Spiralling in Advanced Functions (MHF4U)

I’ve been spiralling my courses for the last few years, but this last semester was the first time I spiralled the Advanced Functions MHF4U course. If you’re new to the spiralling idea check out the blog post from Mary Bourassa and the MHF4U website from Al Overwijk and Janice Bernstein. They’re great resources to get you going.

This post is really to remind my future self on what I did this semester and for anyone else asking spiralling questions.

On Planning

Occasionally I will get an email from a teacher who is interested in trying spiralling and the question they usually ask is, — Where do I start? I think most of us need someone to shine the flash light down the path for us to see where to head. I usually start with a table that shows the strands of the course and where the major skills (overall expectations) fit in. I try to group them by themes. This year I my cycle one was about introducing the functions and focusing on graphing characteristics. Cycle two focused on linking algebraic representations with graphical. See below.

From there I keep an ongoing day-to-day plan.

Click to see the live version

On Homework:

In the past I’ve given out homework in a very traditional way, “Tonight, complete page ___ Questions #__ to ___. Tomorrow we’ll take them up.” And what did homework take-up look like in a grade 12 course? Well, for me, it was always “What problems did you have trouble with? Number 8b? Ok, does anyone have that one completed? Kearra can you put that solution up on the board?” If no one had that question right, then I would put up a solution. And everyone watched, twiddling their thumbs (or more realistically — texted) while I put that solution up….or we all watched Kearra put the solution up. Not a great use our of time.

I’ve changed that process over the last year or so. For me, giving out homework comes in a homework set. I got the idea from Al Overwijk and Mary Bourassa. The sets not only have practice problems from the ideas from that day, but also practice problems from other areas of the course. Each night of homework they are practicing most strands of the course. It keeps concepts fresh in their minds and keeps practice going all semester.

a typical homework set

When students come to class they get a playing card that randomly assigns them a partner. Instead of asking which question we should put up, I choose two or three from the set and the pair has to put them up on the vertical whiteboards/blackboards around the room. They are only allowed one piece of chalk or marker between them. I circulate around the room to give feedback and check for understanding/thinking. I’ll routinely yell out to “switch the marker” which forces students to communicate, error check, and defend their work. A better use of our 10 minute homework take-up time. After, students hand in their homework which allows me to check their understanding and gives me insight on what skills we need to improve on (I choose one or two questions to focus on). Gone are the days where I give out homework and I don’t find out what they really know until test time. Now, I know daily. Is it more work for me? Yes it is. But it’s worth it.

Can’t see the video? Click through to the post

After homework take up.

Whiteboards & Note-taking:

Most of our problem solving and practice work in class this year was done on non-permanent surfaces. For some students, parents, and teachers this is a concern since they are wiped away and there is not a record of that work. Here is an email response I sent a fellow teacher this year to address the concern:

“Do your students need the note? Are they asking to take notes? If so, have a conversation with them about what they need and teach them to take pictures of what they need or make notes for themselves. Or have them summarize what they’ve learned after doing the problems as an exit slip.
I sometimes do “important” solutions on chart paper and then they stay up in the room so we can refer back to them.”


As always I’ll be making changes for the next time I teach the course. I want to include solving equations earlier in the course. This year I didn’t bring it in until cycle 3 and I feel like we could have benefited from more exposure. Also, radians need to be introduced in cycle 1 so that it can fuel all of trig for the rest of the year. I feel like it was crammed into the last cycle.

Day-to-Day Outline and resources for MHF4U

See the outline as a webpage

Get your OWN copy of the Google Sheet to modify.  – You’ll need a Google account

Favourite & Fix: Nov. 11

For the Favourite & Fix series each week I’m posting one idea from my lessons that was my favourite and one topic that I need help on. A topic I hope to fix. I’m hoping that in the comments or on Twitter #Fav&Fix you amazing readers can help me out with some hints, tips, and suggestions.


This week I introduced the unit circle to my MHF4U class. I wasn’t happy with the way I introduced the circle in years past so I made a change. I want students to see that our special triangles are just reflected around the circle. Instead of drawing them, or imagining that they are reflected….I wanted them to physically pick them up and flip them and move them. I wanted them to see that the lengths are the same. So I cut out 30-60-90 triangles and 45-45-90 triangle each having a hypotenuse of 10 cm. I created a circle with radius 10. Now each time you place the triangle on the circle we can easily see the principal angle it creates and the coordinates of the point on the circle….It’s the lengths of the triangle….and since the hypotenuse is one the lengths correspond to the Cosine and sine value of that angle. The physicality of this I believe helped allow the students to grasp what the circle shows.


This week in our MEL3E class used Fry’s Bank from Dan Meyer.

This problem, like many 3-Act Math problems, allowed my class to discuss, question assumptions, and uncover math. The problem helped restore some of our great classroom atmosphere that we’ve been missing lately. I want more! This coming week we’ll keep working through compound interest problems. I’m planning on doing Robert’s Not Cashing the Cheque problem.  After that my resources for compound interest problems are pretty thin. I want to continue posing interesting problems to my students. Can you suggest some? Do you have great compound interest problem that keep students curious and questioning? I’m looking for some!! Share those great problems here or on #fave&Fix on Twitter. I’m looking forward to what you come up with.


