Gamifying the Unit Circle

Introduction

Precalculus students must be able to evaluate trig functions quickly and easily in order for them to be able to handle later trig topics better. To assist students in that respect, throughout the years I have made sure that students compete in groups against one another in a game called UC Bowl. Below is a description of the game and how it is managed in class.

The Details!

• The class was broken onto three competing teams.
• Each team was allowed to have an ID sheet along with a UC chart.
• Everything else including calculators had to be stashed away.
• And each team was given a small whiteboard to write down final answers to shouted out problems.
• Each rounds will involve a different team member acting as the scribe who would write the answer and quickly raise the board when prompted by me.
• I am standing in the from of the room with a TI calculator and a rubber-head hammer as a gavel.
• The TI calculator runs a Trig UC program written by a former colleague of mine (Thank you Eric Kamichke.) The calculator draws random Trig or Inverse Trig questions for students to answer as quickly as I allow for time-wise in any given round.
• I hit announce the problem to be solved, hit the rubber hammer, the students begin solving the problem, I give a warning to write final answer, and then I hit the hammer once again to have the scribe raise the boar immediately.
• Delays in raising the board results in a zero earned by the group and a word answers leads to a zero as well. Right answers are awarded one point each.
• The scores are tabulated to keep track of the scores.
• Between rounds pauses are taken to allow group members to reflect and discuss amongst themselves.
• The winning group earns sweets and cool treats that vary from year to year.
• Below is a sample of images showing some of the items listed in the above bullets.

Table Set Up for a Group in UC Bowl

Another Group's Table Set Up for UC Bowl

More Table Set Up for a Group in UC Bowl

Scoreboard for UC Bowl after some additional rounds!

Final Scoreboard for UC Bowl after the last few rounds were played!

Closing Remarks!

• The engagements was wonderful by all teams.
• The competition was fierce and even one team emerged as the wild posse team. The members dancing whenever they score points.
• The team that came last had a good comeback in the second class day that the UC Bowl was conducted and they still have chances to redeem themselves.
• The importance of UC and knowing values of trig and inverse trig functions began to sink in as an important part of learning trigonometry.
• In later edition of the game, the trig ID sheets with its UC chart will be removed to allow students to do everything in their heads.

Final Note: Please, use the comments area to share your thoughts on this activity and what kind of games you engage your students in math to make their learning fun and competitive in a healthy way. Thank you

The Value of Discovery through Investigations [Part 1: Inverse trig Functions]

Introduction:

After a review of inverse functions, their domains & ranges, and how their graphs are related, Precalculus students were broken onto three groups to investigate inverse trig functions of the basic three trig function (sine, cosine, & tangent) themselves prior to me teaching about them. This process was pursued to achieve the following educational goals:
(a) To offer students the chance to review past material, trig functions, their graphs, and domains and ranges of functions and their inverses.
(b) To help the instructor to gauge how well prior material was grasped by students that other means of assessment may not have captured.
(c) To assist the instructor in figuring out how much transfer of concepts students can do on their own without much intervention from them.
(d) To enable students to use technology as means of verifying and rectifying some misconceptions they may still have held prior to the coverage of inverse trig concepts.

The Prompt & Group Findings:

The following figure shows the smart board with the prompt written on it.

Activity Prompt

The groups then began drafting their own papers in which they addressed the tasks of the prompt. Following are copies of the raw work the groups did without being allowed to use any references of any kind nor any calculators.

Sine Group work shows what they understood and what they misunderstood about sine functions.

Cosine Group work shows that the concept of inverse of a function was understood in terms of reversal of roles of inputs and outputs. But graphically speaking, the idea did not sink in yet in terms of the use of the y=x line and restricted domain.

Tangent Group work shows that the concept of inverse of a function was understood in terms of reversal of roles of inputs and outputs. But graphically speaking, the idea did not sink in yet in terms of the use of the y=x line and restricted domain.

Once the groups were done with the prediction phase, they were instructed to use technology to verify the outcomes of their work and then update their findings regarding inverse trig functions and reflect upon them. Below are a few images that illustrate the checking process followed by a sample revised group results with reflections.

Desmos and a TI graphing calculator were used during the verification phase of the activity.

Desmos and a TI graphing calculator were used during the verification phase of the activity.

