Sarah and I have been meeting once a week to teach ourselves some basic coding. Our first project was a function game. We are currently working on another game involving sequences, which got us thinking about the sequences in this meeting.
Sarah started this meeting by asking the group to consider the following prompt.
We spent some time on our own, then discussed the two sequences. Talking about the first sequence, Maya had a feeling the first number after 32 could be 48, but was pretty sure it was wrong. She saw that the difference between 16 and 32 is 16, so maybe add another 16 to get 48? However, she also so that the differences between the numbers change as the numbers grow, so she knew that the difference between 32 and the next number should be larger than 16.
Cindy looked at the difference between each of the numbers and saw a pattern (+4, +10, +16, …) and decided that the next difference would be +22 since the differences are growing by 6.
Benny described how he found the missing numbers in the second sequence. He saw that the difference between 24 and 34 is +10. Then he looked for a growing sequence of differences that would get from 6 to 24 is three steps.
We then did a notice/wonder activity with the following diagrams:
What do you notice?
- The first rows are our sequences.
- The row below shows the change between each pair.
- The third row is the change of the change.
- The first number in each row is red.
- The third row is the same number repeated.
- When the third row is 6, the numbers grow sequentially higher. When the third row is 2, the numbers grow not as high. The second sequence started with a larger number, but is overtaken.
- The two sequences meet at 16, then the first sequence takes over.
- In the first one, everything is +6. 4 to 10 to 16. The other one has a difference of 2.
- There is a lot of addition going on to get these numbers
- The red numbers are the same numbers are in both sequences.
- All the numbers are even.
- The 3 red numbers uniquely identify the sequence. There is a missing 4th number that should be 0.
- red 2, 4, 6 would be different from 3, 4, 6. First number = position, starting point. Second number = starting speed, change in position over time. Third number = acceleration, change of the change over time.
What do you wonder?
- Why is the first number red?
- What does this mean for numbers? How do I make this meaningful for me? What are the relationships between the numbers?
- What is a picture that would help visual the pattern? Can we see the function pictorially?
- If we put question marks instead of the 2s, could we figure them out? If we replaced the middle numbers with ?s, could we figure them out? How many question marks could we add and still them figure them out? How many numbers could we take out and still recreate the sequence?
- How do we figure out an explicit formula for these sequences?
- Knowing the three red numbers, is that enough to fill out the rest of the sequence (as long as you know the type of sequence)?
- Is three numbers in the sequence enough to continue the sequence?
- Could we pick any three numbers in the diagram to continue the sequence (recreate the diagram)?
- Do the red numbers have to end with a 0? Could the red numbers end with -2, for example?
- Could you put 0, then a number that is not 0 after it?
- Do any 3 red numbers make a sequence?
We spent the rest of the meeting exploring the red numbers, building sequences, finding explicit formulas, and exploring ways of visualizing the growth of sequences. Our Jamboard shows the work of the group.
I started thinking about the red numbers watching a video with James Tanton. (This is the 3rd video in a series, so click Previous to see the others.) He calls them “leading diagonals” and uses them to derive a formula that predicts the nth term in the sequence.
