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1. Fractions
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Pre-Algebra /
Chapter 2
1. 

Fractions

This lesson provides a thorough exploration of fractions, breaking down the roles of the numerator and denominator. It explains how these two components work together to form fractions that represent parts of a whole. Additionally, the concept of equivalent fractions is discussed, which are fractions that may look different but represent the same value. Understanding these fundamentals has real-world applications. For example, fractions are used in cooking recipes, calculating test scores, and even in determining discounts during shopping. The information is presented in a way that's easy to grasp, making it beneficial for both academic and everyday scenarios.

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Student Learning Objectives:
  • Identify rational numbers
  • Understand and simplify fractions
  • Create equivalent fractions
11 Theory slides
8 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
Image Credits expand_more
Fractions
Slide of 11
Many of everyday situations involve numbers that are not whole numbers but are some part, or fraction, of them. This lesson will define fractions, present some facts about them, and show some of their real-life applications.

Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.

Explore

Splitting a Pizza

Jordan invited her friends over for a pizza party. They wonder how to split a pizza so that everyone gets the same number of slices. Try different ways to split the pizza, then click on the slices to decide how much pizza each person will eat.

Pizza that can be split into pieces, which then can be selected. A number that shows which part of the pizza is selected is calculated.
External credits: @macrovector

Jordan can split the pizza into a minimum of 2 and a maximum of 20 slices.
Discussion

Presenting Rational Numbers

We use a certain type of numbers when splitting something into equal parts.

Concept

Rational Numbers

The set of rational numbers, represented by the symbol Q, is formed by all numbers that can be expressed as the ratio between two integers ab, where b≠ 0. - 3, 13, 5, 3145

The set of integer numbers is a subset of rational numbers. Real numbers that are not rational are called irrational numbers.
Discussion

Definition of a Fraction

In the definition of rational numbers, the word fraction showed up but was not explained in detail. To clarify any doubts, its definition will now be presented.

Concept

Fraction

Fractions are a specific type of ratio that compares a part to a whole. Fractions are rational numbers written in the form ab, where the numerator a is the part and the denominator b is the whole.

l part→ whole→ a/b l←numerator ←denominator

There are many possible ways of reading fractions, but one universal method is saying a over b. Fractions where a is less than b are called proper fractions. Fractions where a is greater than or equal to b are called improper fractions.

Applet that shows the names of different fractions and visualizes them using tiles
Fractions are also another way to write a division of the numerator by the denominator. 18/9=18÷9

A fraction like 189 can be simplified to 21, or just 2. It is important to keep in mind that the denominator of a fraction can never be equal to 0 because the quotient of division by 0 is always undefined.
Example

Determining Fractions

Jordan has multiple chocolate bars that are divided into different numbers of pieces.

A) Dark chocolate bar with 10 pieces in total and 5 pieces shaded; B) Milk chocolate bar with 32 pieces in total and 18 pieces shaded; C) White chocolate bar with 6 pieces in total and 4 pieces shaded
External credits: @freepik
She decides to share the chocolate with her friends. The shaded pieces in the diagrams above show how much of each bar the kids ate.

a

What part of the dark chocolate bar A do the shaded pieces represent?

b

What part of the milk chocolate bar B do the shaded pieces represent?

c

What part of the white chocolate bar C do the shaded pieces represent?

Hint
a

Find the total number of pieces the chocolate bar is divided into. Then count the number of shaded pieces.

b

Divide the number of shaded pieces by the total number of pieces.

c

Form a fraction and check whether it can be simplified by dividing its numerator and denominator by their greatest common factor.

Solution
a

Let's start by counting the total number of pieces and the number of shaded pieces in the dark chocolate bar A.

A) Dark chocolate bar with 10 pieces in total and 5 pieces shaded
External credits: @freepik

This chocolate bar has 10 pieces, 5 of which are shaded. We divide the number of shaded pieces by the total number of pieces to find what part of the chocolate bar the shaded pieces represent. Shaded/Total=5/10 Notice that both the numerator and denominator are divisible by 5. This means that the fraction can be simplified. 5÷ 5/10÷ 5=1/2 The shaded pieces represent one-half of the chocolate bar.

b

Now consider the milk chocolate bar B.

