Number - Ratio, proportion, and rates of change

Ratios are used to compare quantities of the same kind, expressed in the form $a:b$ or $a:b:c$. To simplify a ratio, divide all parts by their highest common factor. Visually, a ratio like $2:3$ can be seen as a bar split into $5$ equal segments, where $2$ segments are one color and $3$ are another.

Proportional division involves sharing a total amount into parts defined by a ratio. To calculate this, sum the parts of the ratio to find the 'total parts', divide the total quantity by this sum to find the value of 'one part', and then multiply each ratio number by that value. A pie chart is a helpful visual where the size of each slice corresponds to its ratio share.

Direct proportion occurs when two quantities increase or decrease at the same rate, represented by the equation $y = kx$, where $k$ is the constant of proportionality. On a graph, direct proportion always appears as a straight line starting from the origin $(0,0)$ and extending upwards at a constant gradient.

Inverse proportion describes a relationship where as one variable increases, the other decreases such that their product remains constant, represented as $y = \frac{k}{x}$ or $xy = k$. The visual representation of this is a reciprocal curve (hyperbola) that gets closer to the $x$ and $y$ axes but never touches them.

Unit rates and compound units relate different types of measures, such as speed (distance/time), density (mass/volume), or pressure (force/area). These are often visualized using formula triangles, with the 'total' quantity at the top and the two rates/components at the bottom corners.

Percentage change is a way to express the difference between a new value and an old value as a percentage of the original. If you visualize the original amount as a $100\%$ bar, an increase is an additional segment added to the end, while a decrease is a portion removed from the original bar length.

Map scales and model scales use ratios (e.g., $1:n$) to represent real-world dimensions. In a $1:100$ scale, $1 cm$ on a drawing represents $100 cm$ in reality. This concept can be visualized by zooming into a shape; while the side lengths change according to the ratio, the shape's angles and proportions remain identical (similarity).