Dear Claire,

In a post last week, I described wondering if I was working hard at the wrong thing. Several students were comfortable solving one-step equations like this.

"Subtract one on both sides, then divide by 3."

“Subtract one on both sides, then divide by 3.”

They were less confident, but able to solve, problems like these.

"Add five to both sides and divide by.... minus two."

“Add five to both sides and divide by…. minus two.”


"I minused three on both sides, then I used a calculator."

“I minused three on both sides, then I used a calculator.”

But their methods would prove too weak by the time we got here:

"I got stuck. I don't get fractions."

“I got stuck. I don’t get fractions.”

Claire, it would have been easy to praise their standard-algorithm style on Monday and Tuesday, then give them another standard algorithm on Friday.

Soon, math class is a tool box with a bunch of tools, but students are unable to match the tool to the function.

For most students, it’s just a big box of metal.

image: teresaphillips1965

image: teresaphillips1965

Instead, I wanted them to see why we solve equations the way we do, and Double Clothesline seemed to provide method to the madness.

Naturally, students who had been praised for their use of the standard algorithm were hesitant.

  • I don’t get this way.
  • Do I have to draw the number lines?
  • I like the other way.

The Desmos Activity on Thursday seemed to make some more connections. Some students blew through it quickly, but the questions they asked betrayed the appeal of a formula or rule in math class.

Then on Friday (with no devices, #RealTeacherProbs), I used Desmos Activity Builder to structure the lesson (pulling down those equations as PNG from Google Draw).

Fractions double clothesline 1

“Look, class. Two thirds of x is six. There are two chunks between two-thirds x and zero. How wide is each chunk? Yes, so one-third of x is three, where does x go?”

Fractions double clothesline 2

“If one-fourth of x is 12, where is x?”

Fractions double clothesline 3

“Give me a number, just call it out. [Listens for a number divisible by 3.] Okay, I heard nine, so three-sevenths of x is 9. Talk to your neighbor, where does x go?”

Claire, hopefully you see what I’m trying to do here. Offering a visual cue for equations with fractions. The language that students used to describe their work went like this:

I can see that each of the three chunks takes up three on the number line, so one seventh is three. Then I multiplied that by seven to get 21.

Fabulous. Now, when I show them this:


It makes sense.

I consider it a good day when the standard algorithm makes students say out loud, “Oh! That’s way less work.”

~Matt “Yes, it is. And now you know why.” Vaudrey