In this meeting, we started by looking at the following two images.
Notes from our discussion
Left
- Increasing by 4 (3, 7, 11, …)
- Row of 4s
- Top rows are all primes or multiples of other numbers in the top row.
- Row of 0s from then on
- The “leading diagonal” is 3-4-0-0-0
- We could build another row of numbers on top
Both
- Come to a point
- The second row in each triangle is made of even numbers.
- The 3-4-7 triangle is the same in both figures
- The number on the top left is arbitrary
- zigzags and triangles
- Magic of adding 0 (no change)
Right
- 4,6,8,10 is the increase between the numbers on the top row.
- The numbers are not increasing by the same amount. It’s increasing by 2 more each time.
- Row of 2s
- Row of 0s from then on
- The “leading diagonal” is 3-4-2-0-0
- We could build another row of numbers on top
We annotated the number triangles to show some of the patterns we saw:
I then asked the group…
Can you generate all the missing numbers in this triangle?
How about this one?
Then we started to generate questions:
- What is the significance of the repeated numbers? What does that tell us?
- Can you add a row on top of the triangle?
- Where does the pattern start? How does it grow?
- Why do we keep making zeros?
- What if we took the 3-4-7 triangle and made it the bottom of the triangle, then went up from there?
- Are the numbers being built up or broken down? Do we have to start at the top and go down? Can we also start at the bottom and work up?
- Could we start in the upper right corner? Is it calculated, not imagined?
- Can we put whatever number we want to the upper left (if we build up)?
- What is the significance of the repeated row happening right away (in the second row) or in the third row?
- Is there a simplest case? What if I always choose 1? Is there a parent triangle?
- How would you create a triangle with only 1 zero at the base?
- Do all triangles have a side length of 5?
Answers to the diagonal questions above:
Breakout groups
Then we went into breakout groups to explore some of our questions.
One group made a triangle with a leading diagonal 4-3-2-1-0:
Another group explored what might happen if we used other operations besides addition:
Mark noticed that the number triangles are similar to looking at differences between outputs in a function table. The series below (1, 3, 6, 10, 15, …) are the triangular numbers. The number triangles and the first-, second-, third-, fourth-differences between outputs are the same:
Carol wondered about the series of numbers in other diagonals. For example, what would be the next number in the series 0, 1, 4, 12, 30, … ?
Some things to explore
It is possible to make a number triangle for a series of outputs we would see in a function. For example, the function y = 2x + 3 creates the following in/out table:
| x | y |
| 1 | 5 |
| 2 | 7 |
| 3 | 9 |
| 4 | 11 |
| 5 | 13 |
We could rewrite the outputs as…
5 7 9 11 13
and create a triangle that looks like this:
With a leading diagonal of 5-2-0-0-0.
What would be the leading diagonal of a number triangle made from the function
What would the number triangles look like for these functions? What are their leading diagonals?
y = x
y = x2
y = x2 + x
y = 2x2
Note: The idea of exploring leading diagonals came from James Tanton’s exploration of quadratics in G’Day Math: https://gdaymath.com/lessons/gmp/3-3-aside-a-slide-puzzle/
