--- title: "Mathematics" description: "860 topics for ages 4 to 24, in the order they build on each other." canonical: https://lightmysky.com/learn/mathematics source: https://lightmysky.com/learn/mathematics.md retrieved: 2026-10-19 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Mathematics 860 topics for ages 4 to 24, in the order they build on each other. Page: https://lightmysky.com/learn/mathematics ## Counting & Cardinality 14 topics, ages 4 to 9. https://lightmysky.com/learn/mathematics/areas/counting-and-cardinality - [One-to-one counting](https://lightmysky.com/learn/mathematics/one-to-one-counting-mt_WcfaSfVT33): One-to-one correspondence when counting objects: each object is paired with exactly one number name - [How Many in Total?](https://lightmysky.com/learn/mathematics/how-many-in-total-mt_dmNvjroCPT): Cardinality principle: the last number said when counting a set tells how many objects are in the set, regardless of arrangement or order counted - [One More Each Time](https://lightmysky.com/learn/mathematics/one-more-each-time-mt_sYpKWbq5ra): Each successive counting number represents a quantity that is one larger than the previous number - [Showing numbers many ways](https://lightmysky.com/learn/mathematics/showing-numbers-many-ways-mt_pAcaehday5): Represent numbers using objects, pictorial representations, and the number line - [Counting objects to 20](https://lightmysky.com/learn/mathematics/counting-objects-to-20-mt_yqAL6O5i_v): Count up to 20 objects in structured arrangements, count up to 10 scattered objects, and count out a requested amount from 1 to 20. - [Comparing groups: more or fewer](https://lightmysky.com/learn/mathematics/comparing-groups-more-or-fewer-mt__h7hvT4tEb): Compare two groups of objects to determine which has more, fewer, or whether they are equal, using matching and counting strategies - [Rote counting to 100](https://lightmysky.com/learn/mathematics/rote-counting-to-100-mt_M5PPDJStGm): Rote count forwards and backwards from 0 to 100, beginning from 0, 1, or any given number, by ones - [Counting in 2s, 5s and 10s](https://lightmysky.com/learn/mathematics/counting-in-2s-5s-and-10s-mt_nvdpxAJTBG): Count in multiples of 2, 5, and 10 (skip counting) - [Two written numerals between 1 and 10](https://lightmysky.com/learn/mathematics/two-written-numerals-between-1-and-10-mt__YRJ23GuIK): Compare two written numerals between 1 and 10 to determine which is greater or less - [Counting forwards and backwards](https://lightmysky.com/learn/mathematics/counting-forwards-and-backwards-mt_aHAM29nidj): Count forwards and backwards in steps of 3 from 0 - [Skip Counting (4s, 8s, 50s, 100s)](https://lightmysky.com/learn/mathematics/skip-counting-4s-8s-50s-100s-mt_IzQvs7k_sE): Count from 0 in multiples of 4, 8, 50, and 100 - [Counting in 6s](https://lightmysky.com/learn/mathematics/counting-in-6s-mt_7rJM8eWUfw): Count in multiples of 6, 7, 9, 25, and 1000 - [Counting forwards and backwards (age 6+)](https://lightmysky.com/learn/mathematics/counting-forwards-and-backwards-age-6-mt_OkSJfrmFb_): Count forwards and backwards in tens from any number (not just multiples of 10) - [Counting Within 1,000](https://lightmysky.com/learn/mathematics/counting-within-1-000-mt_M2v1A9OEuM): Count within 1000, including skip-counting by 5s, 10s, and 100s ## Addition & Subtraction 53 topics, ages 4 to 13. https://lightmysky.com/learn/mathematics/areas/addition-and-subtraction - [Addition as combining or putting together two groups](https://lightmysky.com/learn/mathematics/addition-as-combining-or-putting-together-two-groups-mt_OvyoRo47K-): Understand addition as combining or putting together two groups to find the total - [Numbers up to 10 into pairs](https://lightmysky.com/learn/mathematics/numbers-up-to-10-into-pairs-mt_7XcCG43ZZW): Decompose numbers up to 10 into pairs in more than one way (part-part-whole) - [Number bonds to 10](https://lightmysky.com/learn/mathematics/number-bonds-to-10-mt_e8CZ7E5qW7): Find the number that makes 10 when added to a given number from 1 to 9 (number bonds to 10) - [Subtraction as taking away or separating](https://lightmysky.com/learn/mathematics/subtraction-as-taking-away-or-separating-mt_zuKAX6lcYR): Understand subtraction as taking away or separating from a group to find how many remain - [Representing Addition and Subtraction](https://lightmysky.com/learn/mathematics/representing-addition-and-subtraction-mt_PgsHGYJMH-): Represent addition and subtraction using objects, drawings, and mental images - [Addition and subtraction word problems](https://lightmysky.com/learn/mathematics/addition-and-subtraction-word-problems-mt_yJmvUCCym7): Solve addition and subtraction word problems within 10 using objects or drawings - [Fluent adding and subtracting within 5](https://lightmysky.com/learn/mathematics/fluent-adding-and-subtracting-within-5-mt_ghF3Vv6taM): Fluently add and subtract within 5 - [Number bonds](https://lightmysky.com/learn/mathematics/number-bonds-mt_s2mfRBoTal): Recall number bonds (addition and