A Strange Algorithm

(This meeting was based on an underground mathematics lesson, Fawn Nguyen’s post and Michael Lawler’s videos. Thank you all!)

I started the meeting by showing the group the Picture This! web site that turns pairs of numbers into a diagram visualization. I asked for a volunteer to give me two numbers, each less than 10. The first suggestion was 3 & 7. I entered the number into Picture This and this diagram was returned.

I asked everyone to draw this diagram in their notebook and be ready for another pair of numbers. 2 & 8 were suggested next.

After everyone drew this figure for 2 & 8, I asked for another pair of numbers, but this time I asked participants to draw the diagram they thought would appear before I showed it. I then asked partners to share their drawings to see if they drew the same diagram. The suggested pair was 5 & 9. (If you’re seeing this for the first time, try drawing the diagram for 5 & 9 before you scroll down.)

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

I then revealed the diagram for 5 & 9. (Does it look like what you expected?)

The next suggested pair was 7 & 3 (I had asked whether the order of the numbers mattered. This was a test of that question.) Again, I had participants draw what they thought the diagram would look like and share it with a partner to see if they drew the same thing.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Then I revealed 7 & 3.

I asked for one more pair of numbers (6 & 1), asked people to draw predictions, and revealed the diagram.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

I then asked the group to look back at the diagrams for (3&7), (2&8), (5&9), (7&3), and (6&1), thinking about what they notice and what they wonder, then sharing what they noticed and wondered with a partner. In the full group conversation…

We noticed:

• The diagram is broken into even “chunks” if one number is a factor of the other number. If the 2nd number is a multiple of the 1st, then the rectangle is oriented vertically. If the 1st number is a multiple of the 2nd number, the rectangle is horizontal.
• If the 1st number is bigger than the 2nd number, the horizontal length is longer.

And wondered:

• Given two numbers, can you determine how many interior pieces it will be divided into?
• Given two numbers, can you determine how many sizes of squares there will be?

I then asked the group to make some conjectures about how the diagrams were drawn and to talk about the conjectures with a partner. After a few minutes I asked for a couple pairs of numbers that would test the conjectures they had made. The two suggested pairs were 3&5 and 4&6:

I then asked the groups to work together to describe one of these diagrams in words or equations. While they were working on this, I came around and dropped off push and support cards when I thought it would be useful.

As people were working, I brought chart paper around and asked groups to share something they discovered. At the end of the meeting, we put up their posters and each group presented.

In the final moments, I told the group that these diagrams were a visualization of the Euclidean Algorithm. I suggested that they not Google it right away and, instead, try to figure out what its purpose might be. As a clue, I suggested that we think about the following question:

What is the relationship between the length of the smallest square in each diagram and the pair of numbers that produced the diagram?

Here are the fantastic posters produced by each group.

In Attendance: Andrew, Ann, Audrey, Eric, Kevin, Maggie, Sophie, Yi

Programs Represented: DOE Fund, CUNY Start, CUNY LINCT, CUNY Adult Literacy/HSE Program, Brooklyn Public Library