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.'