years feeding happily on the stocked trout.  Since sturgeon are endangered, it was important to return him (or her) to the river, which they were able to do.

This story reminded me that about ten years ago, I was fortunate enough to be present when a team of biologists surveyed a cutthroat trout population on a remote mountain stream.  Biologists typically use a method called electroshock fishing to stun fish so they can be tagged and released. They had a boat with a generator on it and a long metal prod which they poked around in the water.  When the current got near enough to a fish, it would be momentarily stunned, and float to the surface.  The biologists then weighed, measured and tagged the fish before returning it to the stream.  Any fish that had been previously tagged had its number recorded and it was again weighed and measured.  It was a humbling experience to see that there were actually abundant numbers of fish in that stream, because I had been unable to catch any earlier that day with my fly rod, but I digress.

I have used the story of watching the biologists electroshock fish several times in my classes as a ratio problem and asked students determine the population of fish in that stream.  Sometimes I gave my students the number of fish tagged on the first day, and then the number of tagged fish caught on a second run of the same stretch of river later on.  Other times (and probably a better problem), I had my students design an experiment to predict the population of fish before providing them the actual numbers.

Red Deer Advocate story about the fisherman who first caught the fish

Red Deer Advocate story about the biologists capturing and releasing the fish back to the river

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Understanding vs. Doing

At a session I facilitated today, I was going over our old (1996) curriculum as compared to our revised (2010) curriculum.  Many of the outcomes are similar, but the big difference is in the wording of the outcomes.  I asked the participants to compare the two below.

1996

13. Use the Pythagorean relationship to calculate the measure of the third side, of a right triangle, given the other two sides in 2-D applications.

2010

2.  Demonstrate an understanding of the Pythagorean theorem.

The outcomes get at essentially the same thing.  In my opinion, though, the big difference is in the word “understanding” in our 2010 curriculum.  Back in 1996, we don’t appear to have been too concerned about whether or not the students understood it, as long as they could do it.  I threw this thought out in my session, and one person challenged it and said that they were the same thing.  He insisted that a student couldn’t do something unless that student actually understood it.

I disagreed, and he claimed I was arguing semantics (which I had to look up later).  I had to move on, but I certainly thought about what he had said on my flight home.  In the end, though, I stand by my original assertion.  I think that too often we ask kids to “do” things in math class without really trying to get at understanding.  Some kids imitate our algorithms, score well on tests, without having a true understanding of the math involved.

I tried to think of a real-world example of this distinction, and so far the best one I have come up with is as follows.  I know that if I push the gas pedal on my car, it will move forward (if I put that stick thing in the spot marked “D”, which I assume stands for “Go”).  This is something I am confident I can do.  I do not, however, have any understanding as to why the car goes forward or what is going on to make that happen.  I do it without any understanding at all.

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Angle of Elevation and Depression

A few weeks ago, I was asked to go out to a school and teach a demo lesson.  The entire math department was released for the afternoon and they watched the lesson in period 3, and we met to talk about it in period 4.  I was given a Math 10 Applied class (non-academic 10th graders), and instructed to teach a lesson on Angles of Elevation and Depression.  My goal was to present the lesson in a manner that is consistent with the pedagogy of Alberta’s revised program of studies.  As much as possible, I wanted the students active and constructing their own meaning.

To introduce angle of elevation and depression, I first introduced the concept of “horizontal”.  I had the students use tape measures to measure the heights of their eyes.  Then I asked them to circulate around the room and find an object that was at the exact same height as their eyes and label a picture that looked like this:

After that, they were instructed to make a list of objects in the room that they would have to look up to see (elevation),

and objects that they would have to look down to see (depression).

Next, I stole a page from Dan Meyer’s playbook.  I took the class down to the atrium in the middle of the school and asked them to estimate how high above the main floor the railing around the second floor was.  As a motivator, I promised a prize for whichever team of three had the closest estimate to the actual height.

We returned to the classroom, and each group recorded its guess on the SMART board.  Then I had the students construct clinometers similar to this one.   I asked them to return to the atrium and use the clinometer and a tape measure to find the actual height of the railing.  At this point, I made a major pedagogical error.  Instead of letting them go to the atrium and figure out which measurements they needed and how to get them, I diagrammed it all for them on the board.  I helped too much.  I think I did it because the entire math department was there watching and I didn’t want the lesson to flop.  I also didn’t know the kids since it wasn’t my class, and I wasn’t sure how much they would be able to handle on their own.  I should have found out by letting them struggle.

Once the kids had made their measurements, they returned to the classroom to make the calculations.  Each group recorded its calculated height on the SMART board next to the estimate.  One group even self corrected when they realized they were several meters out because they had added the height of the measurer’s eyes in feet to the height of the railing in meters.  The correct answer was revealed, and the group with the best estimate and the group with the best calculated value each got cookies.  I tried to discuss sources of error with them, but it was getting late in a class taught in a different manner than they were accustomed to, so that discussion was not well focused.

Overall, the lesson went well, and I believe the students learned angle of elevation and angle of depression in a more compelling manner than it is normally presented.  I gave exit slips to check this theory, but I forgot to bring them with me when I left.  I would have rushed back to get them, but one of the perks of being a consultant is that I don’t have to mark anymore.

