Posts Tagged ‘Math’

@dandersod posted the following video on his blog last week.

He asked, of course, What Can You Do With This?

I decided that there was a lot I could do with it, so I worked on some editing of the video, and then I rolled it out with a group of teachers this morning to see where it would go.

Before I describe the lesson, I should point out that my first edit of the video involved crossing out some of the measurements the original video contained, as well as beeping out the commentary that mentioned the measurements.  I was limited by my video editing skill, and the software I was using. It ended up being awkward and kind of annoying, so I re-did it simply by deleting the parts I had originally wanted obscured.  The new video was much cleaner, and it had the extra benefit of not being so obvious about what I wanted students to explore.  With the beeps and blocked out numbers visible in the first edit, it was painfully evident what I wanted people to find.  The second edit allows for many more directions to be taken in the exploration, and it’s not even obvious that I have deleted anything.

Here was my lesson plan.

  1. Provide each group a ruler and a 60 g bag of gummy bears.
  2. Play the question video.
  3. Ask the students what they wonder about after seeing the video.  In this edit, they will wonder about a whole bunch of things.  I’m hoping they get to, “How many small gummy bears are equivalent to the giant bear, and what are the  dimensions of the giant gummy bear?”
  4. Elicit guesses, lower bounds, and upper bounds of reasonable answers.
  5. Ask them if they need any clarification or information that might help
  6. Turn them loose.
  7. Work the room.  Help those who are struggling.  Provide extensions for those who hammered through it.
  8. Have students share their answers.  They will be all over the place here.  Those that work with the calorie count will be closest to the “right answer”.  Those that worked with the mass will be a bit farther out.
  9. Play the answer video.
  10. Discuss discrepancies.  Is it measurement error?  Problems with the calculations?  False advertising?
  11. Eat the gummy bears.

The lesson is fun, engaging, and has great curricular fit to Alberta’s Math 20-2 course in the measurement and proportional reasoning units.

At step #3 above, some members of the group I was working with really wanted to go a different direction.  Their suggestions became cool extensions for those that got done quickly.

  1. How tall is the man in the video, who is the same height as the gummy bear from 30 feet away?
  2. How many times would you have to walk around the school to burn off the giant gummy bear?


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After attending a session on our revised curriculum by Dr. Peter Liljedahl, several teachers have told me that they have been inspired to spend the first week of Math 10 Combined on  establishing collaborative and problem solving practices in their classrooms.  Since we will be teaching math in a different manner than teachers and students are used to, it is important to create a safe environment in which students can explore and construct meaning through rich problems.

The idea that we will spend the first week on processes and exploration, rather than on checking off the first few curricular outcomes is frightening to some teachers.  Several have asked me what types of problems they should give students during that first week.  Below are three of my favorites.  They have loose curricular fit, but all allow for rich discussion, debate, exploration and invite multiple methods of solution.  The idea is to throw these out to students before any teaching has happened, and allow them to use any method they can to solve them.  Feel free to give me feedback, and share your own favorite problems.

Problem 1 Fred’s aunt is twice as old as Fred was when she was as old as Fred is now.  How old is Fred and how old is his aunt?

Problem 2 Three businessmen check into a hotel (in the olden days, when hotels were cheaper).  The desk clerk charged them $30, so they each paid $10.  Later on, the desk clerk realized he had made a mistake, and the room charge should only have been $25.  The desk clerk sent a bellboy up to the room with $5 change.  Unsure how to divide the $5 evenly among the three men, the bellboy gave each man $1 back, and kept $2 for himself.  Now each man has paid $9, for a total of $27.  The bellboy has $2 which brings the total money up to $29.  But the men originally paid $30.  Where is the missing dollar?

Problem 3 – The Jail Cell Problem In a prison, there are 100 cells numbered 1 to 100.  On day 1, a guard turns the key in all 100, unlocking them.  On day 2, the guard turns the key in all the even-numbered cells, thereby locking them.  On day 3, the guard turns the key in all the cells numbered with multiples of 3, thereby altering their state.  For example, if the cell was unlocked, he locks it.  If it was locked, he unlocks it.  On day 4, he turns the key in all the cells that are numbered with multiples of 4, thereby changing their state.  He continues this process for 100 days, at which time all the prisoners in unlocked cells are set free.  Which prisoners are set free?

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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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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.


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.


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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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.


  • 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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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.


James Tanton’s Part 1

James Tanton’s Part 2

Jason Dyer’s


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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