B) Milk chocolate bar with 32 pieces in total and 18 pieces shaded
External credits: @freepik

Like in Part A, start by counting the total number of pieces of chocolate and the number of shaded pieces. Total:& 32 Shaded:& 18 Let's write the part of the chocolate bar represented by the shaded pieces as a fraction. The numerator of the fraction is the number of shaded pieces and the denominator is the total number of pieces. Shaded/Total=18/32 This fraction can also be simplified. Since 18 and 32 are even numbers, they both can be divided by 2.

18/32
SplitIntoFactors

Split into factors

9* 2/16* 2
CrossCommonFac

Cross out common factors

9* 2/16* 2
CancelCommonFac

Cancel out common factors

9/16

The shaded pieces represent 1832, or 916, of the chocolate bar.

c

Finally, consider the last chocolate bar, the white chocolate bar C.

C) White chocolate bar with 6 pieces in total and 4 pieces shaded
External credits: @freepik

Count the total number of pieces of chocolate and the number of shaded pieces. Total:& 6 Shaded:& 4 Next, divide the number of shaded pieces by the total number of pieces to find what part of the chocolate bar the kids ate. Shaded/Total=4/6 Both 4 and 6 are even numbers. This means that we can simplify the fraction by dividing the numerator and denominator by 2. 4÷ 2/6÷ 2=2/3 The shaded pieces make up 46, or 23, of the chocolate bar.

Pop Quiz

Identifying Fractions Describing the Graph

Consider a bar that is split into different parts. Find the fraction that describes the relationship between the shaded parts and the bar as a whole. Any shaded parts on the right-hand side indicate that the fraction is an improper fraction. Do not simplify the fractions.

A bar split into different number of parts is randomly generated
Discussion

Equivalent Fractions

Fractions that have different numerators and denominators but represent the same value are called equivalent fractions. The following fractions are equivalent because they are all equal to the same value, even though they look different. 1/2=3/6=5/10 Equivalent fractions can be formed by multiplying or dividing the numerator and denominator by the same number.

Forming equivalent pairs of fractions

Imagine two friends, Emily and Maya, have two identical cakes. Emily divided her cake into four parts and ate one. Maya divided her cake into eight slices and ate two.

Emily's round shape cake is divided into four parts with one shaded, while Maya's round shape cake is divided into eight parts with two shaded.
External credits: @macrovector

We can see from the diagram that Emily and Maya ate the same amount of cake. Therefore, 14 and 28 are equivalent fractions.

1/4=2/8
Discussion

How to Simplify a Fraction

When a fraction has a large numerator and denominator, it can be hard to estimate its value. Simplifying such a fraction and finding an equivalent fraction with a smaller numerator and denominator can be helpful.

Method

Simplifying a Fraction

Fractions of the same value can be written using different pairs of numerators and denominators. This is why some fractions can be simplified to equivalent fractions with a smaller numerator and denominator. Consider the following example. 18/66 There are three steps to follow to simplify this fraction.

1
Determine Whether the Fraction Can Be Simplified
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To determine whether the given fraction can be simplified, split the numerator and denominator into prime factors and see if there are any common factors other than 1. 18 &= 2* 3* 3 66 &= 2* 3* 11 The numerator and denominator share factors 2 and 3, so the fraction can be simplified. If the numerator and denominator of a fraction do not have common factors other than 1, the fraction is said to be simplified or written in its simplest form.
2
Find the Greatest Common Factor
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The greatest common factor (GCF) of the numbers 18 and 66 can be found by multiplying all their common factors. GCF(18,66) = 2* 3= 6 If the numerator and denominator share only one common factor, then that factor is their GCF.
3
Reduce the Fraction
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Finally, divide its numerator and denominator by their GCF to reduce the fraction. 18/66=18/ 6/66/ 6=3/11 As a result, we obtained the equivalent fraction 311. It is the simplest form of the given fraction 1866.

Extra

 Simplification of Different Fractions
The applet below illustrates how different fractions are simplified.

Applet that shows how to simplify different fractions
Example

Pairing Equivalent Fractions

Pair the equivalent fractions.

Hint
Find the greatest common factor (GCF) of the numerator and denominator of a fraction, then use it to simplify the fraction.

Solution
We want to pair equivalent fractions.

6/12 7/16
16/56 1/2
14/32 2/7
20/15 4/3

The fractions in the right column are in simplified form, but those in the left column are not. To find all the corresponding pairs, we can simplify the fractions in the left column.

