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Between 2 Numbers • Chewable Math • Visual Patterns

 

Lessons From the Classroom

Teach Teach

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!

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.

 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.

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.

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

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

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

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

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

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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Teach Teach

When I Got Them to Beg

A quick flip to the quadratics section in each of the six textbooks lying around here I find at least one problem about finding the width of some border. The concrete around a pool, the walkway around a garden, the frame around a picture, the border around a rug. I present to you the collage.

I don't know.

And because I don't quite know what else to do, I come up with a lame lesson idea: make the kids create a frame around a picture given a specified amount of frame, this should drive them bonkers as they won't be able to do it perfectly (not even close!), and then they'll beg me to show them the math to make this task easier. They beg, I win.

I give them the goods. To each kid:

  • scissors

  • ruler

  • any old picture, size 6 cm x 11 cm

  • a white frame, 4 cm x 7 cm (but they get 4 of these because they'll mess up!!)

My one-way conversation with them about the task. I do the talking:

Framing is very expensive. Even if you have the 50%-off coupon, it still costs a lot. For example, last year I took my son's art work that he'd done for his IB Art class to Aaron Brothers to get them framed. I wanted them matted also — you do know that matting is a fancy word for cardboard so they can charge you more, right? — anyway, guess how much the total was? Over four hundred dollars! Four-hundred-dollars-and-that's-with-the-friggin'-coupon.

I told the sales guy, "My son is not Picasso. His drawings are half crap. I don't even want them, you can keep them."

So, your job today is to be a picture framer. Show me which one is the picture... Good. Now, hold up the piece that is the frame... Good. Think of that piece of frame as gold. It's expensive.

How come I gave you 4 frames?... That's right, you'll probably make mistakes, so you get 4 trials.

Your job is to cut your [expensive gold] frame into pieces — strips — that will go around your picture. You have to use the entire 4 cm x 7 cm piece with no leftovers.

But you don't want to cut it into a million pieces either. Fewer cuts means fewer pieces to seal back together to form a frame. And it looks nicer.

Do-you-have-any-questions?-no?-good-begin.

Almost immediately, I hear:

Marissa: What do we do?

Me: I just... e x p l a i n e d...

Malainy: I'll tell her, Mrs. Win. Okay, you cut up the picture to make it fit into...

Me: What?! Cut the picture?! People bring in their most precious picture to you to frame and you cut up their picture?!

Malainy: Oohh noo. Then I'm not sure what we're doing.

Someone: Do the pieces have to be even?

Me: Have you ever seen a frame with different widths?

A Different Someone: I don't get what we're doing.

Me: Should I just speak Chinese to you guys from now on?

Yep. This is pretty much a verbatim snippet of what goes on in my last period today.

After lighting my hair on fire, they manage to work diligently.

Sure enough. About 15 minutes later they grow tired of the frame pieces. A few almost have it, but they know this is not good enough.

They speak up:

There must be a better way to do this!

I'm gonna be fired because I'm wasting all these gold frames and still not getting it right.

My pieces are thick and thin everywhere.

I always have this left over stupid piece!

And here comes the money:

Can you please show us the math for this?

Updated 04/11/13

Thank you to Christopher for sharing with me on Twitter his 3 trials:

Updated 04/16/13

Thank you to Mike Lawler for sharing this video of him working through this problem with his young son.

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Teach Teach

Des-man

Just how fabulous is Desmos.

Following up on last Friday's lesson, I had the kids create "Des-man" on Desmos. I made these very simple sketches, showed them to the kids, and told them the minimum requirements:

  • face and mouth must be parabolas

  • eyes and nose are linear equations

Not only was this so great to reinforce slope, y-intercept, and all the coefficients, it also allowed us to talk about domain and range.

What I heard around the room (that I can remember):

Oh, I get this now! I see what changing this number does!

Oops, I made his face too wide!

His smile is crooked. But I think I'll leave it because he looks cool that way.

Ha!! I see my mistake, I said x had to be greater than 4 but less than 2. Silly me.

I want the eyes to be oval shaped though. My plan is to make 2 parabolas opening into each other.

Can we work on this in 6th period too?

When I saw two students whose graphs were circles for faces, I knew they'd copied these from Desmos gallery as we haven't — and won't — learn circle equations in Algebra 1.

I reminded them of the minimum requirements, but I told the class that they may copy equations and tinker with them to add other features, such as hair and whiskers. (I actually said "whiskers," and Lexi had to tell me, "Whiskers? On a man? You mean beard or mustache?") People don't have whiskers? Good to know.

I made my guys' eyes elliptical and tweeted it, the good folks at Desmos responded.

So cool!

I think we got a lot of mileage from this activity. It's a good sign when teacher instruction is minimal and student engagement and discussion are high.

Just in case you missed the Grand Opening of Daily Desmos about 3 weeks ago, brought to us by Michael Fenton, inspired by Dan Anderson.

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