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Mozart’s Dice Game

I got this idea from a teacher I marked diploma exams with a couple of years ago.  I’d love to credit him, but I can’t remember who he is.  This is a WCYTWT submission, or as I like to call it, WWDDWT (What Would Dan Do With This).  I used it with a couple of classes last year, and they really enjoyed it, but I used it poorly.  I did it at the end of a unit on Permutations and Combinations, so the students knew exactly what to do with it.  It wasn’t truly a problem, because they knew exactly how to solve it when I gave it to them.  I should have used it on day one of Permutations and Combinations, and let them invent the fundamental counting principle on their own.

Mozart’s Dice Game

Mozart is credited with creating a dice game, whereby you roll a pair of dice 16 times to select 16 measures to insert into a minuet section, and then a single die 16 times to select 16 measures to insert into a trio section.  For example, if you roll a 6 for the first measure, you consult a chart to see what measure number to insert into the first measure of your minuet, and so on.  The idea is that no matter what you roll, you always produce a minuet that fits whatever rules go along with a minuet.  This site explains it in a little better detail.  Mozart’s Dice Game.

WCYDWT

You give the students the history, and then you throw this site up on the SMART board.   Play Mozart’s Dice Game has a chart that looks like this:

The drop down menus let the students enter numbers that they roll on dice, so give the kids some dice, and let them enter their rolls.

Notice the link below the chart that says “Make Some Music!” Once the students have all entered their rolls, you click here and the newly created minuet will play.  There’s even a “Generate Score” link below the media player that lets you generate the score for the minuet they created.  My students loved this, and printed theirs off to try to play it on the piano themselves.

After the minuet plays, you tell the kids that even though this was written more than 200 years ago, you are pretty sure that nobody has heard the particular minuet the class just created ever before.  Ask them to discuss this statement.  They’ll say things like, “Why, did they lose the score until recently?” and dance around it until one kid finally asks, “How many minuets could be made in this game?”  Then you’ve got them and you let them play around with it.  They will invent the fundamental counting principal, determine that there are an incredibly large number of possible minuets, and even create their own interesting extensions.

One student even came in the next day with an iPhone app that generates minuets using Mozart’s Dice Game.  The only  problem with the app is that it only randomly generates minuets, and doesn’t allow you to enter your own rolls.

Update: March 11, 2016

My greatest fear was realized. The site I talk about above is no longer functional. Something about Midi Files, Flash players, and the like. I couldn’t get it to work in any browser.

I found a functional site. https://www.mozart-game.cz/

It randomly selects the measures to play, as highlighted below. You can play your own measures by clicking on the un-selected ones, so you could still have your students roll dice and create their own minuet, but you’d have to click your way across the chart. If you can keep time, it’s pretty easy. The vertical columns represent one roll of a pair of dice. So if the first roll was a 2, you’d play measure 96. In the screen shot below, the first roll was 3, the second roll was 6, and so on.

 

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Lecture Method vs. Peer Instruction

Eric Mazur is a Harvard Physics professor.  I came across this video when somebody posted it on Dan Meyer’s blog in response to another person who was demanding data to prove that Dan’s methods would work.  Professor Mazur has data to prove the merit of his shift from being a lecture style instructor to an instructor who facilitates peer teaching.  Peer teaching is definitely a strategy we could employ in our revised curriculum in Alberta.  If a University Physics professor can facilitate learning without having to tell his students everything, then certainly a high school math teacher who wanted to teach for understanding could emulate this process.  This video is about 80 minutes long, but is worth the time if you can spare it.  For those who can’t, I’ve summarized the key points below.

The reason that Professor Mazur switched his style was because his students were not showing gains when he compared pre-test results to post-test results.  Despite going through an entire term of his lecture style physics class, students could not show a significant improvement in their understanding of basic physics.

His Key Messages:

• Students who have recently learned something are better at explaining it to other students than a teacher who learned and mastered it years ago. It is difficult for a teacher who has mastery of a concept to be aware of the conceptual difficulties of the beginning learner.
• Give students more responsibility for gathering information and make it our job to help them with assimilation.
• You can’t learn Physics (or Math or anything) by watching someone else solve problems.  You wouldn’t learn to pay the  piano by watching someone else play.  You wouldn’t train for a marathon by watching other people run.  If you want to learn problem solving, you have to do the problems.
• Better understanding leads to better problem solving.  The converse of this statement is not necessarily true.  Better problem solving does not necessarily indicate better understanding.
• Education is no longer about information transfer.
• He says that in his original methods he covered a lot, but the students didn’t retain much so the coverage was basically meaningless.  In his new method, he has relaxed the coverage a little bit, but increased the comprehension enormously.

The Peer Instruction Process

• Students pre-read the lecture notes or text.
• Class is then used for depth, rather than coverage.
• Depth is attained through what he calls a concept test.

