Chapter Notes: Fractions, Greater than, Less than, Equal

Greater than, Less than, Equal

Example conversions:

• 0.2 = 20/100 = 20%.
• 3/10 = 30/100 = 30%.
• 6/10 = 60/100 = 60%.
• 0.8 = 80/100 = 80%.
• 0.1 = 10/100 = 10%.
• 0.3 = 30/100 = 30%.
• 7/10 = 70/100 = 70%.
• 0.9 = 90/100 = 90%.
• 0.4 = 40/100 = 40%\

Ordering quantities requires converting to a common form (e.g., decimals or percentages):
Example: Order 0.7, 3/10, 1.2, 40%:

• 0.7 = 70%.
• 3/10 = 0.3 = 30%.
• 1.2 = 120%.
• 40% = 0.4.
• Order: 3/10 (0.3), 40% (0.4), 0.7, 1.2.

Adding and Subtracting Fractions

Example: 1/3 + 2/9:

• 1/3 = 3/9 (since 9 is a multiple of 3).
• 3/9 + 2/9 = 5/9.

Example: 7/18 + 5/27:

• Common denominator 27: 7/18 = 7 × 1.5 / 18 × 1.5 = 10.5/27.
• 10.5/27 + 5/27 = 15.5/27.

Diagrams can represent addition and subtraction:

• Example: Adding 1/5 + 1/5 = 2/5 (two shaded fifths).
• Example: Subtracting 3/5 - 1/5 = 2/5 (remove one shaded fifth).

Real-world examples:

• 3/4 + 2/3 (water used): Common denominator 12, 3/4 = 9/12, 2/3 = 8/12, so 9/12 + 8/12 = 17/12 = 1 5/12 litres.
• 3/5 - 2/5 = 1/5 (carrots remaining after eating and selling).

Multiplying and Dividing Unit Fractions by a Whole Number

• Multiplying a unit fraction by a whole number scales the fraction.
  Example: 1/4 × 3(three jugs, each 1/4 full):
  1/4 + 1/4 + 1/4 = 3/4.
• Dividing a unit fraction by a whole number splits the fraction into equal parts.
  Example: 1/2 ÷ 3(half a chocolate bar shared among three):
  1/2 ÷ 3 = 1/2 × 1/3 = 1/6.

Real-world examples:

• 1/2 ÷ 4 = 1/8 (half a tray of cherry cakes into four bags).
• 1/2 ÷ 3 = 1/6 (half a tray of strawberry cakes into three bags).
• 1/2 ÷ 5 = 1/10 (half a tray of carrot cakes into five bags).
• 1/3 ÷ 5 = 1/15 (1/3 metre ribbon into five pieces).
• 1/6 × 5 = 5/6 (lemons for five bottles of lemonade).

Generalization: Multiplying a unit fraction 1/n by a whole number m gives m/n, where the numerator equals the whole number.

Ratio and Proportion

• Ratio compares two or more quantities using the symbol :.
• Example: For every 2 red counters, there are 3 blue counters, ratio 2:3 (red:blue).
• Order matters: 2:3 (red:blue) is different from 3:2 (blue:red).
• Proportion describes a part relative to the whole, as a fraction or percentage.

Example: In a pattern with 2 red and 3 blue counters (total 5):

• Proportion of red: 2/5 = 40%.
• Proportion of blue: 3/5 = 60%.

Comparing three parts in a ratio:

• Example: Ratio of odd to even to prime numbers in a set, or positive to zero to negative numbers.

Real-world examples:

• Spelling test scores: 12/20 = 60%, 14/20 = 70%, 15/20 = 75%, 10/20 = 50%, 16/20 = 80%, 11/20 = 55%.
• Proportion scoring less than 13: 2/6 = 1/3 ≈ 33.33%.
• Proportion scoring more than 13: 3/6 = 1/2 = 50%.

In a class of 30 with 9 nine-year-old boys and 9 nine-year-old girls:

• Proportion of nine-year-olds: (9 + 9)/30 = 18/30 = 3/5 = 60%.
• Proportion not nine years old: 12/30 = 2/5 = 40%.