Lesson Progress:

# Fractions: Part 1

## Overview Fractions are a useful way to describe numbers that aren’t whole. While decimals and percents are useful too, fractions allow us describe how the pieces relate to the whole. Fractions appear in many places, such as finding recipe measurements, knowing the value of money, and understanding how notes fit on a music sheet. Modeling the different types of fractions and ordering them is important to understanding how useful they are. Later on, we can also add and multiply these fractions, but we must crawl before we walk.

## Big Question

How can I model and order fractions so I can perform common operations with them?

## Overview

Fractions are useful numbers that describe the parts of a whole. If we want to describe eating a slice of cake that has been cut into four pieces, we can say we ate “one-fourth” of the cake or 14. The top number, called the numerator, describes how many parts to count and the bottom number, called the denominator, describes how many equal parts the whole is divided into.

Although we want to learn how to work with fractions, it would be a disservice to jump right into how to perform operations with fractions without first understanding what a fraction is. We can do this by modeling fractions.

## Understanding the Meaning of Fractions

Andy and Bobby love pizza. On Monday night, they share a pizza equally. How much of the pizza does each one get? Are you thinking that each boy gets half of the pizza? That’s right. There is one whole pizza, evenly divided into two parts, so each boy gets one of the two equal parts. In math, we write this fraction, 12, to mean one out of two parts.

On Tuesday, Andy and Bobby share a pizza with their parents, Fred and Christy, with each person getting an equal amount of the whole pizza. How much of the pizza does each person get? There is one whole pizza, divided evenly into four equal parts. Each person has one of the four equal parts, so each has 1/4 of the pizza.

On Wednesday, the family invites some friends over for a pizza dinner. There is a total of 12 people. If they share the pizza equally, each person would get 1/12 of the pizza.

## Modeling Proper Fractions

You can use circles and rectangles to model fractions. This will allow you to model and make sense of fractions and their operations.

What does the fraction 2/3 represent? The fraction 2/3 means two of three equal parts.  You can also use your models to make sense of mixed numbers, improper fractions, and equivalent fractions.

## Modeling Mixed Numbers

A mixed number is a number that has a whole number and a fraction. For example, one and two-thirds can be written like 1 ⅔ and means one whole and two out of three equal pieces. ## Modeling Improper Fractions

An improper fraction is a fraction that has a bigger numerator than the denominator. Another way to look at the 1 ⅔ example above is through the improper fraction53or “five thirds,” meaning you have five out of three equal pieces.

Since it only takes three third pieces to make a whole, having the extra two pieces beyond that makes the fraction improper.

Notice how 1 ⅔ and 53 have the same fraction value. Compare the pictures used to model both fractions. They are the same! Keep this in mind as it will become useful when we convert between improper and mixed numbers later.

## Modeling Equivalent Fractions

Fractions can have the same value, but different names. Those are called equivalent fractions. Comparing mixed numbers to improper fractions isn’t the only way to get equivalent fractions. You can do it with any proper fractions too.

Fraction tiles serve as a useful model of equivalent fractions.

Start with a 1/2 tile. How many 1/4 pieces do you need to equal one-half? Since two 14 tiles cover the 1/2 tile, we see that 2/4 is the same as 1/2. So, that means 2/4=1/2.

How many of the 1/6 tiles cover the 1/2 tile? Since three 1/6 tiles cover the 1/2 tile, we can say that 3/6 is the same as 1/2.
So, that means 3/6=1/2.

## Converting from Improper Fractions to Mixed Numbers

Looking at an improper fraction, the denominator tells us how many pieces we need to make one group while the numerator is telling us how many pieces we have. Earlier, we showed 1 ⅔ = 5/3, but let’s try a different problem. Above, we can count that there are 5 half pieces, or halves. We can group 4 halves together to make two wholes; then, we will have 1 half piece left.

We can use division to apply what we modeled.52is asking us to divide 5 into groups of 2. We can make 2 (whole) groups with a remainder of 1 (halves) or 2 ½. This is how we can convert improper fractions into mixed numbers.

Here are the steps for converting an improper fraction to a mixed number:

1. Divide the numerator by the denominator.
2. How many times you fit the denominator into the numerator is the whole number.
3. The remainder over the denominator is the fraction part.

## Converting from Mixed Numbers to Improper Fractions

Now let’s covert our 2 ½ back to 5/2. In the picture above, we can see that one whole can be cut into two halves. If we do that to our 2 wholes, we get the picture below. Now we can count the 5 halves, which we can write as 5/2.

We can use multiplication to apply what we modeled. We turned the 2 wholes into 4 halves by multiplying the whole number by how many halves can fit in a whole (2 wholes x 2 = 4 halves). If we add our left over half, we can say that we have 5 halves or 5/2.

Here are the steps for converting a mixed number to an improper fraction:

1. Multiply the whole number by the denominator.

## Investigate

We can compare the size of proper fractions, improper fractions, and mixed numbers by visualizing and converting between them. Let’s look at an example:

## Example: Finding the Recipe

Alex’s Grandma Marie is known for her trail mix. Alex convinces her to share the family recipe with him. Grandma Marie has trouble remembering the exact recipe, but she remembers the numbers and descriptions of what she needs. She remembers the ingredients: raisins, cherries, almonds, m&ms, chocolate chips, and peanuts.

