Fractions and Decimals - Adding and Subtracting Fractions with Unlike Denominators

Unlike Denominators: When fractions have different denominators, such as $\frac{1}{2}$ and $\frac{1}{3}$, it means the whole has been divided into different sized parts. To add or subtract them, you must first find a common 'language' or size for these parts. Visually, imagine a pizza cut into 2 large slices and another cut into 3 medium slices; you cannot simply count them as '2 slices' because the sizes differ.

Least Common Denominator (LCD): The LCD is the smallest number that both denominators can divide into evenly. This is the Least Common Multiple (LCM) of the denominators. For example, for denominators 4 and 6, the multiples of 4 are $4, 8, 12, 16...$ and multiples of 6 are $6, 12, 18...$, making 12 the LCD.

Creating Equivalent Fractions: Once the LCD is found, you must transform each fraction so it has that denominator. You do this by multiplying both the numerator and denominator by the same number. Visually, this is like taking a rectangle divided into 2 parts and drawing a horizontal line across it to turn those 2 parts into 4 smaller, equivalent parts.

The 'Only Numerators' Rule: After converting fractions to have a common denominator, you only add or subtract the numerators. The denominator remains the same because it represents the size of the pieces, which does not change during the operation. For example, $\frac{2}{10} + \frac{3}{10} = \frac{5}{10}$.

Simplifying the Result: After calculating the sum or difference, always check if the fraction can be simplified to its lowest terms. This involves dividing both the numerator and denominator by their Greatest Common Factor (GCF). For instance, $\frac{5}{10}$ simplifies to $\frac{1}{2}$ because both can be divided by 5.

Visualizing with Grid Models: To add $\frac{1}{3}$ and $\frac{1}{4}$ visually, you can draw a grid that is 3 units wide and 4 units tall. Shading 1 column represents $\frac{1}{3}$ (which is 4 out of 12 squares) and shading 1 row represents $\frac{1}{4}$ (which is 3 out of 12 squares). Combining them gives 7 out of 12 total squares, or $\frac{7}{12}$.

Mixed Numbers and Improper Fractions: If the resulting numerator is larger than the denominator (e.g., $\frac{5}{4}$), the fraction is 'improper'. It should often be converted to a mixed number ($1 \frac{1}{4}$) by seeing how many whole units fit into the numerator.