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


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.

Hyperdocsfree 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!

Grade 10 Math – Resources by Course Expectation

Last year in second semester I taught 1 section of Grade 10 academic math (MPM2D), and 2 sections of Grade 12 data management (MDM4U). Grade 10 math was one of my favourite courses to teach as a student teacher, although I had only taught it at the applied level (MFM2P). The curriculum is so relatable, and it’s concrete enough to constantly have fun 3-act math and other similar activities that I could draw from or make in order to make the math come alive. My class was super eager and into it, and there were many “aha” moments. As a teacher, it was exciting to see my students experience math in a new way.

In order to keep myself organized, I use Google Sheets to keep track of the online resources I use for future reference and to pass on to other teachers looking for ideas.  I made one spreadsheet for each course, and within that, I have one sheet for general resources and information, and one sheet for warm up activities, blogs, and other things like spiralling resources, pedagogy and assessment ideas. I then added a new sheet for each unit: linear systems, analytic geometry, quadratics I, II and III, and trigonometry. I was not spiralling the curriculum this time, but someday I’d like to. Although I don’t like to see math as a bunch of isolated topics, this format helps keep me organized to make sure I am meeting all of the curriculum expectations.


Below are my Grade 10 math resources, organized by unit and topic. Some of the activities I did for one unit I ended up coming back to later in order to teach a different concept. In a spiralled course, these could be blended together over a few days. My resources are geared towards Grade 10 academic (MPM2D) but many can also be done in an applied class and adjusted to meet the expectations, many of which are similar or the same. All the credit goes to the original creators of each activity, lesson or idea. If I forgot to give anyone credit for something, my sincere apologies – email me or comment and I will add it in.

Note that this is not a comprehensive list. Some of the topics I didn’t use an online resource for or haven’t found one yet. I will update this post as I find new information.

Unit 1: Linear Systems

  • Review: slope and y-intercept. Lots of activities for this – here are a couple:
    -Jon Orr’s Reading Relationships
    -Mr. Hogg’s Fast Walker
  • Review: independent/dependent variables
  • Review: graphing linear equations – Jon Orr’s Crazy Taxi (4 representations)
  • Solving systems – substitution and elimination
    -Kyle Pearce pile ups
    -Alex Overwijk – solving systems with manipulatives
  • Solving systems – word problems
    -there are a few types. One is the distance/speed/time problems – I use Jon Orr’s Two Trains

Unit 2: Analytic Geometry

  • Distance formula: Jon Orr’s distance formula without the distance formula (love this one!)
    -review of Pythagorean Theorem – Andrew Stadel’s basketball travel 
    -consolidation: Desmos Zombie Apocalypse by Andrew Stadel
  •  Equation for a circle: no link, but developed my own lesson in a similar style to the “distance formula without the formula” lesson
  • midpoint: Dan Meyer’s best midpoint (similar lessons available for best ________, but I found that too many similar type lessons gets repetitive)
  • review of equation of a line: Kyle Pearce’s paper stacks 
  • equation of the median, altitude, perpendicular bisector
  • characteristics and properties of triangles
  • verifying characteristics and properties of quadrilaterals, circles, triangles, other shapes

Quadratics I: Factoring

  • expanding and simplifying expressions: box method (no link right now)
  • review of algebra tiles: warm up – #29 on Which One Doesn’t Belong
  • common factoring
  • factoring (simple): Jon Orr’s algebra tiles for factoring
  • difference of squares and perfect squares
  • factoring complex trinomials

Quadratics II: Zeros of Quadratics

  • difference between linear and non-linear: find the pattern/group similar things together (is there a Desmos activity for this?)
  • first and second differences: I used Tips4Math’s “Going Around the Curve” activities
  • solving quadratics by factoring: used a video from Legendary Shots with an estimated distance and equation (that factors perfectly) to solve – I will blog this eventually
  • quadratic formula: Dan Meyer’s will it hit the hoop?
  • read and non-real roots
  • problem solving (word problems)

