Spiralling MFM2P Update: Differences of Squares – a lesson idea from the #mtbos

Spiralling MFM2P Update

Another month flies by in my spiralled MFM2P! I don’t know how it’s possible for me to be both bubbling over with energy and totally exhausted at the same time, but seems like that’s teaching for me!

Here’s what October looked like in my class:


That’s right, we start school a week early and get a fall reading week. Jealous? ūüėČ

Some things that happened this month that I didn’t expect:

  • My students had a rough time when they came across a problem they didn’t know how to solve, so we spent some time working on creating a classroom culture of problem solving (read about it here)
  • Doing Peter Liljedahl’s taxman problem was a great choice – thanks Mr. Hogg for the recommendation! It actually ended up working out better than expected as it reviewed factors, which was helpful when we started factoring about a week later
  • My students really struggled with similar triangles. Consequently, I ended up teaching similar triangles three times. Based on formative assessments and my own observations, it seems like they got it, but I haven’t marked their quizzes yet so I’m going to wait and see how those went before I say the third time’s the charm.
  • In our last lesson of the triangles and trigonometry strand, I did Jon Orr’s new 3-act math, Eye-to-Eye. Students struggled with this and a few groups had to try a few different strategies before they figured out what they needed to do. Unfortunately we ran out of time and not every group got to the point where they found the answer correctly, but I was so impressed with my students for continuing to work on the problem and not giving up! I was so proud of them!

Planning lessons with the #mtbos on Twitter

A couple of weeks ago I came across this question on an old test:

I debated showing my students a similar question with different numbers to help “prepare” them for a question like this on an assessment, but I wanted something that involved more thinking. I tried making the question better myself, but I was feeling a bit stuck.¬†So I decided to throw the question up on Twitter and tag some experienced and creative teachers to see if they had any better ideas:


I got some great responses,¬†including connections to optimization, but Jon Orr‘s was my favourite:



That would have been my warm up, but Mr. Hogg asked a question that made me take a second look at the original situation, and I realized that the details of the original wording and numbers chosen, which I hadn’t been too careful about when I started mucking around, were actually pretty important:


Reading the question more carefully, I noticed two very important things:


One was that the question was looking for a binomial expression (expanded and simplified) to represent the remaining area. The other was that the inside shape was not a 3 x 4 rectangle, but a square, which makes this situation literally a difference of squares.

I’m not planning to come back to factoring until December in MFM2P, but I needed to make a lesson plan for one of the modules in my math honour specialist course, so I decided to run with Jon’s idea and use it as an introduction to Difference of Squares. The full lesson plan can be found here, but here are the highlights:

Difference of Squares Lesson

Warm up:¬† this one from Which One Doesn’t Belong:


If when you take it up the students don’t point out that 9, 16 and 25 are all perfect squares and 43 is not, lead them to the idea and help them with the vocabulary of perfect squares and square roots.


Have students draw a big square on their paper, then draw a smaller square in the top right corner, like so:


Label the dimensions of the small square 3m x 3m.

Explain that this is your backyard and the 3 x 3 square in the corner is a shed. The shaded region is covered in grass.

Ask: what dimensions of your backyard are possible? What would be the area of the grass? Send students to the VNPS to work out different dimensions with the restriction that the backyard must be a square (not a rectangle). Have students calculate the area of the grass (outside Рinside).

Bring students back to their seats. Say that now, let the dimensions of the backyard be x by x. Send students back to the VNPS to create an expression for the shaded area (grass) in factored form.

For students who struggle to factor x² Р9, lead them to rewrite the expression as x² + 0x Р9. (This lesson assumes students have already learned factoring simple trinomials.)

Once most groups have factored the expression, explain briefly that a quadratic expression of the form x² Рa² is called a Difference of Squares.

Send students back to the VNPS with the following change: what if the shed were 4m by 4m instead? What would the area of the grassy region be? What would the area of the grassy region be in factored form?  Once they find the factored form, have the students do it again for a shed of 5m by 5m, and again for 6m by 6m. Once they begin to grasp the pattern, have students create a general expression for the area of the shaded region in factored form for a side length of a. Their final result should be:


For students who finish quickly, provide more challenging questions, such as those that require common factoring first (eg. 3x² Р27). For an academic class (MPM2D), you can extend into more complex differences of squares of the form a² Рb² (such as 4x² Р1).


