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:

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

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