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.


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

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 Рemail 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!

Mullets and Chocolate Milk

When I was a student teacher, one of my and my associate teacher’s favourite lessons that we did in our Grade 9 applied (MFM1P) class was the mullet ratio lesson. Quick recap: students learn the index of a ratio by determining who has the best mullet among hockey players, celebrities, and even students in our class and in the school. Some of the students actually had mullets and were happy to be part of the lesson. Students had many ideas about what makes the best mullet (“how long it is in the back”, “how much longer it is in the back than front”, “how greasy it is”). The idea that we were leading into is that the best mullet is one with the highest ratio of party:business (as¬†the saying goes, a mullet is a hairstyle with “business in the front, party in the back”). Hence, the index of a ratio: one number that you can use to compare ratios.

The students loved it! It was something they could relate to since most of them were from rural areas where mullets and country music were very popular. Even the students who were normally completely un-engaged were excited. I couldn’t wait until I had my own class so I could teach it again.

Fast-forward one year: I’m teaching Grade 7 math at an all-girls private school in the suburbs of the Greater Toronto Area. We are starting rates and ratios – my favourite topic! I was excited to break out the mullets again when I realized something: these city girls are not going to be interested in learning about mullets. Some of them might not even know what a mullet is. I needed a different approach. That’s when I came up with the chocolate milk lesson.

We started off with a quick review of the previous day’s material:


And then I posed the question:


We talked about whose chocolate milk is more chocolatey and how you can tell. ¬†Some students tried to guess. Some used similar reasoning to my Grade 9s with the mullet ratios: Sasha’s* has less milk, Kate’s has more chocolate powder. Some found equivalent ratios. I led into the idea of the ratio of chocolate powder to milk in each drink. We found the index of the ratio and determined that Sasha has the higher ratio, and therefore her drink was more chocolatey. We did some practice comparing other chocolate milk ratios:


In that class we only ever talked about ratios in terms of chocolate milk. On the test, I decided to throw in a thinking question to check if my students really understood the concept. Instead of chocolate milk, I asked student whose drink was more lemony. Almost all of them were able to make the transition to thinking about the index of a ratio in a different context.

Sarah and Mia are making lemonade.

Sarah uses 3 scoops of lemonade mix and 8 cups of water. Mia uses 5 scoops of lemonade mix and 11 cups of water.
a) [K/U ‚Äď 1 mark] What is Sarah‚Äôs ratio of lemonade mix to water?

b) [K/U ‚Äď 1 mark] What is Mia‚Äôs ratio of lemonade mix to water?

c) [T ‚Äď 3 marks] Whose lemonade is more lemony? Explain how you know. Show all of your work.

Take away: it’s all about context. Put math into a context that students can relate to and can understand. This will vary depending on the culture and background of your students.

Another example: when I was teaching data management, I included this communication question which I got from an assessment at the school where I was a student teacher:

The price of bread and the price of canola oil both increase sharply after a long period of no rain in the prairies.

Explain the type of causal relationship that most likely exists between the 2 variables. Assume that the first variable is the independent variable, and the second variable is the dependent variable.

When my students got to this section of the test, I got so many questions from students about what canola oil was made of! I didn’t even consider that my suburban students might not make the connection that students who grew up on or around farmland wouldn’t think twice about.


*Names changed.