---
title: "Algebra"
description: "65 topics in Mathematics, in the order they build on each other."
canonical: https://lightmysky.com/learn/mathematics/areas/algebra
source: https://lightmysky.com/learn/mathematics/areas/algebra.md
retrieved: 2026-10-19
---

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# Algebra

65 topics in Mathematics, in the order they build on each other.

Page: https://lightmysky.com/learn/mathematics/areas/algebra

- [Linear number sequences](https://lightmysky.com/learn/mathematics/linear-number-sequences-mt_hBZwbst0ow): Generate and describe linear number sequences, including those with negative and decimal steps; identify the term-to-term rule
- [Systematic Listing](https://lightmysky.com/learn/mathematics/systematic-listing-mt_gu0NPDvlhY): Enumerate possibilities of combinations of two variables systematically (e.g. all ways to choose from a set of options)
- [Using Simple Formulae](https://lightmysky.com/learn/mathematics/using-simple-formulae-mt_Jvg8X0N5u0): Use simple formulae expressed in words or symbols to calculate values (e.g. perimeter = 2 × (length + width))
- [Writing Algebraic Equations](https://lightmysky.com/learn/mathematics/writing-algebraic-equations-mt_i9rJbuFO3p): Express missing number problems algebraically using letters for unknowns; translate word problems into equations
- [Equations with Two Unknowns](https://lightmysky.com/learn/mathematics/equations-with-two-unknowns-mt_YFS7JFk64p): Find pairs of numbers that satisfy an equation with two unknowns (e.g. find pairs (a, b) where a + b = 10 or 2a + b = 15)
- [Algebraic Notation](https://lightmysky.com/learn/mathematics/algebraic-notation-mt_KwdjWEmMNo): Use and interpret algebraic notation including: ab for a × b, 3y for y + y + y, a² for a × a, a/b for a ÷ b, coefficients as fractions, and brackets for grouping; read and write algebraic expressions fluently
- [Coordinates (age 11+)](https://lightmysky.com/learn/mathematics/coordinates-age-11-mt_hVpGOEz2kG): Plot and read coordinates in all four quadrants of the Cartesian plane, using positive and negative x- and y-values to describe positions precisely
- [Modelling Situations with Formulae and Graphs](https://lightmysky.com/learn/mathematics/modelling-situations-with-formulae-and-graphs-mt_z5iwdZyeDr): Model situations or procedures by translating them into algebraic expressions or formulae and by using graphs; move between word problems, algebraic representations, tables, and graphical representations
- [Substituting into Formulae](https://lightmysky.com/learn/mathematics/substituting-into-formulae-mt_vHzVa3SURC): Substitute numerical values into formulae and expressions including scientific formulae; evaluate expressions by replacing variables with given values and computing the result using correct order of operations
- [Generating Sequences](https://lightmysky.com/learn/mathematics/generating-sequences-mt_yXO7lQ9Yn7): Generate terms of a sequence from a term-to-term rule (e.g., 'add 3 each time') or a position-to-term rule (e.g., '2n + 1'), and identify whether a sequence is arithmetic, geometric, or neither
- [Laws of Exponents](https://lightmysky.com/learn/mathematics/laws-of-exponents-mt_UjNba32kkz): Apply the rules of integer exponents to write equivalent expressions: multiply powers by adding indices, divide by subtracting, raise a power to a power by multiplying, and interpret zero and negative exponents (3² × 3⁻⁵ = 3⁻³ = 1/27)
- [Linear Function Graphs](https://lightmysky.com/learn/mathematics/linear-function-graphs-mt_WBdHkc2HTf): Recognise that a linear function produces a straight-line graph, understand the relationship between an equation of the form y = mx + c and its graphical representation, and interpret gradient and y-intercept in context