related subtraction facts) within 20 - [Adding and subtracting](https://lightmysky.com/learn/mathematics/adding-and-subtracting-mt_mr_Vk7FGzK): Add and subtract one-digit and two-digit numbers to 20, including zero - [Early Word Problems](https://lightmysky.com/learn/mathematics/early-word-problems-mt_VAWV_l7J0D): Solve one-step word problems involving addition and subtraction to 20, including missing-number problems - [Addition and subtraction strategies](https://lightmysky.com/learn/mathematics/addition-and-subtraction-strategies-mt_PpWSHA-0kv): Use counting on and counting back as strategies for addition and subtraction - [Addition in any order](https://lightmysky.com/learn/mathematics/addition-in-any-order-mt_QCgbiVrwnp): Understand and apply the commutative property of addition: addends can be added in any order - [Finding a missing number in addition](https://lightmysky.com/learn/mathematics/finding-a-missing-number-in-addition-mt_ezc2m_0dzN): Understand subtraction as finding an unknown addend (e.g. 10 − 8 = ? is the same as 8 + ? = 10) - [Fluent adding and subtracting within 10](https://lightmysky.com/learn/mathematics/fluent-adding-and-subtracting-within-10-mt__we2TDqnJx): Fluently add and subtract within 10 - [Addition and subtraction within 20](https://lightmysky.com/learn/mathematics/addition-and-subtraction-within-20-mt_m1W6nTQJ2b): Add and subtract within 20 using strategies such as making ten, decomposing a number leading to ten, and using known facts - [Grouping numbers to add](https://lightmysky.com/learn/mathematics/grouping-numbers-to-add-mt_T76hKqXf0z): Understand and apply the associative property of addition: when adding three numbers, any two can be added first - [Adding Three Small Numbers](https://lightmysky.com/learn/mathematics/adding-three-small-numbers-mt_Zx1xZM-RbX): Add three one-digit numbers using strategies including looking for pairs that make 10 - [Inverse: addition undoes subtraction](https://lightmysky.com/learn/mathematics/inverse-addition-undoes-subtraction-mt_ehGS_uVSJv): Recognise and use the inverse relationship between addition and subtraction to check calculations and solve missing-number problems - [Fluent adding and subtracting within 20](https://lightmysky.com/learn/mathematics/fluent-adding-and-subtracting-within-20-mt__t4afSyZRm): Fluently add and subtract within 20 using mental strategies and recall single-digit addition facts. - [Reading +, −, and = symbols](https://lightmysky.com/learn/mathematics/reading-and-symbols-mt_8RmpkDxT9L): Read, write, and interpret the symbols +, −, and = in number sentences - [What the equals sign means](https://lightmysky.com/learn/mathematics/what-the-equals-sign-means-mt_oIzycTBeE4): Understand the meaning of the equal sign as 'is the same as' and determine if equations are true or false - [Unknown in Addition & Subtraction](https://lightmysky.com/learn/mathematics/unknown-in-addition-and-subtraction-mt_3JgrHY221M): Determine the unknown whole number in an addition or subtraction equation relating three whole numbers - [Subtracting multiples of 10](https://lightmysky.com/learn/mathematics/subtracting-multiples-of-10-mt_HJTuIGHvcR): Subtract multiples of 10 (10–90) from multiples of 10 using place value strategies - [Adding within 100](https://lightmysky.com/learn/mathematics/adding-within-100-mt_glPPG-kTQY): Add within 100 using strategies based on place value, including adding a two-digit and one-digit number, and a two-digit and a multiple of 10 - [Fluent addition and subtraction](https://lightmysky.com/learn/mathematics/fluent-addition-and-subtraction-mt_3e_PQxwC12): Recall and use addition and subtraction facts to 20 fluently, and derive and use related facts up to 100 - [Mental addition and subtraction (age 6+)](https://lightmysky.com/learn/mathematics/mental-addition-and-subtraction-age-6-mt_AKAtWEwpcj): Add and subtract a two-digit number and ones mentally and using concrete/pictorial representations - [Fluent adding and subtracting within 100](https://lightmysky.com/learn/mathematics/fluent-adding-and-subtracting-within-100-mt_cChv2j_-Da): Fluently add and subtract within 100 using strategies based on place value, properties of operations, and the relationship between addition and subtraction - [Adding several two-digit numbers](https://lightmysky.com/learn/mathematics/adding-several-two-digit-numbers-mt_TiQbi027PE): Add up to four two-digit numbers using strategies based on place value and properties of operations - [Two-Step Word Problems](https://lightmysky.com/learn/mathematics/two-step-word-problems-mt_yTWxkzzoOZ): Solve one- and two-step word problems within 100 using addition and subtraction, with unknowns in all positions - [Adding and subtracting tens mentally](https://lightmysky.com/learn/mathematics/adding-and-subtracting-tens-mentally-mt_I5j1ZWo2cn): Add and subtract a two-digit number and tens mentally and using concrete/pictorial representations - [Adding and subtracting two two-digit numbers](https://lightmysky.com/learn/mathematics/adding-and-subtracting-two-two-digit-numbers-mt_UQnAFPs83F): Add