Category Archives: real world

The coolest (pun intended!) piecewise function that I ever did see! (But didn’t have time to explore).

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In Precalculus we’re covering trigonometric functions (sin, cos, etc).  These functions are periodic in nature (meaning their pattern repeats over time) and can be used to model all sorts of real life scenarios that do the same.  I was planning to do a challenge problem today in class from the book that uses these functions to model ocean tides.  See book problem:

While it’s a fine problem, I had an experience over the weekend that has made me change course (I love when this happens!).  I was shopping at a store with my family that had a sign in the window, “Open 24 hours.”  My 6 year old daughter asked what that meant and I tried to explain to her that there are 24 hours in a day and that the store never closes.  She then asked when does the 24 hours start, which is such an interesting question!, and I was trying to explain that you can start and stop the 24 hours wherever you want:  from 3am to 3am, from 10pm to 10pm, etc. and I realized (because I’ve got trig on my brain) that it’s a lot like a sine or cosine function.  And from there, a much cooler problem was born.  I decided to share this story with my Precalculus students and do our own trigonometric curve modeling to data that is periodic over a 24 hour period:  weather/temperature data. 

I decided to do as little of the problem setup as I could, based in part by a fabulous Ted Talk by Dan Meyer.  I also genuinely prefer class activities that feel less like a perfectly set-up textbook problem and more like a messy, real-life scenario.  I planned to have them collect temperature data for Walnut Creek (via weather.com) over a 24 hour period and to start with a graph of the points. 

Here’s what we came up with:

I asked the students to comment on how this looked like a trig function and how it didn’t.  Let’s start with the ways it looks like a trig function.  Here’s where the math skills kicked in.  We used our knowledge of transformations to build a trigonometric function that matched these observations made by the students:

  • It looked like the cosine function (see small graph on left) shifted over 4 units.  
  • The period was 24 hours (i.e. how long before the pattern repeats)
  • The amplitude was larger (i.e. the difference from average temperature to the maximum or minimum temperature)
  • The average temperature was 49.5 degrees (and they realized they could average the maximum and minimum to find this).  
We incorporated all of these facts to get this beautiful function and curve:
Then we checked a few points to see how well the curve fit our data.  This is where it got interesting.  In some places, the curve did not fit our data well at all.  That’s when we talked again about the ways in which the data did not look sinusoidal:
  • A student spoke up, qualifying that what she was trying to say was hard to explain, that the graph wasn’t balanced.  (Side note:  Isn’t great when our students have to struggle to communicate what they see?  Instead of me directing them with my own language and they then repeating, it was awesome to leave the question open ended and force them to have to develop their own language).  She was absolutely right.  The temperature cooled down much more quickly when the sun was out than overnight.  This is an instance where are data was not sinusoidal. 
  • We also talked about the fact that sinusoidal functions perfectly repeat over time and that doesn’t happen with weather data.  The high one day is not necessarily the high the next day.  
What I had wanted to do, but we ran out of time because the discussion up to this point was so rich, was share what I found when I did a quick google search for whether weather data is really sinusoidal.  I found this little gem.  It’s basically the coolest application of a piecewise function I’ve ever seen!
“Although using hourly weather data offers the greatest accuracy for estimating growing degree-day values, daily maximum and minimum temperature data are often used to estimate these values by approximating the diurnal temperature trends. This paper presents a new empirical model for estimating the hourly mean temperature. The model describes the diurnal variation using a sine function from the minimum temperature at sunrise until the maximum temperature is reached, another sine function from the maximum temperature until sunset, and a square-root function from then until sunrise the next morning. “  Full Article Here.  

I did share this with them and we talked very briefly about the fact that weather data is sinusoidal, but it’s more complicated than that.  Temperature varies in three distinct pieces:  sinusoidal from sunrise to high temperature, a different sinusoidal model from high temperature to sunset and then a third piece to cover from sunset to sunrise that is modeled by a square root function.  It pains me that I couldn’t spend another day on this project.  This year we’ve covered piecewise functions and square root functions they have all of the skills to build this very realistic non-cookie-cutter model.  

And, perhaps this is a topic for a future blog/discussion:  how do we make time for these unexpected projects that require time when we work in a system with courses with already full scope and sequences?  Can (should) we swap quality for quantity?  My own bias tells me that doing deep, complicated projects like these will have a lasting impact, both on their retention of the specific skill (in this case trigonometric functions) but also on their perspective of the world around them.  The perspective that math can model the world around them, but it’s often not simple, often requires more than one equation/formula/function, and often it’s not something that can be solved in a 45 or even 80 minute period.  

Ā”Conduzcamos por la Ciudad!

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I have been taking a course at UC Berkeley for six Saturdays since September, and I finally finished my last course this past Saturday. During these courses, I have begun to move even more away from simply memorizing vocabulary and grammar and truly incorporating the material into real-world applications. 

In Spanish 2, which is a combined Sophomore and Freshman class, students are learning about city vocabulary: streets, signs, turn, go straight, keep going, and also store names: fish market, fruit stand, park, supermarket, bakery, bank etc. I have also heard several of my students chatting about getting their licenses, or taking their permit tests…or failing their license tests.. šŸ™ 
AND THEN I HAD AN EPIPHANY:
I wanted to make city vocabulary as real as possible for my students.
 In groups of 3-5, I had my students open Google Earth and investigate a city in Latin America that they had heard of, or that they were interested in. They basically virtually traveled through all of Latin America, Cuba, Puerto Rico, and Spain and were able to see street names and people’s houses, and a lot of trees. They also got side tracked and searched for their own houses and Carondelet as well…but back to the assignment. 

Once students became excited about looking at cities through Google Earth, I had them pick a city and try to replicate it in the most basic from onto butcher paper. Some students drew Cartagena in Colombia, others drew cities in Mexico and Puerto Rico. 
They drew streets, and labeled the street names and round abouts etc.
Next, they researched common shops based on the vocabulary that they could find in or around the city. They were able to make cultural comparisons and realize that there is a CROCS store here in the US, but also in Colombia,  and the students were surprised to see this. They labeled their stores with the Spanish word, for example: pescaderĆ­a for fish market, and they labeled the name of the market. 
Next students, learned the “nosotros” commands “Let’s drive” “Let’s go” and they also practiced their informal commands as well. Once they were relatively comfortable with their commands, students finished coloring their maps.
When the maps were complete, each group received a “Hot Wheels” car that I had gone out and purchased. They got to pick their cars: the truck, the garbage truck, the race car, and they were excited to drive their cars.
Students took turns filming each other giving one another directions using commands and listening and “driving” their cars according to the directions their peer was giving them to get from one place to another. Students had a blast doing this activity and they were able to integrate culture, grammar, and vocabulary into a fun, innovative, creative project in a group.
What surprised me the most, is that students went completely off script and started to spontaneously incorporate vocabulary from earlier chapters. They decided to make their cars have accidents and they needed to go to the hospital because the driver had an injury. Hospital and injury vocabulary happened way back in chapter 1, however students were still able to recollect the information and use it in an unrehearsed in a spontaneous way to make a story with their maps. This is what pleased me the most. As a language teacher, my goal is to have my students produce the language as spontaneously and unscripted as possible, and they became excited to create their own individual scenarios spontaneously.
Overall, I enjoyed this assignment, and so did my students. I got a lot of great feedback that they were able to use real-world applications in class. 1 week later, I quickly quizzed students on the vocabulary and the commands, and they were all able to produce accurate answers without having studied before.
Here is a sample video: