Y10 Higher Revision Vocabulary

Number Concepts and Operations

The study of Number at the Higher level involves a deep understanding of the properties of different types of numbers and the fluency to convert between them. For decimals to fractions, students must be able to convert both terminating decimals and complex recurring decimals into their representative fractions. For a recurring decimal, standard algebraic methods are used; for example, to convert $0.777...$ into a fraction, one would set $x = 0.777...$, then $10x = 7.777...$, so that $9x = 7$, resulting in $x = \frac{7}{9}$. The product of prime factors is another fundamental skill, where any composite number is broken down into a product of prime numbers using a factor tree. For example, the number $60$ can be expressed in index form as $2^2 \times 3 \times 5$. This methodology is essential for finding the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of sets of numbers.

Financial mathematics or money problems require the application of percentages and decimal calculations to real-world contexts, including profit, loss, and compound interest. Percentages are often calculated using a multiplier method, where an increase of $5\%$ is treated as multiplying by $1.05$. Standard form is utilized to represent very large or very small numbers in the format $A \times 10^n$, where 1 \le A < 10 and $n$ is an integer. Indices follow specific laws: for multiplication, indices are added ($a^m \times a^n = a^{m+n}$); for division, they are subtracted ($a^m \div a^n = a^{m-n}$); and for powers of powers, they are multiplied ($(a^m)^n = a^{mn}$). Negative indices represent reciprocals ($a^{-n} = \frac{1}{a^n}$), and fractional indices represent roots ($a^{\frac{1}{n}} = \sqrt[n]{a}$). Surds involve irrational roots that cannot be simplified to integers. A key requirement is rationalising the denominator, which involves removing the square root from the bottom of a fraction. To rationalise $\frac{k}{\sqrt{a}}$, one multiplies the numerator and denominator by $\sqrt{a}$. For more complex denominators like $a + \sqrt{b}$, one must multiply by the conjugate, $a - \sqrt{b}$, using the difference of two squares identity to clear the surd.

Algebra: Equations, Sequences, and Expressions

Algebraic proficiency is built upon the ability to manipulate expressions and solve equations. The equation of a line is typically written in the form $y = mx + c$, where $m$ is the gradient (calculated as $\frac{y_2 - y_1}{x_2 - x_1}$) and $c$ is the y-intercept. Parallel lines share the same gradient ($m_1 = m_2$), whereas perpendicular lines have gradients that are negative reciprocals of each other ($m_1 \times m_2 = -1$). Coordinate geometry problems may require students to find the midpoint of a line segment or its exact length using Pythagoras' theorem: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. Expanding brackets involves distributing terms; triple brackets, such as $(x+a)(x+b)(x+c)$, are expanded by first multiplying two brackets to form a quadratic and then multiplying that quadratic by the third bracket.

Algebraic fractions involve the same rules as numerical fractions: to add or subtract them, a common denominator must be found, often by multiplying the denominators. Simplifying algebraic fractions usually requires factoring the numerator and denominator first to cancel common terms. Solving linear sequences involves finding the nth term, expressed as $an + b$, where $a$ is the constant difference between terms. Simultaneous equations can be solved using substitution or elimination; Higher level problems often include one linear and one quadratic equation, potentially resulting in two sets of solutions. Forming expressions is the process of translating a worded problem into algebraic terms, while solving identities requires proving that two expressions are mathematically equivalent for all values ($\equiv$). Changing the subject of a formula involves isolating a specific variable through inverse operations, often requiring factoring if the subject appears more than once in the original formula.

Shape, Space, and Measures

Geometry at this level covers the properties of shapes and their measurements. Angles in quadrilaterals always sum to $360^\circ$, and complex problems may require applying properties of specific quadrilaterals like parallelograms or kites. Angle proofs necessitate rigorous logical steps and the citation of geometric rules, such as "angles on a straight line sum to $180^\circ$" or "alternate angles are equal." Volume calculations for prisms are found by multiplying the area of the cross-section by the length. For circles, the area is calculated using $\pi r^2$, where $r$ is the radius. Trigonometry in right-angled triangles uses the ratios sine, cosine, and tangent ($\text{SOH CAH TOA}$). Students are required to know exact values for specific angles: $\sin(30^\circ) = 0.5$, $\sin(45^\circ) = \frac{\sqrt{2}}{2}$, $\cos(60^\circ) = 0.5$, and $\tan(45^\circ) = 1$.

Transformations encompass reflection, rotation, translation, and enlargement. Translation is described by a vector $\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}$, showing horizontal and vertical displacement. Enlargements are defined by a scale factor and a center of enlargement. At the Higher level, this includes the transformation of trigonometric graphs. For a function $y = f(x)$, transformations such as $y = f(x) + a$ (vertical shift), $y = f(x - a)$ (horizontal shift), $y = a f(x)$ (vertical stretch), and $y = f(ax)$ (horizontal stretch) are applied to the sine, cosine, and tangent waves. Vectors are used to represent movement and to prove geometric results, such as whether lines are parallel or whether three points are collinear. Vector addition ($\mathbf{a} + \mathbf{b}$) and scalar multiplication ($k\mathbf{a}$) are standard operations in these proofs.

Ratio, Proportion, and Rates of Change

Proportionality deals with the relationship between variables. Direct proportion means that as one variable increases, the other increases at a constant rate ($y = kx$), while inverse proportion means that as one increases, the other decreases ($y = \frac{k}{x}$, or $y = kx^{-1}$). These relationships are often represented on graphs: direct proportion is a straight line through the origin, and inverse proportion is a reciprocal curve. Solving these problems requires finding the constant of proportionality, $k$, from given data points. Rates of change are also explored through distance-time and velocity-time graphs. In a distance-time graph, the gradient represents the speed. In a velocity-time graph, the gradient represents the acceleration ($\text{m/s}^2$), and the area under the graph represents the total distance traveled. Interpreting these graphs involves analyzing constant speed, acceleration, and deceleration (negative gradient).

Data, Statistics, and Probability

Data analysis involves summarizing and interpreting information. Averages, including the mean, median, and mode, can be calculated from simple lists or frequency tables. For a frequency table, the mean is calculated as $\frac{\sum f x}{\sum f}$, and for grouped data, the midpoint of the class interval is used as an estimate for $x$. Probability trees are used to model the outcomes of combined events; probabilities along any set of branches must sum to $1$, and probabilities are multiplied along a path to find the likelihood of a specific sequence of events occurring. Cumulative frequency graphs are plotted by calculating running totals of frequencies; these graphs allow for the estimation of the median, lower quartile, and upper quartile, which in turn are used to calculate the Interquartile Range (IQR).

Histograms are used for continuous, grouped data where the class widths are not necessarily equal. In a histogram, the area of the bar represents the frequency, and the y-axis represents frequency density, calculated as $\text{Frequency Density} = \frac{\text{Frequency}}{\text{Class Width}}$. Finally, scatter graphs are used to identify correlation between two variables. Correlation can be positive, negative, or non-existent. A line of best fit is drawn through the center of the data points to allow for interpolation (making predictions within the range of data) and extrapolation (making predictions outside the range of data), though the latter is often less reliable.