Quadratics III: Transformations and Completing the Square

  • modelling quadratics: my own performance task – see my post here
  • transformations a, h and k: Laura Wheeler’s Desmos pattern finding
  • comparing y=x^2 and y=2^x: Mary Bourassa
  • graphing y =a(x-h)^2 + k
  • determine the equation of a parabola from the graph
  • transformations activity: Desmos Faceketball
  • transformations activity: Desmos Marbleslides (so much fun!)
  • vertex midvalue method
  • completing the square
    -mathcoachblog: the box method
    -completing the square visual representation
    -whenmathhappens: why completing the square works
  • max area given perimeter: Jon Orr perimeter jumble (tweaked to focus on the Grade 10 expectation)
    -a teacher at my school has his students do the max area given set perimeter by using toothpicks to make the biggest possible area – as a square, using only 3 sides, etc.
  • other max/min problems


  • review – rates, ratios, proportions, angles, congruence/similarity
    -see my post on reviewing angles here
  • comparing congruent and similar triangles: Desmos card sort – congruent triangles
  • similar triangles word problems (shadows, reflection, etc.)
  • tangent – Jon Orr’s trig through slope (love this one)
  • primary trig ratios (SOH CAH TOA): see my tweet here
  • Sine Law: see my post here
  • Cosine Law (and Sine Law): I didn’t use this personally, but maybe you will be able to use some of these visuals
  • problem solving: Dan Meyer’s marine ramp makeover (one of my favourites)


That’s all I’ve got. If you know of any online resources that I missed, please comment and I’ll add it to the list! Happy Friday!

A New Approach to Sine Law

It’s Monday evening of Labour Day, which for most students and teachers in Ontario means tomorrow is the start of a new school year. My school is one of four schools in my board that start a week early, so tomorrow is just another day for us.

I wanted to write about a lesson I’ve taught twice already introducing students to Sine Law. The first time I taught this lesson was as a student teacher, to a Grade 12 college prep math class (MAP4C). The second time I taught it was last year, to a Grade 10 academic math class (MPM2D). For context, we had just finished the primary trig ratios (SOH CAH TOA) and “solved problems involving real-life situations using the primary trig ratios” (also known as word problems).

The lesson went like this:

We did a quick warm up of “find the fake”, an idea I got from my host teacher back when I was a teacher candidate. The students have to figure out which triangle is possible, and which isn’t.


I told students to think of it like shining a flashlight through a hole at each of the angles. At each angle, which light from the flashlight would create the biggest spot of light on the opposite wall? The students quickly realized that the biggest angle would allow the most light to get through – meaning the longest side must be opposite the largest angle. Triangle B is the fake!

We did a brief review of what angle of elevation and angle of depression mean (to help with some of the homework questions). Then we got to the main part of the lesson: developing Sine Law. (We took a minute to talk about how to label sides and their corresponding opposite angles.) Similarly to the lesson I taught when I was a student teacher, I had students work with a partner to complete this table:


By now the students were used to looking for patterns in order to learn new concepts, and I think this format works really well with the Ontario Grade 10 curriculum, where almost everything is concrete enough to be able to learn through patterns.

Within a few minutes, my students figured it out:


Here’s our complete table when we took it up as a class:


The students were able to deduce the 2 forms of Sine Law (sides on top vs. angles on top), and talked about when to use which form. The rest of the lesson we spent going over a few examples, and doing practice questions.

Overall, the lesson went okay and my students understood the idea, but I felt like there was something bothering me about it. Later in the week, I was talking with another teacher in my department, and she told me how she taught Sine Law. The first thing she did was put up a non-right triangle on the board (one that can be solved using Sine Law):


(Note: the above triangle is a random example found online – not necessarily the one my colleague used.)

And all she said was, “Solve it”. At first the students were stumped, but most of them quickly figured out that if they draw a perpendicular line from angle A to the opposite side, they had 2 right triangles, and could therefore use SOH CAH TOA:


Once they had solved all of the angles and sides, she gave the students another non-right triangle, which they were able to solve confidently. She then asked them to look back on what they did and determine how they solved for each missing value in a more general way. They were able to deduce the formula for Sine Law, as proven through the primary trig ratios.