Here’s a template students can fill out to summarize what they learned if you do that in your class:


Many thanks to Jon and everyone else who contributed to helping create this lesson. What a shame to (almost!) waste such a rich problem on an assessment. ūüėČ

I’m so excited for this lesson! Check my Twitter in a few weeks to see how it goes!

Follow my adventures in MFM2P through my spiralled Long Range Plans here.


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!

Spiralling MFM2P: The First Day

What a busy week! For those who are new here or didn’t read my last couple of posts, I am spiralling Grade 10 Applied Math (MFM2P in Ontario) for the first time. (I am also teaching this course for the first time in my career.) I have a new classroom with vertical chalkboards on most of the walls, plus some portable vertical whiteboards. My school is one of four “early start” schools in my board, which means we start a week before Labour Day (and we get a week off in late October/early November). Here’s how my first day went down:


Originally I used Kyle Pearce’s spiralling guide to create my Spiralled Long Range Plans. Later in August, I was at a board workshop, and I met a history teacher in my board who told me that there was a school a few minutes away from mine that was doing, some cool new thing in their math department that I might be interested in, he thinks they called it… “spiralling”? I got the department head’s email and got in touch with him. The department head told me that they spiralled their courses by spending 4 days teaching, with the Friday being review and a quiz. Then on Monday they would start a new strand. I really liked the idea of having a routine that students can rely on so that if they are frustrated or struggling with one concept, they know that they can start fresh on Monday with a new topic. Hopefully the spiralling and constant review will help students remember the concepts for the exam as well. I updated my long range plans to fit this model a bit better (more or less).

My updated Long Range Plans are here. My plan is to continue to add links to resources for each topic throughout the semester. (Eventually I will probably fix up a nice version in Google Sheets, but I work best with a calendar-type LRP, so sticking with that for now.) Review days that seem random are days when I will be away.

EDIT Oct. 30: here are my updated Long Range Plans – because things change.

The First Day

There were a lot of things I wanted to do on the first day and I knew I wouldn’t have time for all of them, so I made a list of everything I wanted to do and then chose the most important ones. Talking Points is a favourite of mine that I’ve used in all of my math classes for the past 2 years. I think it helps establish my classroom norms about growth mindset in math, but this year I decided to change things up a bit. I really hope I get a chance to squeeze in a short lesson about growth mindset later sometime this month. Here’s what I did instead:

I started the class with this warm up from Which One Doesn’t Belong:


I had students discuss it at their tables, then I had each student vote on which number didn’t belong, and explain why. Most of the students agreed that 43 didn’t belong, either because it was prime or because it wasn’t a perfect square (they needed some help with the correct mathematical terminology). The students who didn’t choose 43 were feeling a bit uneasy. I then told the students to choose another number and tell me why it didn’t belong. I tried to enforce the idea that there is no wrong answer here. One student pointed out that the digits of 16, 25 and 43 all add up to 7, but 9 doesn’t, which I had never noticed before.

I then took attendance. A lot of teachers on Twitter suggested forgoing all of the typical “first day” syllabus stuff, but I felt that establishing my expectations and possible consequences was important to do from day 1. I found this article about classroom management from Cult of Pedagogy to be really helpful. One of the strongest points in the article was that for 90% of students, making your class engaging and interesting for them will help deter any classroom management challenges. But for those 10%, you need to have clear expectations and consequences so that students know exactly what is expected of them. As the article recommended, after going over the serious stuff, I told my students that all that being said, I plan to do a lot of fun things this year and I’m looking forward to getting to know them.

I then jumped into Jon Orr’s game of NIM:


The students loved this. I challenged the class to play against me, and then we worked out some of the mechanics of the game. I talked about why I chose to play this game and how it relates to learning math.


Then we did another activity I got from Jon Orr, Graphing your Subjects. We started with the first quadrant only graphing as a review, and went over success criteria for graphing.


We ended up using paper and pencil for this activity instead.


I’m hoping the math point will change by the end of the semester!

Finally, it seems like every teacher swears by Sara Van Der Werf’s name tents. I wasn’t sure if the kids would go for it in high school, but I figured I’d give it a try this semester, and if it doesn’t work, I won’t do it again. Although some students didn’t take them seriously, I found they were very effective for most. Some students told me things that I never would have known otherwise about how they felt about the lesson, questions they had that they didn’t want to ask out loud, things like that. I don’t know if I would do them in a Grade 12 class, but they were great for my 2Ps and my Grade 10 open computer engineering class.


