Unit Descriptor Draft 1 Tony Knox

Title / Understanding Transformations of Trigonometric Functions
Summary / The students will discover how changing the key features of a function affects the functions graph through interacting with a number of GeoGebra files. In some files students will adjust the algebraic representation see how this affects the graph of the function. In other files they will adjust the graphical representation. The learning has been broken down into a sequence of subtopics that build to understanding the effect of changing , , or in the functions and .
Quizzes will be used at various stages to give students the opportunity to test their own understanding. The immediate feedback the interactive files give when adjustments are made is another form of testing i.e. students can spot a pattern, make a conjecture and immediately test if their conjecture holds true for the cases that they are able to check with the file.
Aims / Students will understand the effect of changing , , or in the functions and
Students will relate their understanding of transformations of trigonometric functions to transformations of other functions e.g. quadratic functions
Students will appreciate that trigonometric functions have many applications in the real world.
Learners & Context / Senior cycle student studying leaving certificate higher level mathematics aged approximately 16 or 17 years old. The group would be accustomed to doing some semi-structured discovery learning using premade online manipulatives. They would also be in the habit of verbalising their discoveries and summarising their learning using things like mind maps.
Outcomes / On completion of the learning, learners will be able to:
  • Sketch functions of the form and
  • Write down the equation of a trigonometric function if given a graph of the function
  • Write down the period and range of a trigonometric function
  • Verbalise the effect of changing , , or in the functions and

Activities / During the learning, learners will:
  • Watch a video presentation of….
  • Have the opportunity to revisit content that would be considered prior knowledge (unit circle, circumference of a circle formula, radians,transformations of quadratic functions……)
  • Adjust algebraic representations of functions and see the effect on the graph of the function
  • Adjust graphical representations of functions and see the effect on the algebraic representation of the function

Assessment / To demonstrate the learning outcomes, learners will be asked to:
  • Match algebraic representations of functions with graphical representations.
  • Sequence a set of functions from smallest range to largest range or from smallest period to largest period
  • Select the correct graph from a list of graphs that matches the algebraic representation.
  • Verbalise the effect of changing , , or in the functions and
  • Sketching some graphs and uploading screenshots/photos of the them.

The ARCS Model of Motivation was chosen as a reference tool.

The Attention of the learners will be grabbed at the beginning of the content presentationwith something that will grab the learners’ attention. At present the plan is that this will be a phrase like “Fasten your seatbelts folks and be prepared to see why EVERYONE ALWAYS says you can’t spell “trigonometric functions” without “fun”!” said in a slightly over-enthusiastic voice. This will be followed by a saying “Trigonometric functions have many real world applications, such as” and then stop and say “Hey, wait, I think it’ll be a lot more interesting if YOU go and find some of these applications and share them with the rest of the class. I’ll remind you about that later. First of all let’s just have a look at the work we did on and the other day…”

Relevancewill be used as a motivation in three ways:

  1. It will be said that the concepts and skills are important for the goal doing well in their mathematics exams.
  2. Their own interests will be used as motivation as some may be interested in the real world applications of trigonometric functions in such areas as music, applied maths, biology, chemistry, physics, engineering etc.
  3. The new learning will be linked to prior learning that, hopefully, they are familiar with.

Confidence will be built through quizzes that have questions that should be easy to answer if close attention is paid to the discoveries that are possible on each of the pages. Of course a student has the opportunity to go back and learn more from the interactive files and do the test again if they are unhappy with their performance in the test. It is the learners themselves who will learn from the interactive files and their success is within their own control which builds real confidence.

So far Satisfaction is the part of the model that I have found difficult to plan for. If I ask students to create a summary sheet that describes the effect of changing , , or in the functions and this could be graded fairly under certain criteria that could be shared with the learners before they begin the task.

I also used the ARCS Ten Step Model. (1) The information about the course was found in the Leaving Certificate Mathematics Syllabus for Examination from 2013 from (2 and 3) The students are described above. (4) Existing materials from the Project Maths website were analysed. (5) Objectives that got the heart of the properties of transformations of trigonometric functions were chosen. A variety of assessment tasks were chosen which included methods such as matching, multiple choice and sequencing. The students will be asked to create a summary sheet that describes the effect of changing , , or in the functions and . (6, 7 and 8) Three possibilities for engaging students through the use of social media were thought ofbut haven’t been decided upon yet. All three might be used. They are:

(i) a forum to discuss what they are unsure of,

(ii) students given a project of finding the uses of trig functionsin certain areas and upload them to a space (like an online whiteboard) where all students can access them. If technically possible students star the one they think is the best and comment on why it is the best in their opinion.

