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## Same Problem, Two Presentations

In my quest for a good Three Act Problem for elementary level students, I’ve come up with two more ideas. It’s actually the same idea presented in two different ways. It’s going to hit division at the grade 3 or 4 level, I hope.

Hay Bales – Act I

Cars – Act I

These videos are rough. The car one could be fantastic if a car lot would let one of us come by with a video camera and film them loading a carrier. The hay bales one would be great if we could get a farmer to let us film him loading a truck. In the past week, I have driven to Calgary and back, Stettler and back, and Red Deer and back. I’ve seen a ton of those bales in fields. I haven’t, unfortunately, come across any of them being loaded up or hauled along the highway.

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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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## Let the Kids Ask the Questions

October’s issue of the NCTM Mathematics Teacher magazine included an excerpt from an interesting article in their Media Clips section.  They used a Forbes article that dealt with tween earnings, and the portion they included is:

Hollywood’s 10 top-earning tweens collectively pulled down \$107 million between June 1, 2007 and June 1, 2008.   According to Alloy Media and Marketing, it’s a rabid and often indulgent fan base: America’s 20 million kids aged 8 to 12 spend \$51 billion of their own money annually and influence \$150 billion more in spending by their doting parents. Tied for the top spot: Hannah Montana star Miley Cyrus and Harry Potter lead Daniel Radcliffe. The young talents each banked a cool \$25 million in the last year.

I love this article for the questions it could elicit.  The problem with the Mathematics Teacher Magazine treatment of this is that it follows the article with 8 questions we are supposed to ask kids.  Teachers might see this, and defer to the questions provided, thereby missing a great opportunity.  I’d much rather show students the article, and have the kids decide what would be interesting to explore further.  There are lots of potential rate and ratio questions that could come from student exploration here.

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## Golden Ratio and the Human Face

This is an activity a group of teachers I was working with fleshed out based on ideas on Mark Wahl’s website, and here.  This activity is appropriate for Math 10-3 (Measurement) and Math 10C (Measurement or Real Numbers).

Statues of human bodies that the ancient Greeks considered most “perfect” embodied many Golden Ratios. It turns out that the “perfect” human face also contains many Golden Ratios.  This task allows students to take measurements of an ancient statue, and check whether these measurements approach the Golden Ratio.  Then two celebrity photos are provided and the students check how beautiful the celebrities are based on these calculations.  Finally, students can use pictures of their own faces and GeoGebra to check how they match the Golden Ratio.

Part 1 Students perform the measurements on a photo of a statue, and calculate the ratios requested.  These measurements could be made by printing the picture and physically measuring, or by inserting the picture into GeoGebra and using the software to measure.

Part 2 Students complete the same measurements, either on a provided celebrity photo, or one of their own choice.

Part 3 Students then take their own picture and insert it into GeoGebra.  They can then measure, and check how closely the ratios in their own faces match the Golden Ratio.  Students who are not comfortable using their own pictures can choose other celebrities from the internet. (I didn’t teach Cameron Diaz, but her face is much more likely to be beautiful using this test than my own…)

The complete student package in Word format is available for download here.

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This morning, as I was making my toast and about to put peanut butter on it, I was reminded of something I did early in my career in a Math 31 (Calculus) class.  It was at least 15 years ago, and way before we had digital cameras or Dan Meyer, but I think it definitely qualified as a WCYDWT.

One morning in the early 1990s, I was having peanut butter and toast, and had gotten near the bottom of the jar.  While scooping peanut butter out, I got it all over the handle of the knife, my hand, and then of course, my pants.

It occurred to me that the jar really was taller and narrower than it should be, and in a rare moment where everything came together for me, I realized that the shape of the jar was probably significant and we just happened to be working on Maximization and Minimization problems in Math 31.  Since digital cameras hadn’t been invented yet, the only multi-media artifact I could take into my classroom was the jar itself.

So I took my jar of peanut butter into class, slammed it down on the table in front of the class, ranted for about 5 minutes about how mad I was at Mr. Kraft for getting peanut butter on my hand and pants, and then told the class that we wouldn’t be doing Calculus that day, because I had to write a complaint letter to Mr. Kraft.  I sat down at my desk, and pretended to write, and hoped.  It took a couple minutes, but then it happened.  One student said something like, “Mr. Scammell, there must be a reason it is made in that shape.”  Another jumped in with, “It’s probably the shape that uses the least material but still has 1 kg of peanut butter in it.”  Then they argued for a bit, and decided on their own to figure it all out.  They measured, calculated, argued some more, and then told me to write my letter, because you could hold 1 kg of peanut butter in a container that was wider and shorter so my hand wouldn’t get peanut butter on it.

Again, I pretended to write while I waited and hoped.  And then they extended.  One kid stopped the whole thing, and said that there must be a reason it was the shape it was.  He observed that the lid was thicker than the other plastic, and therefore must cost more.  They re-ran their calculations based on an estimate that the lid cost twice as much as the other plastic, and then concluded that the shape was logical if we assumed the lid cost more.  It’s too bad that this was all done years before the internet hit our schools, or perhaps we could have quickly researched further to find out costs and contact numbers for Kraft.

I tried the Peanut Butter Rant several more times over the years, with varying degrees of success.  Some classes got right to the one important question, while others took some prompting.

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## Favorite Problems

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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## 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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## 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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## 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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