Just Right Conjectures

We are winding up our fractions unit, having ended with adding, subtracting, and multiplying fractions. Today my student teacher Alex led class, and she asked students to think about how operations with fractions are the same or different from operations with whole numbers.

This is a pretty wide and deep question. I watched curiously to see what would emerge.

Students turned and talked with a partner. The two boys next to me sat together quietly. They looked at the board, around the room, and fidgeted. When I asked what they thought, they responded with, "What's the question?"

Around us, pairs were talking animatedly, but I couldn't hear what they were saying, although I caught words like "multiplication" and "denominator." At least they seemed to be talking about math.

"Okay," Alex said, "What did you come up with?"

Vanessa started us off. She had several false starts, and other students kept breaking in to tell her what she was trying to say. Alex quieted them, and I sat poised with the marker, ready to write whatever Vanessa said.

"When you add a fraction to a fraction," she said, "you add the numerators, and the denominator stays the same. Well, if the denominators are the same, that's what happens."

That's not a very sophisticated conjecture, was my first, uncontrolled mental response. Or it sounds like a procedural trick someone taught her. But I knew it wasn't. Vanessa was making sense of fractions for herself, stating a pattern she had noticed. I thought back to the first day we had added fractions, when she had carefully added the numerators and the denominators like this: 2/4 + 1/4 = 3/8, until I asked her to draw a model.

Vanessa continued. "Also, when you add a fraction to a mixed number, you get a mixed number that is greater than the fraction you started with."

Again, my immediate mental reaction was to be unimpressed. I kept my mouth shut and wrote.

Having finished, Vanessa turned around to see who wanted to respond. She called on Amit.

Amit shared a conjecture that some students had started talking about a few days earlier. "When you multiply a fraction by a whole number, your answer is more than the fraction and less than the whole number. But when you multiply two whole numbers, the answer is more than both the numbers."

As I wrote what he said, I asked if I could change the word "more" to "greater." I paused when I got to the word "answer."

"Is there another word we could use for answer, when we are multiplying?" I asked.

"Product," several voices chimed, so I used that instead. I paused again before writing "both the numbers."

"Do you remember that those numbers you are multiplying together are called factors?" I asked. The word factor is familiar enough to our class that they nodded, and I made the substitution.

We didn't dig into this idea more. Amit called on another student, who shared another seemingly simple conjecture: "If you add a whole number to a fraction, the answer will be a mixed number made up of the whole number and the fraction."

I asked if anyone knew what the "answer" was called when you add two numbers. "Product!" Amit said. "No," Jonelle corrected him. "Sum." I wrote sum and went back to add it to the second conjecture on our list as well.

Aliyah shared next. "If you subtract a fraction from a whole number," she began, "the answer will be a fraction."

I mentally paused as I began to write her idea down. Was this true?

I took a second to introduce the word "difference" (mentally chiding myself for never making a chart of these terms for students to refer to).

"But if you subtract a whole number from a fraction," she continued, "you can't really do that."

There was a thoughtful moment of silence, then hands started to wave and voices started to rise.

"Go ahead, Aliyah," I said, "Call on someone."

"You CAN subtract a whole number from a fraction," Joseph said. "If I have 12/8, I can subtract 1."

"Oh, I meant if you were using a fraction that was LESS than 8/8," Aliyah clarified.

"Less than 1," Sandy added.

"So you're saying you can't subtract a whole number from a fraction less than 1," I repeated back.

I wracked my brain for exceptions. On the surface it made sense. But was I going to mess up those middle school teachers if I agreed that you couldn't do it? Should I bring up negative numbers? That was NOT what we were trying to learn about today.

I kept my mouth shut and wrote.

"No!" Sean exclaimed excitedly. "If you subtract a whole number from a fraction less than 1, you get a negative number!"

"Yes! That's right!" several other kids clamored.

"What? What is a negative number?" Skye asked.

Alex jumped in and drew a quick number line on the board. She started to explain, but students excitedly took over her explanation. (They think negative numbers are SO neat!) She wrote in several whole numbers greater than and less than zero and asked what the kids noticed. "It's like a mirror!" someone said.

Alex then added fractions between the whole numbers, greater than and less than zero. Many students started talking at once with questions and observations.

We had to make a choice. I looked at Alex. We stopped them. Back to fractions.

"How should we word this, then?" I asked, and I took suggestions from the class about what to cross out and change. I drew a small number line to illustrate the conjecture. And our time was up -- in fact, math had gone twenty minutes over.

I loved that some students could generalize like this. And I wondered about others, like the two boys I sat with during the turn and talk, who didn't talk at all during this conversation, and mostly fidgeted, heads down.

I spent some time thinking about my mental reactions to these ideas, the fact that my mind kept wanting more sophistication out of these ten year olds. Looking back at the list, I could see that these weren't simple ideas. They were the ideas they were beginning to really solidify for themselves about fractions. They were Just Right Conjectures, for them.

Unsure of what should be my next step (especially considering we are at the end of our time for fractions and have so much to move on to before the end of the school year), I got some advice from Kristin Gray and Jamie Garner.

I'm excited to see what they come up with.

I'll be the teacher in the corner with her mouth shut, writing.