MFM2P Update: Struggling and Learning

Happy Friday! Me and my class just completed the first 5 weeks of spiralling Grade 10 applied math. Here’s what our month looked like:


The colours represent different strands of the curriculum. Gray and pale blue are special events.

Ideally I would have liked my plan to look more like one week per strand, but regular school life has gotten a bit in the way of that. That’s okay. I am learning to be flexible when things don’t go according to plan.

Let’s talk about what happened in the past two days.

The Past Two Days

Yesterday I started the class with Mary Bourassa’s lesson on solving linear systems with substitution:


I suggested students start with a table of values. So far so good – I was looking around the room and most groups were able to start moving in the right direction. One group even wrote down 2  equations for the cost and the profit, but they weren’t sure what to do with them. Some of my students had a hard time wrapping their heads around the idea of revenue – they were trying to subtract the $7 per student cost for the DJ from the $10 per ticket in order to come up with an overall profit. They were struggling, but I was okay with that – it seemed like productive struggling.

And then one group started panicking about not knowing what to do. And suddenly all of them were panicking.

I ended up rushing in to save them – I got everyone in their seats and explained the problem more slowly. Silence. I helped students come up with linear equations for cost and revenue (something that was fresh in memory from our linear relations strand – yay spiralling!). More silence. We went through the whole problem together. I tried to explain it in different ways, and relate it to situations they were familiar with (buying something vs. money made at a job). Dead silence. They were silent the entire period (save for the surprise fire drill). For the first time ever, someone asked me if this lesson was going to be posted on Google Classroom. Yikes.

In retrospect, the problem was that up until now, my students hadn’t seen a problem before that didn’t have a straight answer. I do my best to choose open problems and use 3-act math tasks, and we’ve done warm ups from Would You Rather math and Which One Doesn’t Belong, but I guess none of the questions we had done before had presented them with a problem that they weren’t sure how to solve. But that struggle that students were experiencing, that “getting stuck” – that’s what I want to happen. I don’t want my students to memorize a process and show me the steps – I want them to be problem solvers. I want them to come up with different strategies that might not work or might be wrong, until eventually they get to the solution.

After thinking about it and discussing the situation with a mentor teacher, I decided that, moving forward, I will focus my class on doing more problems where students won’t get the answer right away and will need to try different strategies. Next week we have a quiz. After that I am thinking of taking a short break for a day or two to work on developing a culture of problem solving and productive struggle in my class. I’d like to use one of the Jo Boaler Week of Inspirational Math problems, or one of Peter Liljedahl’s good problems. (If you have a recommendation of a good open problem that works well in MFM2P, please let me know!)

Keeping these ideas in mind, I decided to change up my lesson plan for today. I used this somewhat open-ended problem and we did Jon Orr’s Commit & Crumple:


I saw some great thinking and I loved the peer assessment piece. (And students seemed to enjoy throwing paper balls at me!)

After some more practice with substitution and a short lesson on formal checks, I talked to my students about what happened yesterday. I told them (roughly):

“The problem we did yesterday was challenging, and that’s okay – if you’re struggling, it means you’re learning. All of you were on the right track when you started solving the problem. But when you don’t know exactly what to do, we don’t panic and give up – we try different strategies until we find one that works. In this class – and in life in general – you won’t always know what to do when you’re faced with a problem. I don’t want students who always know exactly what to do and memorize steps – I want you to be problem solvers. If one strategy doesn’t work, we try different things until we figure it out.”

As one of my colleagues says to her students: “I need you to be wrong – it keeps me in business!”

Feedback? Suggestions for good problems? Hit up the comments!


MDM4U: Online Resources by Unit, and Plans for Probability

Happy long weekend!

So far I’ve really been enjoying my 2 classes of Grade 12 Data Management (MDM4U in Ontario). We’ve done several fun activities that involved critical thinking and rich learning. One of the goals I’ve had this year is to really work on making sure my lessons are not only engaging because they’re fun, but also provide good learning opportunities. I’d rather have a less “fun” lesson that involves a lot of rich learning than a super engaging lesson that has very little learning and critical thinking. Some of the lesson activities I’ve done so far:

Combinations and Permutations:

  • for nPr and factorial problems (order matters): Dan Meyer’s Door Lock
  • for combinations “some of” or “up to” problems: my take on Robert Kaplinsky’s Coke Freestyle lesson

Noticings and wonderings from the Coke Freestyle video.

