Fractions Class 6 Notes
Fractions Class 6 Notes
Fractions Class 6 Notes
1. Fraction: A numerical quantity that is not a whole number, representing a part of a whole. It is expressed as a ratio of two integers, with a numerator (top number) and a denominator (bottom number).
2. Numerator: The top part of a fraction that indicates how many parts of the whole are being considered.
3. Denominator: The bottom part of a fraction that indicates the total number of equal parts the whole is divided into.
4. Reading Fractions: In a fraction such as [latex]\frac{5}{6}[/latex], is called the numerator and 6 is called the denominator.
5. Mixed Number (or mixed fraction): A number that combines a whole number and a proper fraction, such as 3[latex]\frac{1}{4}[/latex].
6. Proper Fraction: A fraction where the numerator is less than the denominator, indicating a value less than one (e.g., [latex]\frac{3}{4}[/latex]).
7. Improper Fraction: A fraction where the numerator is greater than or equal to the denominator, indicating a value greater than or equal to one (e.g., [latex]\frac{5}{4}[/latex]).
8. Equivalent Fractions: Different fractions that represent the same value or proportion of a whole (e.g.. [latex]\frac{1}{2}[/latex] is equivalent to [latex]\frac{2}{4}[/latex] ).
9. Fractional Units: When one whole basic unit is divided into equal parts, then each part is called a fractional unit or a unit fraction.
10. Addition Fact: A mathematical statement that shows how fractions can be combined to form a whole (e.g., [latex]\frac{1}{4}[/latex] + [latex]\frac{1}{4}[/latex] + [latex]\frac{1}{4}[/latex] + [latex]\frac{1}{4}[/latex] = 1).
11. Division Fact: A mathematical statement that shows how a whole can be divided into equal parts (e.g., 1 ÷ 4 = [latex]\frac{1}{4}[/latex]).
12. Multiplication Fact: A mathematical statement that shows how many fractional units make up a whole (e.g., 4 × [latex]\frac{1}{4}[/latex] = 1).
13. Fraction Wall: A visual representation used to compare and understand different fractions and their equivalent values.
14. Fraction as equal share: When a whole number of units is divided into equal parts and shared equally, a fraction results.
15. Fractional Lengths: The measurement of parts of a whole expressed in fractions, often used in practical applications like cooking or construction.
16. Three Different Fractional Units: A concept explored in the PDF, referring to the challenge of finding three distinct fractions that add up to one.
17. Lowest terms: A fraction whose numerator and denominator have no common factor other than 1 is said to be in lowest terms or in its simplest form.
18. Brahmagupta’s method for adding fractions: When adding fractions, convert them into equivalent fractions with the same fractional unit (i.e, the same denominator), and then add these equivalent fractions to obtain the sum. This is accomplished by adding the numerators, while keeping the same denominator.
19. Brahmagupta’s method for subtracting fractions: When subtracting fractions, convert them into equivalent fractions with the same fractional unit (be., the same denominator), and then subtract these equivalent fractions. This is accomplished by subtracting the numerators, while keeping the same denominator.
Fractional Units and Equal Shares
Understanding Fractional Units : When we talk about fractional units, we are discussing how we can divide a whole item into smaller, equal parts. Each of these parts is called a fractional unit. For example, if you have a pizza and you cut it into 4 equal slices, each slice is a fractional unit of the whole pizza.
Equal Shares: Equal shares mean that when we divide something, everyone gets the same amount.
Fractional Units as Parts of a Whole
Marking Fraction Lengths on the Number Line
A number line is a straight line that represents numbers at equal intervals. When we mark fractions on a number line, we are showing how fractions fit between whole numbers. This helps us visualise and understand the size of fractions compared to whole numbers.
Mixed Fractions
Understanding Mixed Fractions : A mixed fraction (or mixed number) is a way to represent a number that has both a whole part and a fractional part. For example, if you have 2 whole pizzas and half of another pizza, you can express this as a mixed fraction : 2[latex]\frac{1}{2}[/latex].
Breaking it Down :
- Whole Part: This is the number of complete units you have. In our pizza example, the whole part is 2 (because you have 2 whole pizzas).
- Fractional Part: This shows how much of the next whole unit you have. In our example, you have [latex]\frac{1}{2}[/latex] of a pizza.
So, 2[latex]\frac{1}{2}[/latex] means you have 2 whole pizzas and half of another pizza.
Equivalent Fractions
Understanding Equivalent Fractions :
Equivalent fractions are different fractions that represent the same value or amount. Even though they look different, they show the same part of a whole. For example, [latex]\frac{1}{2}[/latex] and [latex]\frac{2}{4}[/latex] are equivalent fractions because they both represent the same amount.