Desmos and a TI graphing calculator were used during the verification phase of the activity.

Desmos and a TI graphing calculator were used during the verification phase of the activity.

After using Desmos and a TI graphing calculator, the Tangent Group seems to have hit the jackpot of discovery.

To ensure that no group members are left twiddling their thumbs when peers are still finishing their activity, another activity was added. The activity consists of looking at a flier that features the school's organ and then connect what they observe to the subject matter at hand. The organ picture is shown below and the Tangent Group's reflection is shown in the picture above this text.

Cool Math Connection with Organ Pipes

Conclusions:

• The formal coverage of inverse trig functions was preceded by this activity to achieve the goals stated above. The results reported after the technology portion of the activity showed that three quarters of the students made great strides toward understanding inverse trig functions, their graphs, and their domains and ranges. 
• The task of teaching the subject matter went smoother now because it was a mere strengthening of ideas for students who grasped the concepts while it was an eye opener for the ones who missed to understand the concepts initially on their own.
• This activity also made it clear to me that there were still gaps in students' understanding of the basic trig functions (especially their graphs) and the concept of inverse functions in general. So, a nice review of the unit circle and its connection to the basic trig graphs followed. The method of review that was used consisted of a series of class-wide UC-Bowl games that will be featured in another blog post. Inverse functions (especially how to draw their graphs) was also reviewed but now through the official coverage of inverse trig functions.
• The spiraling in tackling concepts was definitely a pleasant outcome of this activity and for this alone I believe the whole activity is worth replicating with other mathematical topics.

Note:

I hope the ideas shared in this post would encourage you to post in the comments area your own wonderful approaches to how you assist your students in internalizing challenging topics such as inverse trig functions. Thank you 

Sinusoid Lab Activities

Nothing helps students get a good handle on mathematical concepts than lab activities that addresses the subtleties and true meaning of the underlying components that accompany such concepts.

Following is a description of a set of lab activities that deal with the sinusoid and its usefulness in trig and in many real life situations that are periodic in nature. 

Day 1: Students were handed a plot, that I obtained in an experiment I conducted as a test run (see still image of video shown below), to model it themselves using the sinusoid (transformed basic sine function) involving the h & k form. First the students had to work alone, then with a partner, and then in a group to assist one another with the modeling process.

Test Run by Mr. Le Nadj! of the Lab the students are about to conduct themselves!

Day 2: Students were handed a data set to enter the first two columns in their respective TI calculators and as a class we worked together to plot the data, model them ourselves as a class, and then the class was broken into three groups whose assignment was to carry out the same process but for columns one and three, one and four, and one and five respectively. The rest of the work consisted of students competing for the best fit equation generated by hand and the group member with the equation that fit the best than her/his group peers will be treated to delicious sweets. No TI sine regression was allowed.

Day 3: Students's regression equations were pitted against one another for each group and the best fit equations were identified and the equations owners were treated to double sweets while the rest of the folks (who did attempt a decent fit of some sort) got single sweets. Afterwards, the sine regression functionality in the TI calculator was introduced and now the students were ready to carry out the lab activities themselves. The setups, the procedures, and all technology matters were introduced. Following, are the pictures of the various setups.

First Setup: The pendulum is facing the motion detector that is resting on a make-shift bubble gum box. The students in this group will alternate in collecting data onto their respective laptops. From the camera perspective, another group is going to use LoggerPro's video capture of the same motions of the pendulum (see picture below.)

Second Setup: The pendulum motion is recorded laterally by the members of another group using LoggerPro's video capture of the same motions of the pendulum that the first group is recording. The idea here is to allow for cross referencing and checking of the various group members' data and results.

Third Setup: An iPad is hanging as a bob from a spring that is attached to a force probe. The oscillatory motion is recorded in two ways, the acceleration of the iPad using SPARKvue app from within the iPad itself (see next pictures) and force using Vernier's dual force probe whose results are fed directly to LoggerPro of the students' laptops. Once again the idea here is to allow for cross referencing and checking of the various group members' data and results using two different means of gathering data related to the same motion.

Day 4: Students did the actual lab; the scene was lively, the graphs were gems (an example is shown below), and the thinking and enthusiasm were Coolism! 

The graph of the oscillatory motion of the iPad shows three parts, the hand picking the iPad to start the recording of the motion, the oscillation, and final the hand picking the iPad again to stop recording the motion.

