Please excuse me while I geek out for a few minutes about Visual Patterns. My love affair with this versatile website has made the transition from autumn to winter as I engage in select patterns with my Algebra classes. I didn’t start using these until a unit on quadratics last trimester, so I was very pleased that a linear pattern could create just as much conversation and mathematical excitement.
For example, this is a replica of pattern #114 that we looked at in class today:
The equation y = 3x + 4 was not terribly difficult for these kids to decipher. But the fun began, as usual, when I asked them to relate their equation back to the figure. Here are some of their findings:
1. Students used the idea of slope and recognized that the slope is the change in the number of squares divided by the change in the step. The y-intercept is the value when the “zero” step is determined.
2. There are always 4 squares in the corner and each “branch” off of that square has a length of x.
3. SImilarly, there is one square in the corner and each branch from that one square has a length of x+1
4. There are always x “sets” of three squares, and four squares left over.
5. The arithmetic sequence formula works nicely here, common difference of 3 and first term of 7.
The final observation deserves its own paragraph, as I was completely blown away by the thought process. The student noticed that if we made each step in the pattern a square, then the formula would be (x+2)^2. He then noticed that the portions that were missing were two sections, each consisting of a triangular number. Recalling the formula we worked out last week (by accident) for the triangular numbers, (.5x^2 + .5x) he took (x+2)^2 -2(.5x^2+.5x) and simplified it. The result is, you guessed it, 3x + 4. Below is a photo of this amazing insight:
What I like most about these visual patterns this time around is that it helps the kids get comfortable having a mathematical conversation. Students build on each other’s thinking and discover new insights by listening to their classmates. This was difficult to do last trimester with a similar group of kids. I think that by starting with a linear patterns, rather than quadratic, the students have acclimated themselves to different ways of approaching the patterns.