- [Number Pattern Relationships](https://lightmysky.com/learn/mathematics/number-pattern-relationships-mt_fxPtngwUfz): Generate two numerical patterns using two given rules; identify relationships between corresponding terms; form ordered pairs and graph them on a coordinate plane
- [Expressions & Equations Vocabulary](https://lightmysky.com/learn/mathematics/expressions-and-equations-vocabulary-mt_g_fkAqmz72): Understand and use the concepts and vocabulary of expressions, equations, inequalities, terms, and factors; distinguish between an expression (no equals sign), an equation (equals sign), and an inequality (inequality sign)
- [Collecting Like Terms](https://lightmysky.com/learn/mathematics/collecting-like-terms-mt_nNNVrLqPW3): Simplify algebraic expressions by collecting like terms: combine terms with the same variable and power (e.g., 3a + 2b + 5a = 8a + 2b) while maintaining equivalence
- [Expanding Single Brackets](https://lightmysky.com/learn/mathematics/expanding-single-brackets-mt_FX4a2Q8XXN): Expand (multiply out) a single term over a bracket using the distributive property, e.g., 3(2x + 5) = 6x + 15; expand expressions involving negative multipliers
- [Solving linear equations](https://lightmysky.com/learn/mathematics/solving-linear-equations-mt_QhFEDyIwSO): Use algebraic methods to solve linear equations in one variable, including rearranging, expanding brackets, collecting terms on both sides, and working with rational coefficients.
- [Inequalities and their solution sets](https://lightmysky.com/learn/mathematics/inequalities-and-their-solution-sets-mt_i2F1nWxJjv): Understand inequalities as statements comparing expressions, represent solutions on a number line, and solve simple linear inequalities using the same inverse-operation methods as equations
- [Factorising Expressions](https://lightmysky.com/learn/mathematics/factorising-expressions-mt_uESbzWCZIq): Factorise algebraic expressions by taking out common factors — identify the highest common factor of all terms and write the expression as a product, e.g., 6x + 9 = 3(2x + 3)
- [Nth-Term Rules](https://lightmysky.com/learn/mathematics/nth-term-rules-mt_HRKzwEQJgO): Find the nth-term expression for an arithmetic sequence by identifying the common difference and the zero-term, and use it to determine any term in the sequence or test whether a given number belongs to the sequence
- [Simple formulae](https://lightmysky.com/learn/mathematics/simple-formulae-mt_3qrCtdoVAU): Understand and use standard mathematical formulae; rearrange formulae to change the subject, performing inverse operations to isolate a different variable
- [Plotting Linear Graphs](https://lightmysky.com/learn/mathematics/plotting-linear-graphs-mt_-3udyo6VyB): Plot linear graphs by generating a table of values, reduce a two-variable linear equation to the form y = mx + c, and calculate gradients from two points on a line
- [Expanding Double Brackets](https://lightmysky.com/learn/mathematics/expanding-double-brackets-mt_KhS7K1Mgrw): Expand products of two or more binomials, e.g., (x + 3)(x - 2) = x² + x - 6, using the grid method or FOIL; simplify the result by collecting like terms
- [Quadratic Graphs](https://lightmysky.com/learn/mathematics/quadratic-graphs-mt_GZuoYaDdWd): Recognise that quadratic functions produce curved (parabolic) graphs, distinguish them from linear graphs, and use plotted quadratic graphs to estimate values and find approximate solutions
- [Reading solutions from a graph](https://lightmysky.com/learn/mathematics/reading-solutions-from-a-graph-mt_Yw27nweoTj): Use graphs of linear and quadratic functions to estimate output values for given inputs, find approximate solutions to equations, and interpret graphical information in real-world contexts
- [Simultaneous equations](https://lightmysky.com/learn/mathematics/simultaneous-equations-mt_mqgu72aCMz): Understand that a simultaneous solution is the pair of values at which two linear graphs meet, and verify that pair by substituting into both equations.