She remembers the numbers:73,46,76,14, 1½ ,23.

She tells you the following:

• There are more peanuts than anything else.
• There are the same amount of m&ms as there are raisins.
• Almonds, peanuts, and chocolate chips are all more than one cup.
• The ingredient you need the least of is dried cherries.
• There are more almonds than chocolate chips.

Alex wants to match the right measurements with the right ingredients using his grandmother’s descriptions.

## Simplifying and Converting

Before we start, it might be more useful to simplify equivalent fractions or convert our fractions over a whole into one form. We are going to work with mixed numbers for this problem, but the work is similar when using improper fractions.

Try simplifying or converting improper fractions into mixed numbers now.

Going back to the list: 73, 46, 76,14, 1½ ,23. Convert73into 2 ⅓ and76into 1 ⅙. Also recognize that46= 23.

This gives us the new list: 2 ⅓ ,46=23, 1 ⅙,14, 1 ½ , 23.

## Ordering Fractions

Grandma Marie’s clues use words like more, less, and same, which are helpful for ordering the fractions.

It is good to use whole numbers to separate our fractions into groups. 46,14, and23are all fractions that can not make a whole, so they are less than 1. The fractions 1 ⅙ and 1 ½ are slightly bigger than 1 but less than 2. This makes 2 ⅓ the largest fraction.

 Numbers less than 1 Numbers between 1 and 2 Numbers more than 2 4/6 (or 23) 1/4, 2/3 1 ⅙ 1 ½ 2 ⅓

Let’s order them one group at a time. Modeling the fractions with bars can be helpful.

Bars are typically easier to hand draw than fraction circles and useful when comparing fractions. Be careful: the whole numbers must be the same size. Looking above,1/4 is smaller than 2/3, and 1 ⅙ is smaller than 1 ½ .

This gives us a list of fractions in order:1/4, 2/3=4/6, 1 ⅙ , 1 ½ , 2 ⅓. Let’s put this all together with Grandma Marie’s clues.

## Using the Clues

Can you use Grandma Marie’s clues to match the fractions with her recipe? Go ahead and try to find out before reading the explanation below.

• More peanuts than anything else = 2 ⅓ is the biggest fraction, so we must need 2 ½ cups of peanuts.
• Same amount of m&ms as raisins =23and46are the same, so either of them can describe how many cups of m&ms and raisins we need.
• For almonds, peanuts, and chocolate chips, you need more than one cup = We already know how much of the peanuts we need, so this means almonds and chocolate chips could be either be 1 ⅙ or 1 ½. We need more information to know which is which.
• The ingredient you need the least of is dried cherries =14is the smallest fraction, so we need14cups of cherries.
• There are more almonds than chocolate chips = this is the information we needed earlier. Since 1 ½ is more than 1 ⅙, we need 1 ½ cups of almonds and 1 ⅙ cups of chocolate chips.

This gives us Grandma Marie’s recipe:

1/4 cup of cherries
4/6 or 2/3 cup of raisins
4/6 or 2/3 cup of m&ms
7/6 (or 2 ⅙) cups of chocolate chips
1 ½ cups of almonds
7/3 or (2 ⅓) cups of peanuts

Use the quiz below to check your understanding of this lesson’s content. You can take this quiz as many times as you like. Once you are finished taking the quiz, click on the “View questions” button to review the correct answers.

## Lesson Resources

##### Lesson Toolbox

Online fraction modeler (pizza, circle, and square)

An online fraction modeler allowing you to model any fraction by inputting how the whole is sliced and selecting how many pieces to count

Fraction bars

An interactive app to play with fraction bars which can be used to find equivalent fractions

Fraction number line

A website allowing you to compare the sizes of common fractions and get equivalent fractions from 0 to 1

## Terms • denominator
the bottom number of a fraction, which describes how many pieces a whole is divided into
• equivalent fractions
fractions that have the same value, but different names
• fractions
numbers that describe the parts of a whole
• improper fraction
a fraction that has a bigger numerator than the denominator
• mixed number
a number that has a whole number and a fraction
• numerator
the top number of a fraction, which describes how many parts to count

#### Lesson Content:

Authored and curated by Kashuan Hopkins for The TEL Library. CC BY NC SA 4.0

Title: Prealgebra – 4.1 Visualize Fractions: Rice University, OpenStax CNX. License: CC BY 4.0

## Media Sources Yellow BarsTEL LibraryPhET Interactive SolutionCC BY NC SA Red CirclesTEL LibraryPhET Interactive SolutionCC BY NC SA Red CirclesTEL LibraryPhET Interactive SolutionCC BY NC SA Red CirclesTEL LibraryPhET Interactive SolutionCC BY NC SA Food Eat DietWikimediaImagesPixabayCC 0 equivalent fractionsOpenStaxOpenStaxCC BY 4.0 equivalent fractions ½OpenStaxOpenStaxCC BY 4.0 mixed numberOpenStaxOpenStaxCC BY 4.0 Two thirdsOpenStaxOpenStaxCC BY 4.0 1/12 pizzaOpenStaxOpenStaxCC BY 4.0 ¼ pizzaOpenStaxOpenStaxCC BY 4.0 ½ pizzaOpenStaxOpenStaxCC BY 4.0