Pentomino Puzzles

A few years ago I was introduced to a series of activities (through my then districts math consultant) that builds a driving need for students to createscreen-shot-2016-09-30-at-8-14-39-am, simplify, and solve linear equations. I used the activity for a few years in a row while I taught grade 9 academic. Since then I had forgotten all about it (funny how that goes) UNTIL NOW!

The activity ran as a series of challenge puzzles around Pentominoes and a giant hundred grid chart.

Activity 1: Explore

Ask students in groups to choose this tile and place it on the hundreds chart so that it covers a sum of 135. The task seems so simple to start but unpacks some great math.

Allow them to determine this sum anyway they like.


I circulate and listen to their strategies. I give them very little feedback at this point. After a few minutes I choose some of those groups I heard interesting strategies to share..then let any other group share out their strategy.

img_2198Activity 2: Keep Exploring

I have them use the same tile and try again. Place the tile so that it covers a sum of 420. Listen to those strategies! Most groups that didn’t have a strategy before will try to adopt a strategy they heard last round. At this point most students will catch the strategy “If I divide the sum by 5, being like the average then I should have the middle number in the shape.”

This is where I stop and have a formal discussion as to why dividing by 5 here works? Will this always work? Will this always work with other shapes? What other shapes will this work with then?

We formalize the strategy.

Our big problem to start is not knowing where to place the tile. Let’s say I label the middle square n. What will the square immediately to the right of n always be? The left? The top? The bottom? Have them check this out by placing the tile repeatedly back on the grid.

Now let’s add all of those expressions up


The middle square must be a multiple of 5!!! I have them try this strategy out by throwing out another sum and have them place the tile.

Look at another tile!


We go back and outline that we could have chosen a different square to label n. Which results in a new equation and solves for different value…..but results in the same placement of the tile!!


We continue by me having them select different tiles, giving them sums, having them create equations and solving them. I love how hands-on this lesson is. Holding the tiles adds some “realness” which I feel drives the need to solve these equations.


this year when I remembered this activity I wasn’t sure I still had the tiles kicking around (I found them later). I immediately made a digital version with Explain Everything.


The digital version gives each student their own copy and while working in groups can chat about what strategy worked and what didn’t. Before on the paper version….only one student could hold the tile. Also, when students have to voice their strategy through Explain Everything they have to have careful thought. They think about the words they want to use. We this careful thought they get to make their thinking visible for me!

One new addition to the activity I get to make here is that they can create their own pentomino…..and then their own puzzle to share with their classmates.



Since then I also created the activity with some help from the team over at Desmos


Click to access and rune the activity

I love their new conversation tools….I get to pause the class and discuss when needed!


Students can even sketch their new tile and create an expression to match! screen-shot-2016-09-30-at-9-24-03-am


Desmos even added some nice extension questions. Love it! screen-shot-2016-09-30-at-9-24-23-am


In the future the next time I explore this lesson I see a blend of hands on tiles with digital support. I think having the best of both worlds here can pack a powerful 1-2-3-4-5 punch!

Pick your favourite!

Download the Explain Everything Pentomino Puzzles .xpl file. 

Access the Desmos Activity



One Best Thing

The MTBoS Blogging initiative has begun! Check out the two options to blog about. 

I choose option 1 which is writing about something good that happened during the day.

Week 1: One Good Thing

Today was a good day! In my mailbox was this little package.


It’s a package created & designed to make students feel amazing!

It’s from Knowledgehook.

Knowledgehook is an ed tech company specializing in  creating “engagement tools to measure and improve student learning“.

My students have been completing practice questions using their Homework product. An added, amazing bonus is when a student completes a mathalon (completing the majority of questions from the course) Knowledgehook sends in the mail a real (heavy duty) medal. Along with it is a pennant we can hang in the room.

Today a student in my class got that medal! We presented it to him in front of the class. IMG_0411
He looked a tad embarrassed, but I could see he was super proud! Big smiles. Later he told me he was going to wear it home to show mom!

That was today’s good thing.

Sneaking in Factoring

I started a series of new warm ups for my MPM2D class today. My goal is to sneak in factoring as warmups throughout the semester. By the time we need to learn it (like when we need to factor to solve equations) we will have mastered it already. I also previously snuck in multiplying binomials when we tackled quadratic patterns as Mary Bourassa did in her 2D class.