Expressing a Fraction in Lowest Terms A fraction is in its lowest terms (or simplest form) when the numerator and denominator have no common factors, other than 1. This means that you cannot simplify the fraction any further. .
How to Express a Fraction in Lowest Terms :
- Find the Greatest Common Factor (GCF): The GCF is the largest number that divides both the numerator and denominator evenly.
- Divide the Numerator and Denomi¬nator by the GCF: This will give you the fraction in its simplest form.
Comparing Fractions
When we compare fractions, we want to find out which fraction is greater, which is smaller, or if they are equal. Comparing fractions can sometimes be tricky, especially if they have different denominators (the bottom number).
How to Compare Fractions :
1. Same Denominator : If the fractions have the same denominator, you can simply compare the numerators (the top numbers). The fraction with the larger numerator is the greater fraction.
Example : Compare [latex]\frac{3}{8}[/latex] and [latex]\frac{5}{8}[/latex] :
Since both fractions have the same denominator (8), we look at the numerators. 5 is greater than 3, so [latex]\frac{5}{8}[/latex] is greater than [latex]\frac{3}{8}[/latex].
2. Different Denominators : If the fraction s have different denominators, you can find a common denominator or convert them to equivalent fractions with the same denominator.
Example : Compare [latex]\frac{2}{3}[/latex] and [latex]\frac{3}{4}[/latex] :
The denominators are 3 and 4. The smallest common multiple of 3 and 4 is 12.
Convert both fractions :
$\begin{aligned} & \frac{2}{3}=\frac{2 \times 4}{3 \times 4}=\frac{8}{12} \\ & \frac{3}{4}=\frac{3 \times 3}{4 \times 3}=\frac{9}{12}\end{aligned}$
Now, compare [latex]\frac{8}{12}[/latex] and [latex]\frac{9}{12}[/latex]. Since 9 is greater than 8, [latex]\frac{3}{4}[/latex] is greater than [latex]\frac{2}{3}[/latex].
Addition and Subtraction of Fractions
Understanding Addition and Subtraction of Fractions :
When we add or subtract fractions, we are combining or taking away parts of a whole. However, we need to pay attention to the denominators (the bottom numbers) of the fractions.
Adding Fractions :
1. Same Denominator : If the fractions have the same denominator, you can simply add the numerators (the top numbers) and keep the numerator the same.
Example : [latex]\frac{1}{4}[/latex] + [latex]\frac{2}{4}[/latex]
Add the numerators : 1 + 2 = 3
Keep the denominator : [latex]\frac{3}{4}[/latex]
2. Different Denominators: If the fractions have different denominators, you need to find a common denominator before you can add them.
Example : [latex]\frac{1}{3}[/latex] + [latex]\frac{1}{4}[/latex].
The smallest common multiple of 3 and 4 is 12.
Convert both fractions to fractions with same denominator :
[latex]\frac{1}{3}[/latex] = [latex]\frac{1 \times 4}{3 \times 4}[/latex] = [latex]\frac{4}{12}[/latex]
[latex]\frac{1}{4}[/latex] = [latex]\frac{1 \times 3}{4 \times 3}[/latex] = [latex]\frac{3}{12}[/latex]
Now add [latex]\frac{4}{12}[/latex] + [latex]\frac{3}{12}[/latex] = [latex]\frac{7}{12}[/latex]
Subtracting Fractions :
1. Same Denominator : If the fractions have the same denominator, subtract the numerators and keep the denominator the same.
Example : [latex]\frac{3}{5}[/latex] - [latex]\frac{1}{5}[/latex]
Subtract the numerators : 3 - 1 = 2
Keep the denominator : [latex]\frac{2}{5}[/latex]
2. Different Denominators : If the fractions have different denominators, find a common denominator before subtracting.
Example : [latex]\frac{5}{6}[/latex] - [latex]\frac{1}{3}[/latex]
The smallest common multiple of 6 and 3 is 6.
Convert [latex]\frac{1}{3}[/latex] to have a denominator of 6 :
[latex]\frac{1}{3}[/latex] = [latex]\frac{1 \times 2}{3 \times 2}[/latex] = [latex]\frac{2}{6}[/latex]
Now subtract:
[latex]\frac{5}{6}[/latex] - [latex]\frac{2}{6}[/latex] = [latex]\frac{3}{6}[/latex] which simplifies to [latex]\frac{1}{2}[/latex].
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