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At first, this is what I thought the student had done:

  • First, the student drew six circles to represent “out of 6 books.”
  • Then, they distributed, one-by-one, the 66 books into each of the 6 circles. (If they just put 11 in each, why tally them?)
  • Then, the student searched for a way to represent the “5 out of” that are non-fiction.
  • It follows that the remaining books are fiction. That makes six sixes, or 36 books.

But then Bridget and Julie came in with a fantastic, different interpretation. Their’s feels like an improvement on my first draft.

We then got to work trying to come up with some activities to address this work. Suppose that your class of 6th Graders try this problem, and a lot of your class has struggles that are similar to the work above. You’re planning tomorrow’s lesson. What activity would you begin class with?

This is what we came up with. Which of these activities do you think would be most helpful? Are there any changes you would make to any of them? Is there a combination and sequence of these activities that you think would work particularly well? (I took a shot at sequencing them below. Some details on activity structures are here.)

5 out of 6 Mistake-page-005

5 out of 6 Mistake-page-003

5 out of 6 Mistake-page-004

5 out of 6 Mistake

5 out of 6 Mistake (1)

 

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When kids are learning to give fractions meaning, I think they often struggle to figure out how the numerator and denominator are coordinated. Here we see a middle step in understanding, maybe: it’s not that the numerator and denominator are totally disconnected. They’re just coordinated in a way that doesn’t really correspond to how they actually work together (i.e. denominator tells you the “unit” and the numerator tells you the “quantity.”)

Maybe the progression of learning looks like this:

  • 2/3 means “2 and 3,” nothing to do with each other. Totally baffling notation.
  • 2/3 means “2 by 3” or “2 times 3,” some more familiar situation where two numbers can be coordinated in a relation.
  • 2/3 means “2 thirds,” which is a productive way to coordinate the numerator and denominator.

Thoughts? Am I overinterpreting this as a middle step in a progression, when it’s actually just a totally uncoordinated interpretation of the fraction?

Fraction comparison for 4th Graders. They’ve been working a lot with representing fractions as circles and as rectangles. They’ve done some basic addition with fractions. Most aren’t generally able to find equivalent fractions.

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What mistakes do you expect to see in the class set?

Make a prediction! Mark it down somewhere. Don’t do that internet thing of just continuously scrolling through a page at half-attention. Take a moment, form a thought. Then scroll on for the full class set of 14.

In the comments, would you please answer this question: Which mistake most surprised you? Why?

Kid 1

6thGrade-copier@saintannsny.org_20150331_124451 (1)-page-001

Kid 2

6thGrade-copier@saintannsny.org_20150331_124451 (1)-page-002

Kid 3

6thGrade-copier@saintannsny.org_20150331_124451 (1)-page-003

Kid 4

6thGrade-copier@saintannsny.org_20150331_124451 (1)-page-004

Kid 5

6thGrade-copier@saintannsny.org_20150331_124451 (1)-page-005

 

Kid 6

6thGrade-copier@saintannsny.org_20150331_124451 (1)-page-007

Kid 7

6thGrade-copier@saintannsny.org_20150331_124451 (1)-page-008

Kid 8

6thGrade-copier@saintannsny.org_20150331_124451 (1)-page-009

Kid 9

6thGrade-copier@saintannsny.org_20150331_124451 (1)-page-010

Kid 10

6thGrade-copier@saintannsny.org_20150331_124451 (1)-page-011

Kid 11

6thGrade-copier@saintannsny.org_20150331_124451 (1)-page-012

Kid 12

6thGrade-copier@saintannsny.org_20150331_124451 (1)-page-013

Kid 13

6thGrade-copier@saintannsny.org_20150331_124451 (1)-page-014

 

Kid 14 

6thGrade-copier@saintannsny.org_20150331_124451 (1)-page-006