I was also planning to have students fill out a short survey and join our Google Classroom, but we ran out of time, so I decided to end there and do the rest tomorrow. Over the next few days, I continued to introduce a bit of syllabus-type information each day for the rest of the week (materials to bring to class, when to get extra help, etc).

Overall, I think day 1 went well, and I would definitely do all of the activities again. Many thanks as always to all of the teachers whose ideas and activities I used.

If you have any feedback for me, hit up the comments! Welcome back to school!

Spiralling MFM2P and Goals for the New School Year

Time is flying by and it’s already August. I will be teaching Grade 10 applied math (MFM2P) for the first time this coming year, and I am very excited! I love the Grade 10 curriculum. I taught Grade 10 academic math (MPM2D) in my first year of teaching, and I have a few weeks’ experience teaching the applied course from when I was a student teacher. I will be teaching the only section of Grade 10 applied math running first semester, so I have a bit of flexibility in terms of pacing and structure, and my department head told me that I can spiral the course!

What is spiralling? If you don’t know, check out Kyle Pearce‘s guide to spiralling¬†here.

I have never taught a spiralled course before, so I am deciding to start small. I plan to blog the process as I go along and share some lessons/ideas/strategies at different points throughout the semester. But first, let me take a step back and share some background information.

This will actually be the first time in my career teaching an applied course.¬†Not only is this my first time teaching MFM2P, but it’s also my first time teaching¬†any applied course as a full-time teacher. However, most of my experience as a student teacher was in applied and college preparation courses, so I do have some experience with the culture and curriculum. I actually requested to teach applied courses when I was asked about my preferences for the upcoming year. I want to help support students who struggle and hopefully change their mindsets and help them be successful.

Some new strategies and tools I plan to use in my classes this year:

  • Laura Wheeler’s course packs for the #ThinkingClassroom: a great way for students to document their learning when most of the classwork is done by solving problems on vertical whiteboards or chalkboards. The classroom I will be teaching in has chalkboards on most of the walls. After 2 years of teaching math in a computer lab, I am very excited to have a vertical classroom!
  • Growth mindset and “rough drafts”: last semester my theme for Grade 12 Data Management was the “bad idea factory” – I encouraged students to share their thinking and ideas even if it might not be “right”. I got this idea at a PD workshop and wrote about it here.
    A few days ago I stumbled upon an excellent post by Andrew Busch about Rough Draft Thinking. Many of his ideas really spoke to me, in particular the idea of valuing the process more than the correct answer. Instead, students are asked to provide a starting point for the conversation that will help us get to the solution.
  • Continue to make my classroom a #ThinkingClassroom, as much as possible.¬†

I think between these things and spiralling, I’ll have my hands pretty full. Other than that, I plan to continue building on what I’ve been doing: using warm ups from Which One Doesn’t Belong and Would You Rather math, 3-act math tasks, one-on-one conferencing to support and get to know individual students, Desmos activities, and frequent formative assessments (no marks, timely feedback) so that the students and I both know where they’re at.

Back to spiralling: my first step in the process was to make a non-spiralled long-range plan for the course separated into units. One of the suggestions in Kyle Pearce’s spiralling guide was to teach 3 days or so of each unit at a time to create a cycle. The next cycle will have the next 3 days of the unit, which builds on the previous 3 days’ material. I want to also try to work in some 3-act tasks that combine skills from different units.

Here are both versions of my long-range plans. Click the links to download the file:

The non-spiralled version: MFM2P non-spiralled Long Range Plans_colourCoded
The spiralled version: MFM2P spiralled Long Range Plans_colourCoded

If you don’t feel like downloading things, here’s the spiralled version:
(Note: if you see some review days that seem random, those are days when I will be away.)




*note on Dec. 11: I’m considering doing trig ratios earlier as a tie-in with linear relations and slope.


The “filler week” is to account for missed classes due to assemblies, snow days, etc. as well as extra time in case some topics take more time than expected.

I tried to plan the schedule so that I would have a unit test every month or so, a smaller quiz about halfway through each cycle, and frequent formative assessments.

I’m currently starting to look for tasks and activities that promote higher-level thinking to use to teach the concepts.

Feedback and suggestions are welcome!

EDIT 8/10/2018: my long range plans are now a Google Doc. Resources I’m using to teach the concepts are hyperlinked (or they will be, as I continue planning the course).

