To launch our explorations, we looked at these two graphs:
What is the same?
- They both have dots connected by lines
- They both have some dots that have 3 lines going into them and some dots with 2 lines
- 3 is the maximum number of lines going into any dot in both figures
- Both have two layers or rings. On the left, the inner ring has 2 dots and the outer ring has 10. On the right, the inner ring has 5 dots and the outer ring has 10.
What is different?
- The figure on the left has 2 curved lines, on the right, all the lines are straight
- There is a different number of regions, dots, and lines
- The shape on the right feels 3D – it was hard for some of us to see flat. The one on the left doesn’t look like it has depth.
- On the right, there is always 1 line between 2 points – on the left, the two points in the middle are connected by 2 curved lines
A few things to count came out of our conversation: dots, lines, regions, and layers/rings.
- The figure on the left has 12 dots, 14 lines, 3 regions, and 2 layers (or rings) – an inner and an outer.
- The figure on the right has 15 dots, 20 lines, 6 regions, and 2 ;ayers (or rings) – an inner and an outer.
Next, Mark told us that each figure represents the same number. Look back at the two figures above: What number do you think they might both represent?
We talked about a few possibilities before settling on the fact that the only number they seemed to share was that both figures had 10 dots in their outer layer/ring.
Then we looked at 2 representations of the number 18.
What do you notice?
- They both have 18 dots in their outer layer/ring.
- They both have 3 layers/rings: an inner ring, a middle ring, and an outer ring
- on the left, the center starts out as an oval with 2 points. On the right, the center starts out as a triangle.
- We noticed the number of dots in each ring is a factor of 18: 2, 3, 6, 9, and 18. We also noticed that if we divide the number of dots in the outer ring, by the number of dots in the middle ring, we get the number of dots in the center ring of the other figure.
- We noticed that the outer layer of both has 18 dots, but the total number of dots is different. 26 dots on the left and 30 dots on the right.
- There are 34 total lines in the figure on the left, 26 lines in the 3 layers and 8 non-layer lines. On the right, there are 42 total lines in the figure on the left, 30 lines in the 3 layers and 12 non-layer lines.
- The total number of lines in the layers of each figure is equal to the total number of dots in each figure – 26 and 34.
- Jeniah noticed that on the right, starting from the triangle, the number of regions grows 1, 1, 3 off each side of the triangle
- Maya saw different kinds of symmetry: Line symmetry and rotational symmetry.
What do you wonder?
- How is it growing out from the center later to the outer layer?
- How can you tell what number one of these is representing? Like if you look at 70, how can you tell it is 70?
- Why does the outer layer, divided by the middle layer give us the inner layer of the other figure?
- Why are there 2 pictures for each number?
- Are the non-layer lines important? Would we still recognize this as 18 without the lines that go in between layers ? Are they mathematical of aesthetic choices? What would it look like without them?
- What is the relationship between the number represented and the number of regions, dots, layers, lines, non layer lines, layer lines, etc?
- We’ve looked at the two figures that represent 10 and 18. How can we build one of these figures for another number?
Mark shared a virtual tool for us to play around with and explore the representations of different numbers.
One thing we noticed right away was the prime factorization of each number at the top of each figure. We see 18 written as 3 x 3 x 2. We also noticed the button to set the largest prime factor as either “inside” or “outside.” Above, we see 3 (the largest prime factor of 18) in the inside layer. Below, we can see the prime factorization is written as 2 x 3 x 3 and the 2 is on the inside.
Take some time and play with the tool for yourself.
What questions come up for you? What discoveries did you make?
Here are some things folks brought back to our whole group share:
- Large numbers are amazing looking – they look almost organic
- Someone else explored 28. The prime factorization is 2 x 2 x 7. On the left we can see the oval with two dots in the inner layer. On the right, we can see the heptagon with 7 dots in the inner layer.
- We can see all of the factors of 28 in the number of dots in the layers: There are 28 dots in the outer layer of both figures. There are 4 and 14 dots in the middle layers, and there are 2 and 7 dots in the center layers. 28, 14, 7, 4, 2 are the factors of 28.
- There are 28 dots around the outer layer. In this figure, we have a butterfly wing with 4 sections – each section has 7 dots. Each section of the wing comes from the square in the middle layer.
- A few folks tried to draw the representation of a number.
Aren didn’t finish, but started to build this representation of 42. (2 x 3 x 7)
One thing that is cool about the tool we used is that it allows us to draw the representation of a number and then check it. Here is what Wolfram Alpha generated for 42 (going 2 x 3 x 7):
Kristy and Jeniah tried to draw the number 6.
One thing we noticed was the representation of 6 shows up in the center of the representation of 42! When we checked the representation of 6 in Wolfram Alpha, it came back looking like this:
Upon reflection, we decided that this picture is the same as what Kristy and Jeniah drew, if you imagine swiveling the oval with two dots inside the perimeter of the hexagon.
Maya was wondering what would happen if we try to build representations where the prime factors are not in size order – for example, What would 42 look like if we went 2 x 7 x 3 or 3 x 2 x 7?
She was also thinking about choosing an aesthetic detail and then trying to come up with a number that generated that detail – for example, she noticed many of the representations looked like flowers and wondered, How could we generate a flower with three petals around the outside?
What else do you want to explore using these prime factorization graphs?
