Adding and Subtracting Fractions
Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that you may be better able to discover with accuracy the peculiar bent of the genius of each. Plato
Although children learn addition of whole numbers with ease, addition of fractions — though conceptually the same as addition of whole numbers — is much harder. It requires knowledge of fraction equivalencies. To add two fractions, you have to know that they must be thought of in terms of like units. We take this for granted when we add whole numbers: is really 3 ones + 5 ones — but not when we add fractions: 3 halves + 5 fourths is, for purposes of addition, 6 fourths + 5 fourths. Why is adding fractions a difficult concept for students to grasp?
Equivalent fractions In the following picture we have ½ of a cake because the whole cake is divided into two congruent parts and we have only one of those parts. But if we cut the cake into smaller congruent pieces, we can see that = Or we can cut the original cake into 6 congruent pieces, a fraction can have many different appearances, these are called equivalent fractions
Now we have 3 pieces out of 6 equal pieces, but the total amount we have is still the same. Therefore, == If you don’t like this, we can cut the original cake into 8 congruent pieces, Equivalent fractions a fraction can have many different appearances, these are called equivalent fractions
then we have 4 pieces out of 8 equal pieces, but the total amount we have is still the same. === We can generalize this to = whenever n is not 0 Therefore, Equivalent fractions a fraction can have many different appearances, these are called equivalent fractions
Let’s Eat Pizza The pizza is currently 8 pieces What if I wanted to eat one eighth of the pizza? One fourth of the pizza? One sixteenth of the pizza? One twelfth of the pizza?
Addition of Fractions The objects must be of the same type We combine bundles with bundles and sticks with sticks. Addition means combining objects in two or more sets In fractions, we can only combine pieces of the same size In other words, the denominators must be the same
Click to see animation + = ? Example: Addition of Fractions
Example: Addition of Fractions + =
+ = Example: The answer is which can be simplified to Addition of Fractions
Addition of Fractions with equal denominators More examples
With different denominators In this case, we need to first convert them into equivalent fraction with the same denominator. Example: An easy choice for a common denominator is 3×5 = 15 Therefore, Addition of Fractions
When the denominators are bigger, we need to find the least common denominator by factoring. If you do not know prime factorization yet, you have to multiply the two denominators together. With different denominators Addition of Fractions
More Exercises: = = = = = = = = =
Subtraction of Fractions Subtraction means taking objects away Objects must be of the same type we can only take away apples from a group of apples In fractions, we can only take away pieces of the same size. In other words, the denominators must be the same.
Subtraction of Fractions Example: This means to take away (Click to see animation) take away equal denominators
Subtraction of Fractions Example: equal denominators
Subtraction of Fractions Example: equal denominators
Subtraction of Fractions Example: equal denominators
Subtraction of Fractions Example: equal denominators
Subtraction of Fractions Example: equal denominators
Subtraction of Fractions Example: equal denominators
Subtraction of Fractions Example: equal denominators
Subtraction of Fractions Example: equal denominators
Subtraction of Fractions Example: equal denominators
Subtraction of Fractions Example: equal denominators
Subtraction of Fractions Example: equal denominators
Subtraction of Fractions Example: equal denominators
Subtraction of Fractions Example: equal denominators
Subtraction of Fractions Example: equal denominators
Subtraction of Fractions Example: equal denominators
Subtraction of Fractions Example: equal denominators
Subtraction of Fractions Example: equal denominators
Subtraction of Fractions Example: equal denominators
Subtraction of Fractions Example: equal denominators
Subtraction of Fractions Example: equal denominators
Now you can see that there are only 8 pieces left, therefore Subtraction of Fractions Example: equal denominators
More examples: Did you get all the answers right? Subtraction of Fractions
Adding/Subtracting == = Fraction Addition/Subtraction Fraction Simplification
= = = Common Denominator = ?????? = Fraction Addition/Subtraction Adding/Subtracting
Fractions: Steps for Success 1. Know the fraction rules and how to apply them 2. Show your work and write out each step 3. Check your work for errors or careless mistakes