Grade 6 Rocks Visual Patterns

I’m always happy to hear how teachers use visualpatterns.org with their students.

Michael Fenton shares how he uses the patterns with Desmos. And this.

Alex Overwijk’s students use the big whiteboards.

 

Bridget Dunbar removes some figures, and kids need to draw them in.

 

Kristin uses the patterns with 5th graders.

 

I do patterns with my students on Mondays as part of our warm-up routine. I’ve already shared 28 pattern talks (and 28 number talks) on mathtalks.net, but I’d like to share a couple more here because my 6th graders have made incredible gains in seeing a pattern in different ways and in articulating an equation to go with each visual.

This is pattern #153. I’m sharing this one because I meant to only use it with my 8th graders, but my printer was acting up and failed to print a different one for my 6th graders, so I just used the same one. Fun challenge!

153

Student 1:

I see these 5 spokes coming out. Each one has n number of hexagons. In between these 5 are Gauss. So, the equation is… five times n, plus five Gausses.

153 marked

 Hexagons = 5n + 5(1+n)(n/2)

 Over time, my students have come to recognize Gauss addition very quickly. They have used Gauss as a verb and a noun, as in, I Gaussed it or I saw two Gausses in the pattern.

Student 2

Each step adds another ring of hexagons on the outside. Looking at the outer most ring, I see three groups of (n+2), plus a leftover. The leftovers are odd numbers. So, the outer ring alone is 3(n+2) + 2n-1.

153 marked 2

And the rings add like Gauss!

Together we write the equation carefully, talking through each step.

Gauss means adding the first and last steps together, then multiply by the pairs of steps. The last step is the outer ring, the first step is the inner ring, which is always 10. So, 10 plus the outer ring, then multiply this by the number of pairs [of rings], which n/2.

Hexagons = [10+3(n+2) + 2n-1](n/2)

We were confident we had the correct answer when both equations simplified to the same equation.

Hexagons = [(5n^2)+15n]/2

 

This is pattern #147. I’m sharing this one because of the many different ways kids tried to see the pattern. Normally, when I randomly call on a kid to share and someone had already shared their same way of seeing, then they just have to come up with a different way.

147

147 marked

Ducks = (n^2) + (2n+1) + n

147 marked 2

Ducks = (n+1) + (3+2n+1)(n/2)

147 marked 3

Ducks = n(n+2) + (n+1)

147 marked 4

Ducks = 2(1+n)(n/2) + (n+1) + n

147 marked 5

Ducks = (n+1)(n+2) – 1

147 marked 6

Ducks = (n+1)^2 + n

I very intentionally do not have kids fill in a table of values for visual patterns. I’m afraid it becomes a starting point for them every time instead of just looking at the pattern itself. For our 8th graders using the CPM curriculum, which I like a lot, there are plenty of opportunities in the textbook to tie all the different representations (table, graph, rule, sketch). These are my 6th graders who are writing quadratic equations without all the fuss right now.

Please continue to share the site. What I love most is learning that the patterns also get used in elementary and high school classrooms.

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