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Archive for the ‘Math 10-3’ Category

I saw this episode of Seinfeld a couple of days ago.  Now that my WCYDWT radar is finely tuned, I realized that it would be a good clip to show in a Math 10-3 class in the unit on rate and ratios.  I know it’s a little dated (who would use a Wizard now?), but it’s still one of my favorite shows.  Click on Morty to play the clip.

The one question kids need to ask will be apparent to them, but the math will be tricky for Math 10-3 students.  In the end, it’s still not a particularly compelling problem, but maybe this is a better way of presenting it to students.

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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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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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