- [Factorising Quadratics into Two Brackets](https://lightmysky.com/learn/mathematics/factorising-quadratics-into-two-brackets-mt_3QXgmWKfoA): Undo the expansion of two brackets for quadratics of the form x² + bx + c by finding the number pair that multiplies to c and adds to b. Check every answer by expanding it back.
- [Difference of Two Squares and Perfect Square Trinomials](https://lightmysky.com/learn/mathematics/difference-of-two-squares-and-perfect-square-trinomials-mt_GSpsHuserT): Recognise the two quadratics that factorise on sight: a² - b² splitting into (a + b)(a - b), and a perfect square trinomial folding back into one bracket squared.
- [Factorising quadratics with a leading coefficient](https://lightmysky.com/learn/mathematics/factorising-quadratics-with-a-leading-coefficient-mt_IdKGCzbH07): Factorise quadratics such as 6x² + 11x + 3, where the x² coefficient is not 1, by splitting the middle term and grouping.
- [Solving quadratic equations by factorising](https://lightmysky.com/learn/mathematics/solving-quadratic-equations-by-factorising-mt_tg_sqRzDjU): Rearrange a quadratic equation so one side is zero, factorise it, then use the fact that a product is zero only when one factor is zero to read off both solutions.
- [Simplifying Algebraic Fractions](https://lightmysky.com/learn/mathematics/simplifying-algebraic-fractions-mt_scDNSoz3eW): Factorise the top and the bottom of an algebraic fraction, cancel the common bracket, and state the values of x the fraction cannot take.
- [Completing the Square](https://lightmysky.com/learn/mathematics/completing-the-square-mt_rHmCfzB4Fm): Write a quadratic as (x + p)² + q by halving the x coefficient and correcting the constant, then solve it by taking the square root of both sides.
- [Function notation and evaluating functions](https://lightmysky.com/learn/mathematics/function-notation-and-evaluating-functions-mt_YWKWzHK_6Z): Read f(x) as a rule that turns each input into one output, evaluate a function at given values, and solve f(x) = k.
- [Composite functions](https://lightmysky.com/learn/mathematics/composite-functions-mt_jqGJHXgYNw): Apply one function to the output of another, writing fg(x) as f(g(x)), and keep the order of the two rules straight.
- [Geometric Sequences and Common Ratios](https://lightmysky.com/learn/mathematics/geometric-sequences-and-common-ratios-mt_NDnbJm2hTY): Recognise a sequence built by multiplying by the same number each time, find the common ratio and the nth term rule, and link it to growth and decay.
- [Inverse functions](https://lightmysky.com/learn/mathematics/inverse-functions-mt_7k7q5ZjMb4): Find the rule that undoes a function by swapping input and output and rearranging, and see the inverse graph as a reflection in the line y = x.
- [Quadratic Sequences and Second Differences](https://lightmysky.com/learn/mathematics/quadratic-sequences-and-second-differences-mt_gyZT3Tcw7U): Recognise a quadratic sequence from its constant second difference, and find the nth term rule in the form an² + bn + c.
- [Rearranging Formulae When the Subject Appears Twice](https://lightmysky.com/learn/mathematics/rearranging-formulae-when-the-subject-appears-twice-mt_Velmpnxfcj): Change the subject of a formula where the new subject shows up more than once: clear any fraction, collect those terms on one side, factorise the subject out, then divide.
- [Solving Linear Inequalities and Solution Sets](https://lightmysky.com/learn/mathematics/solving-linear-inequalities-and-solution-sets-mt_QkG3Cw5VgM): Solve inequalities in one variable with the balancing steps used for equations, flip the sign when multiplying or dividing by a negative, and show the answer on a number line.
- [Graphical Inequalities and Regions in the Plane](https://lightmysky.com/learn/mathematics/graphical-inequalities-and-regions-in-the-plane-mt_sKerzcAcNy): A linear inequality in two variables picks out a half-plane, and several of them together leave a region whose corners answer the question being asked. Drawing the boundary as solid or dashed is what records whether the boundary itself counts.