and subtract two two-digit numbers using concrete objects, pictorial representations, and mental methods - [Mental and written addition and subtraction](https://lightmysky.com/learn/mathematics/mental-and-written-addition-and-subtraction-mt_wzAZ8qFDc4): Solve addition and subtraction problems using mental and written methods, including problems involving numbers, quantities, and measures - [Addition and subtraction strategies (age 7+)](https://lightmysky.com/learn/mathematics/addition-and-subtraction-strategies-age-7-mt_zxST3MarI9): Explain why addition and subtraction strategies work, using place value and the properties of operations - [Addition and subtraction within 1000](https://lightmysky.com/learn/mathematics/addition-and-subtraction-within-1000-mt_ewmuMMPAzP): Add and subtract within 1000 using concrete models, drawings, and strategies based on place value; understand composing and decomposing tens and hundreds - [Adding and subtracting (age 7+)](https://lightmysky.com/learn/mathematics/adding-and-subtracting-age-7-mt_SrrsLiJkr3): Add and subtract numbers with up to three digits using formal written methods of columnar addition and subtraction - [Mentally adding and subtracting ones with 3-digit numbers](https://lightmysky.com/learn/mathematics/mentally-adding-and-subtracting-ones-with-3-digit-numbers-mt_oDU_8zMZjp): Mentally add and subtract a three-digit number and ones - [Missing number problems (age 7+)](https://lightmysky.com/learn/mathematics/missing-number-problems-age-7-mt_XuHmIn2xje): Solve addition and subtraction problems including missing-number problems, using number facts, place value, and more complex methods - [Fluent adding and subtracting within 1000](https://lightmysky.com/learn/mathematics/fluent-adding-and-subtracting-within-1000-mt_HFRYjTb-Z5): Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and the relationship between addition and subtraction - [Numbers on a number line](https://lightmysky.com/learn/mathematics/numbers-on-a-number-line-mt_19j_5AuuQI): Represent whole numbers as lengths on a number line and represent sums and differences within 100 on a number line diagram - [Mentally adding hundreds to 3-digit numbers](https://lightmysky.com/learn/mathematics/mentally-adding-hundreds-to-3-digit-numbers-mt_xPqczp7zPX): Mentally add and subtract a three-digit number and hundreds - [Mentally adding tens to 3-digit numbers](https://lightmysky.com/learn/mathematics/mentally-adding-tens-to-3-digit-numbers-mt_aHQ9kNt3is): Mentally add and subtract a three-digit number and tens - [Adding and subtracting (age 8+)](https://lightmysky.com/learn/mathematics/adding-and-subtracting-age-8-mt_mpS-JK_p_m): Add and subtract numbers with up to four digits using formal written methods of columnar addition and subtraction - [Two-step addition and subtraction problems](https://lightmysky.com/learn/mathematics/two-step-addition-and-subtraction-problems-mt_CDa5AVakLE): Solve addition and subtraction two-step problems in contexts, deciding which operations and methods to use and why - [Fluent addition and subtraction (age 9+)](https://lightmysky.com/learn/mathematics/fluent-addition-and-subtraction-age-9-mt_Yw1_4Nfsql): Fluently add and subtract whole numbers with more than four digits using the standard columnar algorithm - [Estimating by rounding](https://lightmysky.com/learn/mathematics/estimating-by-rounding-mt_QaYfeVL-0C): Estimate the answer to a calculation and use inverse operations to check answers; apply to increasingly large numbers using rounding and inverse reasoning - [Two-Step Equations](https://lightmysky.com/learn/mathematics/two-step-equations-mt_anAe11HAEH): Solve two-step word problems using the four operations; represent problems using equations with a letter standing for the unknown quantity - [Mental addition and subtraction (age 9+)](https://lightmysky.com/learn/mathematics/mental-addition-and-subtraction-age-9-mt_MWXPiaTnEu): Add and subtract numbers mentally with increasingly large numbers, using place-value knowledge and derived facts - [Adding and subtracting (age 9+)](https://lightmysky.com/learn/mathematics/adding-and-subtracting-age-9-mt_-F_Lv_apzH): Solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why - [Checking Answers by Rounding](https://lightmysky.com/learn/mathematics/checking-answers-by-rounding-mt_G3sVFQNCme): Use rounding to check answers to calculations and determine appropriate levels of accuracy in context - [Addition and subtraction strategies (age 10+)](https://lightmysky.com/learn/mathematics/addition-and-subtraction-strategies-age-10-mt_JH_6RpNWjr): Add and subtract decimals to hundredths using strategies based on place value, properties of operations, and the relationship between addition and subtraction; relate strategies to written methods and explain reasoning - [Adding and subtracting (age 10+)](https://lightmysky.com/learn/mathematics/adding-and-subtracting-age-10-mt_M1tnXqmYbn): Solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why, with