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
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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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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.
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
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.
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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.
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
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.
b Now consider the milk chocolate bar B.
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 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.
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
Therefore, the shaded pieces represent 1832, or 916, of the chocolate bar.
c Finally, examine the last chocolate bar, the white chocolate bar C.
External credits: @freepik
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.
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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.
Jordan was worried that she made a mistake. Suddenly, she realized that even though their fractions looked different, they actually had the same values. This means that they both solved the exercise correctly! Pair the equivalent fractions from Jordan's and Emily's 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 2.1
Kriz and Maya are very good friends. One day, they were checking their answers on a math quiz. They noticed that they got different answers for four questions because they wrote different forms of equivalent fractions. Find the equivalent pairs.
{"type":"pair","form":{"alts":[[{"id":0,"text":"56\/78"},{"id":1,"text":"64\/36"},{"id":2,"text":"12\/96"},{"id":3,"text":"3\/7"}],[{"id":0,"text":"28\/39"},{"id":1,"text":"16\/9"},{"id":2,"text":"1\/8"},{"id":3,"text":"69\/161"}]],"lockLeft":true,"lockRight":false},"formTextBefore":"","formTextAfter":"","answer":[[0,1,2,3],[0,1,2,3]]}
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We want to pair the fractions that are equivalent. Let's analyze each fraction in the left-hand column one at a time.
First Fraction
The first fraction is 5678. Let's simplify it. We can do this by dividing the numerator and denominator by the greatest common factor of 56 and 78. First, let's split 56 and 78 into prime factors. 56&= 2* 2* 2* 7 78&= 2* 3* 13 The numbers share only one common factor, which is their GCF. GCF(56,78)=2 Next, we will divide the numerator and denominator by the GCF.
Therefore, the fractions 5678 and 2839 are equivalent.
Second Fraction
We can find an equivalent fraction to the second fraction 6436 by simplifying it. First, let's split the numerator and the denominator into prime factors. 64&= 2* 2* 2* 2 * 2 * 2 36&= 2* 2* 3 * 3 Since 64 and 36 share two common factors, their GCF is the product of these factors 2* 2=4. GCF(64,36)=4 Now, let's divide the numerator and denominator by the GCF.
Therefore, the equivalent fraction is 169.
Third Fraction
The third fraction is 1296. Let's simplify it! We can split the numerator 12 and the denominator 96 into prime factors. 12&= 2* 2* 3 96&=2* 2* 2* 2* 2* 3 We can find the GCF of 12 and 97 by multiplying their common factors. GCF(12,96)=2* 2* 3=12 Now, let's divide the numerator and denominator by the GCF.
Therefore, the equivalent fraction is 18.
Fourth Fraction
The fourth and final fraction is 37. Notice that this fraction is already in its simplest form. Let's instead analyze the remaining fraction from the right-hand side column, 69161. We can check if it is equivalent to 37 by simplifying the fraction. We will start by splitting the numerator and denominator into prime factors. 69&=3* 23 161&=7* 23 We can see that 69 and 161 share only one common factor, 23. This is their GCF! GCF(69,161)=23 Now, let's divide the numerator and denominator by 23.
Therefore, the equivalent fraction is 37.
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Hint & Answer
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Solution
Paulina was planning on baking 36 cupcakes for her friends and relatives. However, when she started baking, she realized that she only had enough ingredients for 24 cupcakes.
What portion of the intended amount of cupcakes will Paulina have to bake later to complete her plan? Write the answer as a fraction in simplest form.
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Let's start by recalling the definition of a fraction. l part→ whole→ a/b l←numerator ←denominator We know that Paulina was planning on baking the total of 36 cupcakes. This is the whole and the denominator of the fraction. However, she was only able to bake 24 cupcakes because she did not have enough ingredients. We want to find what part of the the cupcakes that she still needs to bake. Let's find how many cupcakes this is by subtracting 24 from 36. 36-24= 12cupcakes This is the part and the numerator of the fraction. Let's substitute our numbers into the fraction. 12/36 Finally, we need to simplify the fraction. We can do this by dividing the numerator and denominator by their greatest common factor (GCF). First, let's split 12 and 36 into prime factors. 12 & = 2* 2* 3 36 & = 2* 2* 3* 3 The numbers share three common factors. Let's multiply them to find the GCF of 12 and 36. GCF(12,36)= 2 * 2* 3=12 Now, we can divide the numerator and denominator by 12 to simplify the fraction.
Paulina will need to bake 13 of the planned number of cupcakes to complete her goal.
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Hint & Answer
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Solution
Tiffaniqua recently learned how to make her own stickers and she really enjoys creating new stickers. Over the weekend, she made 5 heart stickers, 8 winking emoji stickers, 9 dog stickers, and 7 octopus stickers.
Later, she realized that to have enough stickers to give to her friends and still have some left for her notebook, she needed to have 9 heart stickers, 7 winking emoji stickers, 13 dog stickers, and 10 octopus stickers. Match the sticker type and the fraction that illustrates the portion of stickers that Tiffaniqua has compared to the number of stickers she needs.
{"type":"pair","form":{"alts":[[{"id":0,"text":"Heart Sticker"},{"id":1,"text":"Emoji Sticker"},{"id":2,"text":"Dog Sticker"},{"id":3,"text":"Octopus Sticker"}],[{"id":0,"text":"5\/9"},{"id":1,"text":"8\/7"},{"id":2,"text":"9\/13"},{"id":3,"text":"7\/10"}]],"lockLeft":true,"lockRight":false},"formTextBefore":"","formTextAfter":"","answer":[[0,1,2,3],[0,1,2,3]]}
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Let's begin by recalling the definition of a fraction. l part→ whole→ a/b l←numerator ←denominator We want to match the fractions that represent the portion of each sticker that Tiffaniqua already has compared to the number of each sticker she needs to have. In other words, we need to identify the parts and wholes of four fractions, one corresponding to each sticker type. Let's start by recalling how many stickers of each type Tiffaniqua already has.
These numbers represent the part, or the numerators, of the fractions we are asked to write. Next, we can identify the whole, or the denominators, by considering how many stickers of each type Tiffaniqua needs to have in total.
Finally, we can write the fractions that match each type of sticker. lcc Heart Sticker &→ &5/9 [1em] Emoji Sticker &→ &8/7 [1em] Dog Sticker &→ &9/13 [1em] Octopus Sticker &→ &7/10 Notice that the second fraction is an improper fraction. It shows that Tiffaniqua has more emoji stickers than she actually needs!
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