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# 5.3: Equivalent Fractions

Difficulty Level: At Grade Created by: CK-12

## Introduction

The Cake Dilemma

At the sixth grade social, one of the activities is a room where students can play different board games. The students make up their own teams and can play games like Chess, Monopoly or Scrabble. Then the teams play against each other to determine a winner.

Each winning team can choose \begin{align*}\frac{1}{2}\end{align*} of a cake as their prize. Christian is in charge of handing out the cakes. The Parent/Teacher Group has baked a bunch of different cakes for the prizes. Because the teams are all different sizes, a Chess team might have two players while a Monopoly team could have a bunch of players, the cakes have been cut into different numbers of slices.

The chocolate cake has been cut into 10 slices.

The vanilla cake has been cut into 6 slices.

The strawberry cake has been cut into 8 slices.

Christian is in charge of handing out the cake slices. He needs to be sure that each team receives the correct number of slices to equal one-half of a cake.

The first team has two people and chooses the chocolate cake.

The second team has three people and chooses the vanilla cake.

The third team has four people and chooses the strawberry cake.

Christian has a tough dilemma. He can see that each cake has been cut into a different number of slices. He needs to be sure that each team receives the correct number of slices so that the portion of the cake equals one-half.

Christian knows that equal fractions are going to be key to solving this dilemma. He just isn’t sure where to start.

This is your task. You need to learn all about equivalent fractions so that you can help Christian pass out the cake. This lesson will teach you everything that you need to know.

What You Will Learn

In this lesson you will learn to:

• Write fractions equivalent to a given fraction.
• Write given fractions in simplest form.
• Describe real-world fractional portions by writing in simplest form.

Teaching Time

I. Write Fractions Equivalent to a Given Fraction

Wow! Christian has got quite a dilemma to solve in the game room at the sixth grade social. He needs to know all about fractions. Fractions are something that you have probably been working with for a while now. You first saw them in elementary school. However, many students often have a difficult time working with fractions. It can be challenging to start thinking about parts instead of wholes. That is what a fraction actually is, a part of a whole.

What is a fraction?

A fraction is a part of a whole. When we work with fractions we think about the relationship between a part of something and the whole thing. Fractions show up all the time in real life. Sometimes, we don’t even realize that we are working with fractions because they are everywhere!

A fraction has two parts. It has a top number and a bottom number. The top number is called the numerator and tells us how many parts we have out of the whole. The bottom number is the denominator. It tells us how many parts the whole has been divided into.

Example

\begin{align*}\frac{4}{5}\end{align*} = means we have four out of five parts.

The four is our numerator, it tells us how many parts we have.

The five is our denominator, it tells us how many parts the whole has been divided into.

We can also show fractions in a visual way by using a picture.

Here our whole has been divided into ten parts. This is our denominator.

Five out of ten are shaded. This is our numerator.

\begin{align*}\frac{5}{10}\end{align*}

We could also write the fraction that is not shaded. In this example it would be the same thing since five out of ten are shaded and five out of ten are not shaded.

Notice that \begin{align*}\frac{5}{10}\end{align*} are shaded and this is the same as \begin{align*}\frac{1}{2}\end{align*} of the whole being shaded.

What? Yes. Look again. Because this whole has been divided into five parts, \begin{align*}\frac{5}{10}\end{align*} is the same as \begin{align*}\frac{1}{2}\end{align*}.

These two fractions are equal or equivalent fractions.

What is an equivalent fraction?

Equivalent Fractions are fractions that have the same value.

For example, \begin{align*}\frac{1}{2}\end{align*} is equivalent to the fractions below. The bars below visually represent why this is true.

If we add up each part then we have a fraction that is equivalent to one half.

Take a look at these.

\begin{align*}\frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{4}{8}\end{align*}

The fractions below are equivalent to \begin{align*}\frac{1}{3}\end{align*}.

The bars below visually represent why this is true. The little numbers above each box show the number of sections that each whole has been divided into. Notice that this number is also the denominator.

We can write an equivalent set of fractions for one-third too.

\begin{align*}\frac{1}{3} = \frac{2}{6} = \frac{3}{9} = \frac{4}{12}\end{align*}

Now that you know what an equivalent fraction is, how can we write them without always drawing pictures?

Anytime that we want to create an equivalent fraction we multiply the numerator and denominator by the same number.

Example

Create a fraction equivalent to \begin{align*}\frac{3}{4}\end{align*}.