numbers up to 10,000,000 and decimals - [Positive and Negative Numbers](https://lightmysky.com/learn/mathematics/positive-and-negative-numbers-mt_9QzSnn8m80): Understand positive and negative numbers as describing quantities with opposite directions or values; use them in context such as temperature, floors in a building, and bank balances - [Adding and Subtracting Negative Numbers](https://lightmysky.com/learn/mathematics/adding-and-subtracting-negative-numbers-mt_qX7X7uMzsY): Add and subtract with negative integers, reading each move as a direction on the number line, and settle what subtracting a negative does. ## Geometry 110 topics, ages 4 to 21. https://lightmysky.com/learn/mathematics/areas/geometry - [2-D shapes](https://lightmysky.com/learn/mathematics/2-d-shapes-mt_KJeEeTutJI): Recognise and name common 2-D shapes (circles, triangles, rectangles including squares) - [3-D shapes](https://lightmysky.com/learn/mathematics/3-d-shapes-mt_Qcp2d_kuta): Recognise and name common 3-D shapes (cubes, cuboids, pyramids, spheres, cylinders, cones) - [Positional Language](https://lightmysky.com/learn/mathematics/positional-language-mt_qeZYF6HZ4o): Describe the position of objects using terms such as above, below, beside, in front of, behind, next to - [Comparing 2-D and 3-D shapes](https://lightmysky.com/learn/mathematics/comparing-2-d-and-3-d-shapes-mt_yGv8doDAmp): Analyse and compare 2-D and 3-D shapes using informal language to describe sides, vertices, and other attributes - [Building & Drawing Shapes](https://lightmysky.com/learn/mathematics/building-and-drawing-shapes-mt_2VR963szuk): Model shapes by building from components (e.g. sticks and clay balls) and by drawing - [Combining Simple Shapes](https://lightmysky.com/learn/mathematics/combining-simple-shapes-mt_XjwUlmxdCT): Compose simple shapes to form larger shapes (e.g. two triangles make a rectangle) - [Flat vs Solid Shapes](https://lightmysky.com/learn/mathematics/flat-vs-solid-shapes-mt_mnEVZNkX3p): Distinguish two-dimensional (flat) shapes from three-dimensional (solid) shapes - [2-D faces on 3-D shapes](https://lightmysky.com/learn/mathematics/2-d-faces-on-3-d-shapes-mt_UooUHC_V7U): Identify 2-D shapes on the surface of 3-D shapes (e.g. a circle on a cylinder, a triangle on a pyramid) - [Building with 3-D Shapes](https://lightmysky.com/learn/mathematics/building-with-3-d-shapes-mt_QNxFnxikCN): Compose three-dimensional shapes (cubes, right rectangular prisms, right circular cones, right circular cylinders) and create composite shapes; build new shapes from component shapes - [Composing Shapes](https://lightmysky.com/learn/mathematics/composing-shapes-mt_MqqR7VUoz1): Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, quarter-circles) to create composite shapes, and compose new shapes from the composite shape - [Defining attributes of shapes](https://lightmysky.com/learn/mathematics/defining-attributes-of-shapes-mt_bfhng6mOuy): Distinguish defining attributes of shapes (e.g. triangles are closed and three-sided) from non-defining attributes (e.g. colour, orientation, overall size) - [2-D shapes (age 6+)](https://lightmysky.com/learn/mathematics/2-d-shapes-age-6-mt_sBcRdUfAzV): Identify and describe properties of 2-D shapes including the number of sides and line symmetry in a vertical line - [Edges, vertices, and faces](https://lightmysky.com/learn/mathematics/edges-vertices-and-faces-mt_Mnodea7mG_): Identify and describe properties of 3-D shapes including the number of edges, vertices, and faces - [Patterns & Sequences](https://lightmysky.com/learn/mathematics/patterns-and-sequences-mt_WXW0hjNhph): Order and arrange combinations of mathematical objects in patterns and sequences - [Sorting 2-D and 3-D shapes](https://lightmysky.com/learn/mathematics/sorting-2-d-and-3-d-shapes-mt_fEuoBYw6bU): Compare and sort common 2-D and 3-D shapes and everyday objects by their properties - [Drawing 2-D shapes and making 3-D shapes](https://lightmysky.com/learn/mathematics/drawing-2-d-shapes-and-making-3-d-shapes-mt_vJUa62bxeR): Draw 2-D shapes and make 3-D shapes using modelling materials; recognise 3-D shapes in different orientations and describe them - [Drawing shapes with given attributes](https://lightmysky.com/learn/mathematics/drawing-shapes-with-given-attributes-mt_zuOGOGFAKb): Recognise and draw shapes having specified attributes (e.g. a given number of angles or equal faces); identify triangles, quadrilaterals, pentagons, hexagons, and cubes - [Classifying quadrilaterals](https://lightmysky.com/learn/mathematics/classifying-quadrilaterals-mt_Xt1cRqaBOW): Understand that shapes in different categories may share attributes defining a larger category; classify quadrilaterals (rhombuses, rectangles, squares) and draw examples of quadrilaterals not in those subcategories - [Lines of symmetry in 2-D shapes](https://lightmysky.com/learn/mathematics/lines-of-symmetry-in-2-d-shapes-mt_hRZyKvz1KN): Identify lines of symmetry in 2-D shapes presented in different orientations; recognise line-symmetric