I began class today showing this pattern on the screen:

The students had been previously exposed to some pattern problems but a partially completed input-output table was taking focus away from the pattern itself. Frankly, it didn’t occur to me that using a table makes the visual irrelevant until I read Fawn’s post on how she conducts her math talks. Another thing I realized was that Fawn is asking her students to read and interpret a pattern. If they are reading a visual, what’s important information for them to solve the problem?

Our math department is scheduled to present the reading strategy determining importance at next month’s staff meeting. Instead of a typical word problem I thought it would be interesting to apply that strategy to a visual pattern.

I asked, “What’s important in this pattern that will help you predict how many dots are in the 100th figure?”

Students stared at the screen for a good ten minutes. Several had no idea where to start, a few had random guesses with no math to support their claim, but one got it.

Student: I saw that the bottom row in the first figure increases by 1.The middle row is always two more than the bottom row. The top row is always one more than the bottom row. So the 100th pattern will have 306 dots.

I didn’t ask for the equation; I probably should have to see how they would have come up with

t = 3n +6.

In any event, at least one student reasoned and persevered through the problem.

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Interesting concept of reading a visual. For the past two years, I have used visual patterns to introduce linearity. And it blows my mind to see the interesting equations they come up with. I intentionally wait to teach them how to simplify because I’m so interested in the unique strategies that their equations represent. An visual pattern whose equation that simplifies to 4n+1=t, students will write as 4(n+1), 4(n+2)-4, n+1+n+1+n+1+n+1=4, 2(n+2)+2n, and my very favorite (n+2)^2 -n^2.

Now that they know how to make a table, they tend to do that more often to help them write the equations, which I think is a shame because I don’t get as many creative equations. Which is why I like to throw nonlinear patterns at them, even though it’s not really part of the 8th core. It forces them to read the visual and think creatively.

Thanks so much for taking the time to write. The student who described her solution so eloquently was able to determine what was important to her–the relationships between the rows. I need to work these in on a regular basis because my 7th grade standard and pre-algebra students would benefit. Thanks again for stopping by.

*equation that simplifies to 4n+4=t

That is a really lovely explanation from your student. Middle school, right?

Yes! Seventh grader!!!

Thanks, Mary, for sharing this post and the shout-out.

If the same student had eventually shared her/his equation as t = 3n + 6, then my questions would be, “Where did you get the 6? Why multiply by 3?…” I ask that because based on the explanation of what step 100 looks like, there was no explicit language of “3 times… add 6.”

I would presume the student to say something like “n + 1, plus n + 3, then plus n + 2.” This is the one time that I ask my [8th grade algebra] kids NOT to simplify their equations simply because I want the equation to match their thinking, their verbal/written description of how they see the pattern.

This is a great pattern to see it in lots of ways. For example, I see a rectangle (I ALWAYS see rectangles, is there a Rectangles Anonymous for me to join?!) with height of 3 and width on (n + 1), plus the 3 dots always left over in upper right. So my equation would be t = 3(n+1) + 3.

You’re fab, Mary!!

Fawn, I learn so much from you, other bloggers like Mr. Dardy, Ms. Moore, and other readers who take time from their busy day to write. You guys are my personal learning network. I mean it.

Mary

Thanks for the kind shout out and for your blog. I’m always happy to see a new post appearing in my email box!