Monday, February 28, 2011

Fractions

Fractions are among the most abstract concepts presented in elementary school math. We need to ensure that a sound conceptual base is established. This means taking time and providing numerous, concrete experiences for children to construct an understanding of these concepts before every formalizing them.

Here's an example of work adding fractions in a 5th grade classroom.

Students were asked to find the sum of the fractions 1/4 + 1/2 + 2/8. Using materials, they were able to find that these three fractions had a sum of 1 whole.


Students also noticed that they could trade in the fourth in for two eighths and the half in for four eighths, leaving them with 8/8, another form of 1 whole. Likewise, they could trade the eighths and half in for fourths, or trade the fourth and eighths in to make a half. Thus proving that it was equivalent to 2/2, 4/4, or 8/8... all equivalents of 1.

Without ever thinking about "common denominators" the students were dealing with the concept. 

In a more challenging problem, they were asked to think about the sum of 4/5, 7/8, and 2/3. Without worrying about the exact sum, they reasoned that 2 < sum < 3. From earlier explorations, they already know that when a fraction was missing a unit fraction, it was as close to 1 as possible. Thus, three fractions close to one added together, must get close to three. It would have to be close to two because each of the fractions was at least 1/2.


Once assembled, children were easily able to see that these three fractions had a sum of about 2 1/3.

Powerful reasoning, with no need for "common denominators."

Yes, they will be exposed to the algorithm for adding fractions, but they will first construct it through many more investigations such as these. In the last example above, the student noted, "I could find the answer by thinking about 3 - 1/3 - 1/5 - 1/8. That's like taking away about 2/3 from three."

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