figures and draw lines of symmetry - [Completing symmetric figures](https://lightmysky.com/learn/mathematics/completing-symmetric-figures-mt_hR2Y7NhMSY): Complete a simple symmetric figure with respect to a specific line of symmetry - [Nets of 3-D Shapes](https://lightmysky.com/learn/mathematics/nets-of-3-d-shapes-mt_vnJEztczji): Identify, draw, and interpret nets of common 3-D shapes — cubes, cuboids, triangular prisms, and square-based pyramids — by predicting which 3-D shape a given flat arrangement of faces will fold into, checking whether a net will close completely, and sketching a net from a description or 3-D model; understand the relationship between the number of faces and the structure of the net - [3-D shapes (age 9+)](https://lightmysky.com/learn/mathematics/3-d-shapes-age-9-mt_wUSbRt3-qw): Identify 3-D shapes, including cubes and other cuboids, from 2-D representations - [3-D shapes (age 10+)](https://lightmysky.com/learn/mathematics/3-d-shapes-age-10-mt_REwgr0d_ss): Recognise, describe, and build simple 3-D shapes, including making nets - [3-D shapes (age 11+)](https://lightmysky.com/learn/mathematics/3-d-shapes-age-11-mt_nRksLqt-iR): Use the properties of faces, surfaces, edges, and vertices of 3-D shapes (cubes, cuboids, prisms, cylinders, pyramids, cones, and spheres) to solve problems, including visualising cross-sections - [Constructions and Loci](https://lightmysky.com/learn/mathematics/constructions-and-loci-mt_yqoXwrKaW3): Construct perpendicular bisectors, angle bisectors and perpendiculars with compasses, and shade the region that fits a set of distance conditions. - [Turns & Directions](https://lightmysky.com/learn/mathematics/turns-and-directions-mt_Y9XKzLrUAZ): Describe movement and direction, including whole, half, quarter, and three-quarter turns - [Position, direction, and movement](https://lightmysky.com/learn/mathematics/position-direction-and-movement-mt_TdV9YGJEoY): Use mathematical vocabulary to describe position, direction, and movement, including straight lines and distinguishing rotation as a turn in terms of right angles (quarter, half, three-quarter turns, clockwise and anti-clockwise) - [Understanding angles](https://lightmysky.com/learn/mathematics/understanding-angles-mt_8OAGVdeTJ_): Recognise angles as a property of shape or a description of a turn - [Right Angles & Turns](https://lightmysky.com/learn/mathematics/right-angles-and-turns-mt_MFfYcnv6Tv): Identify right angles; recognise that two right angles make a half-turn, three make three-quarters, and four make a complete turn - [Parallel and perpendicular lines](https://lightmysky.com/learn/mathematics/parallel-and-perpendicular-lines-mt_u23IGDxOpk): Identify horizontal and vertical lines and pairs of perpendicular and parallel lines - [First Quadrant Coordinates](https://lightmysky.com/learn/mathematics/first-quadrant-coordinates-mt_jBQS-CicNn): Describe positions on a 2-D grid as coordinates in the first quadrant - [Coordinates (age 8+)](https://lightmysky.com/learn/mathematics/coordinates-age-8-mt_i5_HnoFOYw): Plot specified points on a coordinate grid and draw sides to complete a given polygon - [Describing Movements](https://lightmysky.com/learn/mathematics/describing-movements-mt_Uq5vYqboCR): Describe movements between positions as translations of a given unit to the left/right and up/down - [Types of angles](https://lightmysky.com/learn/mathematics/types-of-angles-mt_h0CVtqI2xo): Identify acute and obtuse angles; compare and order angles up to two right angles by size - [Lines, Rays & Angles](https://lightmysky.com/learn/mathematics/lines-rays-and-angles-mt_QysgF57dxh): Draw and identify points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines; identify these in two-dimensional figures - [Classifying shapes by line properties](https://lightmysky.com/learn/mathematics/classifying-shapes-by-line-properties-mt_r8c43QB6wx): Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or angles of a specified size; recognise right triangles as a category - [Transformations on a grid](https://lightmysky.com/learn/mathematics/transformations-on-a-grid-mt_SH7QgFl8-v): Represent and carry out geometric transformations on squared paper or a coordinate grid: reflections (in horizontal, vertical, and diagonal mirror lines, including the axes), translations (described as a vector or as left/right/up/down moves), and rotations (90° or 180° about a stated centre point); describe each transformation precisely using the correct language; identify which transformation maps one shape onto its image by comparing position, orientation, and size - [Reflection and Translation on a Grid](https://lightmysky.com/learn/mathematics/reflection-and-translation-on-a-grid-mt_efOeaGFyGM): Identify, describe, and represent the position of a shape following a reflection or translation using appropriate language; know that the shape has not changed - [What Is an Angle?](https://lightmysky.com/learn/mathematics/what-is-an-angle-mt_3S10OOGPqu): Understand that an angle is a geometric