Focus & Fun with the Array Game, Using Polyhedral Dice

Have you used the array game with your math students?

I was reading the book Mathematical Mindsets by Jo Boaler (so much fantastic research and so many wonderful ideas in her books!) this week, and read about the “array game” (called How Close to 100), which I’ve seen all over Pinterest and thought was very cool.

​I tried it with my 6th grade math classes last year during a little bit of down time, and they liked it. I hadn't really thought of using it in 6th grade again this year, but last week I noticed the baggie of polyhedral dice that I've had for a looooong time and thought it would be cool to use the dodecahedron dice for the array game. With these dice, the students could use numbers up to 12, rather than 6.

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Setting up and Playing the Array Game:

To set up their array games, all students needed to do was outline a 20 by 20 area on their own graph paper.

To play the array game, students paired up and then took turns rolling their dice and drawing arrays to represent the multiplication problem they had rolled.

It was very interesting to observe the way students arranged their arrays:

• Some started in the corner and worked their way out.
  
• Others started on one side and worked their way across.
  
• Some made the arrays touch, if possible, while others left a row between each one.
  
• Some just drew their first few arrays anywhere and then discovered that they didn't have a lot of room to fit additional ones.
  

The "winner" of the array game was the student with the fewest number of boxes left (some did get to zero left). The students really had fun with this!

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Array Game Extensions:

Of course, some finished their games earlier than others. In these cases, I asked students to create arrays that used different numbers than the numbers they rolled, but represented the same area.

• For example, if they rolled 12 and 5, their arrays could be 10 by 6, 15 by 4, or 20 by 3 (not 30 by 2, we discussed, because the grid is only 20 by 20).
  
• If they rolled a number that couldn't be represented by a whole-number array, they could then use an irregular shape, or a triangle - anything they could find the area of. It was interesting to see how some students got stumped when they tried to draw an irregular shape to represent a number like 81.

Most students enjoyed this twist (we continued it the next day so they all got to play this version), but a few complained that it made their heads hurt! That's ok...I know they were really thinking and growing mathematically!

A second extension for early finishers (only a few) was to use the icosahedron (20-sided) dice, and have students create area models to cover their grids and find the answer to the multiplication problems.

• This required a larger grid, so I had them tape 2 pieces of graph paper together and create 20 by 40 grids.
  
• Using the icosahedron dice gave a mix of 1-digit by 1-digit, 1 by 2-digit, and 2 by 2-digit problems to model and solve.
  

Most students didn't get very far with this before we ran out of time, but I think this is a great way for them to visualize what multiplying by a two-digit number means. I'd like to revisit this one!

I'm so glad I thought about using those polyhedral dice! 

​Have you used polyhedral dice or played the array game in your math classroom? If so, please share!​

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