Just before Thanksgiving, I was working with a 3rd grade teacher and her students to extend the idea of subtilizing and how we might leverage subtilizing to help us visualize multiplication. I used an image of 7 dots from Jo Boaler's work. While the students were not surprised that there were 7 dots, they WERE surprised with the number of ways people saw the dots. Students were eager to share ideas and continued to want to share ideas. After describing 16 different ways to see the dots, we moved the conversation from a whole class conversation to partner conversations. Using playing cards and dice to generate values, students had further conversations about how they "see" the arrangements with their partner. Our next thought is to help students arrange how they might see multiple groupings. For example, how might a ten frame help us organize and see 7 groups of 3? We've collected these ideas from Jo Boaler's Fluency without Fear, Christina Tondevold's Build Math Minds fluency webinars, and Graham Fletcher's work with subtilizing multiplication as well as our own experiences observing students' thinking through the years.
We have a professor from Ohio State working with this class who was also intrigued by the number of ways students "see" the dots. She is wondering about the pattern of how many different ways might be possible. For example, with one dot, there is only one way to see them. With two dots, there are 2 ways (as two individual dots and as a pair). How many ways to see 3 dots? Is there a way for us to predict the number of ways to see a pattern of dots? Would this change if the arrangement of dots changes? Seems like a lot of potential for extensions.... I'd love to hear your thoughts about this activity!
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