shape formed by two rays sharing a common endpoint (vertex); recognise angles in real-life contexts and 2-D shapes - [Degrees and turns](https://lightmysky.com/learn/mathematics/degrees-and-turns-mt_qUGMyMYn9m): Know that angles are measured in degrees, where one degree is 1/360 of a full turn; understand that an angle turning through n one-degree angles has a measure of n degrees - [Estimating Angles](https://lightmysky.com/learn/mathematics/estimating-angles-mt_YUJ5pwalqL): Estimate and compare acute, obtuse, and reflex angles in degrees; classify angles by type and order them by size - [Geometric diagram conventions](https://lightmysky.com/learn/mathematics/geometric-diagram-conventions-mt_e4V6hvcuEJ): Use and interpret standard geometric diagram conventions: mark right angles with a small square, equal lengths with single or double tick marks, and equal angles with arc marks; label angles in three-letter notation (∠ABC) and individual angles with a single letter or number; draw diagrams showing angles at a point, angles on a straight line, and angles inside polygons with these conventions; read diagrams with these marks to identify given information and find unknown values - [Angle Sum Rules](https://lightmysky.com/learn/mathematics/angle-sum-rules-mt_oqziWKry-L): Know that angles at a point sum to 360° (one whole turn), angles on a straight line sum to 180°, and vertically opposite angles are equal; use these facts to find missing angles - [Measuring angles](https://lightmysky.com/learn/mathematics/measuring-angles-mt_4MFUAsbx_6): Measure angles in whole-number degrees using a protractor; draw given angles and sketch angles of specified measure - [Regular and irregular polygons](https://lightmysky.com/learn/mathematics/regular-and-irregular-polygons-mt_8H2kO4k2B9): Distinguish between regular and irregular polygons based on reasoning about equal sides and equal angles - [2-D shapes (age 10+)](https://lightmysky.com/learn/mathematics/2-d-shapes-age-10-mt_hdrMoiTgqu): Draw 2-D shapes using given dimensions and angles, using a ruler and protractor accurately - [Classifying shapes by properties](https://lightmysky.com/learn/mathematics/classifying-shapes-by-properties-mt_liIW336odh): Compare and classify geometric shapes based on their properties and sizes; understand that attributes belonging to a category also belong to all subcategories; classify two-dimensional figures in a hierarchy based on properties - [Angles in triangles (age 10+)](https://lightmysky.com/learn/mathematics/angles-in-triangles-age-10-mt_cdMlC7EpTJ): Find unknown angles in triangles, quadrilaterals, and regular polygons using angle sum properties - [Parts of a circle](https://lightmysky.com/learn/mathematics/parts-of-a-circle-mt_xq3YHZ2zeR): Illustrate and name parts of circles, including radius, diameter, and circumference; know that the diameter is twice the radius - [The coordinate plane and ordered pairs](https://lightmysky.com/learn/mathematics/the-coordinate-plane-and-ordered-pairs-mt_oDlduFnemk): Understand a coordinate system defined by two perpendicular number lines (axes) with an origin at (0,0); know that an ordered pair (x, y) specifies a unique point where the first number gives horizontal distance and the second gives vertical distance from the origin - [Plotting points in the first quadrant](https://lightmysky.com/learn/mathematics/plotting-points-in-the-first-quadrant-mt_snlqRCiA1R): Plot and read ordered pairs in the first quadrant of the coordinate plane; represent real-world and mathematical problems by graphing points and interpreting coordinate values in context - [Geometric terms and notation](https://lightmysky.com/learn/mathematics/geometric-terms-and-notation-mt_WtIFJSCQIT): Use conventional geometric terms and notation to describe, sketch, and draw points, lines, parallel and perpendicular lines, right angles, regular polygons, and reflectively/rotationally symmetric polygons - [Properties of triangles, quadrilaterals and circles](https://lightmysky.com/learn/mathematics/properties-of-triangles-quadrilaterals-and-circles-mt_DNYQLahbfa): Derive and illustrate properties of triangles, quadrilaterals, and circles using appropriate language, including interior angles, diagonals, symmetry, and relationships between side lengths - [Properties of rectangles](https://lightmysky.com/learn/mathematics/properties-of-rectangles-mt_KFKUR_gBg_): Use the properties of rectangles to deduce related facts and find missing lengths and angles - [Angles that add up on a diagram](https://lightmysky.com/learn/mathematics/angles-that-add-up-on-a-diagram-mt_89riIKwGYp): Recognise angle measure as additive; find unknown angles by adding or subtracting on a diagram using equations with a symbol for the unknown - [Understanding angles (age 10+)](https://lightmysky.com/learn/mathematics/understanding-angles-age-10-mt_QDTO3GAgcq): Recognise angles where they meet at a point, are on a straight line, or are vertically opposite; find missing angles using these properties - [Understanding