Nice Numbers

Materials needed:

  • Paper or graph paper to draw squares
  • Pencil
  • Partner or partners

The Launch/Opening:

A number is “Nice” if a square can be dissected into that number of smaller squares.

Example:

12 is a “Nice” number.

  • Which numbers are “Nice?”
  • Is there a way to determine whether a number is “Nice” or not?
  • What conjectures can you make?

Work Session:

As students work, observer and ask questions such as:

  • What numbers do you think might be nice?
  • What numbers do you think might not be nice?
  • What strategies have you used to find nice numbers?
  • Could you describe which numbers your strategy(ies) work for?

It may be helpful to provide students with a hundreds chart but guide their focus to the numbers 25 or 30 to start. As they find solutions for specific numbers, they can shade them in.

As students develop strategies for finding “the next” nice number, encourage them to look for patterns in some of their solutions. How can these patterns help them develop strategies that are more efficient than just trial and error?

For example, students may notice that when a square is divided into four more squares, 3 is added to the total number of squares:

Give students multiple opportunities to state their conjectures, have them challenged (and even disproven). One of the main goals of this task is to spark and keep the flame of curiosity burning while discovering, using, and explaining patterns.

Students who are ready can be encouraged to determine which numbers greater than 100 are nice. The goal, here, is for students to refine their strategies to “test” numbers to determine whether or not they are nice, i.e., “I know 12 is nice, and I can add three more by dividing one of those squares into 4, so 15 is also nice, then 18, then 21, etc. They may begin to use multiplication or division to determine the “niceness” of larger numbers.

Closing:

Select student groups to share their thinking and reasoning about this problem. This may include strategies and numbers they found that are nice, new conjectures, and patterns they noticed. Encourage students to ask questions of their peers about the strategies shared and the patterns they used.

Focus the discussion on the patterns found and how those patterns lead to strategies for determining whether a number is nice or not.

Teacher Tips:

  • This task will likely take 2 or more days, depending on how far you wish to take it with your students.
  • Encourage students to:
    • use graph paper
    • keep an accurate record of the nice numbers they find – this will set them up for success as they dive deeper into the task
    • make conjectures and challenge other conjectures
  • Some numbers are more challenging than others. Offer incentives like, “No one has found out anything about 13, yet. Let’s put 13 on the Most Wanted List.”
  • When appropriate, challenge students to look for nice numbers in the hundreds and thousands. This encourages a move away from drawing to more abstract thinking centered around the patterns students discover.
  • Listen to students’ ideas and discoveries. They may not be thinking about this the way you do, but their ideas will lead them to the solution. It may not be the most direct path, but it will be the most meaningful.
  • It turns out that all numbers except, 2, 3, and 5 are nice.