Linking Unit Circle and Trigonometric Functions

In my opinion Mathematics is most powerful when the same concept can be demonstrated in multiple representations. Luckily most concepts satisfy this but I have found that Trigonometric functions are a particularly great example of this. We can plot the values on a Cartesian Plane with a reoccurring cycle or on a unit circle (rotating as the functions move through cycles). I also see the set of algebraic rules as another representation of the concept.
Students often hold onto the first representation that they understand, however being able to translate their understanding into a variety of representations truly shows deeper understanding.

Using diagrams such as the following to superimpose the trigonometry function on top of the unti circle shows the piston like function of 'piston' like nature of the Sine Function.

But then this begs the question, what about Cosine? Well we could subtract pi/2 radians (or 90 degrees), or I have found showing the Cosine function on the Y axis (I.E. x = COS(y)) students can visually connect connect the Unit Circle and Cosine Function.
Clearly it is important to highlight that we generally see the Cosine Function moving Horizontally (I.E. y = COS(x)) but this can atleast link representations.

 Once the Unit Circle and Function are linked then we can review where there may be multiple answers for a certain Sine, Cosine or Tangent value.

Providing student an interactive aid can assist in this being explored by students. They can move the 'a' value to extend the angle.
Then additionally they adjust other variables in "y=b f(cx)" to change the size of the circle, or adjust the impact of the angle.
I have found after providing basic description of the below interactive students learn best by playing with it, and trying to use it to answer questions with a set range of x, but with multiple solutions. Click below to explore further.

 

Algebraic Super Heroes

Introducing algebraic law can be boring for students... and for the teacher too... well it is a rather dry topic.
Students struggle to see the point to remembering which law Associative or Distributive Law, and how do you even say Commutative (are you sure it isn't Communicative or Communitative?)
And the real point of empowering student with tool to do Algebraic Manipulations, is lost to repetitive matching exercises.

As Algebra becomes increasingly complicated understand in higher year levels what can (and can not) be done becomes more important.
Visual aids are very helpful (Mathisfun.com has some good visuals here) but to introduce more visual thinking and introduce some Art into Maths (STEAM instead of STEM), I challenged the students to develop some superheroes with the powers of relevant Algebraic Laws.

The students were given a piece of paper and some pencils, and as we described Algebraic Laws, they were to draw a super hero who demonstrated the law in some way. They had a lot of freedom to use creativity. A few students decided they wanted Sports Stars and others created Super Vehicles while some were happy just sticking with various super heroes.

We started with Commutative Law. We discussed the law, reviewed some visual representation. Some students quickly had ideas, others waited for my example, which swung across the screen... it was Commutative Chimp, from the example many students started creating.

I gave the students a few minutes to get started... long enough to get an idea down, but not to finish the 'pretty' drawings. 

 All students had to pause, and then I presented Distributive Law... this time I saw more students with interesting ideas starting before I finished the visual explanation. I would usually stop them, but they were very keen and clearly running with idea so I just ensured that any other students could hear the instructions.
My second example came bounding across the screen; Distributive Dog.

This time I gave the students slightly less time. All the class had come up with interesting ideas and were very keen to show them off. I didn't want too much sharing (not yet!) so I continued on to our third and final law.. Associative Law.

Again description, then visual representation and finally a sneaky Agent Associative example.

The students were very keen to create attractive and accurate posters to describe their Super Heroes (or similar characters).

I set a timer for 8 minutes for students to see the time counting down and to manage completing their creations.

When the time ran out, I got students into groups of 3 (or less). Each group allocated a member to be A, B & C. 

I explained that we were doing a sharing Gallery (like an Art Gallery). We got all the students to BluTack their posters, with one Wall for A, one for B and another for C.

As a class, we all started at "Wall A", we reviewed the posters and could discuss with the artists. Were there any Super Powers that were incorrect? 

Did we really understand the Laws?

Really the Gallery was for checking understanding, but students were proud of what they created and we keen to share.

At the end of the sharing students were permitted to take their posters home (and many put them in their revision folder for future reference.)

In some ways it may have been better to give all of the laws upfront and let students free with them, but giving one at a time ensured focus and allowed each additional law to be a type of iteration for improvement.

I feel this was a good lesson to give students a creative outlet in Mathematics and differentiate learning for some students could repeat the teacher's examples while other could create truly new and innovative characters.

Angle Chase Quiz

After having spent some time with students learning about relationship between angles, parallel lines and internal angle of shapes, the questions all seem to start looking the same. However when I came across a series of Angle Chases on Cut-The-Knot.org (http://www.cut-the-knot.org/WhatIs/WhatIsAngleChasing.shtml) I found it quite engaging, and *spoiler* so did the kids.

One of the particularly interesting angle chases is embedded below.


I took the fourth diagram and loaded it into an online quiz (using Socrative). The Socrative quiz includes images for the students to work along with as well as give students immediate feedback on what the correct answers are (I think I got them correct). You are welcome to use the quiz by logging into Socrative.com and import the quiz SOC-16913178.

The Angle Chase allows students to build on what they have discovered through the exercise (building on the "Epic Meaning" of the activity) and presents the questions in a manner different to the way textbooks generally structure these type of questions.

