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 stupid 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.”

Just kidding. I didn’t say any of that to him. Even though I wanted to. 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.

## 5 Comments

Hey Fawn,

I just finished doing this (took 2 class periods for my class) and I’ve got some questions:

How much time did you spend considering which dimensions to do and why did you settle on 6cm x 11cm and 4cm x 7cm? (I tried it but my frame was much bigger than the picture, so students just misunderstood the task and tried to put the picture in the middle of the frame, wasting all the precious area behind the picture, yet happy that their picture looked nice…)

Unless my math is wrong, you chose dimensions that give a problem that can’t be factored. Did this lead to you teaching them the quadratic formula or graphing quadratics for solutions or both? I think it would be neat if we could come up with dimensions that are factorable, after setting the equation equal to 0. (I guess the word is composite… oh well.)

What program did you use to make sure that when you printed the picture, it was exactly the dimensions you wanted? I gave up (ran out of time) and so we ended up with a bunch of ugly dimensions (for example, the picture was 8.4 cm x 10.4 cm).

Thanks again for sharing and helping me make a boring day more interesting for the kids!

Hi Jonathan. I can’t remember why I chose those dimensions other than arbitrary whole numbers that seemed reasonable for a picture and its frame. You wrote, “my frame was much bigger than the picture” — not sure what you meant here because the area of picture is 66 cm^2, while the frame’s area is only 28 cm^2.

Right, it was intentionally not factorable to establish a need for quadratic formula. Even then, we need to talk about the precision of the cutting tools that a professional frame shop might have. There’s still a little bit of waste [of the ‘gold’ frame :)] involved, but we can then talk about the need to minimize waste for a company to cut costs and maximize profit. You can certainly make it factorable to establish a need to learn to solve by factoring. I was more interested in just generating a need for mathematics in general.

I just trusted Word to create the rectangles of stated dimensions. Again, even if it didn’t print exactly, the point was that they’d struggle to do this by hand, but with knowing how to set up and solve for quadratic equation, they would be able to figure out the frame width much more quickly. Thanks, Jonathan, for sharing!

Yeah, I meant that it was smart of you to choose the frame to be smaller than the picture–I chose different dimensions and made the mistake of making the picture bigger than the frame so the students just slapped it on and called it a day.

Thanks for answering my questions and that helps. Now I need to be good about the follow-through and keep connecting the math to the scenario we started with. Thanks!

…And I’m also stealing this.

I’ll send you pictures when we give it a try in the spring.

Thanks again for all your write-ups Fawn. I appreciate your willingness to share.

Grace and Peace.

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