When Factors Put on Their Dancing Shoes

What’s the narrative? That question, so fundamental to any novel, may not sound as relevant when applied to mathematics. Take, for example, the topic of factors: 1, 2, 4, and 8 are factors of eight; 3, 5, and 7 are not. Where is the inherent drama in these relationships? In most elementary mathematics curricula, there is nothing remotely exciting or theatrical about factoring.

Not long ago, Keith Devlin, the “math guy” for NPR, proclaimed that You Can Count on Monsters by Richard Schwartz to be “one of the most amazing math books for kids” he had ever seen. At its core, Schwartz’s book is about factors and primes. Oh, and monsters, too.

Schwartz draws charming geometrical monsters for each number between 1 and 100. Each prime monster relates in some way to its numerical value. Three, for example, is represented by a triangle monster whose face, eyes, and mouth are all triangles. Five is a smiling five-pointed star. The composite monsters are more complex. They incorporate elements of those prime monsters that are their factors. So 15, whose prime factors are 3 and 5, is composed of both the triangle and star monster. The book’s progression of 100 monsters offers an engaging narrative arc by embedding the concepts of factoring, primes, and composites into the challenge of analyzing each illustration of a monster to see how it was assembled.

When I saw Schwartz’s book, I began to wonder whether there was a different narrative about factors to be told, one that would involve animation and Sketchpad. I soon settled on a plotline focused on dance. What if a number’s dance partners were its factors? A number like 24 would be incredibly popular, with dancing mates aplenty. But pity those poor primes! Aside from themselves, only 1 would consent to be their partner.

The factors of 12 dance around it in a circle.
The factors of 12 dance around it in a circle.

With this narrative in mind, I set to work on a Sketchpad model. You can download the activity from our Dynamic Number website and watch a video to see how the Sketchpad model works. The number in the big green circle (currently 12) is the number to be factored. It can be assigned any value from 1 to 36. When students press a Dance button, all the factors of 12 (1, 2, 3, 4, 6, and 12) drift onto the dance floor and revolve around it in a circle.

The factors are persistent: If you drag the 12 elsewhere on screen, its factors follow along. And as soon as you change 12 to a different value, say 8, its dance partners react immediately. The numbers 3, 6, and 12 stop dead in their tracks, while 8 comes onto the floor to join 1, 2, and 4 to dance with 8.

Is all this talk about dance really just a cute way to get children to think about factors? To some extent yes, but I would argue that the dancing animations gives students a chance to tell a powerful mathematical narrative about factors. Here are several “storylines” that students can explore:

  • What number from 1–36 has the most dance partners?
  • Does any number dance with everyone?
  • Does every number have itself as a dance partner?
  • What is true about numbers that have just two dance partners?
  • What is true about numbers that have 2 as a dance partner?

And what stories can students tell after completing these explorations? The number 1 is a tireless dancer. It is a factor of every number. Even the unpopular numbers (the primes) can count on 1 to join them. Numbers that share 2 as a dance partner belong to a unique club—they’re even. And so on. By anthropomorphizing the numbers, young learners can tell stories in ways that resonate with them and are every bit, if not more, mathematical than the staid approaches of textbooks.

Can you think of other mathematical narratives relating to numbers and their factors?

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