From a teacher's point of view keeping a track of student's results is vital. Socrative automatically tracks the student's attempts. This allowed me to export the results and check where the students are up to (with names removed below is the report). 


The class results showed that up to angle "h" most students performed well, and then students had mix results. Angle "j" and "r" both achieved less than 30% of correct answers so as a class we discussed these angles, allowing the high performing students to explain their thought process and teach the students which did not correctly solve those angles.

Running of the angle chase took around 20mins, after which some students had completed all angles, and others were beginning to lose focus. At the end of the activity many students asked for more, so I allowed them to work through the other diagrams in Angle Chase diagram.

Online canvas for students to explore Probability

The majority of these come from Cambridge University's Nrich Maths Resources (http://nrich.maths.org/)


For best effect view the Sway using Full Screen:

Dance Dance Transversal

Dance to learn angle related to transversals

I have seen a few different teachers using this dancing method to practise learning related angles between parallel lines.

Algebrainiac did it here :https://algebrainiac.wordpress.com/2013/10/22/dance-dance-transversal/

St Mary Star of the Sea did it here: https://www.youtube.com/watch?v=QMBTPuNMHvM

Communicating Mathematically has a guide here: http://jennvadnais.com/2015/05/17/dance-dance-transversal/

And I understand the activity originally come from a series of discussions at a #msmathchat

Overall the concept is that the students use masking tape (or something similar) to draw 2 parallel lines, and then a transversal. The teacher instructs the names of each pair of angles (Corresponding, Alternate, etc) and then sets up a presentation flashing up each type of angle and the students dance so that their feet land on the relevant angles.

It seems our curriculum is somewhat different in the naming of certain angles, so I have taken primarily from MathIsFun.com to name my angles, then set some music for students to dance to the Angles.

Dance Dance Transversal - Introduction Level (Australian)

The first video has a nice slow song, and plenty of time to get to know the angles and introduces using the related angles.

Dance Dance Transversal - LEVEL UP (Australian)

Once the students are warm and know the angles well, they get some challenges to extend their understanding of angle pairs (and have a little fun). The Level Up video is faster, and ends with a very fast section intended as a joke (although 2 students reportedly succeeded all the way to the end...but I am not sure if I would)

The students had fun, and were very active for a final period on a Friday.

Teaching Logarithmic Graphs with Desmos

Teaching Logarithmic Graphs

Exponential Functions lead to many discussions of Moore's Law, Dominos and Paper Folding, however Logarithms are somewhat less immediately apparent.

Exploring how Logarithmic Functions transform is valuable example of transformation of functions more generally, so making it playful and accessible will aid the students understanding.

To enable this I created some challenges for students:



Answers for these challenges are linked below.



Once students are confident with understanding Logarithmic (and Exponential) graph transformations, let them test their vocabulary using Desmos' Polygraph. I have created a series of Log and Exponent Graphs, the students need to ask questions like Guess Who, to identify which graph they have picked.

Have a look at the activity here (student.desmos.com Class code: zhad)

or you can run the activity in your classroom through here.

Through these activities students can develop a strong understanding of Transformation of Functions (in particular Logarithmic Functions) while essentially playing with computers (and thanks to a great foundation set by Desmos)

Online Canvas for revision

The test is coming up... how do I give my students the best way to revise the content?

For previous revision lessons we prepared mind maps, flash cards and revision sheets, but this time we will be using an online canvas covering all of the content, with links to various informative and interactive sites to support the students preparation for the test.
This was something different for the Year 8 Class, and they seemed to really enjoy it.

When we got into class, I shared the link with the students. Different students gravitated to different aims of the sway. As I circulated the classroom, I could see where each student was focussed.

Some went straight to the chapter review test, others diligently created their own revision pages in their workbook from the canvas and some went straight to the interactive games and websites that I had linked to. This allowed for a differentiation based on student preference.

As I circulated the classroom, I encouraged student to focus on areas that they were not sure of and I knew that some students had missed parts of certain topics. Interestingly most had naturally tended towards areas that they had missed.

For homework, I suggested that there was a mistake in one of the pictures on the canvas- they needed to find it. I do this to encourage the students to question and ensure they truly understand the content.

The students interestingly found two, and returned with a range of great questions (highlighting where further clarity is required).

Feedback from students was that they found Sway useful and for me it took limited effort to create.


You can see my Sway below or per the link.
https://sway.com/XtWfvGwyrV3Fp4SO



Creating the Sway was incredibly easy. I logged into Sway.com (with my free Microsoft account) and named the Sway "Year 8 Semester 1 Revision", set headings for each topic, then I clicked Insert. Sway was able to search the internet and recommended pictures, videos and even wikipedia snippets that related to the content.

Working through the topics was as simple as drag and dropping the images, videos and embedded interactive objects about the content into the canvas. As I identified specific aspects of the topic, I was able to search and find precisely the material I wanted. 

Once I was satisfied that all content was covered, I turned to the look and feel of the canvas.

There is always the option to leave it chance and click "Remix", but I wanted to be a little more directive. I arranged the navigation to be vertical and selected the font, colour and background that I wanted.

Overall, I found Sway to be quick to create, valuable to use and easy to share with my students. I have already used this with my year 10 class as well as year 8 classes, all with plenty of success. I certainly will be using it again.