How do you cut your Pizza?🍕

*This lesson is based on an activity I tried out in my Intermediate Mathematics course at uOttawa.

Slightly adapting it and integrating some technology in the process, I introduced it to my MFM2P students during practicum via Pear Deck, an interactive presentation tool, here. It was my choice of tech for this lesson because it had all the features I needed to set-up the activity with the students and collaboratively showcase our understanding of the actual task before working on it smaller groups. With features such as adding images, text/number responses, drawing/sketching, and the anonymous grid display of answers, I was able to breakdown the activity and gather that diagnostic assessment all through whole-group discussions based on student responses.

First, I displayed this image:mfm2p_pizza_problemThen asked:pizza-problem

We focused our discussion on,”Not equal slices” and “The way it’s cut.”  Both pizzas used only 3-straight cuts, but the one on the left gave us 6 pieces, and the one on the right gave us 7. Why would you cut it differently? My idea here was to provide some meaning as to why we’re about to cut our pizza in such a way …perhaps it depends on whether we’re interested in producing equal size pieces vs. maximum number of pieces.

Next, we introduced 4-straight cuts. Using Pear Deck’s student response and answer display features, we estimated the maximum number of pieces possible, showcased all responses, and consolidated by having a student volunteer to come-up to the board to present his answer —11 pieces.
In my opinion, I found Pear Deck to be extremely complementary to the introduction of this activity before breaking it down to working in groups via hands-on:

  1. It allowed for students’ initial engagement with the task to be on an individual level, yet in a collaborative environment where they can see each other’s anonymous responses and formulate whole-class discussions around their understanding.
  2. In a sense, it also allowed for differentiation because it created a safe learning space for students to engage with the task—even if unsure of their response—followed by the opportunity to self-assess/reflect by gathering feedback from peer responses and class discussions.
  3. Provided for better time-management through the collective presentation of student responses as opposed to directly introducing it through hands-on experimentation and travelling from group to group to gather diagnostic assessment.

The hands-on portion for this activity was introduced through the main challenge, which was now to find the maximum number of pieces possible using 20-straight cuts. Again, we estimated some numbers together, then they worked on it in random groups of two using the materials provided. Some students took to it and wanted to sketch it out on their boards, even though it wasn’t an easy sketch to draw —kind of left me wondering if this request was a reflection on their preference and level of comfort in working on Vertical Non-Permanent Surfaces (VNPS).


With each answer they came up with for maximum number of pieces, there was still room to find more. Only one group started collecting their data using a table of values, and so we directed our discussion towards this idea and discussed a possible next step —noticing a pattern in the data collected. Once we identified the constant second difference and the quadratic relation, Desmos Graphing Calculator  was the math tool of choice to finally help us solve this question —211 pieces!


Overall, the students seemed really interested and engaged in the task, however, I think I could have definitely consolidated it better. I dedicated a bit more time than needed before stepping in to provide some support during productive struggle and thus had less time than originally planned to spend on interpreting the actual graph we produced. My plan was to use Pear Deck’s drawing features to collectively gather and discuss student responses by labelling some points on the graph together.
Looking back on it now, however, I think a better choice would have been to design the graphing and answering portion of this lesson using Desmos Activity Builder and possibly introducing it as a consolidation activity for the following day. Not rushing consolidation is just as important as providing the necessary time for productive struggle.

My preference in using Pear Deck was mainly for what it’s actually designed for —an interactive presentation for my activity; however, when it comes to solving and graphing, Desmos is the right tool for that task. I need to see precise responses for better assessment and feedback, while allowing students the ease in answering —given the different features offered by Desmos as the interface design is math directed.

Desmos: Your Smart Graphing Calculator

“Do I have to buy a graphing calculator?”

This was the first question asked by a student in a grade 10 math class during my final practicum. A very honest question. I know when I start reading any course syllabus, the first question on my mind is, do I have to buy a textbook?

