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.”

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

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

“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.”

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

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).

“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?”

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

“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

*Related*

Dude! The Desmos Activities are very effective. I particularly like the way you elucidate the 9(7/3), showing why we invert and multiply. I am always curious how #MTBOS ideas pan out with data, so please keep us posted on assessment outcomes.

Hey Matt,

I finally had a chance to play around with the desmos activities and really like them. I think you are onto something. I’m bookmarking them and would love to see them in action with students.

I’m curious what you think about adding a slide or two to the Kraft activity (with fractions) that asks students about their thinking. I’d want to know more about student understanding (or misconceptions) when they’re presented with a fraction of a number equaling another number. Especially the improper fraction.

Keep up the great work! Those students (and the teacher) are lucky to have you stepping in!

Yep. Improper fractions and non-integer answers are on the agenda for this week. I’m interested in getting some devices or some more lab time so I can utilize teacher.desmos.com as the powerful tool it is.

Teaching with that tool is like using a roman candle to conduct an orchestra.

Great way to visualize fraction coefficients. I was also thinking start more concrete with the classic pizza model. If 3/4pizza is 21 ounces, what is a pizza? (show image of pizza of 3 slices of a pizza that was cut in fourths) Ss notice that each slice is 7(divide by 3) and the whole pizza is 28(multiply slice by 4). Then go to the Desmos. Next get more abstract with 1/5p = 8 (refer back to the pizza), Then 5/3x=40 and the algorithm.

I like the movement from super-concrete to concrete to representational to abstract here.

The Double Clothesline – Equations is sweet! It gives students a great visual for the “divide both sides of the equation by a/b” or “multiply both sides of the equation by the reciprocal of a/b” steps of the algorithm. Lightbulb moment for my students.