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| An example of one done together in a guided group. |
- Are some fractions easier to place than others? Why? Can we use that to carefully pick our fractions, rather than throwing numerators and denominators together randomly?
- What about improper fractions? Allowed?
- How can benchmarking help us accurately place the fractions?
After working through some together, Rich and I decided to set up a center where the kids could collaborate and try some on their own. I created a sheet for them to record their work, but also asked them to come up with a consensus answer on the whiteboard.
One group started off with these 5 fractions:
One group started off with these 5 fractions:
"Decide how to arrange these fractions, and get them up on a number line on the whiteboard. I'll be back in a few minutes to see how things are going," I told them.
I went off to check in on some kids who were working in another center. When I came back, here's what I saw:
I went off to check in on some kids who were working in another center. When I came back, here's what I saw:
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| Nothing. |
Me: "What's wrong? How come there are no fractions up on the number line?"
Them: "We're having trouble."
Me: "What kind of trouble?"
Them: "We're pretty sure about 0/8, 1/9, and 2/6. But we can't decide on 3/4 and 5/7. We're not sure which comes first."
I asked them to elaborate. They knew that 0/8 was equal to 0, and that 1/9 was smaller than 2/6 because 1/9 was smaller than 1/6. They knew that both 1/9 and 2/6 were smaller than 1/2 because their numerators were less than half their denominators. And they knew that both 3/4 and 5/7 were greater than 1/2 because their numerators were more than half of their denominators. Yet the inability to make a comparison between 3/4 and 5/7 left them paralyzed, staring at an empty number line. It was time to step in.
Me: "First off, I love your thinking about 0/8, 1/9, and 2/6; how you're looking at the relationships between their numerators and denominators. And you're not the first to be stumped by 3/4 and 5/7. Those two fractions are hard to compare because they are really close together. Here's a way you can compare them."
Them: "They are only 1/28 apart. That's not very much."
Me: "Carry on."
As I reflected on what had happened, it occurred to me that showing the kids how to compare two fractions by converting them each into equivalent fractions with common denominators within the context of the Open Middle task was a good example of filling an intellectual need, a concept I was introduced to several years ago in this Dan Meyer post. The kids knew the skill; they had been finding common denominators and generating equivalents, but in the context of adding and subtracting fractions. It had not occurred to them to use it to help them with this task.
"For students to learn what we intend to teach them," write Evan Fuller, Jeffrey M. Rabin, and Guershon Harel, "They must have a need for it, where 'need' means intellectual need, not social or economic need." Many of the Open Middle tasks are activities that Fuller, Rabin, and Harel would describe as "problem-laden": mathematical activities that stimulate intellectual need, and that's what makes them worthwhile. The kids needed a strategy to compare 5/7 and 3/4, not because it was going to be on a test, or because it would come in handy later in their lives when they grew up and got jobs, but because it would help them clear a path through the open middle of an intellectually stimulating task. In the real world of the elementary school math classroom, the teaching and the needing don't always intersect. But when they do, they create a nice corner on which to stand.
Them: "We're having trouble."
Me: "What kind of trouble?"
Them: "We're pretty sure about 0/8, 1/9, and 2/6. But we can't decide on 3/4 and 5/7. We're not sure which comes first."
I asked them to elaborate. They knew that 0/8 was equal to 0, and that 1/9 was smaller than 2/6 because 1/9 was smaller than 1/6. They knew that both 1/9 and 2/6 were smaller than 1/2 because their numerators were less than half their denominators. And they knew that both 3/4 and 5/7 were greater than 1/2 because their numerators were more than half of their denominators. Yet the inability to make a comparison between 3/4 and 5/7 left them paralyzed, staring at an empty number line. It was time to step in.
Me: "First off, I love your thinking about 0/8, 1/9, and 2/6; how you're looking at the relationships between their numerators and denominators. And you're not the first to be stumped by 3/4 and 5/7. Those two fractions are hard to compare because they are really close together. Here's a way you can compare them."
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| I threw this up on the whiteboard. |
Them: "They are only 1/28 apart. That's not very much."
Me: "Carry on."
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| This student remembered to benchmark the 1 whole... |
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| ...and this one benchmarked 1/2 as well. |
"For students to learn what we intend to teach them," write Evan Fuller, Jeffrey M. Rabin, and Guershon Harel, "They must have a need for it, where 'need' means intellectual need, not social or economic need." Many of the Open Middle tasks are activities that Fuller, Rabin, and Harel would describe as "problem-laden": mathematical activities that stimulate intellectual need, and that's what makes them worthwhile. The kids needed a strategy to compare 5/7 and 3/4, not because it was going to be on a test, or because it would come in handy later in their lives when they grew up and got jobs, but because it would help them clear a path through the open middle of an intellectually stimulating task. In the real world of the elementary school math classroom, the teaching and the needing don't always intersect. But when they do, they create a nice corner on which to stand.
It's so great to read about that happened around this problem. I especially love the questions that arose. Now way to answer them without involving MP3 and MP6. Thanks so much for this inspiring reflection into your practice.
ReplyDeleteThank you Robert. The activities that you have collected at Open Middle and on your site have provided the students at my school with many opportunities to put the practice standards into action.
DeleteI must look into Open Middle!
DeleteI was wondering yesterday about decimal or fraction charades. Have a metaphorical hat full of rational numbers and students take turns trying to get other students to say them. Leave up previous tries nd label them after they're scored. (Will probably try it on Wednesday.)
ReplyDeleteThat sounds like a very intriguing activity. How did it work out? What kinds of charade actions did the kids employ? I'd love to hear about the experience.
DeleteStill waiting - our other activity went over (for worthwhile reasons!)
DeleteI love how open the task is - chosen fractions, chosen position - with such a simple but firm constraint.
ReplyDeleteI might well have shown the way to see which is the bigger fraction too. But, reflecting, I wonder if it might be possible in a similar lesson to say, look, just mark on what you know and put the other two on top of a question mark where you think they are roughly and we'll talk about it later? And give it a bigger moment in the life of the class?
That would have been a great move. It would have allowed the kids to move on with the task on their own. Reminds me of when I taught writing and there were kids who, because they were afraid to spell a word incorrectly, had a hard time putting their thoughts to paper. So I would say, "Spell it the best way you can and circle it, and move on." Allowed them an acknowledgement that they might not be right.
Delete