This explains why the mixed number equivalent to 7/3 is 2 + 1/3, also written 2

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1 Chapter 28: Proper and Improper Fractions A fraction is called improper if the numerator is greater than the denominator For example, 7/ is improper because the numerator 7 is greater than the denominator A fraction is proper if it is not improper For example, / and 2/7 are proper fractions A mixed number consists of a whole number added to a proper fraction For example + / is a mixed number It is common to skip the "+" in mixed numbers Thus + / is commonly written This notation is very unfortunate because in nearly every other situation in mathematics, when two things are put side-by-side like this, it represents multiplication and not addition However, this is common usage and thus must be accepted and explained to students Students should be taught how to convert improper fractions into mixed numbers and vice-versa This will help them to emphasize that fractions are just another way to represent division This will also give students a feel for the size of a fraction, relative to the nearest whole numbers The best place to begin is on a number line For example, look at the improper fraction 7/ on a number line that has been divided into thirds: The usual way to look at this number is as 7 third-steps from start However, we could also reach this same mark by first taking as many whole steps as possible, and then as many third-steps as necessary In this case, we can reach this mark with 2 whole steps and then third-step: This explains why the mixed number equivalent to 7/ is 2 + /, also written 2 9

2 When we multiply, divide, and compare fractions, it is easiest if our fractions are written in improper form This is why the word "improper" is a bit unfair It would be better to call such fractions "working fractions," much like the work clothes that we might put on to do yard work When we want to display a fraction in its prettiest form, however, we use mixed and proper fractions, like the proper clothes we wear to weddings and formal occasions The fraction 7/ is explained with its two numbers there are 7 pieces and it takes of these pieces to make a whole The equivalent mixed fraction 2 is explained with three numbers This mixed fraction emphasizes that the number is a little bit bigger than 2 We could use measuring cups to illustrate the conversion of an improper fraction to proper form Suppose a recipe calls for / cups of sugar We certainly could measure this out by filling the / measuring cup with sugar times: However, we know that of these / cups makes a whole cup: Instead of filling a / cup times, we could fill a -cup and a /-cup instead This shows why / = 92

3 The best short-cut for converting a fraction from improper to proper form is division For example, suppose we want to represent the improper fraction 7/ as a mixed number We know that this number has 7 parts, and it takes of those parts to make a whole To figure out how many wholes we can make from 7, we divide 7 With long-division, this looks like: We began by dividing the 7 tens by We were able to put aside group of tens, with 2 tens left over We then converted the 2 tens into 20 ones, added them to the ones, and got 2 We then divided these 2 ones by and got groups of with a remainder of At this point in the process, we should divide the ones by, if possible Before we had fractions, we did not know how to divide by, and so we would say that 7 =, "with a remainder of " Now, however, we know that = / Thus, we can say that 7 = This illustrates how division can convert an improper fraction like 7/ into its equivalent mixed number The improper fraction 7/ describes a number formed by 7 pieces which are each / of a whole, while the mixed-number makes it clear that this number is a bit more than, but not quite With a little practice, students ought to be able to mentally convert improper fractions into mixed numbers For example, 22/7 should not require setting up a long-division problem If we try to divide 22 by 7, we will get with a remainder of because 7 = 2 is shy of 22 Thus, we can immediately see that 22/7 = 7 Similarly, / = 2 because goes into twice, with a remainder of It is important to emphasize that the denominator will never change We are putting together as many wholes as possible, but the remaining pieces will stay the same size If we start with sixths, for example, then the remainder will be in sixths as well It is also useful to be able to convert back from mixed numbers to improper fractions This sounds a bit strange when we use the word "improper," because children are generally discouraged from behaving improperly However, if we look at improper fractions as "working fractions," it makes more sense When we want to multiply or divide fractions, for example, it will be much easier when the fractions are written in improper form In advanced mathematics, in fact, fractions are almost always kept in improper form 9

4 To convert a mixed number into improper form, it is only necessary to remember the unwritten "+" sign For example, 2 really means 2 + / When we represent these two numbers on a symbol board, it will look like: Some students get confused when trying to view a whole number like 2 as a fraction However, this really need not be difficult The two is not being divided by anything, and thus there is nothing in the denominator Of course, by "nothing" we mean the number, the number which "does nothing" when multiplying and dividing When we perform the "make the bottoms the same" game, we need only put a in the top and bottom of the 2 fraction: We can now add the two fractions: 0/ + / = / This is the improper fraction equivalent to 2 We can illustrate what has just happened by representing the number 2 with areas If a circle represents a whole, then we have two wholes and then / of a whole: 9

5 When we put the in the top and bottom of the 2 fraction, we split everything into equal pieces This divided each of the whole circles into fifths: The end result is that 2 has been converted into / Whenever a mixed-number is converted into an improper fraction, the process will be remarkably similar to this For example, let us convert 6 into an improper fraction by using the pencil-and-paper version of the symbols game: Here, we convert the 6 wholes into 2/ by chopping each whole into equal-sized pieces These 2/ can then be added to the / to make a total of 27/ Each time we convert a mixed number, we will have to break the wholes up into pieces of the same size as those in the fraction This will result in as many pieces as the whole times the denominator As a short-cut, we can do this directly: multiply the whole number by the denominator and then add the numerator: 9

6 2 For example, to convert 9 into an improper fraction, we first multiply 9 =, next we add + 2 = 7, and finally we put this new numerator over our old denominator: 7/ What we are really doing, of course, is converting the 9 into / and then adding the 2/ As always, make sure your students really understand what is happening before teaching them this short-cut With a little practice and a lot of emphasis on the underlying reasons and meaning, students ought to be able to convert between proper and improper fractions without much difficulty Questions: () Give examples of three proper fractions, three improper fractions, and three mixed numbers (2) Illustrate how a number line can convert /2 into a mixed number () Show how measuring cups can illustrate the conversion of 8/ into a mixed number () Use long division to convert 22/7 into a mixed number () Use the symbols game to convert 7 6 into an improper fraction (6) Use the pencil-and-paper symbols game to convert into an improper fraction (7) Use areas to show how to convert into an improper fraction (8) Compare the short-cut method and the pencil-and-paper symbols method for converting 9 into an improper fraction 96