Hello! We are just wrapping up week 3 of the second semester. This semester comes with all kinds of exciting things for me. For the first time in my teaching career, I have the amazing opportunity of getting to teach a course I’ve taught before! The course I’m teaching is Grade 12 Data Management (MDM4U in Ontario). Having taught the course before makes planning a lot easier! Here’s what I’ve been doing to plan each lesson:

- Look at my Long-Range Plans and the curriculum expectations
- Find my lesson on the same concept from last year
- Search the #mtbos search engine and use my own ideas to make my lesson better
- Keep existing parts of my old lesson that I liked, and use what I’ve learned in the past year about differentiation for students with IEPs and English Language Learners to make my lesson easier for my students to understand

Recently, my class had started the unit on Combinations: choosing r items from a group of n items without replacement, where the order doesn’t matter (“n choose r”). The next topic was Combinations “some of” or “up to” questions: how many ways there are to choose at least 1 item from a group (up to n items)? I searched the #mtbos search engine and found Robert Kaplinsky‘s (of Open Middle Fame) Soda Combos Coke Freestyle lesson. This fit perfectly with what I wanted to do. I modified the lesson to fit the “some of” questions in the curriculum. I sent Robert Kaplinsky this tweet:

He replied back asking me to share my lesson, so here it is!

I started off the lesson by giving a mini whiteboard and marker to each pair or small group of students, and told them to divide their whiteboard into two sections: I notice and I wonder. I played the Coke Freestyle video a couple of times and had students write down their observations. Then I told them to switch the marker to their partner:

Then I asked students to estimate: how many different drinks are possible, if you can have as many different flavours in your cup as you want? You must have at least one flavour. For simplicity, I told them to disregard the second step and only use the flavours from the original panel (not the 7 variations per flavour that the machine offers). I did this to make the numbers a bit less overwhelming, although the problem could have worked with all of the sub-flavours as well. I encouraged students to use “too high, too low, best guess” to help them estimate.

I asked each group to share either their too high, too low or best guess. Most of the groups were hesitant to share their best guess (we’re working on that!), but they shared some great “too high” answers:

Then I gave them some more information:

At first I let the students struggle a bit to figure out a strategy and gave them a couple of minutes to talk about it with their groups. Then I revealed the hint. As mathematicians, we are constantly looking for patterns. We did the first line in the table together, and then I let them do the rest in their groups:

Most students were able to figure out the pattern: the number of choices for up to n flavours is 2^n – 1. This would make the total number of possible combinations for the coke machine 2^14 – 1 = 16 383 different drinks!

Then we talked about what the formula meant:

We did a simple practice question, and then moved on to the formula for “some of” problems with some identical elements. Like in the previous example, I encouraged students to make a table of values, and gradually add items to their pool of objects to choose from.

For the green shirt, you have two choices: donate, or don’t donate it.

For the blue shirt, there are three choices: donate 2 shirts, donate 1 shirt, or donate none.

For the red shirt, there are four choices: donate 3, 2, 1 or no shirts.

We have to donate at least one shirt, so we subtract 1 from the total to eliminate the option of not donating any shirts at all. So the formula in general becomes **(p+1)(q+1)(r+1) – 1**.

After that, we did a similar practice question and then I gave students some time to get started on their homework and ask questions about the homework from yesterday. Thanks Robert Kaplinsky for the original problem, and for asking me to share my lesson!

Questions? Feedback? Tried this in your own class and want to let me know how it went? Hit me up in the **comments**!

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