Sarah Lonberg-Lew of the Adult Numeracy Network and SABES joined us from Gloucester, MA to lead this meeting with me (honestly, I did very little). We explored a diagram that Play With Your Math calls factor graphs. They got the idea from Things to Make and Do in the Fourth Dimension, by the mathematician and educator Matt Parker. (Check out Numberphile for some of his videos.)
The week before the meeting we sent out this teaser:
And got a few responses:
Is it possible to connect five houses on a flat surface without crossing the lines?
We started the meeting with the same four house problem and got another solution for connecting the houses.
Is it possible to connect all four houses without crossing the lines, without moving the houses, while also keeping the lines straight?
We then moved on to a notice/wonder with the following image.
We took a few minutes on our own to take note of what we noticed and wondered about the 8-factor graph above. After everyone had some time to think on their own, we shared what we noticed and wondered. Eric took notes on a split screen so the group could see the graph and the notes as we spoke. Here are the notes from our notice/wonder:
Here are the rules that were used to draw the 8-factor graph that we looked at:
- List a number and its factors
- Each number within the graph connects to its factors (each number also needs to be connected to its multiples)
- Lines can’t cross (unless you change the rules)
Sarah prefaced the rest of the meeting by explaining the four house problem is from the field of graph theory, which uses a different definition of the word “graph.” In graph theory, a “graph” involves the ideas of “vertex” (a point), “edge” (a line connecting two vertices), and “degree” (the number of edges that connect to a particular vertex).
After the discussion, we moved into individual thinking time for about 5 minutes. Sarah and Eric experimented with muting the group for enforced silence, so that everyone had some uninterrupted time to consider our questions on their own.
We then moved into two breakout groups for about 40 minutes of small group work. After the small group work, we came back together to discuss what the group discovered.
Maya and Patricia worked on making a factor graph for 16. They started by thinking about a cube and wondering if it would be possible to place factors on different vertices of the cube. They thought to remove lines when they weren’t needed. They realized 1 had to connect to all the other factors of 16. And 16 also had to connect to every other number.
Then they imagined laying strings around a cube to connect the factors and multiples. It seemed possible to connect all the factors and multiples in this way.
They then tried to make a factor group of 16 on a flat surface. They were able to make every connection except the one between 2 and 8 (shown by the red line). However they arranged the numbers, there were always two numbers that couldn’t be connected.
Maggie, Amy, Violeta, and Nicholas also tried to make a factor graph for 16. They looked at a few different arrangements including this one.
In this graph, 1 is connected to 2, 4, 8, and 16. Then 2 is connected to 4, 8, and 16. Then 4 is connected 8, and 8 is connected to 16. However, there isn’t a way to connect 4 to 16 without crossing a line.
An aside: The group noticed a connection to the handshake problem. In the 16-factor graph, each number has to “shake hands” with each of the other numbers.
The group then created factor graphs for other numbers like 25, 12, and 18.
In our discussion, Sarah wondered about the graph of 16. Do you think it can be done and we just haven’t found it, or do we think it can’t be done? If you think it can’t be done, how do you decide that? Maggie said it is impossible, maybe a bit tongue-in-cheek, since seven of us have worked on it and haven’t been able to find a solution. Nicholas said he was mostly convinced that it is impossible, but would need more time with it. Patricia talked about how it seemed possible with a globe and string, in three dimensions.
We noticed that the two factor graphs for 12 and 18 seem to be identical…
- They both have 6 vertices.
- You could substitute the numbers in one for the numbers in the other.
… which made us wonder if there is some kind of equivalence between 12 and 18. And what other numbers would have the same factor graph.
Other questions that came up at the end of our conversation included:
- What is the smallest number where a factor graph isn’t possible? (Maybe greater than 16, since it seems that its graph isn’t possible.)
- What is it about 16 that makes it not work? (Something about how every number has to connect to every other number. If 5 elements all have to connect to each other, we’re in trouble.)
You can watch the full meeting here:
