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.
| Student Learning Objectives: |
|---|
|
| | 11 Theory slides |
| | 8 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
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.
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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.
External credits: @macrovector
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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.
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Discussion
Fractions
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 has multiple chocolate bars that are divided into different numbers of pieces.
External credits: @freepik
a
What part of the dark chocolate bar A 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{1}{2}","\\dfrac{5}{10}"]}}
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
Let's start by counting the total number of pieces and the number of shaded pieces in the dark chocolate bar A.
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 is shaded. Shaded/Total=5/10 We can see 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, let's count the pieces in the milk chocolate bar B.
Here, we have 18 shaded pieces out of 32 pieces in total. Next, we can write the fraction that represents the shaded pieces. Shaded/Total=18/32 Since 18 and 32 are even numbers, the fraction can be simplified. To do this, we divide both 18 and 32 by 2. 18 ÷ 2/32 ÷ 2 = 9/16 The shaded pieces make up 1832, or 916, of the chocolate bar.
c
Lastly, let's count the pieces in the white chocolate bar C.
External credits: @freepik
In this chocolate bar, 4 out of 6 pieces are shaded. Now, we can write the fraction of shaded pieces to the total number of pieces. Shaded/Total=4/6 Both 4 and 6 are even numbers, so we can divide them by 2 to simplify the fraction. 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. 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.
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.
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
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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 There are three steps to follow to simplify this fraction.
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, 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
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, 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 FractionsThe applet below illustrates how different fractions are simplified. 
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Example
Pairing Equivalent Fractions
Pair the equivalent fractions.
{"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
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 |
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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
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.
- 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.
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: Tests, exams, and homework assignments are generally scored with fractions, like 18/20.
- Money: A dime is 110 of a dollar. A quarter is a 14 of a dollar.
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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.
How many items of each type of clothing does Magdalena have?
{"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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