#JulyChallenge2014: Modeling With Trig Functions

Link to Geogebra Book with all 4 applets

Last year was the first year we taught trig graphing in Algebra II, and so it was a brand new teaching topic for a couple of our teachers.  Perfect opportunity for collaboration!

We decided to introduce the topic using Dan Meyer's Ferris Wheel.  This introduction built on intuition and gave students a day to practice reasoning with a context that involved sine and cosine graphs.  The equation for a sine graph was introduced at the end of the lesson, and students used sliders in Geogebra to fit the sine graph to the ferris wheel data of time versus height.

Towards the end of this unit students were asked to model real world situations using trigonometric functions.  We picked a few problems involving tides and ended with a problem from Illustrative Mathematics called Foxes and Rabbits 2.  We used a series of scaffolded Geogebra presentations to help students get started.  The first presentation is below, and is an embedded applet that you can play with.  Type a function into the f(x) input bar and press enter to see it on the graph.  You should try this right now. It is fun! The goal is to get the sine graph to pass through all of the data points.  



We introduced this to students by first talking about the data and how tides are measured.  You can talk about how the moon impacts tides and how the data is roughly periodic.  At this point students should know the amplitude and period for y=sin(x). Next, have a class conversation about how to transform this graph to pass through the given points.  The power of a premade interactive model is that you can pause during the conversation to allow students time to process.  I make use of the think-pair-share structure with the expectation that students may be called on to explain what they know or discussed with a partner.   One of the teachers on the Algebra II team started writing and sharing the questions that she used with these presentations, and the other teachers found this to be very helpful.  Below is a list of questions that can be used with this presentation after the initial conversation about the data.  Feedback  on this list of questions would be great!

1. What is the midline for the Santa Cruz Tides data? (Discuss first, then reveal the midline by selecting the midline box.)

2.  How can I change (transform) the function f(x)=sin(x) so that it has this midline? (Discuss, then type in the correct function and reveal the change in the graph.)

3.  What is the maximum value of this function?  What does it represent with respect to the tides?  Find the minimum value as well.  Discuss and then reveal the max and min lines by selecting the appropriate box.

4.  What is the difference between the highest and lowest tide measurements?  Does this help me find the amplitude, period, or vertical shift of my function? 

5. Now that I know that the amplitude is 1.5, how can I change my function f(x)=sin(x)+2.5 to account for an amplitude that is not 1?

6.  Discuss with your partner what one period of y=sin(x) looks like.  My current graph has been shifted horizontally from the parent graph y=sin(x).  Can you find a new starting point?  How can we change our function y=1.5sin(x)+2.5 so that it has been shifted horizontally to your new starting point? (There are multiple answers here, which can be discussed now or later depending time.)

7.  What about the period?  Note: Most classes will have a formula to use to account for a change in period.  I haven't taught trig in years, so I just use horizontal stretch/shrink reasoning.  y=sin(x) has a period of 2pi, and Santa Cruz tides has a period of 12.  Since 12 is larger than 2pi, I will multiply x by a factor of (2pi)/12.  My new function is f(x)=1.5sin(2pi/12(x-9))+2.5.

One of the teachers from this team made a worksheet for students to use as we modeled the thinking.  I find this to be an important step so that students can refer back as they practice and study.  Please let me know if you are interested in having this worksheet, and I can get you a copy.  

Below is a list of all the tide problems that students worked on.  

Santa Cruz Tides
San Mateo Bridge Tides
Bay of Fundy Tides

Students also worked on Foxes and Rabbits from Illustrative Mathematics.

Illustrative Mathematics Foxes and Rabbits 2 Problem
Illustrative Mathematics Foxes and Rabbits Geogebra Tool

Crowdsourcing I: Concept Intro for CCSS Lessons

I observed a class this past year using the M & M experiment where students posted the results to a Google Spreadsheet.  Recently I participated in a similar lesson using 10-sided dice to conduct the experiment.  In both cases the math world confirmed the results of an experiment in the real world in a powerful way, and served as the introductory and anchor lesson for exponential functions.  Below are the steps of the activity performed with 10-sided dice.

1.  Roll 40 ten-sided dice.  Count the number of dice that DO NOT have a zero showing on the face and record in your table.  Remove the dice that are showing a zero.

2.  Repeat for 20 rolls.

3.  Enter your data into the shared Google Spreadsheet.

4.  Enter the average number of dice without a zero showing on the face into the GeoGebra spreadsheet. Note: It is often easier for the teacher to calculate the average, or to have the spreadsheet pre-programmed to find the average.

4.  Graph the set of points generated in the GeoGebra spreadsheet.

5.  After a class discussion, graph the function that represents the theoretical number of dice left after each roll that DO NOT have a zero showing on the face.

-Enter the function into the Input bar at the bottom of the page

If you cant see the points, select the "Move Graphics View" tool.  You can drag the axes to resize.

Clearly, this function isn't the best fit for the data.  That is because I don't actually have the original data from the three groups.  What I can say is that with even as few as three samples of data, the function fit the average value of the data extremely well.  This activity is one of many that can serve as an anchor activity for exponential functions.  Additionally it provides an authentic reason for students to use technology to model data.  The process of inputting data points, graphing them, and graphing a model function against the data is a process that will prove valuable for preparation for the SBAC test.  Many thanks to Henry Picciotto for his No Limits Workshop this summer at The Urban School of San Francisco.  This workshop provided inspiration for many new lessons, including lessons that integrate technology in a meaningful and innovative way.