And so came the introduction of Desmos —a free online graphing calculator.  But far from the fact that it’s free and doesn’t need any fancy equipment to use in class (students can access it from any personal device), Desmos has a lot more to offer. Think of the smartphone you’re using now, and then think of what you were using before that. That’s Desmos. It’s a smart graphing calculator, and graphing is just one of its features.

What I found to be most helpful before using it; however, was observing an experienced teacher put it into action. This is definitely something I would recommend when it comes to integrating new technology in the classroom. Reading about it or watching tutorials online on how to use it is extremely helpful, but watching someone put it into action or even having the opportunity to engage with it offers you a full view of that desired outcome.

The big picture; what does Desmos offer you:

  1. Online graphing calculator —no account needed; if you have internet access, simply go to and start graphing from any personal device. If you want to save your work; however, you need to be signed in under your account (free). You can also download it as an App on your device.
  2. Pre-made Classroom Activities that you can use as-is or adapt/edit to fit your own classroom needs —simply sign up for a free account to access them.
  3. Custom activities via Desmos Activity Builder —design and create your own digital math activity online to integrate into your lesson.

Classroom activities or custom activities are simply a series of interactive slides that students can work their way through either at their own pace, while still engaged with their peers (see below), or using teacher pacing (whole class working on the same slide).

Here is a screenshot of a classroom activity offered by Desmos that I tried with my students during Some features you can add into your digital activities include; text, notes, hidden folders (from students), student input, multiple-choice questions, images, videos, and graphs. Students can also interact with their graphs, sketch their thoughts, type their response, or have their work carried forward from a previous slide to continue solving.

What does it offer your classroom:

  • Allows you to focus on the math in an engaging way
  • Allows students to see how their graph changes in real-time depending on their input
  • Supports social interaction in and out of the classroom
    • Students are able to see peer responses and provide them with instant feedback or critique their work
    • Teachers can build a strong PLN with other educators using Desmos to share/discuss ideas and/or collect feedback
  • Safe learning environment: you can choose to anonymize student names when sharing responses and have them take on famous mathematician’s names instead
  •  Teacher/student pacing: have the whole class working on the same slide together while facilitating a discussion or allow them to work at their own pace while still getting that feedback and support from their peers -students can also continue to work on the activity from home
  • Pause the class: grab your students’ attention at once by pausing their work to showcase a solution or start a discussion that can help them carry forward
  • Formative assessment: check-in on responses while your students are working to provide them with instant feedback or use it to guide your next move.

Here are some screenshots of what the teacher can see as students are working. These responses can also be shared with the class as a focus on discussions. Activity link.

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Breaking it down; how to use it:

Note: One of the best things I like about Desmos is that it’s not just designed with students in mind, but teachers as well. The instructions and support offered for this tool are unlike any I have seen; albeit I’m still starting off my career in teaching, but I’m sure experienced teachers who have used it would argue the same.

Team Desmos offers video tutorials on its graphing calculator features and classroom activities. A full PD package is also ready for download if you’re looking to share the learning with others. You can access them here or you can also check out their user guide loaded with visuals.

Classroom Example:

Here is an example of a lesson I adapted from my intermediate mathematics course at uOttawa that integrated Desmos into Pear Deck into hands-on.

Task: What is the maximum number of pieces you could get using 20 straight cuts?


First, we experimented with this:

Then, after noting the quadratic relation, we used Desmos to find our solution:screen-shot-2017-01-29-at-3-24-16-pm

In this lesson, I integrated Desmos online graphing calculator into my original plan and provided a link on Pear Deck for students to easily access it. Overall, I think they enjoyed the activity in itself and going through that productive struggle in trying to figure out the solution really paved the way for Desmos as a math tool to help us solve problems. I think what students like most about using Desmos is that it’s easy to use; you don’t spend the majority of the time explaining how to use it, but in actually using it.

