Number - Operations with Fractions

Simplifying Fractions: To simplify a fraction, divide both the numerator and the denominator by their Greatest Common Factor (GCF). Visually, this means expressing the same portion of a whole using fewer, larger pieces. For example, if you have a circle where $\frac{4}{8}$ is shaded, you can see it is the same area as $\frac{1}{2}$ of the circle, showing the fraction is in its simplest form when no more common factors exist.

Equivalent Fractions: Fractions that represent the same value or part of a whole are equivalent. You can find them by multiplying or dividing the numerator and denominator by the same non-zero number. On a number line, equivalent fractions like $\frac{1}{2}$, $\frac{2}{4}$, and $\frac{4}{8}$ all sit at the exact same point halfway between $0$ and $1$.

Adding and Subtracting Fractions: Fractions can only be added or subtracted if they have a common denominator, representing parts of the same size. If denominators differ, find the Least Common Multiple (LCM). Visually, this is like taking two different grids (one divided into 3rds and one into 4ths) and subdividing them both into 12ths so the 'slices' are identical and can be counted together.

Multiplying Fractions: To multiply fractions, multiply the numerators together and the denominators together. Visually, this represents taking 'a part of a part.' For example, $\frac{1}{2} \times \frac{3}{4}$ can be seen as an area model where a rectangle's width is $\frac{1}{2}$ and its height is $\frac{3}{4}$; the overlapping shaded area represents the product, $\frac{3}{8}$.

Dividing Fractions: Dividing by a fraction is equivalent to multiplying by its reciprocal (flipping the numerator and denominator). This is the 'Keep-Change-Flip' method. Visually, $\frac{1}{2} \div \frac{1}{4}$ asks 'How many quarters fit into one half?' By looking at a diagram of a half-circle, you can see that $2$ quarters fit inside it, so the answer is $2$.

Mixed Numbers and Improper Fractions: An improper fraction has a numerator larger than or equal to the denominator (e.g., $\frac{5}{3}$), while a mixed number combines a whole number and a fraction (e.g., $1\frac{2}{3}$). Visually, $\frac{5}{3}$ is shown as one full shape divided into 3rds plus 2 additional 3rds from a second shape.

Comparing Fractions: To determine which fraction is larger, either convert them to a common denominator or use cross-multiplication. Visually, if you compare $\frac{2}{3}$ and $\frac{3}{4}$ on two identical bars, the bar representing $\frac{3}{4}$ will have a slightly larger shaded region than the one representing $\frac{2}{3}$.