- [Solving Quadratic Inequalities](https://lightmysky.com/learn/mathematics/solving-quadratic-inequalities-mt_dukJtB_0i8): Solve inequalities such as x² - x - 6 > 0 by finding the roots, sketching the parabola, and reading off the x values where the curve is on the required side of the axis.
- [The Quadratic Formula](https://lightmysky.com/learn/mathematics/the-quadratic-formula-mt_C-4brnRYHY): Solve any quadratic with x = (-b ± √(b² - 4ac))/(2a), including ones that do not factorise, giving answers exactly or to a stated accuracy.
- [The Discriminant and the Number of Roots](https://lightmysky.com/learn/mathematics/the-discriminant-and-the-number-of-roots-mt_uj95V0aLVS): Use b² - 4ac to decide whether a quadratic has two roots, one repeated root or none, without solving it, and link that count to the graph crossing the x-axis.
- [Sketching Quadratic Graphs from Roots and the Turning Point](https://lightmysky.com/learn/mathematics/sketching-quadratic-graphs-from-roots-and-the-turning-point-mt_RcrJIbbN2x): Sketch a parabola from its roots, y-intercept, line of symmetry and turning point, reading the turning point off the completed square form.
- [Simultaneous Equations with One Quadratic](https://lightmysky.com/learn/mathematics/simultaneous-equations-with-one-quadratic-mt_D89ql2sJvc): Solve a linear and a quadratic equation together by substituting the linear one into the quadratic, and read the answers as the points where a line meets a curve.
- [Transformations of Graphs](https://lightmysky.com/learn/mathematics/transformations-of-graphs-mt_691_c-7Z5M): Describe and apply the effect of f(x) + a, f(x + a), -f(x) and af(x) on a graph, and give the coordinates of a marked point after the change.
- [Recognising cubic, reciprocal and exponential graphs](https://lightmysky.com/learn/mathematics/recognising-cubic-reciprocal-and-exponential-graphs-mt_2IJJ51rmq9): Describe the shapes and key features of y = x³, y = 1/x and y = kˣ, calculate their values, and match fixed-multiplier situations to exponential models.
- [Domain and Range of a Function](https://lightmysky.com/learn/mathematics/domain-and-range-of-a-function-mt_b_4zoHz8Jc): Describe the set of inputs a function accepts and the set of outputs it can return, and say what forces a restriction: a division, a square root, or the situation being modelled.
- [Modulus Functions and Modulus Equations](https://lightmysky.com/learn/mathematics/modulus-functions-and-modulus-equations-mt__W-aDpT5jz): Read the modulus as distance from zero, sketch the V-shaped graph it produces, and solve equations and inequalities that contain one.
- [Polynomial Division and the Factor Theorem](https://lightmysky.com/learn/mathematics/polynomial-division-and-the-factor-theorem-mt_haL2UkGa5C): Divide a polynomial by a linear expression and read the remainder, then use the fact that a root gives a factor to break a cubic apart.
- [Sketching a Curve from Its Factorised Form](https://lightmysky.com/learn/mathematics/sketching-a-curve-from-its-factorised-form-mt_x4tZLdH8fh): Turn a factorised polynomial into a sketch: roots give the crossings, the constant term gives the y-intercept, the highest power decides what the ends do, and a repeated factor touches instead of crossing.
- [The Exponential Function and the Number e](https://lightmysky.com/learn/mathematics/the-exponential-function-and-the-number-e-mt_2hyjN4JfJJ): Treat y = aˣ as a family of curves: always positive, multiplied by a fixed factor per unit step, with a horizontal asymptote. One member, y = eˣ, has a gradient equal to its own value everywhere.
- [Logarithms as the Inverse of Exponentials](https://lightmysky.com/learn/mathematics/logarithms-as-the-inverse-of-exponentials-mt_skLQLUan0T): Read logₐ x as the answer to the question 'what power of a gives x'. The log graph is the exponential graph reflected in y = x, and each function undoes the other.
