Mathematical Skills

Identifying Significance: The first significant figure is the first non-zero digit when reading from left to right. All digits following it are significant, including zeros that fall between non-zero digits.

Trailing Zeros: Zeros at the end of a decimal number are significant as they indicate the precision of the measurement, whereas leading zeros in a decimal (e.g., $0.005$) are merely placeholders.

Rounding Rules: When rounding to $n$ significant figures, look at the $(n+1)^{th}$ digit; if it is $5$ or greater, round up the $n^{th}$ digit. For large numbers, replace non-significant places with zeros to maintain the magnitude.

Calculation Consistency: In multi-step calculations, it is best practice to keep extra digits during intermediate steps and only round to the appropriate number of significant figures in the final answer.

Ratios: A mathematical comparison of two or more quantities, expressed as $a:b$. It describes how many times one value contains another or the relative size of parts within a whole.

Direct Proportion: A relationship where two variables increase or decrease at the same rate. If $y$ is directly proportional to $x$, then $y = kx$ for some constant $k$.

Inverse Proportion: A relationship where one variable increases as the other decreases. This is often seen in scenarios where a fixed resource is divided among a changing number of recipients.

Scale and Magnitude: Scale represents the ratio between a distance on a map or model and the corresponding distance in reality (e.g., $1:50,000$). Magnitude refers to the absolute or relative size of a quantity.

Percentiles: Values that divide a data set into 100 equal parts. The $n^{th}$ percentile is the value below which $n\%$ of the data falls.

Quartiles: Specific percentiles that divide data into four equal quarters. The first quartile ($Q_1$) is the $25^{th}$ percentile, the second ($Q_2$) is the median, and the third ($Q_3$) is the $75^{th}$ percentile.

Interquartile Range (IQR): The difference between the third and first quartiles ($Q_3 - Q_1$), representing the spread of the middle $50\%$ of the data.

Cumulative Frequency: A running total of frequencies that shows how many data points fall below a certain value, often used to estimate percentiles graphically.

Check Units: Always ensure that units are consistent before performing calculations. If a ratio involves different units (e.g., cm and km), convert them to the same unit first.

Show Working: Examiners often award marks for the correct method even if the final numerical answer is wrong due to a calculation error.

Sanity Checks: Evaluate if the answer makes sense in context. For example, a percentage decrease cannot exceed $100\%$ unless the value becomes negative.

Precision Matching: Ensure your final answer uses the same level of precision (significant figures) as the data provided in the question.