Cup Stacking 3-Act Math Extension: PoI

If you’re a teacher in Ontario, welcome to semester 2! (If you’re not a teacher in Ontario, you are also welcome ūüôā ) 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: how tall is Ms. Kozai in cups?

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:


Note: it’s a bit difficult to tell, but the measurement of 13.6 cm for the red solo cup includes the rim.

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:



There is a mistake in one of this group’s equations – I addressed it when we consolidated without calling them out/embarrassing them.





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 ūüėČ


Questions, comments, feedback? Comment below!


Spiralling MFM2P: The End!

I am way late on this post, as our second semester already began this Monday! In any case, I wanted to post one last update on how the last few weeks of my spiralled Grade 10 applied math class went.

I was on a bit of a tight schedule with exams approaching, so the rest of the semester went pretty much exactly as planned:


The Last Strand: Trigonometry

Our last strand of the semester was trigonometry of right triangles (commonly known as the primary trig ratios, or “SOH CAH TOA”). Our assessment covered two strands: trigonometry, and measurement from before winter break (surface area and volume). I felt that this strand went really well for most students and it’s a challenging one, so I was very happy about this! Trigonometry is a brand new topic for my students so I wanted to do some good thinking tasks, as well as really teach students to understand what they’re doing rather than memorize formulas and steps. A brief summary of my lessons for the week:

  • Day 0: we reviewed a bunch of odds and ends, mainly ratios and proportions. I had students think of proportions as linear equations, and solve for the unknown using opposite operations.
  • Day 1: Jon Orr’s Trig through Slope lesson to introduce the tangent ratio. Students were very engaged when I taught this in Grade 10 academic math 2 years ago. I wasn’t sure all the students would understand the connection¬†between the triangles of different sizes and the idea that they were all similar triangles, but I think at least a few of them did and I heard someone say, “that’s so cool!” They enjoyed the slope guessing game.
  • Day 2: I introduced the remaining two ratios, sine and cosine, and we worked on solving for unknown sides in right triangles. To help students discover the formulas for sine and cosine, I showed students a 3-4-5 triangle and had them calculate tan(theta) on their calculators. Students solved for the hypotenuse using the Pythagorean Theorem. I then showed them the values for sine(theta) and cosine(theta), and had students figure out the formulas:


I had also considered a few different strategies to use for solving proportions when the variable is in the denominator: flipping the fraction, cross-multiplying, etc. I thought about this carefully; I wanted students to really understand what they were doing when they were solving and not see the strategy as a “trick” (#nixTheTricks). I put the question up on Twitter to see what other teachers were doing. Based on the responses I got, I decided that the best way to teach trig was to remind students that the fractions are also ratios. Because they are ratios, it doesn’t matter if we say “2 cups of flour for every 1 cup of sugar” or “1 cup of sugar for every 2 cups of flour”. It’s okay to flip the order of the ratios, and therefore it’s okay to flip the fractions (it’s not just a trick!).

  • Day 3: solving for missing angles using the primary trig ratios. I didn’t really do anything exciting for this lesson. I showed students a right triangle with 2 sides labelled with values and asked what question they thought I would ask about this triangle. Some students solved for the hypotenuse. I did a bit of direct instruction to show them how to find unknown angles using sine inverse on the calculator. We did some practice questions on VNPS in random groups.
  • Day 4: Dan Meyer’s marine ramp makeover 3-act math. It was Friday afternoon and students were a bit tired and unfocused. They told me there was no problem in the image of the dock without a ramp because they could jump that far. Improvising, I said, “Not with my baby cousin in the stroller!” That seemed to get their attention.

Me and my “baby cousin” in the stroller need a ramp to cross to the dock.

The last couple days we did review on VNPS and ended with our last quest (quiz/test) of the semester. The quest was no less difficult than any of the others and covered two strands instead of one, but most of my students did really well.

Wrapping up my first Spiralled Course…

As I’ve mentioned, this was my first time spiralling a course – and also my first time teaching an applied class aside from student teaching. Some of my observations from the experience:

  • All of my students were successful in the course – and all of them achieved a final mark above 60%. This was not the case at the beginning of the semester and it was a wonderful experience for me watching my students learn and improve.
  • Peter Liljedahl’s Building Thinking Classrooms workshop really helped me figure out how to use VNPS effectively. By the end of the course we were using them almost every day and I found it really made a difference in students’ learning.
  • One day after class had ended, I was taking pictures of my students’ work and I noticed that some of the solutions the groups found had mistakes. The students were already long gone and they had an assessment the next day! I posted full solutions on Google Classroom, but I doubt any of the students checked. To fix this, I started including an answer key at the bottom or side of the slides for when we do practice problems on VNPS. As much as I try to check each group’s solutions while they’re working, there’s only one of me and twenty students or more, so sometimes I miss things. Once most groups have finished, I reveal the solutions and instruct students to “check that this is what you have”. If their solution is incorrect, they go back and fix their mistakes.
  • The retention aspect of spiralling when we came back to concepts was kind of hit or miss. Next time I want to have smaller spirals where it makes sense and throw in small lessons and ideas here and there throughout the semester so that we never go more than 5 or 6 weeks without seeing all of the big ideas from the course.

Looking ahead: what I’m doing differently next time

I think the spiralling was effective and I’ll be doing it again with my new MFM2P class this semester. Since this was my first run through, I didn’t spiral as much as I could have – we only really had 2 long cycles through the 7 units of the course. By the time we came back to some concepts the second time around, most students had forgotten what we had done earlier in the year. Now that I’m a bit more familiar with the curriculum, I’m seeing more connections between units so I can “sneak in” some of those connections when I teach those topics. For example, our first strand was linear relations, which focused mainly on writing equations and graphing lines. Our third strand is solving linear systems through graphing, substitution and elimination. I’m planning to introduce finding the Point of Intersection graphically in the first strand when we do graphing, and then coming back to it in the linear systems strand. I also liked the structure of having quizzes and quests near the end of the week rather than large unit tests a few times a year. I feel like it gave students a small goal to work towards that was less intimidating than a big test.

