1. Fractions
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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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Lesson Settings & Tools
| | 11 Theory slides |
| | 8 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Fractions
Slide of 11
People deal with numbers in countless daily situations. Many of these 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 different real-life applications.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
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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. 
Note that Jordan can only split the pizza into a minimum of 2 and a maximum of 20 slices.
External credits: @macrovector
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Discussion
Presenting Rational Numbers
In order to state what part of the pizza each person will eat, a certain type of numbers should be used.
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 All four of these examples are rational numbers. Note that integers are also rational because they can always be written as fractions with a denominator of 1. -3 = -31, 5= 51 In other words, integer numbers is a subset of rational numbers.
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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
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.
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Example
Determining Fractions
Jordan's friends know how much she loves chocolate, so three people brought her chocolate bars as a gift when they came for her pizza party. Each chocolate bar was divided into a different number of pieces. 
Jordan decides to share the chocolate with her friends. The shaded pieces in the diagrams above show how much of each bar the kids ate.
First, count the total number of pieces and the number of shaded pieces.
Total:& 10 Shaded:& 5
To find what part of the chocolate bar the shaded pieces represent, divide the number of shaded pieces by the total number of pieces.
Shaded/Total=5/10
Therefore, the shaded pieces represent 510 (five-tenths) of the chocolate bar. Notice that both the numerator and denominator are divisible by 5. This means that the fraction can be simplified by dividing its numerator and denominator by 5.
5÷ 5/10÷ 5=1/2
This means that the shaded pieces represent one-half of the chocolate bar.
Like in Part A, start by counting the total number of pieces of chocolate and the number of shaded pieces.
Total:& 32 Shaded:& 18
The part of the chocolate bar represented by the shaded pieces can be written 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.
Therefore, the shaded pieces represent 1832, or 916, of the chocolate bar.
Again, 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
Since both the numerator and denominator are divisible by 2, the fraction can be simplified.
4÷ 2/6÷ 2=2/3
The shaded pieces make up 46, or 23, of the chocolate bar.
External credits: @freepik
a What part of the dark chocolate bar A do the shaded pieces represent?
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b What part of the milk chocolate bar B do the shaded pieces represent?
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["\\dfrac{18}{32}","\\dfrac{9}{16}"]}}
c What part of the white chocolate bar C do the shaded pieces represent?
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["\\dfrac{4}{6}","\\dfrac{2}{3}"]}}
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 Start by analyzing the dark chocolate bar A.
External credits: @freepik
b Now consider the milk chocolate bar B.
External credits: @freepik
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
c Finally, examine the last chocolate bar, the white chocolate bar C.
External credits: @freepik
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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.
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Discussion
Equivalent Fractions
Fractions that have different numerators and denominators but represent the same value are called equivalent fractions. For example, 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 Imagine two friends, Emily and Maya, who have two identical cakes. Emily divided her cake into four parts and ate one. Maya divided her cake into eight slices and ate two.
External credits: @macrovector
Emily and Maya ate the same amount of cake. Therefore, 14 and 28 are equivalent fractions. 1/4=2/8 Equivalent fractions can be formed by multiplying or dividing the numerator and denominator by the same number.
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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
In order to simplify this fraction, there are three steps to follow.
1
Determine Whether the Fraction Can Be Simplified
expand_more 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. Therefore, the fraction can be simplified. If the numerator and denominator did not have common factors other than 1, the fraction would be said to be simplified or written in its simplest form.
2
Find the Greatest Common Factor
expand_more 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
expand_more Finally, to reduce the fraction, divide its numerator and denominator by their GCF. 18/66=18/ 6/66/ 6=3/11 As a result, an equivalent fraction of 311 was obtained. It is the simplest form of the given fraction 1866.
Extra
Simplification of Different FractionsThe applet below illustrates how different fractions are simplified. 
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Example
Pairing Equivalent Fractions
After having a lot of fun with her friends on Saturday, Jordan woke up rested and energized the next day and decided to do her math homework. When she finished, she texted her answers to her friend Emily and asked if she got the same results.
{"type":"pair","form":{"alts":[[{"id":0,"text":"6\/12"},{"id":1,"text":"16\/56"},{"id":2,"text":"14\/32"},{"id":3,"text":"20\/15"}],[{"id":0,"text":"1\/2"},{"id":1,"text":"2\/7"},{"id":2,"text":"7\/16"},{"id":3,"text":"4\/3"}]],"lockLeft":true,"lockRight":false},"formTextBefore":"","formTextAfter":"","answer":[[0,1,2,3],[0,1,2,3]]}
Hint
Find the greatest common factor (GCF) of the numerator and denominator of a fraction, then use it to simplify the fraction.
Solution
Notice that Emily's fractions are written in their simplest form, but Jordan's fractions are not simplified. Therefore, to find the equivalent fractions, each of Jordan's fractions will be analyzed and simplified, one at a time.