I realized that I liked my colleague’s idea better: while there is definitely value to students figuring out rules by finding patterns, a few examples that fit a pattern is not a proof. As a math student, I remember my university professors stressing that just because a rule works in a few cases, it doesn’t mean it will always work. My colleague’s technique was stronger because the students derived Sine Law using formulas that they already knew – a true proof. Because they derived it themselves, these students had a deeper understanding of why Sine Law works, and if they ever forget it, they know how to derive it again. This is how I plan to teach Sine Law next time around. It’s always better that students have a deeper understanding of why formulas work rather than simply memorizing them.

Big Ideas for this Year

These past few weeks I’ve been getting ready to start the school year at a new school teaching new courses. This semester I will be teaching Grade 9 business technology (BTT1O), Grade 10 computer science (ICS2O), and Grade 11 computer science (ICS3U/3C). I have a lot of ideas for things I want to try and I’m hoping to get to as many of them as possible. For anyone else looking for some new ideas for the new school year, here are some of the things I want to try:

  • Processing: I’m planning to use Processing as the programming language/environment for both of my computer science courses (Grade 10 open and Grade 11 college/university). Processing combines programming with animation. I decided to use it because it makes programming much more exciting and applicable because you can see the results of your code right on the screen! Here is an example of a simple image I put together using a few lines of code. I’ve been having a lot of fun playing around with Processing the past few days, and I’m very excited to introduce it to my students.
  • Assessment through conferencing: earlier in the summer I attended a conference on inquiry-based learning in intermediate math. One of the teachers who I met gave me some great ideas for teaching computer science, including assessment through conferencing. Instead of having a test at the end of each unit, the teacher meets with each student for 5-10 minutes and ask them questions about what they learned. The students have the option of preparing a product, such as a Powtoon or presentation, to show what they know. All assessment is done on the spot and the students receive instant feedback (plus, less piles of marking to take home!). Conferencing can be done in combination with a summative project or as an assessment on its own. In project-based courses like the ones I’m teaching, I feel like it makes more sense to have a conversation about what students know rather than a pen-on-paper test. Conferencing also helps eliminate common problems in these courses like plagiarism.
  • #OntarioClassMatch: today I had the pleasure of chatting with an awesome colleague about collaborating with other classes in Ontario! Class collaborations can be anything from working together on a project, students writing blogs about something and responding to each other, or sending each other problems to solve. I have no idea what this is going to look like for my classes yet, but I’m super excited! If anyone reading this is teaching BTT1O or any computer science class and would like to collaborate, feel free to reach out to me! I’m also happy to collaborate with anyone who is doing surveys at any point in the year (I’m also on Twitter!)
  • High School Genius Hour: another great idea from a colleague that I am hoping to try with my business tech class. I am envisioning something like the show Shark Tank, where the students have to come up with an idea that somehow solves a problem or improves a product. The problem they solve can be big or small – anything from improving the environment to making something more convenient within their own class or home. Students will be given time throughout the semester to work on their projects, and it will tie in with some of the units – for example, I will have them write out their pitch formally in MS Word during the unit on Word Processing, I may have the students model their survey around their invention when I teach Excel, and their electronic presentations at the end of the year will be the finalized descriptions of their product or invention.
  • Collaborating with primary students: this is something I’ve been interested in for a while. I also attended a conference workshop about collaborating with elementary school students. I am hoping for my computer science classes to do a field trip in early December to one of their feeder schools to do the Hour of Code with an elementary class. Very fun, and very cool for my students to help the younger students. At the conference workshop, the speaker spoke about having the elementary students come up with ideas for games for my students to implement, and for my students to write programs to model one of the strategies that Grade 3 students use to add two-digit numbers. I have a colleague who is teaching a Grade 1 class at a school nearby, and I’m hoping we can coordinate to do something together at some point in the year. If any other elementary teachers are interested in collaborating, feel free to contact me!

That’s all for now. I hope to post an update later on in the year with the results. If you have any feedback or ideas, or you are a teacher looking to collaborate, please let me know! It’s going to be a great year!