- [Arithmetic Series and Sigma Notation](https://lightmysky.com/learn/mathematics/arithmetic-series-and-sigma-notation-mt_bAIPng87JO): Add the terms of a linear sequence with the pairing argument behind Sₙ = n(a + l)/2, and read and write sums in sigma notation.
- [Geometric Series and the Sum to Infinity](https://lightmysky.com/learn/mathematics/geometric-series-and-the-sum-to-infinity-mt_HHxFOv6vXS): Sum a geometric sequence with the standard formula, and see that when the common ratio is between -1 and 1 the running total settles on a finite limit however many terms are added.
- [The Binomial Expansion](https://lightmysky.com/learn/mathematics/the-binomial-expansion-mt_z16HZkNZ8c): Expand (a + b)ⁿ with coefficients from Pascal's triangle or the nCr key, pick out a single term without writing the rest, and use the first few terms as an approximation when one term is small.
- [Hyperbolic Functions and Their Inverses](https://lightmysky.com/learn/mathematics/hyperbolic-functions-and-their-inverses-mt_gCUl1sWi9M): Build sinh and cosh from the exponential function, prove the identity that replaces the Pythagorean one, and write the inverses as logarithms.
- [Rational Functions, Asymptotes and Curve Sketching](https://lightmysky.com/learn/mathematics/rational-functions-asymptotes-and-curve-sketching-mt_U-f9YTNb2o): Sketch a quotient of polynomials by finding its zeros, its vertical asymptotes and what it does far from the origin.
- [The Imaginary Unit and Complex Arithmetic](https://lightmysky.com/learn/mathematics/the-imaginary-unit-and-complex-arithmetic-mt_Faemafn550): Quadratics with no real roots stop being dead ends: i is defined so that i squared is -1, and numbers of the form a + bi add, multiply, conjugate and divide by the ordinary rules of algebra.
- [The Argand Plane, Modulus and Argument](https://lightmysky.com/learn/mathematics/the-argand-plane-modulus-and-argument-mt_Wq01c5UN9e): A complex number is a point: real part across, imaginary part up. Its distance from the origin is the modulus and its angle is the argument, so multiplying becomes scaling and turning.
- [Euler's Formula and the Exponential Form](https://lightmysky.com/learn/mathematics/eulers-formula-and-the-exponential-form-mt_RX2IxsnSwG): e to the i theta traces the unit circle, tying the exponential function to sine and cosine. In exponential form, multiplying complex numbers means adding angles, and De Moivre's theorem falls out for free.
- [Roots of Polynomials and Roots of Unity](https://lightmysky.com/learn/mathematics/roots-of-polynomials-and-roots-of-unity-mt_XAISygqdQr): Every polynomial of degree n has n complex roots, with non-real roots of real polynomials arriving in conjugate pairs, and the nth roots of unity sit evenly spaced around the unit circle.
- [The Laws of Logarithms](https://lightmysky.com/learn/mathematics/the-laws-of-logarithms-mt_5vHdWlxaqY): Turn products into sums, quotients into differences and powers into multipliers, and combine several logarithms into one before solving or simplifying.
- [Solving Exponential Equations with Logarithms](https://lightmysky.com/learn/mathematics/solving-exponential-equations-with-logarithms-mt_DaxxpSbDOn): Take logarithms of both sides to bring an unknown down from an exponent, and solve equations that are quadratic in eˣ by substitution.
- [Straightening Data with Logarithmic Graphs](https://lightmysky.com/learn/mathematics/straightening-data-with-logarithmic-graphs-mt_TVar2jxRMI): Take logarithms of a model to make it linear: y = axⁿ gives a straight line for log y against log x, and y = abˣ gives one for log y against x. The gradient and intercept then return the constants.