With time management in consideration, I think next time I try this lesson, I will use Desmos Activity Builder for the graphing portion. Linking it to Pear Deck for students to access on their own doesn’t allow you to see their work once finished to be able to assess/offer feedback.

I still have a lot of learning and experimenting to do in integrating Desmos into the classroom, but it’s definitely a tool I would recommend to try and introduce to your students.  

3-Act: Super-Sized Coffee

This was the first lesson I did to start off my practicum. It’s entirely based on a 3-Act Task by Dan Meyer. Originally, I had my lesson set-up inline with the task to target the question: How many gallons of coffee would it take to fill up the super-sized cup? 

Walking through the plan with my mentor, however, she noticed that the students might be quick to solve this one—given that they had some similar practice before—and suggested that we start with the extension question instead: How many regular-sized cups of coffee would it take to fill up the super-sized cup? This would also offer them a new challenge during the conversion process.


What I appreciate the most about great mentors is that feedback is always offered with choice. Even though my plan was fully written-out and I was now playing around with my slides on Pear Deck 20-minutes before I walk into my lesson, the decision to change the plan was still mine to make. Of course knowing that your mentor will be there to back you up when you need the support makes risk-taking much easier.

Moving on, I presented the entire task via Pear Deck here (full credit to my AT, for showing me how to properly set-up a 3-Act Math Task on Pear Deck and choice of wording).

ACT 1: Gourmet Gift Baskets Video (making/filling-up the super-sized cup) followed by: wonder

So, “why are they wasting coffee?” To try and set a new world record! But before finding out if they were successful or not, we focused on, “how many average cups could fill the large cup?” We estimated numbers that are too high, too low, made a best guess, and then it was time to solve.

ACT 2: What information do we need to help us solve?

I presented the image of the super-sized cup along with its dimensions on the screen, and acting on another pro-tip, I simply placed my Starbucks coffee cup on a high table—front and center—and left it up to the students to get their own measurements. Most measured in centimetres, but one group went with inches instead. They worked on this task in VRG on VNPS.solving

All groups applied the right formula to find the volume of the super-sized cup and average cup but ended-up with different numbers. With support from my AT, here’s what we discussed:


1. The top diameter of the Starbucks coffee cup is 8cm and the base diameter is 6cm. We had to take this into consideration to get a more accurate sense of how much coffee can actually fit inside the small cup. One of the students suggested that we take their average and so we did.

2. Most students aimed to convert the volume of the large cup into centimetres, but did a length conversion instead. My AT highlighted the difference and then we talked a little about the method of operation they used to solve (subtraction vs. division). It was then time to wrap-up the lesson. error-conversion

The following day I asked two of the students to use the portable keyboard available in the classroom to explain and model the steps for converting volume before sending them back to their boards to finish solving.

Here are some of the answers we got:answer_coffee20161108_115943

ACT 3: Video result (available in Pear Deck link): were they able to set a new world record/how many gallons of coffee is that?

We didn’t actually get around to watching the video result mainly because it took me sometime to get a good handle on time management during practicum, but we worked on a consolidation handout to allow for individual reflection/practice on conversion—which is the area I found they struggled with the most.

Overall, I really enjoyed this task and one that I would definitely try again. I felt it was really rich in its content and offered students a nice visual to develop their conceptual understanding of volume and conversion. Time management and better preparation to be able to offer strategic support during productive struggle will be my areas of improvement for next time.

Trig Investigation: Planning for a Productive Struggle

My first attempt in trying to introduce Trigonometry to my MFM2P class didn’t get off to a good start. Yesterday, however, things were looking much better.

My main issue was in trying to plan for a lesson inline with one of my learning goals for this year —designing a student-led learning experience. Much easier said than done, but trying to choose the right problems/activities that will allow me to comfortably step back and watch as my students learn through productive struggle proved to be more challenging than I expected.

In the process, I went through some productive struggle of my own, but it was well worth it! The idea of learning a new concept by allowing yourself time and space to first struggle with it is something I strongly believe in, and hence the reason why I’m keen on learning how to incorporate it in my designs.

