# Fractions Simplified: Relationships

When your goal is to make fractions simple for your child, one of the things you need to focus on is the relationships between the types of fractions. (You can read more about the types of fractions in the first post in this series.)

When working with fractions, especially in upper elementary, it is important to be able to understand how fractions work in relationship to each other. Your child needs to understand that ^{5}⁄_{5} is the same as 1 whole. They will need this understanding when they begin to rename when adding and subtracting fractions. Understanding the relationships between mixed numbers and improper fractions will be important especially when you child begins multiplying and dividing fractions.

## Fraction Relationships in Pictures

The best way to allow them to really grasp these relationships is to start by using lots of hands on examples and pictures. For fractions equal to 1, give them a piece of paper folded into fourths, and have them write down a fraction that includes the whole paper (^{4}⁄_{4}). Then fold the paper even smaller and have them give you a new fraction (^{8}⁄_{8}). You can continue doing this until you can’t fold the paper anymore. Point out to your child that all of those fractions still equal 1 piece of paper. Show them that in each fraction, the numerator and denominator was the same. You can also practice this more by drawing shapes divided into sections. Color in the whole shape and have your child give you a fraction that shows the colored part.

For mixed numbers and improper fractions, draw several shapes the same size and divided into the same number of sections. (For example, draw three rectangles all divided into 8 sections.) Then color in several whole shapes and part of the last shape. (In our example, you might color 2 whole rectangles and 3 sections of the last rectangle.) Have your child write an improper fraction for the picture (^{19}⁄_{8}) and a mixed number as well (2 ^{3}⁄_{8}). Do this many different times always reminding them that the two numbers are worth the same amount.

## Converting Mixed Numbers to Improper Fractions

Once your child has grasped the concept that these two types of fractions are simply different ways to name the same number, you can teach them a simple way to convert them. It is as simple as multiplying and adding. You simply multiply the denominator by the whole number. Then add your answer to the numerator. Your answer becomes you new numerator, and your denominator stays the same. For example: 4 ^{3}⁄_{5} – Multiply 5 x 4 = 20. Add 20 + 3 = 23. Your improper fraction is ^{23}⁄_{5}. This simple 2 step method works for every mixed number and with practice is easy to remember and use.

## Converting Improper Fractions to Mixed Numbers

You children should also be able to change improper fractions back into mixed numbers without drawing pictures. The easiest way to do that is by dividing. We will use the same fraction example from above: ^{23}⁄_{5}. You just divide 23 ÷ 5. Of course, you get 4 as an answer with 3 remaining. The 4 becomes your whole number, the remaining 3 becomes your numerator, and your denominator stays 5. This results in the mixed number 4 ^{3}⁄_{5}.

## Simplifying Fractions

When working with fractions, there may be times when your child begins to end up with large fractions as answers (like ^{28}⁄_{44}). It is important for him to know how to simplify those large fractions. Simplifying a fraction involves dividing the fraction by a fraction equal to 1 to get smaller numbers. Let’s simplify ^{28}⁄_{44} as an example. Both 28 and 44 are divisible by 2 so we can divide ^{28}⁄_{44} by ^{2}⁄_{2}. When we divide 28 ÷ 2 and 44 ÷ 2, we get ^{14}⁄_{22}. Because both of these can be divided by 2 again, we will simply find ^{14}⁄_{22} ÷ ^{2}⁄_{2} which gives us ^{7}⁄_{11}. Because 7 and 11 cannot be divided by any other numbers, we have now put our fraction into lowest terms, the lowest possible numbers. When your child starts adding, subtracting, multiplying, and dividing fractions, he should always put his answers into the lowest terms possible by simplifying.

While it is important for your child to understand why these simple conversions work, it is also crucial that they be able to quickly and easily convert between mixed numbers and improper fractions as they move into more difficult fraction problems. What tricks do you have for showing the relationships between types of fractions?