Amazing Race Hay Bale Roll 3-Act Math

I just came back from the Teaching Math through Problem Solving 2 Conference which featured keynote presenters Jon Orr and Kyle Pearce (which was awesome!), and I am feeling newly inspired to do new and exciting things in my classroom in September, including spiralling a course for the first time (but more on that later).

This past semester I was happy to be teaching Grade 12 Data Management (MDM4U) for the second time. I focused as much as I could on using rich math tasks to help my students become critical thinkers. One of the best things I stumbled upon was Bob Lochel’s blog. While there are many resources out there for Grade 9 and 10 math concepts, it was harder for me to find good posts about higher-level discrete math, statistics and probability, and it was a data management teacher’s dream to find a stats teacher with blog posts for days!

This 3-act task comes from the Amazing Race problem in¬†this post¬†by Bob Lochel – scroll to the heading “Statistical tales of the improbable”.¬†This lesson focuses on the hypergeometric distribution. My students had already finished a unit on probability, and in previous lessons learned about the binomial and geometric distributions. Students should have some prior knowledge of probability and a good foundation on counting techniques (combinations in particular).

Here’s how this problem would look if it was in the textbook:

hay word problem.PNG

How many of your students have tuned out?

Now here’s the 3-act version:


First, I played this video up until about 1:14 and had students write down: what do you notice? What do you wonder?


Here’s what one of my students came up with:


I wrote down some of my class’s ideas on the board:



Next, I revealed some information to confirm what my students saw and heard in the video, and posed a challenge for them to solve:


My students had seen questions like this before in the Probability unit, but if you’re spiralling, all you really need is an understanding of combinations and some basic probability to be able to solve the problem. For those who finished quickly, you can offer follow-up questions:

  • If a team unrolls 6 bales, what is the probability of finding exactly 2 clues?
  • If a team unrolls 6 bales, what’s the probability of them finding¬†at least one¬†clue?

I then had students make a probability distribution table and graph:


Students then answered some questions about the problem they had just solved. You can do this as a handout, but my school does not use handouts so I displayed it on the board:


I used variables r, n, and a because this is what the textbook uses. In other places I’ve seen variables n, N and M used instead.

We took this up together and I classified these characteristics as defining features of the Hypergeometric Distribution. We talked about the importance of the trials being dependent, which was different from other distributions we had seen (binomial, geometric, uniform). Next, I challenged students to use the variables we defined to come up with a general probability formula P(X=x) for the Hypergeometric Distribution.


I didn’t have an Act 3 for this problem at the time, but if I were to do it again, I would print off 270 game cards which include 20 clues and 250 blanks and have a student draw 6 cards without replacement.

To wrap up my class, I had students check out this simulation:


Data Management is one of my favourite courses and I was a bit disappointed to find out I won’t be teaching it again next year, but I’m excited for new challenges. I will be teaching Grade 10 applied math (MFM2P) for the first time, and I will be spiralling! Will post more about it as I go!


Thanks again to Bob Lochel for the idea and problem for this post!

Oh and here’s my friend Julia (@MsFolino) and I at the conference with Jon Orr and Kyle Pearce!

EDIT: Meg Craig has corrected me on the rules of the Amazing Race and also suggested an idea for an extension to the problem, which I made into a Would You Rather math warm up:


Introducing Probability Distributions with “Clear the Board” in MDM4U

Time is flying by in my Grade 12 Data Management class! I can’t believe we are already on our last unit, which is also my favourite unit: Probability Distributions.

I draw a lot of inspiration from other math teachers I’ve discovered on Twitter and through the MTBoS database. One of them is Sarah Carter, better known as Math Equals Love. If you’ve never heard of her, go check out her blog now! Sarah has lots of great lesson and classroom ideas and I borrow from her regularly (thanks Sarah!).

Last year was my first year of teaching and also my first time teaching Data Management (a discrete math, statistics, and probability course in Ontario, Canada). When I was first teaching the Probability unit, I used one of Sarah Carter’s lessons, Blocko! That day one of the teachers in my department and her student teacher decided to stop by my classroom to see what I was up to with the linking cubes. I had them join in the lesson and gave them some linking cubes to play the game with. After I explained the instructions, I was walking around the room to make sure everyone knew what to do, and my colleague asked me, “This is about Probability Distributions, right?” Actually it wasn’t. It was about theoretical and experimental probability. That comment got me thinking, and the more I thought about it, the more it made sense in the context of probability distributions. This semester, I am teaching Data Management again, and I decided to try out Blocko! for probability distributions! Here’s what my lesson looked like:

We started with this warm up question:

warm up

We had done questions like this before in the probability unit. Most students used a tree diagram. I encouraged students to think about how they could use counting techniques to answer the question (combinations and permutations).

