Playing with Blocks: Three Dimensional Visual Sequences

During this school year, we now have occasional 90 minute blocks with our classes. I was trying to decide what to do a couple weeks ago with my precalculus class, and stumbled upon the embryo of a good idea. Kids playing with blocks to create 3D sequences. (This idea was inspired by Fawn Nguyen’s site Visual Patterns.)

I got blocks from our lower school math coach. I told kids (either working individually or in pairs) to play around with them until they found a pattern that looked interesting to them. I didn’t want them thinking about the sequence yet… I wanted them to create patterns that looked neat. The only restriction I put on them is that the pattern had to be three dimensional. If it could be represented in two dimensions, I didn’t want to see it.

They made some really nice sequences! Here are a random set of 4 to look at:

I then had students work on filling out this form. It asks them to articulate their “rule” (for building up the sequence) and has them attempt to come up with both explicit and recursive forms to get the nth term. I make it clear to them that if they can’t get the formulae, I’ll give them full marks as long as they show a serious attempt. (Some of the sequences they built involve some mathematical hoops they might not be able to traverse… for example, one group needed to find $1^2+2^2+3^2+...+n^2$ which is lovely, but not something they are going to easily figure out.

[.docx version here]

If I had time, I’d love to do two more things with this.

(1) I think it would be neat to take the photographs of one person’s sequence and give them to another person, to see what they figured out for the explicit and recursive definitions for these sequences. Why? Not only is it sharing more publicly the sequence the kids created, but many of them got a bit stuck on an explicit formula that they do have the capabilities to find, but couldn’t. I think a fresh pair of eyes, and a conversation, could be beneficial for both the original sequence creator and the new person approaching the sequence. (Additionally, there are often many ways to look at these sequences, so even if both got the same formula, there is a good chance they came up with it in different ways.)

(2) Students created a table with the first 5 terms of the sequence in it. I’d love for students to extend the table to 7 or 8 terms in the sequence, and then have students work on finding the first differences, the second differences, the third differences, etc. If students understand that having the same first difference means they have a linear relationship, having the same second difference means they have a quadratic relationship, having the same third difference means they have a cubic relationship, etc., then students who got stuck will have a new tool in their arsenal to find the explicit formula for the sequence. If, for example, they had 5, 9, 15, 23, …, and saw a common second difference, they could do the following:

Since they suspect the relationship is quadratic, they could say: $t(n)=an^2+bn+c$. And then they’d be hunting for the $a,b,c$ to make this the correct quadratic for our sequence. And then use the following three equations, they could come up with the $a,b,c$.

$5=a+b+c$

$9=4a+2b+c$

$15=9a+3b+c$.

In fact, this is an awesome thing to revisit when we get to matrices to solve systems of three variables!!!

UPDATE: One more thought before I lose it! What if I gave students the numerical sequence (e.g. 5, 9, 15, 23) expressed either written out as a list, written out as an explicit formula, or written out as a recursive formula, and had them generate a visual sequence to match it. I’d love to see how many different and interesting sequences might be created that go along with a single sequence!