A Visual Comeback

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.

Nrich – Factors and Multiples Puzzle

Nothing gets me more excited about teaching mathematics than a task that can engage my lower level students while simultaneously challenge my high achieving students. The Factors and Multiples Puzzle from Nrich did just that. (Thanks to @drrajshah for posting this on twitter.)

I’m glad I used this in multiple classes because if nothing else, it gave students the opportunity to learn about triangular numbers! What a testament to the fact that we don’t allow students to explore with numbers nearly enough: I’ll bet only one student out of 60 had any idea what triangular numbers were.  A fantastic, interesting set of numbers, arithmetically and visually, was unbeknownst to 99% of my students.

My math recovery students were intrigued by the puzzle portion of it. In fact, I have one student in particular who is not particularly motivated by much . He’s a ‘too cool for school’ kind of kid, and he’ll tell you as much. When I bust out a puzzle, he’s all in. And when I say ‘all in,’ I mean 100%, until he solves it. It’s pretty awesome stuff to have been able to catch his attention and see how cleverly he thinks through things. Amazing.

I also gave this task to a group of advanced students. An interesting strategy these students developed was to grab a whiteboard to work out some patterns in groups of numbers.  I loved walking around and hearing their strategies.  As some groups finished, they started walking around and giving tips (not answers) to other groups.  It was wonderful.

One of my particularly eager students taped his together uniquely.  I appreciated his humor.  🙂