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Fractions and equivalence

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Notes de leçon

What is a Fraction?

  • A fraction shows a part of a whole or a group. For example, \frac{3}{4} means 3 parts out of 4 equal parts.
  • The numerator is the top number. It tells how many equal parts you have. In \frac{3}{4}, the numerator is 3.
  • The denominator is the bottom number. It tells how many equal parts make a whole. In \frac{3}{4}, the denominator is 4.
  • The line between them is called the fraction bar. It means 'divide' or 'out of'.
  • A fraction can also mean division. \frac{3}{4} is the same as 3 ÷ 4.

Equivalent Fractions

  • Equivalent fractions are different fractions that name the same amount. For example, \frac{1}{2} and \frac{2}{4} are equivalent.
  • You can make an equivalent fraction by multiplying the numerator and denominator by the same number. For example, \frac{1}{2} \times \frac{2}{2} = \frac{2}{4}.
  • You can also make an equivalent fraction by dividing the numerator and denominator by the same number. For example, \frac{4}{8} \div \frac{4}{4} = \frac{1}{2}.
  • When you multiply or divide the top and bottom by the same number, the fraction's value does not change.
  • Use fraction strips or pie charts to see that \frac{1}{2}, \frac{2}{4}, and \frac{3}{6} all cover the same area.

Simplifying Fractions

  • Simplifying a fraction means making it as simple as possible. You divide the numerator and denominator by their greatest common factor (the biggest number that divides both evenly).
  • For example, to simplify \frac{6}{8}, divide both by 2: \frac{6}{8} \div \frac{2}{2} = \frac{3}{4}.
  • A fraction is in simplest form when the only common factor of the numerator and denominator is 1.
  • Simplifying is also called reducing a fraction.
  • Always check if you can simplify your answer after adding or subtracting fractions.

Comparing Fractions

  • When two fractions have the same denominator, the one with the bigger numerator is larger. For example, \frac{3}{5} > \frac{2}{5}.
  • When fractions have different denominators, you can compare them by finding equivalent fractions with a common denominator.
  • For example, to compare \frac{1}{2} and \frac{3}{8}, change \frac{1}{2} to \frac{4}{8}. Now \frac{4}{8} > \frac{3}{8}, so \frac{1}{2} > \frac{3}{8}.
  • You can also compare fractions by using cross multiplication. For \frac{a}{b} and \frac{c}{d}, compare a \times d and c \times b.
  • Use fraction bars or number lines to help you see which fraction is bigger.

Adding and Subtracting Fractions with the Same Denominator

  • To add fractions with the same denominator, just add the numerators and keep the denominator the same. For example, \frac{2}{7} + \frac{3}{7} = \frac{5}{7}.
  • To subtract fractions with the same denominator, subtract the numerators and keep the denominator the same. For example, \frac{5}{8} - \frac{2}{8} = \frac{3}{8}.
  • The denominator does not change when you add or subtract fractions with the same denominator.
  • Always check if your answer can be simplified. For example, \frac{2}{4} + \frac{1}{4} = \frac{3}{4} (already simple), but \frac{1}{4} + \frac{1}{4} = \frac{2}{4} which simplifies to \frac{1}{2}.
  • When adding or subtracting mixed numbers, you can add the whole parts and the fractional parts separately, or convert to improper fractions first.

Improper Fractions and Mixed Numbers

  • An improper fraction has a numerator that is bigger than or equal to the denominator. For example, \frac{7}{4} is improper.
  • A mixed number has a whole number and a proper fraction. For example, 1\frac{3}{4} is a mixed number.
  • To convert an improper fraction to a mixed number, divide the numerator by the denominator. The quotient is the whole number, and the remainder becomes the numerator of the fraction part. For example, \frac{7}{4} = 1\frac{3}{4}.
  • To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and put that over the original denominator. For example, 1\frac{3}{4} = \frac{1 \times 4 + 3}{4} = \frac{7}{4}.
  • Improper fractions and mixed numbers both represent the same amount. You can choose which one to use.

Finding a Fraction of a Quantity

  • To find a fraction of a quantity, divide the quantity by the denominator, then multiply by the numerator.
  • For example, to find \frac{3}{4} of 20: first divide 20 by 4 to get 5, then multiply 5 by 3 to get 15. So \frac{3}{4} of 20 is 15.
  • You can also multiply the quantity by the fraction: \frac{3}{4} \times 20 = 15.
  • This works for any quantity, like money, length, or number of objects.
  • Practice with real-life examples: 'What is \frac{2}{3} of 12 apples?' Answer: 8 apples.

Fractions on a Number Line

  • You can show fractions on a number line by dividing the space between 0 and 1 into equal parts.
  • For example, to show \frac{3}{4}, divide the line from 0 to 1 into 4 equal parts and count 3 parts from 0.
  • Equivalent fractions appear at the same point on the number line. For example, \frac{1}{2} and \frac{2}{4} are at the same spot.
  • Number lines help you compare fractions and see which is larger or smaller.
  • Improper fractions are placed beyond 1 on the number line. For example, \frac{5}{4} is between 1 and 2.

Key Vocabulary

  • Fraction: a number that shows part of a whole.
  • Numerator: the top number in a fraction; it tells how many parts you have.
  • Denominator: the bottom number in a fraction; it tells how many equal parts make a whole.
  • Equivalent fractions: fractions that name the same amount.
  • Simplify: to make a fraction as simple as possible by dividing the top and bottom by the same number.
  • Improper fraction: a fraction where the numerator is greater than or equal to the denominator.
  • Mixed number: a number made of a whole number and a proper fraction.

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  1. 1.In the fraction 3/4, what does the denominator 4 tell us?

    Easy
    • AHow many equal parts make up a whole
    • BHow many parts are being described
    • CThe total number of parts altogether
    • DThe number of wholes

  2. 2.Which fraction is equivalent to 1/2?

    Easy
    • A2/3
    • B2/4
    • C3/4
    • D1/3

  3. 3.The fraction 6/8 is equivalent to 3/4.

    Easy

    True or false?

  4. 4.Which of these fractions is the largest?

    Medium
    • A3/4
    • B2/3
    • C5/8
    • D1/2

  5. 5.Which of the following are equivalent to 2/3? (Select all that apply)

    Medium
    • A4/6
    • B6/9
    • C3/4
    • D8/12
    • E5/7

  6. 6.Order these fractions from smallest to largest.

    Medium
    • 1/2
    • 3/4
    • 2/3
    • 5/8

  7. 7.Match each fraction to its equivalent fraction.

    Medium
    • 1/2
    • 2/3
    • 3/4
    • 6/8
    • 4/6
    • 3/6

  8. 8.A pizza is cut into 8 equal slices. Tom eats 3 slices and Jerry eats 2 slices. What fraction of the pizza is left?

    Medium
    • A3/8
    • B5/8
    • C2/8
    • D1/8

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