So today I gave them this slide and said I want you to solve a puzzle!

Screen Shot 2015-09-29 at 9.04.00 PM

They broke out their iPads and used the Algebra Tile app to put together the rectangle. The kids worked away and you could see them trying to put tiles in a way to make the rectangle

….and they soon found out that they had to fit a certain way!! 
On take up we made sure everyone had either my rectangle or a rotated version.

Then we did this one…..

Screen Shot 2015-09-29 at 9.04.25 PM


After we were done I asked the class: “If the combination of squares and rectangles makes up the area, what are the dimensions of the rectangle?” They had a little bit of a hard time here, but finally could see the x + 4 and the x + 2 as the length and the width. I then wrote …

 And then I heard some “aaah”s. We had previously seen both versions of the quadratic expressions and discussed why the factored form helped us out quite a bit if we wanted to find the x-intercepts.

We stopped there….It only took us 15 minutes. Tomorrow we will do a few more…..always writing the factored form after. I will also try to get students to notice efficient strategies to make the rectangles.

  • Why did you put 4 x terms along the width and 2 x terms along the length?
  • How does that relate to the number of singles?

Where I hope to go with these warm ups is to factor all types of trinomials:

  • Perfect Squares

    This time…..make a square

… and get this…



  • Trinomials of the Type ax^2 +bx + c


  • Completing the square too!!!!


This time…make a square

We’ll be definitely working our way out of the app and onto paper with area diagrams…





Completing the square


Completing the square

I think working with these puzzles for the next few weeks first will give us a strong base when it’s time to factor to help solve equations and then complete the square. I think I’ll track all the warm ups we do like this and I’ll post them all!

Desmos Challenges in iTunesU, Multi-Touch Book, and Web Version

For many years now my classes have been completing a course wide project on Picture Modelling. Before Desmos (B.D.) I use to send home copies of Geometer’s Sketchpad for students to generate a picture using only functions.Screen Shot 2015-08-28 at 12.41.47 PM
After Desmos showed up it was now super easy for students to generate art and access graphing software from any device.

Screen Shot 2014-05-13 at 9.53.09 PM

The project has been so successful at engaging students to learn about various functions and their transformations I extended it to all grades! For the last few years the project spans grades 9 through 12. Each year learning new functions and creating art.

Screen Shot 2015-06-10 at 1.49.50 PMThis summer while at the Apple Distinguished Educator Institute in Miami I started a project that would create a digital resource that would link the Modelling Functions with Art Project with function challenges created by Michael Fenton, Dylan Kane, and myself.

If you are in an one-to-one iPad room or have access to iPads the resources are in an iTunesU course and multi-touch book for iPad  otherwise they are linked on this site for any device (see below).

Each chapter starts with linking patterns, tables, graphs and equations in pre-made Desmos graphs or in pre-made Desmos activities made using Activity Builder.Screen Shot 2015-08-17 at 7.49.53 PM
Following that, activities ask students to match functions to specific criteria like Michael Fenton’s Match My Line or in my Match My Trig Function. Again the teacher can choose to use the activities in the Multi-touch book or from the pre-made Desmos activity.

Every so often in the challenges students are asked to show their thinking by uploading a picture of their work on a Padlet page. Students can crowd source different ways to solve the same problem. 

Screen Shot 2015-06-25 at 10.04.52 AM

Finally, at the end of each chapter students are to create a working piece of art and share it on a Padlet gallery page! Students can see each others work and comment.

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Each chapter covers different functions but many chapters can be done in the same course:

Ontario curriculum suggested chapters:

  • Chapter 1 – Linear Functions  – grade 9 & 10
  • Chapter 2 – Quadratic Functions – Grades 10 & 11 & 12
  • Chapter 3 – Various Functions (function notation, cubic, square root, reciprocal, non-functions).  – Grades 11 & 12
  • Chapter 4 – Trigonometric Functions  – Grades 11 & 12
  • Chapter 5 – Exponential & Logarithmic Functions – Grades 11 & 12 (Coming soon!).

The project page has more details on how to access the course, book, and web resources.