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.