Overall: glad I did it, and looking forward to making improvements in my new MFM2P class this semester!

Check out my spiralling posts from MFM2P last semester here and my long range plans here.

Check out my long range plans for my new MFM2P class this semester here.

Thanks for coming along for the ride! Check back here for more posts about my second go at spiralling coming soon!

Spiralling MFM2P: Double Update!!

Happy winter vacation! I have fallen a bit behind on blogging, so here is a two-month update on how spiralling in my MFM2P class is going!

(If you’re new here: I am a third year teacher in Ontario spiralling Grade 10 Applied Math for the first time. This is also my first time teaching an applied course. Follow my spiralling long range plans here.¬†Follow my spiralling journey here.)

Here’s what our November looked like:


The colours represent each of the different strands in the course.

And December:


Overall we’ve had a great two months with more good days than bad days. Some of our best days so far have been:

Spiralling Update

“How’s spiralling going?”¬†Most of my department is aware that I am spiralling MFM2P (a sure sign that I talk about it way too much). I am teaching the only section of MFM2P that is running this semester, so I have some liberty to follow my own schedule in the course. Whenever anyone has asked how spiralling is going, up until about mid-November, I haven’t had any conclusive results to share, since we hadn’t spiralled back to any topics yet… In mid-November, we started our second round of each strand.

How spiralling is going: In our first linear systems strand, we did Point of Intersection graphically and solving systems by substitution. In the¬†second linear systems strand, we reviewed solving by graphing and substitution, and added special cases of solving systems by graphing (parallel lines, overlapping lines) and elimination. When we started the second linear systems strand, I was disappointed that my students’ retention of concepts from earlier in the course wasn’t as good as I’d expected. Lots of blank stares when I first mentioned solving systems by substitution after a 7-week gap. It took a bit of prompting, but it came back to them. Students struggled with substitution the first time, but I think our problem solving and skills from the other strands helped, and it went much better in the second strand.¬†In our quest (cross between a quiz and test) at the end of the strand, almost all of my students got full marks on the substitution and elimination questions. I was so proud of them!

Many of the other strands were review of previous years (linear relations, solving equations), and retention wasn’t as much of a problem. Their memory of factoring was also a bit rusty, though not as much as linear systems. When it came to assessment time, most students did really well on factoring. About two or three students completely forgot factoring trinomials on the assessment and tried to common factor all of them ūüė¶ I’m still not sure what happened there and what I could have done differently to help those students. Maybe we went through the second round of quadratic expressions too quickly and we should have spent more time reviewing. Maybe I spent too much time on expanding and simplifying and not enough time on factoring. Or maybe the way I taught factoring the first time was not effective – students had a hard time seeing the connection between algebra tiles and factoring. If I were to do things differently, I think I would use “expand and simplify but backwards” as a teaching strategy instead, and send students to the VNPS to work through problems together with increasing difficulty, then talk strategy after when we consolidate.

For quadratic relations, we did a quick review of key features, and then launched into quadratics problem solving with Dan Meyer’s baseball projectile situation, which I turned into a quadratics problem using Desmos.¬†This strand was met with mixed results from the students. I think part of it was that it was less than 2 weeks before the holidays and all of my classes were working at about half speed.¬†Given my students’ new vacation mode, I decided to switch up the order of my last 2 strands: we did measurement before the break (surface area and volume), which was mostly review from Grades 7-9, and after we come back we will finish off the year with trigonometry of right triangles. I’m hoping my students will come back well-rested and ready to tackle one last new topic before exams.

Spiralling and Assessment

I’ve deviated away from having a quiz every week to doing more “quests” (cross between a quiz and test) every other week. We lost a bunch of days in the last couple weeks of November (I was sick one day, away for PD, other regular school interruptions like fire drills) and I was worried about being able to finish the curriculum with enough time for exam review and a final project. The quests cover about 2 weeks of material, or 2 strands. I mix the questions for both strands in all 4 achievement categories. Normally my quizzes would take half a period and would assess 2 or 3 achievement categories (Knowledge/Understanding, Application, and either Thinking or Communication). My quests assess all 4 achievement categories and I give students the whole period. Most students don’t need the full 75 minutes, but giving them 60 minutes wouldn’t leave much time to do much else, so I just give them the whole time and post some review questions for the next strand for students to work on if they finish early.

Was it successful?

It’s hard to say I’ve seen any big changes, but I’ve seen several small successes in my class. The biggest thing I was going for in my first attempt at spiralling was retention of concepts, and I believe that’s been mostly successful. The one aspect of spiralling I didn’t really do and would like to do more of next time is build in more connections between concepts from different strands through spiralled tasks. I plan to do some of this in my MFM2P class this semester in our final project.

Looking Ahead…

Here’s what January is supposed to look like in my MFM2P class:


Right now I am in search of a spiralled task that covers at least 3 of the following concepts that I can use as a final project:

  • similar triangles
  • trigonometry (primary trig ratios, Pythagorean Theorem)
  • surface area, volume, and unit conversions
  • solving linear equations
  • graphing and equations of lines
  • solving linear systems (graphing, substitution, elimination)
  • quadratic expressions (expanding and simplifying, factoring)
  • characteristics of quadratic relations
  • graphs of quadratic relations

If you have any ideas for a final project (or feedback on anything else), please share them in the comments! Happy holidays!

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