The Concept Test:

1. A question is posed.
2. The students think silently about the question for a minute or so and it must be completely silent in the class.
3. Students answer individually and vote by show of hands or by SMART Response systems.
4. Peer discussion.  Defend your answer.
5. Revised group answer.
6. Explanation

Then he will “lecture” for a couple of minutes, and repeat the process with his class.

Benefits of his process:

• This process promotes active engagement.  “It is impossible to sleep through class, because every few minutes, your neighbor will start talking to you.”
• He can continually assess where his students are.

Challenges:

• Teachers have to find the right questions to ask in their classes.
• Students will write on their evaluations that professor Mazur isn’t teaching them anything and that they have to learn everything themselves.

My take on all this:

I see this process as being one we could employ in our high school classes.  I don’t think we’d expect kids to pre-read sections of the textbook, but we could certainly present problems to them in this manner and allow them to explore and construct solutions without having to tell them so much.

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Contest Entries

For your viewing pleasure, here are what I believe to be all the entries in the binomial expansion contest.  They are presented without bias, and in no particular order other than putting the best one at the end.

Eric’s

James Tanton’s Part 1

James Tanton’s Part 2

Jason Dyer’s

Mine

Edit (May 30):  Jason Dyer’s was judged the winner in the competition.  See here for the rationale for the decision.  Congratulations, Jason.

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Binomial Expansion Contest

A few weeks ago, Kate Nowak posted on her blog that she was frustrated at the lack of online resources to support the teaching of the binomial expansion.  She threw out a challenge asking people to create a better way to teach the binomial expansion.

Here’s her challenge:

“Objective : Present the binomial expansion in a way that makes sense. Bonus points if students are able to as a result completely expand a power of a binomial and find a specific term in an expansion.”

Our revised curriculum in Alberta includes a pedagogical shift that asks teachers to create opportunities for students to establish their own meaning through exploration, problem solving, investigation and developing personal strategies.  One of the common concerns I hear from teachers is that we can’t teach higher level mathematics this way, because the material is so hard that students won’t be able to construct their own meaning.

Kate’s challenge gave me an opportunity to try to teach in this manner in one of our academic courses.  In Alberta, binomial expansion shows up in a pre-calculus course called Math 30 Pure.  Students go from that course into Math 31, which is a calculus course covering limits, derivatives, application of derivatives, and basic integration.  Binomial expansion, and the binomial theorem are two pretty dry topics.  I took the challenge and attempted to create a lesson that was both engaging, and allowed students to construct their own meaning through looking at patterns.  A colleague was nice enough to let me try the lesson in his class.

My lesson went over fairly well with the students, and I think I met all of Kate’s objectives.  The introduction/hook was more compelling than anything I had done in the past on that topic.  Students were able develop the patterns in the binomial expansion on their own, and were able to apply those patterns to expand more complicated binomials.  What surprised me was that they were able to come up with the formula to find any term in an expansion all on their own.

I’d love to hear some feedback on the lesson.

An 11 minute compressed version of the lesson can be watched here:  Binomial Expansion Lesson

If you’d prefer to just read the lesson plan, here it is.

Lesson Plan

Introduction:  I showed them a mathemagician video from TED.

Hook:  I claimed that I was a mathemagician, too. I wrote (a + b)^2 on the board, and asked them to expand it. Then I wrote (a + b)^3 on the board and had them expand it. Then I wrote (a + b)^4 and (a + b)^5 and challenged them to a race. They could use friends, pencil, paper, calculators or whatever they wanted, and I would just use my brain. I pretended to struggle, and then wrote down the answers as quickly as I could. I had to hurry because one kid was darn quick, and was getting at (a + b)^4 by multiplying the answer to (a + b)^3 by (a + b) rather than expanding the whole thing as I had expected them to do.  If you watch the video, we appear to tie on that one, but he already started while I was blabbering.

Students look for patterns:  I explained that I am not really that smart, and that I was cheating using a pattern. I wrote “In the expansion of (a + b)^n, ” on the board, and asked them to spend a few minutes together coming up with ways to complete that statement. They got that each term had the same degree as the exponent on the binomial, and that there were n+1 terms in the expansion. It took a little longer and some direction from me for them to notice that the a’s started at a^n and decreased to a^0, while the b’s did the same thing in the opposite direction. One girl put Pascal’s triangle on the board when she noticed the pattern of coefficients.

Expand a binomial:   I asked them to use all that they had learned to expand (x + 2y)^5. They worked together and there was much discussion about how to handle the 2y part. They managed to figure it out.

Develop a formula for a general term:  I gave them one like (x + 2y)^12 and asked them if they could figure out the 8th term without writing out the first 7. I gave them three blanks to fill in: coefficient, “a”, and “b”. They got the a and b part just fine, but struggled with whether the coefficient should be 12 choose 7 or 12 choose 8. They figured it out by counting.   I told them that they had just figured out the formula. I wrote the formula tk+1 = nCk x^(n-k) y^k on the board.

(At this point I fell back into my old bad lecture style, and did two examples with them using the formula, rather than making them do it themselves.)

Example: Find the term containing y^7 in the expansion of (x – 3y)^7.

Example: Find the constant term in the expansion of (3x – 2/x^5)^12.

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