Grid Power

What mathematical questions can you ask of a blank piece of grid paper?

Facilitator(s): Eric Appleton
Date of Meeting: July 14, 2016
Problem: · url

After CAMI was recently accepted as a member of the Math Teachers’ Circle Network, a few of us started exploring their resources, which include a series of videos of mathematicians leading teacher circles. Eric was inspired to share today’s problem after watching Tatiana Shubin’s Grid Power.

Eric started the meeting by asking participants to look at a blank piece of graph paper for 7 minutes and write down questions that came up. 7 minutes?! Yes, 7 minutes.

After the time was up, Eric asked participants to talk in pairs about their questions and brainstorm further questions. When sharing, Eric asked participants to consider the following:

  • What do you like about this question?
  • Do you have questions about this question? Would you like more explanation of the question?
  • Can you put the question in your own words?
  • Can you think of a simpler version of the question? An extension of the question?

After discussion, Eric then asked pairs to choose their favorite questions, which we then put on the board.

drunken-squares-our-questions
Some of our favorite questions after looking at blank grid paper

After a discussion, Eric told the group that he would like to explore the question, How many squares are there in 7×7 grid?, with the additional clarifying question, How many different kinds of squares are there?

Before letting participants go on the problem, Eric asked, “Aren’t there just 49 squares? 7 x 7 = 49. Could you give me an example of a square that wasn’t counted?” We talked about how this counts only the 1×1 squares. What about the 2×2 squares? The 3×3 squares? Etc.

Parvoneh's demo through Twitter
Parvoneh’s demo through Twitter

Participants worked on their own, then discussed solutions to the problem. The following notes show the different kinds of squares in the first column and the number of each kind in the right column, with a total of 140 squares.

drunken-squares-finding-the-number-of-sober-squares
Kinds of regular squares and how many there are on 7×7 grid

Eric then asked if we had found all the squares possible on a 7×7 grid. “Is it possible to draw a square on the grid that isn’t one of these squares (1×1, 2×2, 3×3…)?” He then asked, “What other questions occur to you? Talk to your partner.”

In small group discussions we found other examples of squares, which we called “drunken” squares. We could see that they are squares by showing the length of each side was the same, and they were clearly different that than the regular squares we had before, which we started calling “sober”.

This led to a new question: How many different kinds of “drunken” squares are there on a 7×7 grid?

drunken-squares-defining-the-question
Examples of “drunken” squares

We then talked about how to categorize the “drunken” squares we found, so that we could make sure that we were finding them all.

drunken-squares-categorizing
Thinking about how to categorize “drunken” squares
drunken-squares-finding-area
Categorizing “drunken” squares by area
drunken-squares-formula
Finding an area for the “drunken” square –> deriving the Pythagorean theorem

drunken-squares-how-many-squares-can-be-drawn-on-any-grid

drunken-sober-squares
All the squares on a 7×7 grid, “sober” and “drunken”

In attendance: Eric, Jane, Maggie, Mark, Solange, Tyler

Programs represented: BMCC, CUNY PD Team, Fifth Avenue Committee, York College

Location: BMCC

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