Chapter Notes: Fractions and Division

Fractions and Division

• A fraction represents a part of a whole, where the numerator indicates the number of parts and the denominator indicates the total number of equal parts.
• Dividing a whole into equal parts creates fractions:
  • Example: Dividing one sheet of paper into 10 equal parts results in each part being 1/10 of the whole.
  • Division sentence: 1 ÷ 10 = 1/10.
• A unit fraction has a numerator of 1 (e.g., 1/10), with the denominator indicating the number of equal parts the whole is divided into.
• Dividing multiple wholes:
  • Example: Dividing 3 sheets of paper into 10 equal parts each results in each part being 1/10 of one whole.
  • Total fraction: 3 ÷ 10 = 3/10.
• Dividing amounts into equal parts:
  • Example: Dividing 1 liter of water equally among 6 containers gives 1 ÷ 6 = 1/6 liter per container.
  • Example: Dividing 1 liter among 8 containers gives 1 ÷ 8 = 1/8 liter per container.
  • Example: Dividing 3 liters among 3 containers gives 3 ÷ 3 = 1 liter per container.
• Dividing into 100 equal parts:
  • Example: Dividing $1 into 100 equal parts gives 1 ÷ 100 = 1/100 dollar, or 1 cent.

Equivalent Fractions

• Equivalent fractions represent the same proportion of a whole but have different numerators and denominators.
• Example: 1/2 = 2/4 = 4/8 are equivalent fractions, as they represent the same position on a number line.
• Equivalent fractions may look different but shade the same proportion of a shape:
  • Example: 2/3 = 4/6 = 8/9 (if 2/3 of a shape is shaded, 4/6 or 8/9 of a different shape can represent the same proportion).

Improper Fractions and Mixed Numbers

• An improper fraction has a numerator larger than or equal to the denominator (e.g., 5/3).
• A mixed number combines a whole number and a proper fraction (e.g., 1 2/3).
• Converting improper fractions to mixed numbers:
  • Example: 5/3 = 1 2/3 (since 5 ÷ 3 = 1 remainder 2, so 1 + 2/3 = 1 2/3).
• Converting mixed numbers to improper fractions:
  • Example: 1 2/3 = (1 × 3 + 2)/3 = 5/3.
• Improper fractions and mixed numbers are equivalent:
  • Example: 7/4 = 1 3/4 (since 7 ÷ 4 = 1 remainder 3, so 1 + 3/4 = 1 3/4).

Fractions as Operators

• Fractions can act as operators to find a part of a whole amount.
• Finding a unit fraction of an amount:
  • Example: 1/10 of $5 = $5 ÷ 10 = $0.50 (or 50 cents).
• Finding a non-unit fraction of an amount:
  • Method 1: Divide by the denominator, then multiply by the numerator.
  • Example: 3/10 of $5 = (5 ÷ 10) × 3 = 0.5 × 3 = $1.50.
  • Method 2: Multiply by the numerator, then divide by the denominator.
  • Example: 3/10 of $5 = (5 × 3) ÷ 10 = 15 ÷ 10 = $1.50.
• Finding the whole from a fraction:
  • Example: If 1/8 of a set costs 40 cents, the whole set costs 40 × 8 = 320 cents.
  • Using factor pairs: 40 × 8 = (10 × 4) × 8 = 10 × (4 × 8) = 10 × 32 = 320.
• Bar models can represent fractions of amounts:
  • Example: 1/2 of a set = 75 cents, so the whole set is 75 × 2 = 150 cents.
  • Example: 1/10 of a set = 60 cents, so the whole set is 60 × 10 = 600 cents.
  • Example: 1/3 of a set = 30 cents, so the whole set is 30 × 3 = 90 cents.

Adding and Subtracting Fractions

• Fractions with the same denominator can be added or subtracted directly:
  • Example: 2/7 + 3/7 = 5/7.
  • Example: 7/8 - 2/8 = 5/8.
• Fractions with different denominators require equivalent fractions with a common denominator:
  • Example: 1/5 + 1/10:
    • Convert 1/5 = 2/10.
    • Add: 2/10 + 1/10 = 3/10.
  • Example: 4/5 - 1/8:
    • Common denominator is 40 (since 5 × 8 = 40).
    • Convert: 4/5 = 32/40, 1/8 = 5/40.
    • Subtract: 32/40 - 5/40 = 27/40.