Q&A #3589

Multiplying fractions

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From: Pat Ballew (for Teacher2Teacher Service)
Date: Apr 04, 2000 at 18:08:52
Subject: Re: Multiplying fractions

First I want to suggest that you don't view multiplication of fractions as if it is separate from all the other mathematical experiences of the student. The foundation for teaching multiplication of fractions should have been a constant process during the earliest introduction to multiplication and the meaning of fractions. Nothing that happens here should be a stark departure from what they have understood about multiplication in the past, or you are, in my opinion, doing damage rather than good. Assuming all that is foundation... For a first introduction you can build a square out of a number of tiles that has easy factors. 12x12 is usually a reasonable size and good to work with.. With a ruler or strait edge that allows the student to partition the square, illustrate that 2/3 or some other easy fraction can be divided off along one edge and use the straight edge to separate out three rectangles that are each 1/3 of the original amount (whether we think of this as 1/3 of a unit square or 1/3 of 144 tiles, the process is the same, and that is worth lots of talk with and between kids... multiplying fractions has no meaning unless the "of what" has a meaning) Now repeat with another fraction in the perpendicular direction... 1/2 would be an easy one. Be sure to divide the entire square (now in three rectangles) because we want the product of the denominators to be visible as sixths of the square. We have in one corner, 1/2 of 1/3 of the object we divided. Two of these would be 1/2 of 2/3 (which is also easy to show to be 2/2 of 1/3 making some abstract cancellation properties make sense later) Have them draw a picture of each result after the division to lay a framework for the figural approach and write out the results in proper fraction notation. We want the physical idea, the figural image, and the abstract relationship to become associated so that in the future, the fractions will stimulate a mental picture. After several examples of these done by the teacher and the students and time for students to talk and process this,,, you may want to move to a figural model. Now the square is just a representation and no divisions are shown. The student draws in the division by thirds with vertical lines, and the same kind of conversation should occur as with the physical. If you have taken the time during the physical manipulatives to have them draw a "picture" of the physical model, they have already seen examples of what they are doing, so we suggest that sometimes we can draw the picture without actually having the physical model at all. At this early stage would be a good time to illustrate some BIG ideas that will come up later... commutativity is easy to show (does it matter which fraction we use first to divide the shape??? try both ways) as is equivalent fractions(If we take 4/6 of the shape is that different than taking 2/3 ?) Hopefully when you were multiplying 8x4 you used an area model as one of the representations of multiplication so that that is already a familiar idea, and when you introduced fractions you returned to the area model to portion denominate pieces and enumerate the number that were required.. and the idea of area and fractions are neither a new idea which distract from the focus on unification of two familiar concepts, multiplication and fractions. Good luck, I hope some of this rambling has helped give you some ideas to work with. -Pat Ballew, for the Teacher2Teacher service

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