This was the first day of a new unit, so I reminded students of some terminology and introduced the idea of a random variable: a variable that has a single value for each outcome of an experiment. I had students create a Probability Distribution Table and Probability Distribution Graph for the event of rolling one fair six-sided die.


After that I gave students a modified version of the Blocko game board with spaces for numbers 1 through 6. Each group of 2 or 3 got one game board and 12 linking cubes.


The modified Blocko! game board.

I called the game “Clear the Board” and told my students to place the 12 cubes wherever they wanted on the board, so long as each cube falls in only one section of the board. I explained that I would be rolling one die 12 times, and whichever group is left with the fewest cubes on the board at the end of the game would win. Now that they knew what the game was about, some groups quickly decided to rearrange the cubes. I instructed each group to take a picture of their original game board so that if they won, we would know what the winning game board looked like.

blog4   blog3

After playing a couple of rounds, I chose one of the winning game boards and drew it on the board at the front of the room (I purposely chose a game board where the cubes were distributed fairly evenly).

Next I introduced the full version: 2 dice, 12 rolls, new board for the sums of the rolls:


This game ran the same way as Sarah Carter’s. Experimental probability is unpredictable. In the first round I didn’t roll any 7s! After a few rounds of playing I chose a winning board and drew it at the front of the room underneath the drawing of the winning board for one die. I then asked students to discuss with a partner: what are some of the differences between the winning boards in the one-die and two-die games? They quickly realized that, as they remembered from the previous unit, when you roll one die all outcomes are equally likely, but when you roll two dice, the sums are not all equally likely.


The winning game boards, overlaid with the probabilities of rolling each value or sum.

I then had students create a probability distribution table and graph for the two-dice game. We compared the probability distribution graphs for one die and two dice: in the first graph all outcomes had the same probability, while in the second they didn’t. I introduced the terminology of a Uniform Distribution and Non-Uniform Distribution to describe the 2 different patterns.


We then did a similar example to the warm up where students practiced creating a probability distribution table and graph.


I like to end my lessons with a summary of the learning goals we covered. I find that announcing the learning goals at the beginning ruins the fun!


Questions? Feedback? 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

Coke Freestyle Soda Combos: Secondary Edition

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:

  1. Look at my Long-Range Plans and the curriculum expectations
  2. Find my lesson on the same concept from last year
  3. Search the #mtbos search engine and use my own ideas to make my lesson better
  4. 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:


Some of my students’ notice and wonder about the video.

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:


(Note: “Raspberry-Lime” is an entirely different flavour and is not “Raspberry” and “Lime” mixed together.)

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:


Some student work.

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!

NTIP Learning Tour: Modern Learning and Ed Tech

Recently, I attended a PD session run by my board for new teachers called the NTIP Learning Tour. During the workshop I got to observe two teachers team-teaching a class and learn more about modern learning and education technology.

One of our board’s initiatives this year is modern learning. The school I went to visit had a modern learning classroom, with whiteboard surfaces on large shared desks and a wall between the two classrooms that could be opened for team teaching. One thing I took away from the experience was that you don’t need to have any specific furniture or materials in order to make learning relevant and skills-driven (as opposed to content-driven).

The lesson I got to observe was a Grade 11 university prep English class (ENG3U). Before the workshop I was a bit skeptical about how I would apply what I learned from an English class to my teaching as a math and computer science teacher, but the ideas were very applicable and transferable to other subjects. As teachers, it’s rare that we get to sit and observe a colleague teaching, and I think this is something that needs to be done more often because we can really learn a lot from each other. In this post I’m going to share some of the things I learned.

Know your students

The session started out with me and the other new teachers learning about the background of the class we were going to see: who the students are, what their strengths and weaknesses are, what their needs are, the prior knowledge they had coming into the lesson. I think most of us do this without realizing it, but when we plan lessons, we always have to think about whether this lesson idea would work for this particular group of students. Jon Orr once mentioned somewhere that he’s never taught the same course the same way twice – how can you, when you don’t have the same students?