I decided to go with my AT’s (Associate Teacher) advice and instead of trying to put together a plan from scratch, why not try out some of the lessons she previously designed to allow myself room to focus on the experience itself and what the targeted outcome should feel like first. This proved to be extremely successful because I was able to focus my energy on planning and visualizing how I will engage the students in the lesson, orchestrate the discussion so that it would last longer than the usual question-answer period, support them in the learning process without actually interfering, and consolidate key concepts through feedback on their own work.

For this lesson, the curriculum expectation was to be able to determine the measurements of a missing side-length and angle in a right triangle using Pythagorean theorem and trig ratios.

First, students logged onto Desmos to complete a short activity that allowed them to come up with the targeted questions on their own and then estimate the measurements before solving for them. The entire activity is based on a lesson design by Laura Wheeler. Here is a screen shot of some of their answers:



I gave them some time to work on it then highlighted the questions they came-up with (length of missing side + measure of angle theta).

Next, I instructed them to go to their boards with their groups to find the answers. I didn’t want to separate the question into two parts (first this, then that) because I wanted them to practice problem-solving on their own and come-up with their own reasoning for which part of the question they wanted to solve for first. Most of the groups started with the missing side-length using Pythagorean theorem —they were applying prior knowledge for this part, but some experienced a little bit of trouble when solving for it. I guess this is what’s great about spiralling; you introduce a concept and keep referring back to it throughout the semester to build on it. Another group started solving for the missing angle by subtracting the 90 degrees from the total of 180 and then assuming that it’s an isosceles right triangle (which it was), but I instructed them to justify that assumption.

I circulated the room a bit more and engaged in some conversations with the students about the thought processes they used. One thing I planned on trying out differently in this lesson, was to pick-out which groups I wanted to call on to share some of their strategies and then provide them with a little bit of a heads-up and choice for sharing. I picked up on this idea from Jo Boaler and Cathy Humphreys book on Connecting Mathematical Ideas. Before, I would usually just throw it out as a general question during our discussion, “who would like to share?” and it didn’t really get much conversation going. But this time, I strategized better and asked the groups I was interested in if they would like to share their strategy/process with the class (hinting that it would benefit their peers) and left the choice of the ‘speaker’ up to them.

The lesson was going great so far; I wasn’t doing any direct teaching, yet still watching them learn. I called on my group and they shared their thought process for their work on finding the missing side-length.


I reiterated the points they shared and then instructed the class to polish their work—if needed—and move forward to the next step (finding the angle).

Trig tables and course-pack notes started coming out and most of the groups were able to find the missing angle accurately. I was glad to see that they used different ratios (and some used all) to find the angle because it added to our discussion later. Had it not been for a previous discussion with my AT about how I wanted to approach this part of the question, I probably would have directly asked them to find the angle using all three trig ratios and wouldn’t have had this much diversity in their answers. Pro-tips are awesome 🙂

Again, as I walked around talking to students, I picked out my groups and was happy to see that some went as far as re-polishing their work to be able to explain their process better.


We shared solutions from three different groups that used different ratios to find the angle and then highlighted what this ‘ratio’ they found and looked up in the trig table actually represents. I asked them to incorporate the meaning of this number as they explained their answer. Looking back on the lesson, however, I would have liked to ask them if this ratio would be equivalent in a similar triangle.



I got some thumbs up at the end of the lesson for how they’re feeling about trig ratios, but what really counted was how much they engaged with the lesson and in our discussion. The feedback I picked up on their learning is what distinguishes my good from not so great lessons.

Overall, I was extremely satisfied with how this lesson went and glad I had the chance to experience the process of designing and running a lesson from a problem-based approach.

I don’t expect that one great lesson will provide me with an ‘ah-ha’ learning moment for all my future ones, but this will definitely serve as an important motivational point of reference for when I need it.