To: Teacher2Teacher Public Discussion
Subject: let's put it all together....
The link which was previously suggested had a useful model for area diagrams but did not lend itself very well to what dividing fractions is really about. The suggestion to look at Liping Ma's book (fabulously written and very enlightening) is one of the turning points for me to understand student misconceptions about division. I use the model of soup and ladel with my students, much like someone had suggested previously--division asks "how many of these are there in that?" For example, if you have 2/3 cup of soup and your ladel is 1/3 cup large, how many servings would you get? If you had 6 cups of sugar and you needed 1/2 cup for every batch of pancakes, how many batches could you make? The mystery of dividing fractions is actually more mysterious when students are taught the mechanical rules (invert and multiply) which generate the correct answer, but lose the meaning of the division process. The rectangular area model can work if you mark off partitions along the width for the first denominator, and then mark partitions down the length to correspond to the second denominator. This model is used for multiplying fractions in many textbooks. If you then look at the shaded regions for the dividend and the divisor, you can simply divide the numerators since you now have a common denominator. For example, 2/3 divided by 1/5 would be diagrammed as a rectangle which is 3x5, with 2 of the 3 colums shaded (a total of 10 pieces). There would also be 1 row shaded (a total of 3 pieces). You now can compare the 10 pieces to the 3; how many 3's are in 10? In this way, you do not "see" the invert & multiply procedure, instead, you really are dividing after you have a common denominator. I hope this helps someone :)
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