Speed Bump #1: Fractions Many students find fractions frustrating. While they understand the visual representations, the mathematical operations, particularly adding and subtracting, remain a challenge for some students throughout their math studies. Hopefully this review will provide those students with the support they need to conquer this fundamental skill. Many of us learned fractions using representations like the one above. We had no problem understanding that the colored sections represented the number on the top of the fraction (the numerator), while the total number of sections represented the bottom (the denominator). Our challenge was how to combine these pieces. If we took the rectangular purple pieces and combined them with the rectangular pink pieces would we have a whole rectangle? If not, how much do we have? The purple rectangle represents 2/5 and the pink rectangle represents 4/8. These two fractions do not have pieces of the same size. Their denominators are different. In order to combine these to fractions we need to make the size of the pieces the same. We do that by multiplying by fractions equal to one. Remember that anything divided by itself is one and fractions are just division problems. So any fractions with the same numerator and denominator equals 1. 2 8
4
5
× +
× 5 8
8 5
16 20
36
+ = 40 40
40
The final answer 36 over 40 can be reduced. Both 36 and 40 can be divided by 4 !
leaving our final answer as !" . It is important to remember that in order to add or subtract fractions we MUST have common denominators. To keep from working with very large numbers it is advisable to reduce fractions BEFORE you multiply to !
get common denominators. In the problem above, ideally you would have reduced ! !
to ! before beginning. The problem would have then become 2 2
1 5
× + ( × ) 5 2
2 5
4
5
9
+ = 10 10
10
While this process of finding a common denominator must be used for subtraction and addition, it is not necessary for multiplication or division. The size of the pieces doesn’t matter when you going to have twice as many (multiplication) or half as much (division). It only matters when you are combining or taking away pieces as in addition and subtraction. For multiplication you simple multiply the numerators (top numbers) and the denominators (bottom numbers). Notice that the denominator will change in both of the processes, while in addition it did not. 3 2
6
3
× = = 4 7
28
14
Division is just like multiplication with ONE additional step, you must first flip the second fraction over. 3
4
3 7
21
1
÷ = × = = 1 5
7
5 4
20
20
When the numerator of a fraction is greater than the denominator, it is an improper fraction. To write this fraction as a mixed number you must divide the numerator by the denominator and use the remainder as the numerator of your new fraction. Your whole number will be the number of times the denominator goes into the numerator. There is one final type of fraction I would like to present, complex fractions. They have fractions as either or both the numerator and the denominator. For example: 3
4 1
2
Remember that fractions are just division problems so this is really 3
1
÷ 4
2
so we treat it as a division problem and change the division to multiplication and flip the second fraction 3 2
6
3
× = = 4 1
4
2
Remember when I suggested that you simplify the fractions before multiplying or dividing? You can simplify using the numerator and the denominator of either fraction in the multiplication problem. See below, 3 2
3 1
3
× = × = 4 1
2 1
2
even though the 2 and the 4 were not in the same fraction they still could be reduced. Now try some example problems. The answers are provided.