Exploration 6: Cutting Squares

Make a square out of paper, and then trace around it so you can remember the size. Cut the original square in half. Then cut one of the halves in half. Then cut that half of a half in half. Keep going!

  • With your first cut, you produced a half of the square. What part of the square did you produce with the next cut? In other words, how many of these smaller pieces would it take to cover the whole big square?
  • What happens with the next cut? The next?
  • How long can you keep going? What would happen if you could keep going after that?
  • What shapes did you produce with your cuts? Could you have done the task in a way that would produce other shapes? Would that have changed any of your answers?
  • Change the task a bit and try it again — here are a couple of suggested variations, try one of them on for size (or do something different that interests you)
    • Make another square. This time cut the square in thirds. What shapes do you get? Is there more than one way to do this cutting? What part (or what fraction) of the square do you get? Another way to think of this is how many same-size pieces do you need to cover the whole? Cut one third into thirds. What shapes do you get and what fractions? Keep going if you like!
    • What if the shape was different, like a triangle or hexagon — could you get interesting things from repeated cuts?

Make comments about your exploration, including answers to some of the questions above and bring it to class!

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