Whiteboard surfaces for better problem solving

I’ve said this before and many others have too, but having students work on erasable surfaces – whether it’s jotting down ideas in an English class, problem solving, or writing code – opens up student thinking in a way that writing on paper doesn’t. Students are much more willing to try new things and write down whatever ideas they have when it can easily be erased, modified, changed in one swipe. In my computer science class we don’t have any whiteboard surfaces or markers, so my students do rough problem solving on scrap paper, and I’m finding that the students don’t get the same value or deep thinking out of problem solving as I’ve seen in my math classes where we had whiteboards. (In Ms. Folino’s tweet here we had a chat about DIY whiteboards.)

Bad Idea Factory

The teachers I observed had a policy that no idea is a bad idea. They encouraged students to write down whatever they were thinking or feeling, even if they weren’t sure if it was “right”. As I observed the class, I found that the students were very open about sharing their ideas – even in front of a group twice the size of a regular class!¬†The kind of openness that this class had is what I aspire to do in my math and computer science classes: whenever I can, I’ll do a warm up from Would You Rather Math, Which One Doesn’t Belong, or another source where there’s no single correct answer: it gets students to think about things in different ways and to think critically and creatively.

Student reflections on feedback to improve

One of the ideas we talked about after the lesson observation was how to get students to use feedback to inform their learning. I sometimes have students fill out a reflection about an assignment, test or ticket out of class answering the questions:

  • What did I do well?
  • What do I need to improve?
  • What did I learn?

In the PD session, one of the teachers leading suggested incorporating the reflection¬†into the assessment: after the assessment has been returned with feedback, students use teacher feedback to answer these questions and hand these in as part of their mark for the assignment. I love this idea because it motivates students to take reflection seriously and really think about where they can improve and what they learned. This is kind of similar to Jon Orr’s take on tests. I used this strategy in my data management classes last year and my students loved it. One told me that if it weren’t for the mark upgrading, he never would have understood a certain concept. Sometimes the learning happens¬†after the test, and there’s nothing wrong with that.

Since attending the workshop, I had a chance to try the post-assignment reflection with my Grade 10 and 11 computer science classes. I was impressed with their insightful reflections! Here are some examples of the comments they made (each from a different student):



Ed Tech Stuff

At the end of the Learning Tour, the teachers showed us some tech resources we can use in our classes. Here are some of the ones I haven’t heard of and am looking forward to trying:

Screencastify: an extension to Chrome that allows you to record yourself speaking along with a video of your screen. Great for anything from giving video feedback on student work to flipping the classroom to anything else.

Hyperdocs:¬†free resources for all subjects that open as a Google Drive folder that you can copy to your own drive. They only thing they ask is that when you make a copy, each resource says “Copyright _______” in the footer. When you use it, add “adapted by ________” to the footer.

4C’s for Ted-Talks: works with any Ted-Talk as a way for students to analyze the message. You can split up students into 4 different groups, one for each “C” to focus on in this Ted-Talk, and then share thoughts as a class.

DocAppender: an add-on to Google Forms that allows you to grab all the information from all the Google Forms assessments that you’ve done and create one Google Doc per student with all of the information about that student stored in the same place.

Have feedback for me? Post it in the comments!

End-of-Year Check-in on My Goals for this Year

The first semester is winding down, and 2 of my 3 classes have started final projects. After the break we only have a couple weeks to go before we start exams. I felt like this was a good time to reflect on some of the goals I had for this semester and share how they worked out.


Recap:¬†Processing is a platform that combines coding with art to create animated sketches. My main reasons for using it in my computer science classes were that, from an educator’s perspective, it’s very gratifying for students who are learning to see the results of their code visually on the screen, plus it gets students to think more creatively in a subject that is usually viewed as very mechanical and rigid when you don’t know a lot about it.

How it turned out:¬†I think overall, Processing was a success. As I had hoped, it was a good learning tool. I felt like students could really see the usefulness of variables, conditionals and loops. In terms of creative thinking, I was amazed at some of the programs my students created. I also got to see them admiring each other’s work in class, both through their blogs and by looking at what other people around the room were working on. I thought that was pretty cool – definitely not something that would happen if we were using another platform that didn’t have a visual aspect.