angles (age 11+)](https://lightmysky.com/learn/mathematics/understanding-angles-age-11-mt_tedML_iu4Y): Apply the properties of angles at a point (360°), on a straight line (180°), and vertically opposite angles to find unknown angles in multi-step problems - [Angles with parallel lines](https://lightmysky.com/learn/mathematics/angles-with-parallel-lines-mt_H_pNJ3ZI_S): Understand and use the relationship between parallel lines cut by a transversal: corresponding angles, alternate interior angles, and co-interior (same-side interior) angles; use these to find unknown angles - [Coordinates (age 10+)](https://lightmysky.com/learn/mathematics/coordinates-age-10-mt_R4AY0LKxfl): Describe positions on the full coordinate grid (all four quadrants); use coordinates with negative values - [Translating and reflecting shapes](https://lightmysky.com/learn/mathematics/translating-and-reflecting-shapes-mt_lxaM6iVpdr): Draw and translate simple shapes on the coordinate plane; reflect shapes in the axes - [Area and volume formulae](https://lightmysky.com/learn/mathematics/area-and-volume-formulae-mt_tAJH5BrpOx): Derive and apply formulae for the area of triangles, parallelograms, and trapezia, and for the volume of cuboids and other prisms (including cylinders), connecting each formula to its geometric reasoning - [Surface Area of Solids](https://lightmysky.com/learn/mathematics/surface-area-of-solids-mt_hdhmurUndc): Find the surface area of prisms and pyramids by unfolding them into nets of rectangles and triangles, working out each face's area and summing them; apply this to practical problems like wrapping boxes or painting walls - [Circles: Circumference & Area](https://lightmysky.com/learn/mathematics/circles-circumference-and-area-mt_svFa6_mjO_): Calculate the circumference and area of circles using the formulae C = πd (or 2πr) and A = πr², and solve problems involving perimeters and areas of composite shapes that include circular parts - [Volume and Surface Area of Prisms and Cylinders](https://lightmysky.com/learn/mathematics/volume-and-surface-area-of-prisms-and-cylinders-mt_JTLJ5LDrYh): Find the volume of any prism as base area times length, and its surface area as the two ends plus the wrapped rectangle, including the cylinder. - [Angle sums in triangles and polygons](https://lightmysky.com/learn/mathematics/angle-sums-in-triangles-and-polygons-mt_J03RFlVdas): Derive and use the angle sum in a triangle (180°), use it to deduce the angle sum in any polygon ((n−2) × 180°), and calculate interior and exterior angles of regular polygons - [Coordinate Transformations](https://lightmysky.com/learn/mathematics/coordinate-transformations-mt_K0Y15w48SY): Identify properties of translations, rotations, and reflections; describe and perform these transformations on given figures, and understand that the image is congruent to the original - [Similarity and enlargement](https://lightmysky.com/learn/mathematics/similarity-and-enlargement-mt_y-BuQAfw4B): Understand similarity as a relationship where one shape is an enlargement of another; construct similar shapes by enlargement with a given scale factor and centre, with and without coordinate grids - [Vectors: Notation, Addition and Scalar Multiples](https://lightmysky.com/learn/mathematics/vectors-notation-addition-and-scalar-multiples-mt_NkqmrL7pLe): Write vectors as columns or as bold letters, add and subtract them along a route, and multiply by a scalar to change length or reverse direction. - [Accurate drawing and scale diagrams](https://lightmysky.com/learn/mathematics/accurate-drawing-and-scale-diagrams-mt_Jf8xcX4UTq): Draw and measure line segments and angles accurately using ruler and protractor, and interpret scale drawings to extract real measurements - [Ruler and compass constructions](https://lightmysky.com/learn/mathematics/ruler-and-compass-constructions-mt_ZrsqGVG-Wt): Use ruler and compasses to perform standard constructions: perpendicular bisector of a line segment, perpendicular to a line from or at a given point, and bisecting an angle - [Triangle congruence criteria](https://lightmysky.com/learn/mathematics/triangle-congruence-criteria-mt_167X6Ax8P7): Know and use the criteria for triangle congruence (SSS, SAS, ASA, RHS), use standard labelling conventions for sides and angles of triangle ABC, and determine whether two triangles are congruent - [Bearings and Scale Drawings](https://lightmysky.com/learn/mathematics/bearings-and-scale-drawings-mt_9q32jC0Fk_): Describe a direction as a three-figure bearing measured clockwise from north, and use a scale drawing to move between a plan and real distances. - [Congruent Triangles and the Congruence Conditions](https://lightmysky.com/learn/mathematics/congruent-triangles-and-the-congruence-conditions-mt_it8t7mY71J): Use SSS, SAS, ASA and RHS to decide whether two triangles are congruent, and write the decision as a short reasoned argument. - [Similar Triangles and Proving Similarity](https://lightmysky.com/learn/mathematics/similar-triangles-and-proving-similarity-mt_NPRVlr4lrs): Decide