# IGCSE Mathematics Foundation: Calculations with Fractions

## Scheme of work: IGCSE Foundation: Year 10: Term 4: Calculations with Fractions

#### Prerequisite Knowledge

1. Equivalent Fractions
• Concept: Understanding how to create equivalent fractions by multiplying or dividing the numerator and the denominator by the same number.
• Example: $\text{Equivalent fractions for } \frac{1}{2} \text{ are } \frac{2}{4}, \frac{3}{6}, \text{ and } \frac{4}{8}.$
2. Converting Between Fractions, Decimals, and Percentages
• Concept: Knowing how to convert between fractions, decimals, and percentages.
• Example: $\text{To convert } \frac{1}{4} \text{ to a decimal, divide 1 by 4 to get } 0.25.$ $\text{To convert } 0.25 \text{ to a percentage, multiply by 100 to get } 25\%.$ $\text{To convert } 25\% \text{ to a fraction, write } 25\% \text{ as } \frac{25}{100} \text{ and simplify to } \frac{1}{4}.$

#### Success Criteria

1. Use Common Denominators to Add and Subtract Fractions and Mixed Numbers
• Objective: Accurately add and subtract fractions and mixed numbers by finding a common denominator.
• Example:

Add $$\frac{2}{3} + \frac{1}{4}$$.

Find a common denominator: $$\frac{2}{3} = \frac{8}{12}$$ and $$\frac{1}{4} = \frac{3}{12}$$.

Add: $$\frac{8}{12} + \frac{3}{12} = \frac{11}{12}$$.

Subtract $$1\frac{1}{2} – \frac{3}{4}$$.

Convert mixed number: $$1\frac{1}{2} = \frac{3}{2}$$.

Find a common denominator: $$\frac{3}{2} = \frac{6}{4}$$.

Subtract: $$\frac{6}{4} – \frac{3}{4} = \frac{3}{4}$$.

2. Understand and Use Unit Fractions as Multiplicative Inverses
• Objective: Understand that the reciprocal of a unit fraction is its multiplicative inverse, and use this property in calculations.
• Example:

The reciprocal of $$\frac{1}{5}$$ is $$5$$ because $$\frac{1}{5} \times 5 = 1$$.

Use the multiplicative inverse to solve $$\frac{1}{5} \times x = 3$$.

$$x = 3 \times 5 = 15$$.

3. Multiply and Divide Fractions and Mixed Numbers
• Objective: Accurately multiply and divide fractions and mixed numbers by applying the appropriate rules and simplifying the results.
• Example:

Multiply $$\frac{2}{3} \times \frac{4}{5}$$.

Multiply the numerators and denominators: $$\frac{2 \times 4}{3 \times 5} = \frac{8}{15}$$.

Divide $$\frac{7}{8} \div \frac{2}{3}$$.

Multiply by the reciprocal: $$\frac{7}{8} \times \frac{3}{2} = \frac{7 \times 3}{8 \times 2} = \frac{21}{16}$$.

Multiply mixed numbers: $$1\frac{1}{2} \times 2\frac{2}{3}$$.

Convert to improper fractions: $$\frac{3}{2} \times \frac{8}{3} = \frac{3 \times 8}{2 \times 3} = 4$$.

#### Key Concepts

1. Use Common Denominators to Add and Subtract Fractions and Mixed Numbers
• Concept: To add or subtract fractions, they must have the same denominator. The common denominator is found by identifying the least common multiple (LCM) of the denominators.
• Example:

To add $$\frac{2}{3} + \frac{1}{4}$$, find the LCM of 3 and 4, which is 12.

Convert $$\frac{2}{3}$$ to $$\frac{8}{12}$$ and $$\frac{1}{4}$$ to $$\frac{3}{12}$$.

Add the fractions: $$\frac{8}{12} + \frac{3}{12} = \frac{11}{12}$$.

2. Understand and Use Unit Fractions as Multiplicative Inverses
• Concept: A unit fraction (a fraction with a numerator of 1) has a multiplicative inverse that is a whole number. The product of a unit fraction and its multiplicative inverse is 1.
• Example:

The multiplicative inverse of $$\frac{1}{5}$$ is 5 because $$\frac{1}{5} \times 5 = 1$$.

To solve $$\frac{1}{5} \times x = 3$$, multiply both sides by 5:

$$x = 3 \times 5 = 15$$.

3. Multiply and Divide Fractions and Mixed Numbers
• Concept: To multiply fractions, multiply the numerators together and the denominators together. To divide fractions, multiply by the reciprocal of the divisor.
• Example:

Multiply $$\frac{2}{3} \times \frac{4}{5}$$ by multiplying the numerators and denominators:

$$\frac{2 \times 4}{3 \times 5} = \frac{8}{15}$$.

Divide $$\frac{7}{8} \div \frac{2}{3}$$ by multiplying by the reciprocal:

$$\frac{7}{8} \times \frac{3}{2} = \frac{21}{16}$$.

#### Common Misconceptions

1. Use Common Denominators to Add and Subtract Fractions and Mixed Numbers
• Common Mistake: Failing to find the correct common denominator or not converting all fractions to equivalent fractions with the common denominator.
• Example:

Incorrect: Adding $$\frac{2}{3} + \frac{1}{4}$$ without finding a common denominator.

Correct: Finding the common denominator (12) and converting: $$\frac{2}{3} = \frac{8}{12}$$ and $$\frac{1}{4} = \frac{3}{12}$$.

Then add: $$\frac{8}{12} + \frac{3}{12} = \frac{11}{12}$$.

2. Understand and Use Unit Fractions as Multiplicative Inverses
• Common Mistake: Misidentifying the multiplicative inverse or not applying the inverse correctly in equations.
• Example:

Incorrect: Identifying the multiplicative inverse of $$\frac{1}{5}$$ as $$\frac{1}{5}$$.

Correct: The multiplicative inverse of $$\frac{1}{5}$$ is 5.

To solve $$\frac{1}{5} \times x = 3$$, multiply both sides by 5:

$$x = 3 \times 5 = 15$$.

3. Multiply and Divide Fractions and Mixed Numbers
• Common Mistake: Failing to multiply numerators and denominators correctly or not converting mixed numbers to improper fractions before multiplying or dividing.
• Example:

Incorrect: Multiplying $$\frac{2}{3} \times \frac{4}{5}$$ by adding the numerators and denominators: $$\frac{2 + 4}{3 + 5} = \frac{6}{8}$$.

Correct: Multiply the numerators and denominators: $$\frac{2 \times 4}{3 \times 5} = \frac{8}{15}$$.

Incorrect: Dividing mixed numbers without converting: $$1\frac{1}{2} \div 2\frac{2}{3}$$.

Correct: Convert to improper fractions first: $$\frac{3}{2} \div \frac{8}{3} = \frac{3}{2} \times \frac{3}{8} = \frac{9}{16}$$.

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