Using Visual Representations to Find New Patterns 

           This past week we started practicing the thinking habits that mathematicians use. The students were given this problem:

The Painted Cube

A 3 x 3 x 3 cube is dipped into paint. How many of the 1 x 1 x 1 cubes 

would have 3 sides painted? 2 sides painted? 1 side painted? 0 sides painted?

What if the cube was a 4 x 4 x 4? A 5 x 5 x 5?

           

           By the end of the first session most students were able to come up with the number of cubes for each of the categories, and some students had creative ways to track how many cubes would have 3 sides, 2 sides, 1 side or 0 sides painted. Most could find some patterns, but for some of the numbers the patterns were challenging to discern.

         The next day, we talked about creating and recording visual representations and connecting them to numerical or symbolic representations. We created these visuals  in order to look for patterns that would help us come up with Generalizations for a 4 x 4 x 4, a 5 x 5 x 5 or a cube even larger. The challenge was to come up with the patterns that would help students to determine how many cubes would fit in each of the categories without building and counting cubes. 

         Different teams came up with some helpful visual representations that helped everyone continue looking for patterns. 

Student Voices Describing Patterns

After creating more systematic visual representations, students were able to describe and Justify the patterns they were seeing and how they would grow to larger cubes. Listen to Emily and Jonathan

share their arguments.

        Listen to Matthew as he describes an equation he sees for any size cubed. If y = a side length, then (y-2) x 12 will calculate all the cubes that have only 2 sides painted. 

         Ask your child to share some pattern they saw in the cube patterns. I have included a visual representation to help them share their thinking.

Leadership Day Challenge Activities

Leadership Day in the Woods

On Thursday, we went out to the woods and engaged in a series of challenges. For this day we split the boys and girls into two groups. It was a great opportunity to learn about the communication styles in the classroom and to watch how the students were able to negotiate with their classmates to solve problems collaboratively. It wasn't easy, and some tasks took more times than others. 


The following day, the students watched a video clip of themselves working on one of the tasks and reflected on what they might have done to contribute to the group and what they might have done that did not contribute to the group. What we learned about ourselves and how we work together and communicate as a community will now be incorporated into the classroom work. Students will offering their voice into the group more or holding back their voice and allowing room for others to add ideas. In the end all students walked away smiling...even those with some new bruises.