Grade 6 Rocks Visual Patterns

I’m always happy to hear how teachers use 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, 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!


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 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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  1. Posted March 7, 2015 at 2:46 pm | Permalink

    Love this! Thank you for sharing these great ideas. :)

    Elizabeth (@cheesemonkeysf)

  2. Chris
    Posted March 8, 2015 at 2:48 pm | Permalink

    I’m curious, how much time do you spend on this? If it is your warm up the I suppose a smallish fraction of your period. I’ve been trying to do something similar as warm ups once a week but they always seem to take so long that the “real” focus of the day gets shortchanged. I’m wondering if you have any tips.

    Thanks for the great post!

    • Fawn
      Posted March 8, 2015 at 7:26 pm | Permalink

      Hi Chris. Normally it takes no more than 10 minutes, but yes, some days it stretches close to 20 minutes. I am mindful of the clock, but I’ll let it go this long when the discussion is worthwhile. And there are days when I just tell them that we need to stop and move on. They groan a bit but I’ll also say, “You may share with us tomorrow if you’d like.”

  3. Shannon
    Posted March 12, 2015 at 11:56 am | Permalink


    I am always excited to read your blog and I have been using some of the visual patterns with my “math support” students. I have been making tables with them and sticking to just the linear patterns.

    How do you start visual patterns with your students? What questions do you ask to start and then encourage the conversation?

    I want to try a different approach and see what happens. :)

  4. Russell
    Posted March 22, 2015 at 1:38 pm | Permalink

    Hi Fawn. I’m using CPM for Math 8 as well. After chapter 5, my warm ups became two linear patterns. Instant system of equations!! We then solve the system using the tables, the graphs, and algebraically.

    • Fawn
      Posted March 23, 2015 at 9:31 pm | Permalink

      Hi Russell. You rock! Yeah, CPM consistently employs visual patterns throughout their Course 3, so it’s good for kids to see the connections among tables, graphs, equations. Thank you for dropping in.

      • Russell
        Posted March 23, 2015 at 10:33 pm | Permalink

        I have a couple questions for you…

        1) How you used the CPM Course 2? And if so how do you feel about it?

        2)When you are first introducing patterns to students, like your sixth graders for example, how do you explain using a variable? I’ve tried to get students to work on relating sections of the pattern to the step number but I still get a ton of confused looks.

  5. David Stack
    Posted December 3, 2015 at 5:10 pm | Permalink

    This is a very good idea. Pattern recognition is absolutely essential in mathematics and is a skill that is not forgotten even in upper level mathematics. Having students come up with equations to describe patterns like this is a great way to promote critical thinking and the ability to make use of patterns. What kind of other patterns do you look at with your students? How long do you spend with them on Mondays doing patterns?

  6. Dan Kenley
    Posted January 10, 2016 at 10:48 pm | Permalink

    Thank You!!

  7. Posted May 5, 2016 at 7:30 am | Permalink

    My students enjoy Visual Patterns once each week. I often videotape their comments about the patterns. I do this as part of a number talk/warm up each day of the week so they only have 5-7 minutes. I blog three positive events each work day of the school year and on Wednesdays and Thursdays, the Visual Pattern often makes my blog as one of the highlights of MY day. I’m glad the students like it too!

One Trackback

  • By Number Talk 1 – mbrunnermath on August 5, 2016 at 6:12 pm

    […] blog about Number Talks last week and found it really inspirational. Also, I ran across Fawn Nguyen’s blog post from last year, and while it talks about Visual Patterns rather than Number Talks I think it speaks […]

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