The original idea for this lesson came from Al Overwijk. Thanks again Al!

The possible Ontario overall curriculum expectations covered in the activity:

Grade 10 applied:

graph a line and write the equation of a line from given information

Grade 9 applied & academic:

solve problems involving proportional reasoning;

apply data-management techniques to investigate relationships between two variables;

demonstrate an understanding of constant rate of change and its connection to linear relation

Grade 8:

solve problems by using proportional reasoning in a variety of meaningful contexts.

Grade 7:

demonstrate an understanding of proportional relationships using percent, ratio, and rate.

Grade 6:

demonstrate an understanding of relationships involving percent, ratio, and unit rate.

Act 1: Turbo Texting:

I started with “I was with my brother one afternoon and I needed to text my wife. After texting her, my brother informed me that I was a ‘terrible texter’. He said I was soooooo slow. I on the other hand disagreed. Then we decided to settle this once and for all—- race!!!”

ME: “Use any method you choose to determine: Who is the faster texter?” I allowed them time here to work on a strategy. I watched carefully what strategies they used or didn’t use.

Seeing the different strategies gave us a nice discussion the importance understanding what rate we are determining and how to interpret it to answer the problem.

I showed this picture next:

and this piece of info…

Students completed this problem and we discussed the assumptions we needed to make.

Texting Time

How do your students compare to Jon and Kevin? Have them time each other while texting the 165 character message. Have them determine their texting speed to see who the fastest texter is in the class.

Linear Modelling

ME: “Now you may have texted that message in 18 seconds, but would you do this all of the time? Would you keep that same rate for a shorter message? Longer message? We better keep this experiment going.

I set them off to text various messages of different lengths using this handout (I modelled the handout format after Mary Bourassa’s Spegettini and Pennies handout – thanks Mary).

Click to download a copy

Students used Desmos and the regression tool to create a linear model. They used that model to predict how long it would take to text 140 characters, 200 characters, and this message: “Dear Mom and Dad I promise to never text and drive.” They finally timed themselves to compare the calculated time and the actual time.

Extension: Compare the relationship between the number of words in a message and the time to text the message. How would the equation change? Is it still proportional?

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.

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.

For the Favourite & Fix series I’m posting one idea from my lessons that week that was my favourite and one topic that I need help on. Something 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: The Cheating Quiz

This week I gave a quiz to my grade 9 applied students. It consisted of 4 questions – Two on linear relations and two on reading distance-time graphs. After the quiz was over I said “It’s time to do a little cheating.” Each student is to find another student they were comfortable sharing their work with. I said, “For question 2 only, share your work with each other. Discuss what you notice about each other’s solution. Do you have the same? If you have different solutions who is more right? After you discussion go back and adjust your solution if you need to. Hand in after.

I really enjoyed listening to them share. It was interesting to see how they defended (or didn’t defend) their answers. After reviewing their new work on that question it not only gave me insight into that one students thinking, it gave me some insight into what their partner was thinking too. For the student below I can see some really good thinking about how the linear relation changes. But now I know for both of these students we need to have a discussion how the increase of 100 every 5 people affects the equation. Looking at each students paper in the room now tells me a lot more about my class’ understanding compared to not having a “cheating quiz”

Fix (just a comment)

My MEL3E class is coming off a two week themed activity where we designed, built and launched rockets. Today we were completing the Sugar sugar Desmos Activity and a student says to me: “When are we going to do something fun?” I relply, “Fun?”….he says, “yeah, like watch a movie.”

I’m not one to show movies in class.

Why do students always equate fun in class with movie watching? How does the student who just smiled through two weeks of math class, built and launched rockets, helped me fix the launcher numerous times, and today, yes today, defended his choice on which sugary cereal was the best choice not know he was having fun?

I guess enjoying class does not equal “having fun”.

A major expectation for our grade 9 applied class is to “connect various representations of a linear relation, and solve problems using the representations.” Early in the spiralled grade 9 course I bring in Fawn’s Visual Patterns website as warm ups. We routinely continue the patterns, create tables, equations, and graphs to show the representations. Students also create their own patterns.

More and more I notice that grade 9 applied students don’t see what I see when looking at patterns (which is definitely not a bad thing). I love hearing all about how students see the patterns. However, I always see the patterns as growing/shrinking…..what I mean is that I see that one shape morphing into a bigger/smaller version. What I’ve heard from some students though is that they see each figure as a separate object, separate things that looks slightly different. I wanted to explore if students seeing the patterns morph instead of seeing them as separate objects could help them with seeing connections among the different forms of the relation.

To start the class I showed this video:

I asked: What do you notice?
Students described the pattern to each other while sitting in pairs. We decided that if the first set of shapes represented figure 1….then every figure after that showed two more shapes being added in. I asked them to go ahead and find out how many shapes were in figure 108.

I gave out the following set of instructions:

Create your own animated pattern video

Create a tough pattern for your classmates to discover. Ex: Show how the pattern changes in other ways than figure 1 then figure 2 then figure 3. Maybe show how your pattern changes from figure 1 to figure 3 then figure 5.

Create a question for your fellow classmates to solve about your pattern.

Display your video around the room for a gallery walk. In your display hide the table and equation and answer to your question.

They went to work on building & shooting their patterns. Having them skip figure numbers made them really think about how to create their patterns and how the equations related. Since they were invested in their own patterns they worked hard at creating the tables and equations.

After they created their video they were to create a display for a gallery walk. The gallery walk gave us a purpose to practice finding rates of change, determining equations, generating equations and solving problems. We wanted to see the creative patterns from our classmates and see if we could solve each others problems. Like a challenge! Each display showed the video and then under flap of paper was an answer to a problem with a table and equation. Students left their display and visited each others displays with a recording sheet.

We spent two class days working on building the videos/patterns and the gallery walk. There are a variety of stop animation apps on the app store. My students used various different ones. Some students used iMovie.

I felt students were stronger on knowing why we need to find the rate of change for our equations and not just take the first difference value. The one-two combo of actually building the patterns and then making them move through animation built a deeper understanding of the representations than just completing a worksheet!!