Some of the challenges I had with Processing:

  1. The school computers did not have Processing installed, and I was told it would take about 6 months for it to get approved (if it got approved). I thought I could work around this by having each student bring in a USB and run Processing off a USB. This worked for my laptop, but it didn’t work on the school computers. In the end I managed to solve it by using P5.js – Processing together with JavaScript. No extra installations needed. It was not as flexible or as good for debugging as Processing in Python, but in the end it was still effective.
  2. Processing does not have an easy way to do keyboard input. After some research online, I ended up abandoning Processing for a few weeks to teach keyboard input.

Some student work from one of the practice exercises in Processing.

Assessment through Conferencing

Recap: in addition to or in place of another summative assessment such as a test, have a conversation with each student about what they learned. They would answer questions about the project they worked on, as well as overall reflections on the unit as a whole (what they found interesting, what they found challenging, etc).

How it turned out:¬†as planned, I did a conference together with a project for the first summative assessment in each of my computer science courses. I quickly found that this strategy was not the most effective – or at least, it didn’t work very well for my classes. One of the biggest problems I found was that¬†it takes forever.¬†Since I was teaching lessons for at least part of every class, with both of my classes at capacity, it took me weeks to get to every student, Student absences, my own absences and late assignments all made the process take even longer. For the next assignment, I adjusted my strategy: instead of a conference, this time each student would write a short reflection on their blog about the project that would answer the same questions (what did you find interesting? What did you find challenging?). I found that the post-unit reflection worked better as a written activity. Much less time-consuming, although I missed the one-on-one time I got with each student in the first assignment conference.

What I found worked the best was formative conferencing. In my Grade 9 class, students first did a formative activity on whatever skill we were working on. They worked on that for a couple days, then I would post the summative. While the students worked on the summative in class, I went around the room and conferenced with each one about their formative work. I gave them feedback on what they did well and what they needed to improve. This saved me a lot of marking at home, and the students found the in-person feedback helpful.

The next time I teach data management (MDM4U), I am planning to conference with each student about their ideas for the final project. Last year, I found that some students constantly had to revise their ideas, and I found it worked best to come up with a plan as a conversation rather than putting it into words in a structure that didn’t really fit.

Overall, I’m glad that conferencing was something I tried, even though it didn’t turn out exactly as I thought it would.


Recap: connect with another class somewhere in Ontario (or around the world) and collaborate in some way. I learned about it from Heather’s post here – check it out!

How it turned out: I reached out on Twitter to anyone I could find who had tweeted with the hashtags #ICS2O, #ICS3U or #ICS3C recently (the courses that I am currently teaching). I managed to connect with a few teachers who shared some great resources with me, although unfortunately none of them were teaching computer science this semester. Then, one of my friends from teachers college told me he would be teaching Grade 10 computer science this semester, so we decided to connect our classes through class blogs. In the end we didn’t communicate with each other’s classes as much as I would have liked. We did manage to connect our classes once. Both of us are new teachers teaching three different courses for the first time, and we were both too busy to really invest in it. I think it was still a good experience for our students to see that there are other classes out there, and that we can learn from each other. Next time, I would aim for one blog a month for the semester – a total of five posts. I think having a concrete goal would help keep the initiative on track, despite how busy life is otherwise.

In the future, I’m excited about connecting my math classes with other classes around Ontario in some way. Computer science is an elective and math is for the most part a mandatory course, so I think it would be easier to find another class to connect with – or even multiple classes. For data management, I think it would be cool to have my students and another class fill out each other’s surveys and analyze the data. In any class I teach, we could make up and share questions with another class. Lots of different options. Next semester I am hoping to try #OntarioClassMatch with another class.

High School Genius Hour

Recap: students get to work on a passion project throughout the semester that is connected to the course that I am teaching Рin this case, BTT1O, Grade 9 Business Technology Today.

How it turned out:¬†this one didn’t end up happening at all, for a couple of reasons:

  • no time – we were a bit pressed for time to finish the curriculum and didn’t have time for any extra projects
  • the business department at my school does a similar project in the Grade 10 intro to business course, and in the interest of students continuing to take business courses, I didn’t want the courses to be too similar

 Collaborating with Primary Students

Recap: do something together with elementary students. In my case, I wanted my students to do the Hour of Code with an elementary class.

How it turned out: this actually happened, and I think it was successful! I am planning to write another post about how it went in detail later on (but no promises). If I do, I will link it here.

Questions? Comments? Ideas? Feedback is always welcome. Hit up the comments!

Happy holidays!