The Big Ideas of Trigonometry (2)

In order to consolidate what they had previously explored on sinusoidal graphs, we gave the students the Marbleslides Periodics activity. Half of our students were with us and the other half joined us via Zoom (their teacher provided them with the class code). 

Engagement was not a problem: all students got stuck into the tasks straight away and we could use the teacher dashboard to see which students needed prompts and which ones did not. If a student at home needed the prompt, the teacher sometimes did this quickly through Zoom.

We had set the activity up so that we were both co-teachers which meant either one of us could use the feedback button to give feedback to the students working at home.

In addition to the conversational and observational evidence that we were getting we could also use the students' responses to some of the 'Predict' questions (shown below) to provide feedback.

By using the 'Snapshots' tool, we could take some responses for a particular question and look at these with the whole class (those in the room and those working at home). We could then give feedback as to how we could improve these answers by using more precise mathematical terminology.

It was during the Challenge Slides that we really could see that the students had a solid understanding of sinusoidal graphs: conversations were littered with suggestions such as 'Change the amplitude', 'Shift it up', 'Change the b value and it will stretch the graph out'.

Challenge 4 was a sticking point though: 

Students had no problem getting a graph that went through the stars but with the marbles being dropped vertically from the point (0,12), they just settled in a dip without collecting any stars.

At this point, I was a bit stumped too so we stopped the class to try and think our way out of this. I made the point of telling the students that I was stuck and that this often happens when you do good problems. I wanted to model what I do when I get stuck so that they could develop these strategies too.

Firstly, we confirmed that we had pretty much exhausted all the potentially useful transformations.

Then we mused: why don't these work? We agreed that it was the position that the marbles were being dropped from. This of course begged the question: 'if we could change something, what would we change?'

Well, in this case we would like to change the position that the marbles are dropped from. I do confess that I did try to drag the launch point but to no avail!

So if we can't change the position of the drop, what else could we do?

Then the penny dropped (or rather, the marbles): We could divert the marbles.

How could we divert the marbles? With a second sinusoidal and by using domain restrictions!

I then noticed the instructions on the page:

In the rows below, type as many equations of periodic functions as you need to collect all the stars.

Well played, Desmos. Well played.

After a quick exchange of ideas as to what this second graph could look like, the students came up with some great solutions:

It is one thing to hear the cheers of the students in the class as they are successful at each challenge, but it is another thing when one of the students working at home sent this message via Zoom:

Of course, this is all very well but if the students are just being entertained by marbles sliding down ramps and collecting stars, does that actually mean they have learned anything?

After the lesson, the teacher stated quite simply this:

'They get it. They totally get it.'

'It' being the overall expectation we were focussed on:

Demonstrate an understanding of periodic relationships and sinusoidal functions, and make connections between the numerical, graphical, and algebraic representations of sinusoidal functions.

'I can give them all level fours get it and not worry about giving them a test.'

This was important: with face-to-face time at a premium, any time spent on tests takes away from this. There will be situations where a product (such as a quiz or a test) might be a better way to assess or evaluate our students, but my sense is that if we look more closely at our overall expectations, we will see that many of them lend themselves just as nicely to activities such as this where we can use our observations and conversations to assess and evaluate.

More importantly, the first line in Ontario's Growing Success document should be guide our assessment and evaluation practices:

'The primary purpose of assessment and evaluation is to improve student learning.'

The Big Ideas of Trigonometry (1)

Teachers around the world are facing many challenges as we get to grips with new schedules involving online learning, cohorts, quadmesters and even octomesters. For many, the reduction of face-to-face time with their students (the most valuable relationship) has teachers feeling increased pressures in effectively delivering any particular curriculum. I wonder if, now more than ever, we need to focus on the big mathematical ideas of each course.

Last week, I worked with a teacher of a Grade 11 class who were about to begin trigonometry. There are many specific expectations in the Ontario curriculum which (if you only see your students face-to-face for two or three mornings every two weeks) can be overwhelming: where to start?

Instead, we looked at one overall expectation:

Demonstrate an understanding of periodic relationships and sinusoidal functions, and make connections between the numerical, graphical, and algebraic representations of sinusoidal functions.