To do this, we need to multiply the numerator and denominator by the same number. Let’s choose 2. Two is always a good place to start.

\begin{align*}\frac{3 \times 2}{4 \times 2} &= \frac{6}{8}\\ \frac{6}{8} &= \frac{3}{4}\end{align*}

We could create another equivalent fraction by choosing a different number. Let’s try four.

\begin{align*}\frac{3 \times 4}{4 \times 4} = \frac{12}{16}\end{align*}

These fractions are also equivalent.

It is time for you to try this out. Create an equivalent fraction for each fraction below.

1. \begin{align*}\frac{1}{4}\end{align*}
2. \begin{align*}\frac{2}{3}\end{align*}
3. \begin{align*}\frac{4}{5}\end{align*}

Answers will vary so check your work with a peer. Be sure that you multiplied the numerator and denominator by the same number.

II. Write Given Fractions in Simplest Form

One of the trickiest skills involving equivalent fractions is being able to determine whether or not they are equivalent. Look at this example.

Example

Are \begin{align*}\frac{3}{6}\end{align*} and \begin{align*}\frac{4}{8}\end{align*} equivalent?

This is tricky because we can’t tell if the numerator and denominator were multiplied by the same number. These fractions look like they might be equal, but how can we tell for sure? This is where simplifying fractions is important.

How do we simplify fractions?

You can think of simplifying fractions as the opposite of creating equal fractions. When we created equal fractions we multiplied. When we simplify fractions, we divide.

What do we divide?

To simplify a fraction, we divide the top and the bottom number by the Greatest Common Factor.

Let’s simplify \begin{align*}\frac{3}{6}\end{align*}. To do this, we need to divide the numerator and denominator by the GCF.

The GCF of 3 and 6 is 3.

\begin{align*}\frac{3 \div 3}{6 \div 3} = \frac{1}{2}\end{align*}

Let’s simplify \begin{align*}\frac{4}{8}\end{align*}. To do this, we need to divide the numerator and the denominator by the GCF.

The GCF of 4 and 8 is 4.

\begin{align*}\frac{4 \div 4}{8 \div 4} = \frac{1}{2}\end{align*}

We can see that \begin{align*}\frac{3}{6}\end{align*} and \begin{align*}\frac{4}{8} = \frac{1}{2}\end{align*}. They are equivalent fractions.

We can use simplifying to determine if two fractions are equivalent, or we can just simplify a fraction to be sure that it is the simplest it can be. Sometimes you will also hear simplifying called reducing a fraction.

Simplify the following fractions by dividing by the GCF of the numerator and the denominator.

1. \begin{align*}\frac{4}{20}\end{align*}
2. \begin{align*}\frac{8}{16}\end{align*}
3. \begin{align*}\frac{5}{15}\end{align*}

Take a minute to check your work with a friend.

## Real Life Example Completed

The Cake Dilemma

Now that you have learned about equivalent fractions and simplifying fractions, you are ready to help Christian. Here is the problem once again.

At the sixth grade social, one of the activities is a room where students can play different board games. The students make up their own teams and can play games like Chess, Monopoly or Scrabble. Then the teams play against each other to determine a winner.

Each winning team can choose \begin{align*}\frac{1}{2}\end{align*} of a cake as their prize. Christian is in charge of handing out the cakes. The Parent/Teacher Group has baked a bunch of different cakes for the prizes. Because the teams are all different sizes, a Chess team might have two players while a Monopoly team could have a bunch of players, the cakes have been cut into different numbers of slices.

The chocolate cake has been cut into 10 slices.

The vanilla cake has been cut into 6 slices.

The strawberry cake has been cut into 8 slices.

Christian is in charge of handing out the cake slices. He needs to be sure that each team receives the correct number of slices to equal one-half of a cake.

The first team chooses the chocolate cake.

The second team chooses the vanilla cake.

The third team chooses the strawberry cake.

First, go back and underline the important information.

Now let’s look at figuring out how many slices of cake each team should get based on getting one-half of the cake.

