They've done a lot of work around linear table patterns, the idea of rate of change, graphing, and function diagrams. Side note: I learned about function diagrams from the amazing Henri Picciotto, who is actually presenting on them next month at the CMC North conference at Asilomar, if you are interested in developing this additional way to represent functions. I highly recommend it!
So then I learned about VisualPatterns.org, and I gave them Pattern 109 for homework last week. They liked it, saying that at first it was strange to only get the images, but then they figured it out and it was easy (y=3x). Success at dealing with the online homework, but no rich math discussion. So then I gave them Pattern 82 and Pattern 111. Pattern 82 they handled pretty well (y=2x +1), again no discussion.
Then we got to Pattern 111.
MUCH DISCUSSION. You see, they are not given Figures 1, 2 and 3 (which makes it easy to figure out how much the pattern is increasing by every time). Instead, they have to use these examples to figure it out. We had a lot of different ways:
She made a table including the missing figures, and then looked at the "jump" in the table from 6 blocks to 12, then 12 to 18. It was immediately obvious to her that there were 2 more blocks for every new figure, so she filled in the table in purple. It took us a little side discussion to figure out how you could find the "2" if it didn't jump out at you (in red). Then she said that meant the equation must start with "2n" and then she tried 2*2 and 2*3 and realized she needed to add 4 each time to get to the correct number of blocks. Hence, her rule!
Parker took a different approach:
See Parker also many a table, where the change in x was 1. She didn't see that the y-value went up 2 when x went up by 1, so she did guess and check and found that it went up by 1. She, along with the rest of the class, assumed that it was a linear pattern - I'm glad next week they'll have to deal with some quadratics.
She used the y-intercept, or the "zero method" to first find the "+ 4" at the end of the rule. Revealingly, she previously learned "y = mx + b" but didn't realize that the "+2" on her table actually was the m in the general equation. She had to substitute in the values from the first figure to solve for m. So I stepped in and asked the class how else she could have found it. Others pointed out the connection to the rate of change she found on the table.
Finally, Lucas delighted me by his embrace of function diagrams!
Interestingly, he only needed the given values in his table. Part of function diagram is the "magnification factor." We ask "what did we do the the 3 to get 6?" A.k.a. "∆x times what = ∆y?" Sometimes we even abbreviate magnification factor as m. It's very exciting when they realize that's the SAME m as they met in middle school in y = mx + b. "Oh, that's why it's the letter m!"
That gave him "2n + ?" and he went to the 4th figure's value to figure what he needed to add to 2•4 to get 12. Then he verbally checked in with 7 and 18.
Okay, thanks for reading! I was really excited to have so many different approaches to discuss, and appreciative of VisualPatterns.org for the inspiration for my students!