# 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.”

### Changes:

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

# Appointment Clock

In class today we practiced, error-checked, discussed solutions, got peer feedback, got teacher feedback, smiled, laughed, and cringed. Today’s class was supposed to be boring. We were supposed to just practice solving polynomial and rational inequalities. Boring right?

A few years ago I saw an activity structure called Appointment Clock from an English teacher in my district. It was one of those structures you see at a PD day and think… “that’s kinda cool” and then the weekend happens, and by Monday it’s gone. For some reason, this weekend, years later….it popped back into by brain.

To start all students got an appointment clock handout.

They were given two to three minutes to circulate around the room and schedule “an appointment” at the indicated times.

Next, they were given ONE inequality (list of inequalities) and about 7 or eight minutes to solve it. They were to write the solution to their inequality on the handout and keep it hidden from the other students. They were to check their solution using Desmos. I circulated to help anyone who needed it. “Now, this inequality is YOUR inequality….you are the master of this one.” Once everyone was ready, I announced, “Get up, and move to meet with your 2 o’clock appointment. Show your new partner your inequality. Complete their problem in your notes and check with them to verify your answer.” I gave them 7 minutes. This is where great stuff happens. They check with each other to find mistakes, get feedback, improve. After the 7 minutes or so, I announced, “Now, meet with your 10 o’clock appointment and repeat the procedure.” The structure is very much like Speed Dating

We did this for the entire class. Every minute was worth it!

At no time was practicing solving polynomial and rational inequalities boring. Not today!

# Puzzling Dimensions

I wanted to grab some thoughts from you on a possible lesson idea.

On the weekend Jules and I worked on some puzzles. After we dumped the puzzles out I would ask her how big will the puzzle be? She would look up, and with that wondering look say …”pretty big” or “not too big,” but nothing exact….she’s only 6. I narrowed the question down. How many pieces would be along the bottom? Along the side? And she would make a guess. We would do the puzzle and then find out.

This got me thinking.

I was thinking about factors of numbers and how that relates to the dimensions. I also thought about optimal dimensions of rectangles given a set area.

If a puzzle had 60 pieces what could the dimensions be? 100 pieces? 1000 pieces?

Take Elsa for example. With 48 pieces do you know what the dimensions will be? Think of some possible combinations. Got them?

And….. bam! Did you think of 8 pieces by 6 pieces?

The puzzle we worked on had 100 pieces and it was a 10 piece by 10 piece puzzle.

I feel like there is a lesson here but I’m not 100% sure where it fits. It may fit in many places.

If Act 1 is a short clip of us putting a puzzle together like below, then how does the rest of the lesson go? How do you see the rest of the lesson play out if you teach Kindergarten? Primary grades? Middle school? High school?

I would love to hear your ideas on the lesson goals as well as the lesson format! Together we can do it!

# 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.

### Favourite:

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.

### Fix

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.

### [Update]

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?

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!