First Fraction

Let’s start with the first fraction in the left column. 6/12 Let’s find the greatest common factor (GCF) of the numerator and denominator by first splitting 6 and 12 into their prime factors. 6&= 2* 3 12&=2* 2* 3 As we can see, 6 and 12 share two common factors. The product of these common factors is the GCF. GCF(6,12)=2* 3= 6 Now we divide the numerator and the denominator by the GCF. 6/12 = 6 ÷ 6/12 ÷ 6 = 1/2 We found that 612 is equivalent to 12.

Remaining Fractions

For the remaining fractions, we can find the GCFs of the numerators and denominators in a table.

Fraction Prime Factorization GCF
16/56 2* 2* 2* 2/2* 2* 2* 7 GCF(16,56) = 8
14/32 2* 7/2 * 2 * 2 * 2 * 2 GCF(14,32)= 2
20/15 2* 2* 5/3 * 5 GCF(20,15)= 5

Next, we simplify the fractions by dividing the numerator and denominator of each fraction by the GCF we just found.

Fraction GCF Simplify Equivalent Fraction
16/56 GCF(16,56) = 8 16÷ 8/56÷ 8 2/7
14/32 GCF(14,32)= 2 14÷ 2/32÷ 2 7/16
20/15 GCF(20,15)= 5 20÷ 5/15÷ 5 4/3

We successfully found each equivalent fraction pair.

Original Fraction Equivalent Fraction
6/12 1/2
16/56 2/7
14/32 7/16
20/15 4/3

Pop Quiz

Simplifying Fractions

Consider the given fraction. Can it be simplified? If yes, write the given fraction in its simplest form. If the fraction is already simplified, write it as it is.

Random fractions are shown
Closure

Real Life Applications of Fractions

Here are a few more real-life applications of fractions.

  • Recipes: Cooking is full of fractions. For example, a recipe for four servings might suggest using 12 teaspoon of vanilla extract and 34 tablespoon of sugar. If someone wants to cook for only two people, they would need to use fractions to adjust the ingredients accordingly.
Two girls preparing ingredients for cookies
External credits: @brgfx, @pikisuperstar
  • Sports: Fractions are frequently used to analyze the performance of a particular player and team or determine statistics like shooting percentages. Note that although percentages often appear in such statistics, they are calculated with fractions.
A screen of a phone with different statistics about the half-time football game between two teams
External credits: Image by brgfx
  • Shopping: When there is a sale, fractions can be used to calculate the reduced price of a product. Sales tax and coupons also use fractions.
A dress with a discount of 15%, trousers with a tax of 7.5% and a t-shirt with a coupon
  • Tests and exams: Tests, exams, and homework assignments are generally scored with fractions, like 18/20.
Test sheet with a score 18/20
  • Money: A dime is 110 of a dollar. A quarter is a 14 of a dollar.
A dollar, a quarter, a dime, a nickel
External credits: Wikipedia, Wikipedia, Wikipedia, Wikipedia
These are just some of the situations where fractions are really useful, but there are so many more! Look around and try to find where else fractions might appear in day-to-day life.



Level 1
Level 2
Level 3
Fractions
Exercise 1.1
Write the fraction corresponding to the given diagram. Do not simplify the fraction.
A rectangle split into different number of parts
A rectangle split into different number of parts
A rectangle split into different number of parts

Let's start by recalling the definition of a fraction. l part→ whole→ a/b l←numerator ←denominator We can see that a fraction is a ratio that compares a part to a whole. Now, let's analyze the given diagram and try to identify the part and the whole.

There are 6 bars in total on the left-hand side of the diagram, which is the whole. Only 5 of the bars are colored, which represents the part. We can substitute these numbers into the ratio to write the desired fraction. 5/6 Therefore, 56 is the fraction that corresponds to the given diagram.

Let's now examine the second given diagram in a similar fashion.

This time there are also colored bars on the right-hand side of the diagram. Since the lines are dashed and there is no rectangle, these bars are additional, so to speak. The rectangle on the left-hand side, consisting of 5 colored bars, represents the whole. If we add the 3 extra colored bars from the right side, we get a part of 8. 8/5 This is an improper fraction. This result makes sense because there are more colored bars than just the ones in the rectangle representing the whole.

