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 create, 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.

Activity 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.

However,

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

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!

Desmos even added some nice extension questions. Love it!

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.

## 10 thoughts on “Pentomino Puzzles”

Excellent activities. A nice extension to this would be to place any pentomino anywhere on the hundreds chart and tell me what the remainder will be if the sum that is covered is divided by 5. Some very nice invariants begin to emerge.

Hi Jon,

Great post. I love how you took this to so many levels. I will be featuring this in an upcoming Global Math Department newsletter.

Thanks Andrew I see this tool as quite vertical…can go down to lower grades for great number talks.

Just stumbled across your post and used to do similar work on a 100 grid learning carpet in grade three when I taught primary grades. Kids loved the challenge and the multiple patterns. I then purchased a set of the pentominoes as a hands on set so that the students could work in small groups to investigate patterns. I really like this post and will be sure to try it out with my grade 8 group this year. Thanks for sharing so many great math challenges and investigations.

This is a great activity and has some really deep thinking for math students as all of your tasks doing. Thanks for sharing and I’ll let you know how this goes tomorrow in class.

Just used the hands on version in my class and it was awesome! They finally understood the need to use expressions and to simplify expressions! Thanks so much, my grade 9 applied class and I loved this activity.

As practice, I had them come up with expressions for the sum of numbers covered by a few different pentomino shapes.

Thanks for sharing your experience. Interesting discussion pop out when two or more students have different expressions for the same tile.

Where did you get the see thu pentominos?

That just went straight into two programs for me this year. Thank you.

Awesome to hear.