Intro to trig graphing with Desmos Activity Builder

Background info: I co-taught this Desmos Activity to 11-12th graders taking Trig/Math Analysis.  They are going to start the chapter on graphing trigonometric functions on Monday, so this is an introductory activity to help students develop some of the vocabulary and background knowledge that they will use for the chapter.  This also happened to be Spirit day, with shortened classes and a spirit rally for that night's football game against the rival school.

Despite the somewhat challenging circumstances, the activity went great.  Students were engaged from bell to bell.  We heard comments like "thanks for making math fun" and "when can we do this again?"  The biggest hit of the activity was the ferris wheel screens (11-12) with one girl claiming that she couldn't even focus on the rest of the activity because she was too distracted by those  screens.  

                                   Note: The ferris wheel looks way better in the activity.  
                                   Check it out here on screens 11-12.

What I like most about an activity like this is that students begin by using informal language to describe how a graph changes, and then we build the formal vocabulary from their descriptions.  I gave maybe a 5 minute lecture in the middle about how the amplitude, period, and vertical shift are connected to the standard form of the sine graph (y=a*sin(b(x-h))+k).  We didn't look at horizontal shifts on this day as this was an introductory activity.  Students spent the period checking their understanding of both the vocabulary, and how it related to the equation for a given graph.  They even had a chance to apply their knowledge of the sine function by looking at how to model a ferris wheel rider's height versus time.  The other teachers and I used the teacher dashboard all throughout to see which groups of students needed help on a particular screen.  

The only challenge we ran into throughout the day was when students deleted elements on the graphing pages.  Some of them deleted the slider on the ferris wheel screen (11), so they weren't able to do the task at all.  We recommended that they try the following similar problem, or log in a second time.  Being able to reset a graphing screen would be helpful.  I've also been telling students to stay out of any construction folders, explaining to them how easy it is to mess up some aspect of the page, in which case they won't be able to complete the task.  

Overall, a very successful activity.  Thanks to #mtbos for giving me the ideas for the screens in this activity.  Two resources in particular that were helpful are here (by Steve Phelps) and here (by John Golden).

#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

Geogebra Series I: Angles in Standard Position

This past year my math department started using Dropbox to share files by course team, and the way it improved collaboration was very exciting.  One of the course team leads set up a folder for each unit, with subfolders for the normal items like assessments and lesson plans, but she also added a subfolder for common core materials.  The teachers for this course team (Algebra II) started posting Geogebra files in the common core folder, and we would discuss how to use the presentations at course team meetings.  Feedback after these lessons was mixed.  Some teachers thought the presentation was extremely helpful, and others thought the students still needed more practice.  Our team knew that it would be ideal to observe each other using the presentation, but that doesn't always fit into the schedule.  We started using screencasts and teacher guides to help out with this situation.  In this series I will share some of the teacher guides and screencasts for Geogebra tools I have used over the years.  I think the questions that we choose to ask can make all the difference in how useful the tool is, so if you have a question to add to this list, please share.

I chose angles in standard position first because it was the first tool I used to introduce a concept to students, and it made such a big difference in level of understanding from the previous year.  For this lesson it is helpful if each student has a calculator so that they can all participate (any type is fine).




A student version of this tool is available at the link below.
Angles in Standard Position Tool

Our team went through the process below when introducing angles in standard position.

1.  Enter an angle in degrees into the positive or negative angle input box, then hit enter.  Do not put the negative sign for negative angles.  You also don't need to include the degree symbol.  We'll start with a 200 degree angle.

2.  From here the teacher can talk about the definition of an angle in standard position, and identify the initial ray and terminal ray by color.  A quick review/reminder of the quadrants and the 90, 180, 270 and 360 degree angles fits well at this point in the lesson.

3.  Type 300 degrees (or similar) into input box, but don't hit enter yet.  Have students predict what the angle will look like, and discuss how they know.  Then hit enter and call on a few students  to give the formal name for the blue and yellow rays, as well as to describe the procedure for graphing an angle in standard position.

4.  Now we are ready for the fun stuff.  Type 2000 degrees into the positive angle input box, and let students predict what will happen.  Where will the angle terminate? How do you know?  How do you graph an angle bigger than 360 degrees?  Discuss with your partner.  I'll give a couple of minutes for students to discuss in pairs, and then we will share out before we reveal the answer.

5.  Spend some time talking about why Geogebra graphs a 2000 degree angle as a 200 degree angle.  Have students predict some other angles that will be graphed the same as a 200 or 2000 degree angle.  Write some predictions on the board, have students justify answers.  Introduce positive coterminal angle vocabulary.

6.  Show students how negative angles are graphed.  I start with -160 degrees on the same diagram from step 5 so that they can see that we can have negative coterminal angles as well.

7.  At this point in time I hand out this problem set, and I model the first problem with them so that they know how I want the work to be shown.  This includes showing work for how to find a positive and negative coterminal angle, as well as a diagram of the angle.  We get fancy with the diagram and color code, because I am not convinced that they really understand the concept until I see initial and terminal rays along with the number of rotations and the direction of rotation.

8.  I also model the second problem with the class, which asks students to graph a -1160 degree angle in standard position.  The worksheet asks for a positive and negative coterminal angle, and so I encourage students to find the one that GeoGebra will give (between 0 and 360 degrees for all cases).  Students then complete the rest of the problem set, using Geogebra to verify answers.

This tool shows a positive angle larger than 360 degrees being sketched dynamically.  I am guessing there is a better tool out there to demonstrate this visual, and if you know of one please share!

In addition to being a powerful visual aid, Geogebra is a great tool to check for understanding.  I try to keep the emphasis on the formative assessment aspect of a presentation when possible, and I love that the tools can be accessed quickly for review at the end of a class or on another day.

Useful Links:

How to graph a 2000 degree angle with Geogebra
How to graph a -1160 degree angle with Geogebra
Geogebra Tools from Making Math Visual blog.
Geogebra Wiki.  Includes tutorials on how to get started with Geogebra.
Some Things I Wish I Knew When I Started Using Geogebra.