Let's now analyze the final diagram.

First, we need to count how many bars there are in the rectangle. We can see that there are 9 bars in the rectangle. This is the number of the whole. Also, there are 3 colored bars, so 3 is the part. Now we have enough information to write the fraction. 3/9

E
Hint & Answer
S
Solution

Tearrik's parents bought 14 tangerines. Tearrik ate 3 of them. What fraction describes the amount of tangerines that Tearrik ate?

Three tangerines

Let's start by recalling the definition of a fraction. l part→ whole→ a/b l←numerator ←denominator To write a fraction representing the situation, we need to identify the part and the whole. We know that Tearrik's parents bought 14 tangerines, so this is the whole. Of these, Tearrik ate 3 tangerines, which represents the part. Let's substitute these values into the fraction. 3/14 Therefore, 314 describes the part of the tangerines that Tearrik ate.

E
Hint & Answer
S
Solution
Simplify the given fractions. 10/24

15/40

36/28

Let's start by analyzing the first given fraction. 10/24 We can simplify it by dividing both the numerator and denominator by their greatest common factor (GCF). To find the GCF, we first need to split the numbers into their prime factors. 10 & = 2* 5 24 & = 2* 2* 2* 3 We can see that 10 and 24 share only one common factor, 2. This is their GCF. GCF(10,24)=2 Let's divide the numerator and denominator of the fraction by 2 to simplify it.

The simplified fraction is 512.

We will begin by considering the second given fraction. 15/40 To simplify it, we need to divide both the numerator and denominator by their GCF. First, we need to split the numbers into their prime factors. 15 & =3* 5 40 & =2* 2* 2* 5 We can see that 15 and 40 share only one common factor, 5. This is their GCF. GCF(15,40)=5 Let's divide the numerator and denominator of the fraction by 5 to simplify it.

The simplified fraction is 38.

Let's analyze the last given fraction. 36/28 We will simplify it by dividing both the numerator and denominator by their GCF. First, we will split the numbers into their prime factors. 36 & = 2* 2* 3* 3 28 & = 2* 2* 7 We see that 36 and 28 share two common factors, 2 and 2. The product of these factors is the GCF of 36 and 28. GCF(36,28)=2* 2=4 Let's divide the numerator and denominator of the fraction by 4 to simplify it.

The simplified fraction is 97.

E
Hint & Answer
S
Solution

Pair equivalent fractions.

To find the equivalent fraction pairs, let's analyze each fraction in the left column one at a time.

First Fraction

The first fraction is 1612. To find an equivalent fraction, let's simplify 1612 by dividing the numerator and denominator by their greatest common factor (GCF). First, let's split 16 and 12 into their prime factors. 16&= 2* 2* 2* 2 12&= 2* 2* 3 The GCF of two numbers is the product of their common factors. Since 16 and 12 share two common factors, their GCF is 2* 2=4. GCF(16,12)=4 Next, we will divide the numerator and denominator by the GCF.

Therefore, the equivalent fraction is 43.

Second Fraction

We can find an equivalent fraction of the second fraction 1976 by simplifying it. First, let's split the numerator of 19 and the denominator of 76 into their prime factors. 19&= 19* 1 76&= 19* 4 Since 19 and 76 share only one common factor, their GCF is 19. GCF(19,76)=19 Now, let's divide the numerator and denominator by the GCF.

Therefore, the equivalent fraction is 14.

Third Fraction

The third fraction is 3577. Let's simplify it! First, we can split the numerator of 35 and the denominator of 77 into their prime factors. 35&=5* 7 77&= 7* 11 Since 35 and 77 share only one common factor, their GCF is 7. GCF(35,77)=7 Now, let's divide the numerator and denominator by the GCF.

Therefore, the equivalent fraction is 511.

Fourth Fraction

The fourth and final fraction is 8448. We can find an equivalent fraction by simplifying it. Let's start by splitting the numerator and denominator into prime factors. 84&= 2* 2* 3* 7 48&= 2* 2* 2* 2* 3 We can see that 84 and 48 share three common factors. We can find the GCF of the numbers by multiplying these factors. GCF(84,48)=2* 2* 3=12 Now, let's divide the numerator and denominator by 12.

Therefore, the equivalent fraction is 74.

E
Hint & Answer
S
Solution

Fractions

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