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.
pizza-cuts-estimatespizza-cuts
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).

pizza-estimtes

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!

solving2_pizzascreen-shot-2017-01-29-at-3-24-16-pm

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.
pizza_responses
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 desmos.com 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 practicum.digital-activites 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?

pizza-problem

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.