First Fraction
Begin by considering the first fraction from Jordan's list. 6/12 To simplify this fraction, its numerator and denominator should be divided by their greatest common factor (GCF). To find it, split 6 and 12 into prime factors. 6&= 2* 3 12&=2* 2* 3 As shown, 6 and 12 share two common factors. Their product is the GCF of these numbers. GCF(6,12)=2* 3=6 Finally, divide both the numerator and denominator by 6 to simplify the fraction. Therefore, 12 is an equivalent fraction to 612.Second Fraction
Now, consider the second fraction from Jordan's list. 16/56 Again, find the GCF of the numerator 16 and the denominator 56. To do so, write each number as the product of prime numbers. 16&= 2* 2* 2* 2 56&= 2* 2* 2* 7 There are three common factors between 16 and 56. Multiply them to calculate the GCF of 16 and 56. 2* 2* 2 = 8 Next, simplify the fraction by dividing its numerator and denominator by 8. Therefore, 1656 and 27 are equivalent fractions.Third Fraction
Similarly, examine the third fraction that Jordan wrote. 14/32 Just as before, determine the GCF of 14 and 32 by factoring the numbers into prime factors. 14&= 2* 7 32&= 2* 2* 2* 2 * 2 The numbers share only one common factor, 2. This means that the GCF of 14 and 32 is 2. Next, simplify the fraction by dividing the numerator and denominator by 2. It can be concluded that 716 is an equivalent fraction to 1432.Fourth Fraction
Finally, consider the last fraction from Jordan's list. 20/15 Start by rewriting the numerator and denominator as the product of prime factors. 20 & =2* 2* 5 15 & =3* 5 The common factor between 20 and 15 is 5. This is their GCF. Divide both the numerator and denominator by 5 to simplify the fraction. It can be concluded that 43 is an equivalent fraction to 2015.
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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.
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Closure
Real Life Applications of Fractions
On top of the topics mentioned in this lesson, there are many other real-life applications of fractions. Here are some of them.
- 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.
- 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 the usage of fractions.
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.
- Tests and exams: Scores of tests, exams, and homework assignments are generally expressed as fractions, like 27/30.
- Money: A dime is 110 of a dollar. A quarter is a 14 of a dollar. Understanding fractions makes adding money quick and easy.
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Fractions
Exercise 3.1
Magdalena is cleaning out her closet. She sets aside 8 T-shirts, 2 pairs of jeans, 5 dresses, and 4 sweaters that she does not wear anymore to be donated. The following fractions show what part the donated clothes represent of each type of clothing in her wardrobe.
{"type":"pair","form":{"alts":[[{"id":0,"text":"T-Shirts"},{"id":1,"text":"Jeans"},{"id":2,"text":"Dresses"},{"id":3,"text":"Sweaters"}],[{"id":0,"text":"14"},{"id":1,"text":"4"},{"id":2,"text":"7"},{"id":3,"text":"5"}]],"lockLeft":true,"lockRight":false},"formTextBefore":"","formTextAfter":"","answer":[[0,1,2,3],[0,1,2,3]]}
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Let's find how many items of each type of clothing are in Magdalena's wardrobe. We will do this one at a time.
T-Shirts
We are told that Magdalena decided to donate 8 T-shirts. This number is 47 of the total number of T-shirts that she owns. If we knew the total number of T-shirts Magdalena has, we could easily write the fraction for the ratio of donated shirts to total number of shirts. 8/Total The numerator of this fraction is 8 because Magdalena is donating 8 shirts. However, the numerator in the given fraction is 4. Let's rewrite the given fraction so that it also has a numerator of 8. We can do this by expanding the fraction by a factor of 2. 4/7=4 * 2/7 * 2=8/14 Notice that the fraction we wrote and the given fraction represent the same thing — the ratio of donated T-shirts to the total number of T-shirts Magdalena owns. 8/Total=8/14 We can say that Magdalena has 14 T-shirts in total!
Jeans
Let's use a the same process to find the total number of pairs of jeans that Magdalena owns. It is given that Magdalena is donating 2 pairs of jeans. This is 12 of the total number of pairs of jeans she owns. Let's write what the fraction for the ratio of donated items to total items would be if we knew the total number of items. 2/Total Since Magdalena is donating 2 pairs of jeans, the numerator is 2. Notice that the numerator in the given fraction is 1. Let's rewrite the given fraction so that it also has a numerator of 2 by expanding the fraction by a factor of 2. 1/2=1 * 2/2 * 2=2/4 The fraction we wrote and the given fraction represent the same thing — the ratio of donated pairs of jeans to the total number of pairs of jeans. 2/Total=2/4 This means that Magdalena owns 4 pairs of jeans.
Dresses
Now we will try to determine how many dresses Magdalena has. We know that she wants to donate 5 dresses. We also know that this number makes up 57 of all of her dresses. 5/Total=5/7 The numerators already match! This means that we can conclude that Magdalena owns 7 dresses in total.
Sweaters
Let's now consider the final type of clothing, sweaters. Magdalena will donate 4 sweaters. We also know that this number makes up 45 of all of her sweaters. 4/Total=4/5 The numerators already match, so Magdalena has 5 sweaters in total.
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