11-23-2009Multiplying and Dividing FractionsIn this section we discuss the operations of multiplication and division in the rationalnumbers.Multiplication. Given two fractionsabandcdwe know that the product is computedby:ab·cd=a · bc · d.That is, we just multiply the numerators and the denominators. Let us investigate whythis simple formula works. We can do this with using the area model. Recall that tomultiple whole numbers m and n in the area model, we compute the area of the rectanglewith side lengths m and n. To multiply the proper fractionsabandcd:1. Draw a square with side length 1.2. Cut the square into b rows and d columns.3. Shade in the rectangle which has a rows for its height and c columns for its length.4. Count the number of all the subrectangles (b · d).5. Count the number of rectangles in the shaded rectangle (a · b).See the diagrams on page 376. Similar procedures will show us how to model themultiplication of improper fractions.Division. Given two fractionabandcd6= 0 we can compute:ab÷cd=abcd=ab·dc.There are several ways to understand the division of fractions. One way is to use themissing factor model. To solve the problemab÷cd=? we can instead askab=cd· (?).We can see that the answer to this new problem is(?) =dcabsincecd·dcab=cddcab= 1ab=ab.This establishes the formula.We recall that in our discussion of division of whole numbers, we often viewed divisionin terms of grouping. A similar idea will work for fractions. (See example 6.11 on page380).Another option is to think of division as a partitioning. (See example 6.12 on
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