Last week, to help review mental math strategies and all of the number operations, I read the students 365 Penguins, one of my all-time favourite math books. We read it like a chapter book over three days, stopping to discuss the math and doing one of the problems as a collaborative problem each day.

The students loved this book and I think the illustrations really sparked their imaginations and supported their mathematical thinking.

The first problem the students recorded in their notebooks was one I have done several times with other classes as a way to think about multiplication and division. In the story, the father decides he needs to "organize" the penguins and puts them in 4 groups of 15, stacked in penguin pyramids! The problem asked of the students was to find as many different ways as they could to organize the penguins in equal groups.

Many students represented the numbers of penguins in each group with their drawings (using circles or Xs for the penguins, or tally marks) and others labelled their groups of penguins with a number to represent the quantity of penguins in that group...a subtle difference but this gives some insight into their mathematical thinking and level of abstraction.

Some of the students soon realized there were related groups and looked for all the possibilities. For example, if there were 5 groups of 12 penguins, they realized there would also be 12 groups of 5 penguins.

Later in the book, the father creates a penguin cube with the dimensions of 6x6x6. I asked the students to work in their table groups to create the cube using blocks and to figure out how many penguins were in the cube.

Some groups worked together to create the horizontal layers of the cube, making 6x6 squares while other groups made vertical towers of 6 and then stacked them up together in rows of six.

The materials the students selected affected how their cubes came together. The centimetre and unifix cubes only snap together one way so those cubes were made from trains or towers of 6 while the interlocking cubes were often snapped together more randomly to make up the large 6x6x6 cube. Every group made a solid cube except for one group that chose to make a hollow cube so they wouldn't use so many cubes. I thought this group of girls really demonstrated some big mathematical thinking as they visualized how their cube would come together as a hollow shell only.

Representing their cubes on paper proved to be a bit of a challenge, as did the mathematics involved in figuring out how many cubes/penguins made up the large cube and we did end up working through the 36 x 6 part together with me doing a "math aloud" using all sorts of mental math strategies such as decomposition into parts, working with tens, etc.

I'm thinking it will be interesting to do a related task with smaller cubes like 2x2x2, 3x3x3 etc.

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