The first team has two people and chooses the chocolate cake. The chocolate cake has 10 slices. If they are going to get one-half of the cake, how many slices should that team receive? To do this, we need to find a fraction equivalent to one-half that has ten has a denominator.

\begin{align*}\frac{1}{2} = \frac{}{10}\end{align*}

What is one-half of ten? That is five. Let’s try that and see if it works.

\begin{align*}\frac{1}{2} = \frac{5}{10}\end{align*}

To check our work, we can simplify \begin{align*}\frac{5}{10}\end{align*}. If we get one-half, then we know that the team should receive five slices of cake.

\begin{align*}\frac{5 \div 5}{10 \div 5} = \frac{1}{2}\end{align*}

Having checked his work, Christian gives the first team five slices of cake. Wow! That is a lot of cake for two people to eat.

Next, Christian moves on to the second team. They chose the vanilla cake which has been cut into six slices. If they are going to get one-half of the cake, how many slices out of the 6 should they receive? Here is an equivalent fraction to solve.

\begin{align*}\frac{1}{2} = \frac{}{6}\end{align*}

What is one-half of 6? Let’s try 3 and see if that works.

\begin{align*}\frac{1}{2} = \frac{3}{6}\end{align*}

Go ahead and simplify three-sixths in your notebook. Did you get one-half?

Yes. Christian gives the second team three slices of cake.

The third team has chosen the strawberry cake. It is cut into 8 slices. Go ahead and work this one through in your notebook first. See if you can figure out how many slices the third team will receive.

Christian gives the third team 4 slices of cake.

All of the teams have received their cake. Christian feels great about figuring out equivalent fractions.

## Vocabulary

Here are the vocabulary words that are found in this lesson.

Fraction
a part of a whole.
Equivalent
means equal
Numerator
the top number in a fraction
Denominator
the bottom number in a fraction
Simplifying Fractions
dividing a numerator and a denominator by the GCF to create a fraction that is in simplest form. An equivalent fraction is created.
Reducing
another way to say simplifying

## Technology Integration

This video introduces the concept of equivalent fractions.

Other Videos:

1. http://www.mathplayground.com/howto_fractions_reduce.html – a blackboard style video on reducing/simplifying fractions

## Time to Practice

Directions: Write an equivalent fraction for each fraction listed below.

1. \begin{align*}\frac{1}{2}\end{align*}

2. \begin{align*}\frac{1}{3}\end{align*}

3. \begin{align*}\frac{1}{4}\end{align*}

4. \begin{align*}\frac{1}{5}\end{align*}

5. \begin{align*}\frac{2}{3}\end{align*}

6. \begin{align*}\frac{2}{5}\end{align*}

7. \begin{align*}\frac{3}{4}\end{align*}

8. \begin{align*}\frac{3}{10}\end{align*}

9. \begin{align*}\frac{2}{9}\end{align*}

10. \begin{align*}\frac{2}{7}\end{align*}

Directions: Determine whether or not each pair of fractions is equivalent. Use true or false as your answer

11. \begin{align*}\frac{1}{2}\end{align*} and \begin{align*}\frac{3}{6}\end{align*}

12. \begin{align*}\frac{2}{3}\end{align*} and \begin{align*}\frac{4}{9}\end{align*}

13. \begin{align*}\frac{2}{5}\end{align*} and \begin{align*}\frac{4}{20}\end{align*}

14. \begin{align*}\frac{3}{7}\end{align*} and \begin{align*}\frac{9}{21}\end{align*}

15. \begin{align*}\frac{5}{9}\end{align*} and \begin{align*}\frac{25}{45}\end{align*}

Directions: Simplify each fraction. If the fraction is already in simplest form write simplest form for your answer.

16. \begin{align*}\frac{8}{10}\end{align*}

17. \begin{align*}\frac{4}{10}\end{align*}

18. \begin{align*}\frac{2}{10}\end{align*}

19. \begin{align*}\frac{2}{12}\end{align*}

20. \begin{align*}\frac{3}{12}\end{align*}

21. \begin{align*}\frac{4}{9}\end{align*}

22. \begin{align*}\frac{5}{20}\end{align*}

23. \begin{align*}\frac{12}{24}\end{align*}

24. \begin{align*}\frac{12}{36}\end{align*}

25. \begin{align*}\frac{11}{44}\end{align*}

26. \begin{align*}\frac{20}{45}\end{align*}

27. \begin{align*}\frac{18}{20}\end{align*}

28. \begin{align*}\frac{12}{30}\end{align*}

29. \begin{align*}\frac{22}{40}\end{align*}

30. \begin{align*}\frac{35}{63}\end{align*}

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Date Created:
Feb 22, 2012
Jul 19, 2016
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