(iii) Students make a summary diagram of what they have learned e.g. they choose whatever numbers they want e.g and they must explain where they see the 4 and the 3 in the graph. This is either created electronically and uploaded somewhere or is made on paper and a photo of it is uploaded somewhere. If technically possible students star the one they think is the best and comment on why it is the best in their opinion.

The tactic of students interacting with files and spotting relationships and testing conjectures was chosen as the main tactic. This type of discovery approach will be used throughout the unit of learning.

(9) I’m at the stage where the activity materials and assessment materials need to be developed. Some samples of them have been developed so they can be evaluated by others so I can get some feedback from my peers.

The 12 Practical Principles of Multimedia Learning were referred to throughout the design process.

Signalling is used on the homepage (and will also be used on other pages).

The learning has been broken into a number of pieces e.g. Period, Range etc. as segmenting the learning is important. The friendly voice, which it goes without saying that I haveis used for the content presentation to adhere to the voice principle. I endeavoured to make the website coherent and planned to have no extraneous noise or distractions on the website. The website is designed to be used with a group of learners who have learned about the key concepts of period and range of sin(x) and cos(x) in class within days of being made aware of the website. This would fit in with the pre-training principle. Other principles (RedundancyPrinciple, Spatial Contiguity Principle, Temporal Contiguity Principle, Modality Principle, Multimedia Principle, Personalization Principle, Image Principle) will also be kept in mind as more content is added.

Elements of the Content Presentation

  1. The unit circle
  2. The work on graphing sin(x) and cos(x) by plotting lots of points that was done in recent days (which includes the concepts of period and range).
  3. The work on radians that was also done in recent days which include the vine of how many radians are in a circle.
  4. (show the table representation??)
  5. Communicating the goal of understanding the effect of changing , , or in the functions and .
  6. A demo of the type of work that is expected to be done with the interactive files i.e. what they should be able to adjust in the various files, questions they should be asking themselves and whether they should have a pen and paper when working or not.
  7. Language of Transformations. Stretch, compress, translate, reflect, shift, scale
  8. Communicating the other goals(researching real world applications of trig functions and creating a summary sheet of the learning)

Have chapters for the presentation so students can easily click to the part of the presentation they want if they want to refer back to a particular part.

Accessibility

Choose background colour and font type and font colour carefully

Provide a transcript of the video or subtitles.

Use website checker website.

Question types

Sequence the algebraic functions from smallest to largest in terms of (i) period (ii) range.

Matching graphs to algebra

Matching Algebra to graphs

MCQ graphs to algebra

MCQ algebra to graphs

Would like to be able to link to hints easily in the quiz.

Journal

A lot of playing around with how the quiz would look (the program is designed for publishing presentations so there were a lot of unnecessary features in the standard way of publishing a quiz e.g. navigation at the side of quizzes).

Recording ideas on phone.

Extreme headaches with html and embedding the first quiz I tried. Navigation buttons and text that was put under the quiz wasn’t shown. The suggested method on the ispringsolutions.com website left out the tags for closing the iframe. A lot of trial and error and looking up eventually got it working. It wasn’t going to be very good if it was only possible to have text and navigation buttons or links above each quiz. It doesn’t seem to be possible to upload files to the backend of a Weebly site and it didn’t seem possible on a Wordpress.com website either, only on a Wordpress.org one that is self-hosted so for each quiz I ended up saving each quiz, which is a folder of files, to a public folder in Dropbox and linking a file in that public folder for each quiz. There was the possibility of signing up to a trial period with an online service from iSpring.

Pushed to combine old skills in GeoGebra and learn new skills in GeoGebra.

Mathjax to be looked at for writing mathematics.

Problems with having more than one GeoGebra file on the same page

References

Leaving Certificate Mathematics Syllabus for Examination from 2013 from

Resources from and Learning Plan on the Unit Circle, Teaching and Learning Plan on Trigonometric Functions, Exploring Trigonometric Graphs)