One-variable Statistics:

  • for measures of central tendency: Crazy Math Lady’s mean, median, mode
  • for sampling techniques: Mr. Waddall’s take on the Jelly Blubbers Colony experiment
  •  for measurement bias: Bob Lochel’s opener on the effect of a leading question.
    I did this as a Google Form: I assigned half the class a Google Form with one of the questions to fill out, and the other half of the class the other question (I split the class in half by their student numbers). I did the demo in real-time, which was terrifying, but it more or less worked! (I showed my class Bob Lochel’s data as well.)
  • for standard deviation: John Scammell’s celebrity guessing game. Here is my celebrity slide deck (as of March 9, 2018)

Celebrity guessing game!

Two-variable Statistics:

  • for linear correlation: Bob Lochel’s Friendship Compatibility Test.
    I loved this one. I wrote a Python program to match students up randomly (I try to get my students to get to know each other outside of their chosen table groups). This lesson was scaffolded really well and it went smoothly.
  • for least-squares regression: I roughly based my lesson on Fawn Nguyen’s Vroom Vroom and Jon Orr’s take, Vroom. I modified it to teach Least Squares Regression to find a good line of best fit for the linear data.
    Comment from one of my students about this lesson: “Miss, class was so fun today! I mean, it wasn’t that fun because we had to do math, but it was still kinda fun.”

Collecting data from the pullback cars in “Vroom Vroom”.

As usual, all the credit goes to the incredible teachers who take the time to share their resources on the internet FOR FREE! Thanks to all of you, me and my students both win.

Now we have reached what I consider to be “my least favourite unit”: Probability.
But why?! Probability is so fun! I agree. Probability is fun. We get to play with dice and cards and spinners. When I taught the unit last year, students had a lot of fun. My problem with the Probability unit is that, although students were engaged, I didn’t feel like the critical thinking aspect was very strong. Even if we have to skip out on some of the fun, this year I am determined to make sure that a lot of deep thinking happens and that my students remember the learning and not just the fun.

Here is my tentative plan for the unit. This is a work in progress and I am still looking for ideas. If you have interesting inquiry-based lesson ideas, please comment or message me and I will happily add them and give you credit! Here goes:


  • DAY 1: Intro to Probability: I’m actually going to be away that day, so my plan is to leave students with a handout of probability definitions, including introducing the idea of “odds”, and some practice questions. (Boring, I know, but I won’t be there so I don’t want to leave anything too complicated.)
  • DAY 2: consolidate intro to probability and odds:
    Warm up: jumbled note review of the definitions from the previous day.
    Lesson: play Bob Lochel’s Jolly Ranchers game. Tie in connections to probability, sample space, favourable outcomes, and odds.
  • DAY 3: review of permutations and combinations:
    Warm up: ticket (formative quiz) on previous two days’ material.
    Lesson: we will review permutations and combinations as prior knowledge for probability with counting techniques, coming up tomorrow. I’m still not sure what I’m going to do for this lesson yet, so I’m open to ideas. I’d like to do something involving a lot of problem solving on whiteboards.
  • DAY 4: probability with counting techniques:
    Warm up: one of these from Would You Rather Math.
    Lesson: going to tweak Dan Meyer’s Starburst 3-act math to be about counting techniques. Maybe change the problem around a few times to address various concepts of the permutations/combinations unit.
  • DAY 5: independent and dependent events:
    Warm up:
    Lesson: play Skunk Redux. Key takeaway: no matter how many times you roll the dice and do not roll a 1, the probability of rolling a 1 on the next roll is still 11/36.
  • DAY 6: conditional probability:
    Warm up:
    not sure yet. Maybe a formative quiz.
    Lesson: Bob Lochel’s Egg Roulette lesson. Focus on tie-ins to conditional probability. Might come back to this for the hypergeometric distribution in the next unit.
  • DAY 7: mutually exclusive and non-mutually exclusive events:
    Not sure about this one yet. Maybe something with John Scammell’s free throws for the win.
  • DAY 8: wrap-ups:
    Warm up: the famous Monty Hall problem.
    Lesson: tie up loose ends, start test review.
  • DAY 9: test review
  • DAY 10: unit test

That’s pretty much all I have for now. After this is my favourite unit, Probability Distributions. I love this unit because there are many interesting problems that involve higher-level thinking and are also really fun! I will hopefully find time to post about that later on. In the meantime, I’m turning the floor over to you:

What lesson ideas and activities do you have for Probability?
What would you add to my unit plan?
What would changes would you make to my unit plan?

Write me in the comments!

I’m on Twitter! Come say hi! #mtbos