I wanted to share a fun little pattern of the day. This is an extension of other visual patterns we've worked on, but this one right now stomps my 8th-graders, and it's kind of fun to see them stomped. Only about one or two kids have managed to figure out the second equation. A girl asked me if she can model the irregular shape by chopping it up into two regular rectangles, finding each quadratic equation separately, and then recombining them. YES! That's brilliant.
Questions to ask your students:
1. Can you conjecture what the next two stages will look like?
2. Which pattern is linear and which is quadratic? Which geometric property likely affects the type of function that the pattern will have?
3. For each pattern, find an equation that helps you predict the size (number of blocks) in stage x.
4. Can you figure out how many blocks will be in stage 50 of each pattern?
5. Can you figure out when (in which stage) there will be exactly _____ blocks?
Let expansion and factorization fall out as a matter of necessity. I think that's how we can help kids really sink their teeth into those skills, even before we get to "real world" problems.