Tips for Teachers by Craig Barton

Adding and Subtrating fractions with different denominators V2

Adding and Subtracting Fractions with Different Denominators
Atoms of Knowledge
(F) What a fraction represents – A fraction \(\frac{a}{b}\) represents \(a\) parts out of \(b\) equal parts, where \(a\) is the numerator and \(b\) is the denominator.
(C) Equivalent fractions – Fractions that represent the same value but have different numerators and denominators
✗\(\frac{2}{3}\) and \(\frac{3}{5}\). How do I know? Because \(2 \times 5 = 10\) but \(3 \times 3 = 9\), so they’re not equal.
✓\(\frac{1}{2}\) and \(\frac{2}{4}\). How do I know? Because \(1 \times 4 = 4\) and \(2 \times 2 = 4\), so they’re equal.
✓\(\frac{3}{5}\) and \(\frac{6}{10}\). How do I know? Because \(3 \times 10 = 30\) and \(5 \times 6 = 30\), so they’re equal.
✓\(\frac{2}{3}\) and \(\frac{8}{12}\). How do I know? Because \(2 \times 12 = 24\) and \(3 \times 8 = 24\), so they’re equal.
✗\(\frac{4}{5}\) and \(\frac{8}{9}\). How do I know? Because \(4 \times 9 = 36\) but \(5 \times 8 = 40\), so they’re not equal.
(T) Converting to an equivalent fraction – Multiply both numerator and denominator by the same number

Example: Convert \(\frac{2}{5}\) to an equivalent fraction with denominator 20.

Steps:

1. We need to change 5 to 20, so we multiply by \(\frac{20}{5} = 4\)

2. Multiply both numerator and denominator by 4: \(\frac{2 \times 4}{5 \times 4} = \frac{8}{20}\)

Commentary: When we multiply both the top and bottom by the same number, we create an equivalent fraction with a larger denominator while keeping the same value.

(R) Finding the Least Common Multiple (LCM) – The smallest number that is a multiple of both given numbers

Example: Find the LCM of 4 and 6.

Steps:

1. List multiples of 4: 4, 8, 12, 16, 20, 24…

2. List multiples of 6: 6, 12, 18, 24, 30…

3. Find the smallest number in both lists: 12

Therefore, LCM(4, 6) = 12

Commentary: The LCM gives us the smallest common denominator we can use when adding or subtracting fractions with different denominators.

(T) Adding and subtracting fractions with the same denominator – Add or subtract the numerators, keep the denominator the same

Example: Calculate \(\frac{5}{12} + \frac{3}{12}\)

Steps:

1. The denominators are already the same (12)

2. Add the numerators: \(5 + 3 = 8\)

3. Keep the denominator: \(\frac{8}{12}\)

4. Simplify: \(\frac{8}{12} = \frac{2}{3}\) (dividing both by 4)

Commentary: When fractions have the same denominator, we can directly add or subtract the numerators because we’re working with the same-sized parts.

(F) Simplifying fractions – Divide both numerator and denominator by their highest common factor (HCF) to write the fraction in its simplest form.