Thank to all of you who commented through Twitter on great compound interest problems. Here is one from Diana,

Here’s where our class went on Monday:
We started off with Robert Kaplinksy’s How Much Did Patrick Peterson Lose By Not Cashing His Check problem. Go ahead and read his lesson plan.

What made this problem great for our class was the discussion that occurred before any math happened. An amazing argument bubbled up with one side saying “Who cares….what’s the big deal” and the other side saying “That’s just super insensitive…..I could use the interest off that account”. My class from the beginning of the year was back! They had put away the drama that had happened between them and focused on the problem. We guessed at the interest he was losing daily. And then using the info from Robert’s site calculated the interest in the first couple of days. Then broke out the Finance Solver to determine how much was lost for the 27 days.


The class wanted to know more! They wanted to know what he would lose if he didn’t cash it for the year, 2 years, 5 years!! So we did that too.


Next, we investigated Robert’s How Much Should Dr. Evil Demand?

Read Robert’s post to see the plan.

Again, with this group, we didn’t use exponential functions…but the Finance Solver to determine what $1000000 would be worth 3o years later with average inflation of 5.33% per year. We also extended to find how long we would have to wait for $1 million to be worth $100 billion.

Thanks for the help!


It begins! [Day 1]

So….let’s do this! Day 1 is done!

This semester I’m teaching grade 9 applied period 1, MEL3E (grade 11 workplace) period 2, and MHF4U (Advanced Functions) period 3. I haven’t decided yet which course yet to blog about consistently, but I’m planning on picking either MEL3E or MHF4U; which would you like to read about more? I will be spiralling both of these courses for the first time and I’m pretty excited about it. Here’s how day 1 looked:


After grabbing a playing card to determine who their partner and where their seat would be… I did a 5 min talk on my strong beliefs around curiosity, growth and team work we did a couple activities — No talk on course outline, marks, or rules:
Activity 1: Math is Like…. 

I ask students to fill in the statement “Math is like…” I love the variety of statements that come out of this. Backgrounds and experiences play a huge part and they shine here. I have them read their statement to their partner and then take volunteers to read to the big group! I also read some of the grade 9 responses out loud to the seniors….which they find hilarious!

Activity 2: Graph Your Subjects

I saw this tweet a few weeks ago from Jen McAleer.

Here are a few of our pictures:




Activity 3: Pyramid of Pennies
I showed them the Pyramid of Pennies 3-act math problem from Dan Meyer. They wrote noticings and wonderings.

Then using the vertical non-permanent surfaces (VNPS) around the room they worked on solving the problem of How much money is that?




After taking this problem up and discussing that a they’ll be working together vertically to solve problems daily I assigned some review questions for homework.

You can follow my day to day on this spread sheet which I’ll update as we go through the course.


This class ran through the same activities as MHF4U except for some minor differences: In activity 2 they did a warm up graph with placing Dad, Daughter, and Granny on Height vs. age graph. We reviewed what a point would look like for a baby, and for Michael Jordan. After getting the hang of placing points on the graph they graphed their subjects just like the grade 12s.


In activity 3 they noticed and wondered about the Pennies. They made estimates on too high, too low for the number of pennies. I then revealed the answer.
You can follow my day to day on that course here.

Which would you rather read about in the future? The grade 11 workplace? or the grade 12 advanced functions? Help me make a decision!

Match My Graph & Crowd Sourcing Challenges

Here’s a quick synapsis of an activity from my Advanced Functions class with transformations of trig functions.

We used a Custom Polygraph from Desmos to generate talk/discussion on key properties of trig functions (Students have previously dealt with trig functions in grade 11).

I overheard students asking questions about x-intercepts, period, and amplitude. Awesome!

We took a note on key properties of the sine function and cosine function (We ran out of time for Tangent). 

Let the struggle begin!

Students were then asked to work through this Match My Trig Function Activity built using Desmos’ Activity Builder.

Screen Shot 2015-11-19 at 7.44.10 PM

Each slide is set up as a challenge. They are to write a sinusoidal function that “overlaps” the black target function. Students will have to use their memory or trial and error to discover how the parameters change the graph.

Watching the dashboard I can ensure their struggle is productive. I can jump in with feedback when I see they need it.

Screen Shot 2015-11-19 at 7.46.23 PM

Here’s the best part, once they completed all 12 challenges they created their own trig function matching challenge and shared it out on a Padlet board. We had crowd sourced a bank of challenges to work through!  The students didn’t hold back either… They wanted to create hard ones to push their friends.

See the challenge – Live Board Below

Can’t see this board? Click here

That’s where class ended. When we came in the next day  and they all choose at least 5 peer challenges to complete…. And that’s when the taunting began!

To end it off we took a note based on their discoveries of how the parameters changed the graphs.

Click here to create your own Custom Activity Builder or here to create your own Padlet board.