A B

Notes for Session 9, Part A

 Note 2 You probably remember most, if not all, of the rules (or algorithms) for the multiplication and division of fractions. But can you actually remember why those rules work? As you examine the models used to demonstrate these operations, think about how the models are connected to paper-and-pencil computations.

 Note 3 In this case, we are thinking of division as a missing-factor problem. We know the product and one of the factors; we need to find the other factor.

 Note 4 We can, however, show you why the algorithm works: As you can see, we first wrote the division problem as a fraction with fractions for both its numerator and its denominator. Next, to change the messy denominator to a nice, tidy denominator, we multiplied by 1 in the form of (4/3)/(4/3). We then showed the multiplication problem as multiplying numerators and multiplying denominators. We computed the denominator, which was 1, and then divided by 1, which didn't change the numerator. We are left with 2/3 • 4/3, exactly as the algorithm tells us: The division problem 2/3 3/4 can be changed to the multiplication problem 2/3 • 4/3, and both will produce the same answer (8/9).

 Note 5 We often forget that terminating decimals are fractions too. And actually, the word "decimal" is something of a misnomer; we should not call them decimals unless we are referring specifically to the digits to the right of the decimal point. They should really be called decimal fractions. (Did you know that fractions that were not decimal fractions used to be called vulgar fractions? Perhaps our forebears didn't like fractions with denominators that were not powers of 10!)