Students and Staff at J.L. Ilsley High School recently returned from a March break trip to Italy. Their stories about Rome and pizza and gelato inspired this "Would You Rather?" math question. Most students are pretty familiar with pizza and have strong opinions to share on their favourite type and restaurant for pizza.
Would you rather have a slice of pizza from New York or from Rome? The New York pizza costs $2.75 US per slice. The Rome pizza costs 1,50€ per 100 grams.
In Rome, pizza by the slice or "pizza al taglio" is typically sold in rectangular pieces by weight. Prices are often listed per 100 grams. Prices can vary greatly depending upon the type and location of restaurant. Restaurants close to major tourist attractions in Rome are often much more expensive. The price I quote above is from Pizza Florida in Rome. Estimating the weight of a typical slice of pizza might be difficult for students. How much does a typical piece of pizza weigh? According to Pizza Pizza, a 1/10 slice of a 14 inch diameter pizza is approximately 110 grams. There is also the issue of currency conversion. You could ask for 3 Euros worth of pizza, but how much will that cost you in Canadian dollars? An online currency conversion website or app can help with currency exchange.
The Nova Scotia Mathematics 10 curriculum has outcomes on both currency exchange and SI to imperial unit measurement conversions so I thought this would be a nice warm up question to be used in that course.
In case you were wondering where you should go to eat pizza, here are the 14 top cities for pizza, as identified on the Conde Nast Traveler Best Pizza in the World list. Note that a Canadian city, Edmonton, made the list.
Nova Scotia Mathematics Curriculum Outcomes
Mathematics 10 - M02 Students will be expected to apply proportional reasoning to problems that involve conversions between SI and imperial units of measure.
Mathematics 10 - FM01 Students will be expected to solve problems that involve unit pricing and currency exchange, using proportional reasoning.
Revisiting the Classic Ferris Wheel Problem
This type of pseudo-context word problem robs students of the opportunity to explore and analyze real-world problems in much depth. Dan Meyer has written quite quite a lot about pseudocontext. My concern with the Ferris wheel problem is not that you can't model the height of a seat on a Ferris wheel with a sine function, it is why would you do it?
Instead of doing a textbook problem with a fictional Ferris wheel, I decided to use a real Ferris wheel from a nearby amusement park that some of my students would be familiar with. I visited the park to take a video of the Ferris wheel in action. Below is a 30 second clip of the "Big Ellie" Ferris Wheel at Atlantic Playland.
Notice and Wonder
I started by asking students what they noticed in the video. After brainstorming and recording the students observations I asked students what they wondered about in the video. They asked questions like "how fast is the ride going?", "how tall is this Ferris wheel?", "how far can you see from the top of the ride?", "how long does the ride last?". In order to investigate these questions further we needed to estimate some values such as the radius of the wheel, how long it takes to make one revolution, and the height of the central axis about the ground. I asked students to estimate these values using the clues in the video we watched. We watched it several times in order to get some good estimates.
I also talked about some of the mental math required to operate a ride like this. Because it is belt driven, you have to load the Ferris wheel so that it is equally balanced around the wheel. Otherwise, one side of the wheel would become too heavy and the drive cable would slip in the rim and the wheel wouldn't be able to turn! This requires a lot of on the fly estimates of weights of the riders as it is being loaded.
In order to get a see how good we did with our estimations we turned to the internet in order to try to hunt down some of these values with a Google search. This lead to a discussion about what keywords we could use to hunt down this information. A search of "height of the central axis of the Ferris wheel at Atlantic Playland" was not very fruitful... an essential skill to solve a problem like this is to translate mathematical language into common terms that you can use for a Google search. Ve Anusic has a great blog post where he discusses a similar problem and the discussion with his students about the information you need and the information you might find online. First we did a search to find Atlantic Playland's website and found that they called their ride "Big Ellie". A search for this name lead us to believe that this Ferris wheel is a No. 5 Big Eli wheel made by Eli Bridge (I later emailed the park and confirmed that this is indeed the model of their Ferris wheel). Eli Bridge's website gave us some interesting information but not exactly what we were looking for. A bit more searching and we were able to find a pdf of the owner's manual for this ride that included a helpful diagram.
It is only after we were able to answer some of the students' questions regarding the video of the Ferris wheel did we start to talk how we might mathematically modeling the height of a person riding the wheel over time and the periodic nature of this function. Students were much better able to make sense of this visual model once they had a good grasp of the context of the problem.
Nova Scotia Mathematics Curriculum Outcomes
Mathematics 12 - RF03 Represent data, using sinusoidal functions, to solve problems.
Pre-calculus 12 - T04 Graph and analyze the trigonometric functions sine, cosine and tangent to solve problems.