Wednesday, March 25, 2015

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.



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.

Sine Group work shows that the concept of inverse of a function did not gel yet for it was confused with -sin(x).

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.



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 now teaching the subject matter went smoother 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 concepts' tackling 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 


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