Once the data collection process was completed and every student obtained her/his data and graphs on their respective laptops, the task of switching LoggerPro's form to theform. This part ended up being the homework that was due next day; every student had to complete the form change and make the predictions requested in parts B, C, & D listed in the lab sheet (shown at the end of this post).

Day 5: Students showed their form changes and their predictions, compared their respective constants, and then discussed their predictions and how they would be checked empirically.

As a whole the activity was not just fun but also led to many insights. The experiment is still in progress in that a followup lab that will make the oscillations dampened shall ensue when exponential functions come into the picture.

Thank you for reading the post and please provide some feedback, suggestions, or variations you deem worth consideration. 

Thank you and take great care! :-) 

Hand Outs:

Use of QRs in Math Class to add a bit of Fun

My schedule has been very hectic as of late; busy taking MOOCs and taking care of school work.

The current MOOC that I am taking through coursera is "Emerging Trends & Technologies in the Virtual K-12 Classroom". The topic of using QRs came up and I recalled that two years ago I prepared a little game a la scavenger hunt to help my students review for the Precalculus test. 

Following is a brief description of the said activity:

1) I used classtools.com (I strongly recommend this cool site; Coolism!) to generate the scavenger hunt for this activity. The site guides you through the whole process.

2) I entered my five questions, generated the corresponding QR codes (try them out by pointing your own mobil device that has a QR reader), printed them, cut them up, and then posted them around the building in conspicuous places that the students will have to find. 

3) The students were split into groups and the adventure started.

4) The five questions were appropriate for the 50-minute class. The group that completes all solution processes first and correctly got sweets.

5) The session was fun as the students attested and they got a bit of a review.

QR Code of Question 1

QR Code of Question 2

QR Code of Question 3

QR Code of Question 4

QR Code of Question 5

6) These are just a few examples of how such a technology may be used. Adding games and fun to any learning activity is a big plus.

Please, reply to this post and make sure you add your own game and fun-filled activities in the comments area. Thank you 

When Trig Meets Series and Sequences

Combining the ingenuity of one of the problems posted by Maths Challenge site, "Corner Circle" and the versatility of Desmos this problem led to further neat sub problems that end up connecting trigonometry to topics of series and sequences. Below is the link the investigation "Mr. Le Nadj! Desmoses "Corner Circle" Problem." 

Please, give this activity a whirl in your class and report back as to how it goes for your students. Thank you

Updated Unit Circle Investigation in Desmos

Once my students worked on Unit Circle Investigation 2, new cool things emerged that I decided to add to the first two investigations and committed them to Unit Circle Investigation 3 within Desmos. Please, check out this new investigation and provide feedback and how did it work for your students? Thank you

Unit Circle Desmos Investigations

I have just posted two Unit Circle investigations at desmos.com, Unit Circle Investigation 1 & Unit Circle Investigation 2.

Please, take them for a spin, assess their educational value, and then post some feedback in the comments area. If you use these with your students report back regarding its success or lack thereof. Thank you

Invitation to a Twitter Chat #precalcchat

All Pre-Calculus teachers, instructors, and professors are cordially invited to the launch of a series of twitter chats that would address everything Pre-Calculus. The first chat is Thursday, August 22nd starting at 8:30 PM. 

Event Title: "1st Week Matters," & it will discuss topics such as syllabi, pre-tests, technology, TIs, opening activities, etc.

Event Time: Thursday, August 22nd starting at 8:30 PM

Event Hosts: @MrLeNadj & @untilnextstop

Event Sponsor: Global Math Department with Cool Initiative of @jreulbach

Event Twitter Location: #precalcchat

We look forward having you and happy new Pre-Calculus school year!

Cool Free Trig Apps for You & Your Students

I have been covering trig as of late and there are three free apps that I recommend for everyone, students and teachers alike. Following are links to these three apps in order of Coolism rating.

Coolism: FunTrig

Cooler: Trig Wizard

Cool: iUnit Circle

3 Trig Functions Animated

Today in class students were treated to an animation that illustrates how the motion of a point on the unit circle that generates the usual graphs students see in their calculators when they graph the sine, cosine, and tangent functions. The artistic beauty of math shows up when the animation is left to proceed beyond a given set of cycles.