What activities could we give our students so that they could demonstrate an understanding of all of these especially given the scheduling constraints (a 2.5 hour lesson on Wednesday and Friday for one cohort and a 2.5 hour lesson on Thursday for the other cohort)?

For me it boiled down to this:

• Mathematicians need to describe how things move in circular paths or behave in regular cycles.
• The Unit Circle is key to understanding trigonometric ratios and can be connected to what they know about right-angled trigonometry.
• The Unit Circle gives us the sine and cosine graphs, and we can see how these are connected.
• We can transform these graphs in a number of ways similar to what we have learned with other functions.

These are the activities that we did and (spoiler alert) allowed the students to demonstrate their understanding.

1) Graphing Stories

As a minds on we gave them two videos from the excellent Graphing Stories site.

Plot the height of the waist above the ground:

The distance of the person on a roundabout from the camera: 

Both videos are by Adam Poetzel (Twitter: @adampoetzel).

I made sure that students had the chance to watch each video four times so as to allow them to improve their sketch graphs each time. Once we had done these, I could point out that straight line motion results in graphs with straight line segments. Circular motion, on the other hand, results in a wavy graph that repeats itself.

I could now tell the students that we were going to learn about the second type of graph.

2) Redefining Sine and Cosine

As an interlude, I asked them to write down a definition of what they understand 'sine' means. This puzzled them for a bit, so I rephrased it and said, 'Draw a right-angled triangle and use this to describe what we mean by the sine of an angle and the cosine of an angle'. 

This worked much better: they were confident in telling me that:

 sin θ=opp/hyp and  cos θ = adj/hyp.

This begged the question: what is the largest value that θ can have?

After a bit of debate, they agreed that θ had to be less than 90˚.

I told them that if this is the case, then our current definitions of sine and cosine would not help us with this triangle:

We could use the cosine law, of course, to find the missing side but this means that we have to find the cosine of 140˚. To do this we have to redefine the sine and cosine functions.

3) The Unit Circle

I used a Desmos graph to show the students what a unit circle and by seeing the right-angled triangle formed in the first quadrant, we can now define the cosine of an angle to be the x-coordinate associated with that point and that the sine of the angle is the y-coordinate of that point.

From here, we can see straight away the values for the sine and cosine of 0˚, 90˚, 180˚, 270˚ and 360˚.

The unit circle is a powerful mnemonic device so I recommended that they get used to sketching it!

 

4) Connecting the Unit Circle to the Graphs of Sine and Cosine

I asked the students to sketch a graph and plot the known values for sin θ between 0˚ and 720˚. They got something like this:

Now, it was a case of asking them to 'fill in the gaps' and sketch the complete graph. They all did this successfully and I reckon that the graphing stories they did earlier helped them see this. I did the same for the known values of cos θ between 0˚ and 720˚:

Again, they were successful. Now I could show them this lovely Desmos graph to confirm what they told me:

With these two graphs established, the students could tell me:

• that the maximum and minimum values for sin θ and cos θ are 1 and ¯1.
• that the cosine graph is just a sine graph shifted to the right (and vice versa)
• that these neverending waves (which we can now call sinusoidal) are made up of repeating periods, and that the length of one of these periods is 360˚

5) Transforming Sinusoidal Graphs

Next, I grouped the students into threes and had them stand (socially distanced, of course) by a whiteboard, with each person having their own marker. They had downloaded Desmos on their own devices and I quickly showed them how to scale the axis so as to allow us to work in degrees.

Their task was to find out the effects of each of the following transformations:

• y=sin x +c
• y=a sin x
• y=sin (x – d)
• y=sin b x

Since they had already explored transformations of other functions, they were able to tell me what each of these effects were. With a bit of further provocation, they could tell me that the 'a' value gives us the amplitude of a sinusoidal graph (a measure of how 'tall' the waves are) whilst the 'b' value can be used to calculate the 'length' of one period of any wave.

I finished the lesson by asking them to make sure that they summarised what they had learned in a short note. I told them that this note would be useful in the next lesson when they would combine two or more of these transformations.

Afterwards, I chatted with the teacher and we both agreed that we were pretty convinced (through conversations and observations) that the students had a solid understanding of the overall expectation. The Desmos activity that I had planned for the next session would confirm for us how solid this understanding was.