If you’re a teacher in Ontario, welcome to semester 2! (If you’re not a teacher in Ontario, you are also welcome 🙂 ) Last semester I tried spiralling my 2P class for the first time (read about my spiralling adventures here). This semester I have the amazing opportunity to teach the same course again, and I’m always looking for ways to improve. One of my goals for this semester was to improve my spiralling by including more spiralled tasks and connections between different strands in the course. I am still following the rough model of one strand per week with a small assessment at the end of each week, but with a bit more spiralling here and there. Follow my new semester 2 Long Range Plans here!

I started off the semester similarly to last semester: my first day activities were very similar to last semester’s, except last time I was a bit short on time, so I decided to postpone name tents to the second day and end with students completing my first day survey on Google Forms. On the second day, we reviewed linear relations with a visual pattern. I had students represent how the figure is growing using a table of values, equation, and graph, then answer the questions: how many tiles will be in figure 43? Which figure has 255 tiles? This lesson was largely based on Laura Wheeler‘s lessons, which I borrow from frequently. We consolidated as a group and I had students try 2 more patterns on their own at their seats.

On day 3 we did Dan Meyer’s Cup Stacking, which was very successful and a lot of fun. It was a bit chaotic – the stack of cups kept falling when we tried to stack them up to the top of my head, but the learning was great and students had fun.

## Cup Stacking Extension: The Point of Intersection

As I mentioned, one of my goals this semester was to tie in more connections between strands. As we were approaching the end of our linear relations strand, I decided to include a quick intro to linear systems with finding the Point of Intersection graphically. At the same time, I had 200 Styrofoam cups left over from the previous week’s cup stacking activity, and that gave me this idea.

I remember seeing some red solo cups lying around in my first semester classroom. When I checked, they were still there, so I borrowed a few and took this picture:

I started the lesson by asking students: What do you notice? What do you wonder? They recorded their ideas on mini whiteboards or on their tables (we have a couple of whiteboard tables in the room). Here are some of the things they came up with:

- I notice that there are white and red cups
- I notice the white cups have a thicker rim
- I notice the red cups are taller
- I notice there are 3 cups in each stack
- I wonder why are they stacked?
- I wonder why there are 2 different types of cups?
- I wonder will both stacks ever be the same height?

Next, I had students look at the picture again and predict: if we must have the *same number* of each type of cup, how many of each type of cup do we need in order for the stacks to be the *same height*? (Note: I had a bit of trouble trying to find a clear way to phrase this question – if you can think of a better way, please let me know.)

I asked: what information do you need to solve this problem? The students told me they needed the height of each type of cup and the width of each rim, so I showed them this:

I put students into random groups of 2 or 3 and sent them to the VNPS to solve the problem. Originally, I was going to have students represent the problem using various representations (table of values, equations, graph) and then solve, but then I thought it would be better to leave the question open and let students solve it any way they want first. Some groups started with a table of values, while others started with a graph. When I noticed a group was finished, I asked if they could think of another way to solve the problem, which got students to turn to the other representations. Some groups wrote equations to show how the heights of the two stacks were changing, and one group even started to verify the equations by checking if the point of intersection worked for both lines:

I consolidated by numbering off the boards and going through the problem using the two approaches students took: table of values and graph. We talked about the importance of the point (8, 15) that appears in both the table of values and graph, the Point of Intersection, and what it means about the stacks of cups. I then had students go back to their seats and we did a couple of examples of graphing lines to find the Point of Intersection graphically.

## What I would do differently next time

Honestly, the only critique I had was that students had a hard time telling if the picture of the red solo cups includes the rim or not, so I would try to make that image more clear. And I didn’t make an Act 3 for this task (although one group drew a messy sketch of what Act 3 would have looked like on the board). Otherwise, I’d say this was one of our most successful lessons so far, so I’d chalk it up as a win. If you use this lesson with your class, please comment and/or Tweet me and let me know how it goes!

In the meantime, I still have a lot of Styrofoam cups… If you have any ideas of what to do with them that’s related to math, I am open to suggestions 😉

Questions, comments, feedback? Comment below!