whether two triangles are similar from equal angles or matching side ratios, set the reason out in one line, and use the scale factor to find a missing length. - [Similar Shapes: Area and Volume Scale Factors](https://lightmysky.com/learn/mathematics/similar-shapes-area-and-volume-scale-factors-mt_v37c_AeLQa): Use the fact that areas scale by the square of the length scale factor and volumes by its cube, and work the relationship in both directions. - [Exact Values of Sine, Cosine and Tangent](https://lightmysky.com/learn/mathematics/exact-values-of-sine-cosine-and-tangent-mt_SNJEbXYOrR): Know and use the exact values of sin, cos and tan for 0, 30, 45, 60 and 90 degrees, with the acute values 30, 45 and 60 taken from a halved equilateral triangle and a half square, and the endpoints 0 and 90 read off the unit circle. - [The Sine Rule](https://lightmysky.com/learn/mathematics/the-sine-rule-mt_cTt_-9KZho): Use a/sin A = b/sin B in triangles with no right angle to find a missing side or angle, and check whether an obtuse second answer also fits. - [The Cosine Rule](https://lightmysky.com/learn/mathematics/the-cosine-rule-mt_yy73gK5CEP): Use a² = b² + c² - 2bc cos A when the sine rule does not apply: three sides given, or two sides with the angle between them. - [Triangle Area with Half ab sin C](https://lightmysky.com/learn/mathematics/triangle-area-with-half-ab-sin-c-mt_PiwizT4D2b): Find the area of any triangle from two sides and the angle between them, and run the same rule backwards to find an angle from a known area. - [Pythagoras' Theorem](https://lightmysky.com/learn/mathematics/pythagoras-theorem-mt_1VmTUxBrNd): Apply Pythagoras’ Theorem (a² + b² = c²) to calculate unknown side lengths in right-angled triangles, including in real-world and coordinate-geometry contexts - [Pythagoras and Trigonometry in Three Dimensions](https://lightmysky.com/learn/mathematics/pythagoras-and-trigonometry-in-three-dimensions-mt_7yKxYqQHB2): Find lengths and angles inside cuboids, prisms and pyramids by picking a right-angled triangle out of the solid and solving it on its own. - [Choosing a Triangle Method](https://lightmysky.com/learn/mathematics/choosing-a-triangle-method-mt_nubZzZjVvn): Decide from what a question gives you which tool a triangle needs: Pythagoras, right-angle ratios, the sine rule or the cosine rule. - [Arc Length, Sector Area and Segments](https://lightmysky.com/learn/mathematics/arc-length-sector-area-and-segments-mt_z9LIwmfKYJ): Find arc lengths and sector areas as fractions of a whole circle, and find a segment by subtracting a triangle from its sector. - [Bearing Problems with the Sine and Cosine Rules](https://lightmysky.com/learn/mathematics/bearing-problems-with-the-sine-and-cosine-rules-mt_YQFdU4aiFP): Turn a journey described by bearings into a triangle, then solve it with the sine or cosine rule instead of drawing it to scale. - [Circle Theorems: Angles at the Centre and in a Semicircle](https://lightmysky.com/learn/mathematics/circle-theorems-angles-at-the-centre-and-in-a-semicircle-mt_kOwb61sfUR): Use the fact that the angle at the centre is twice the angle at the circumference on the same arc, and that the angle in a semicircle is a right angle. - [Circle Theorems: Cyclic Quadrilaterals and the Same Segment](https://lightmysky.com/learn/mathematics/circle-theorems-cyclic-quadrilaterals-and-the-same-segment-mt_11GlmkiqB1): Use the equal angles subtended by the same arc, and the opposite angles of a cyclic quadrilateral adding to 180 degrees. - [Tangents, Radii and the Alternate Segment Theorem](https://lightmysky.com/learn/mathematics/tangents-radii-and-the-alternate-segment-theorem-mt_Y-Xf0BOb1C): Use the right angle between a tangent and the radius at the point of contact, the equal tangents from an outside point, and the alternate segment theorem. - [Volume and Surface Area of Cones, Spheres and Pyramids](https://lightmysky.com/learn/mathematics/volume-and-surface-area-of-cones-spheres-and-pyramids-mt_s4K5YMBSzG): Apply the standard formulae for pyramids, cones and spheres, and find a slant height when only the perpendicular height is given. - [Radian Measure and Sectors](https://lightmysky.com/learn/mathematics/radian-measure-and-sectors-mt_SvAYAu6mQ4): Measure an angle by the arc it cuts on a circle of radius 1, so a full turn is 2π. In radians the arc length is rθ and the sector area is ½r²θ, with no fraction of 360 to carry. - [Vectors in Three Dimensions](https://lightmysky.com/learn/mathematics/vectors-in-three-dimensions-mt_o97GclnNC-): Extend position, addition and scalar multiples to space, with distance and the standard unit vectors. Later topics in this sequence use this notation. - [Forms of the Equation of a Straight Line](https://lightmysky.com/learn/mathematics/forms-of-the-equation-of-a-straight-line-mt_dhIIxdFwcp): Move between y = mx + c, y - y₁ = m(x - x₁) and ax + by + c = 0, and produce the equation of a line from a point and a gradient or from two points. - [Parallel and Perpendicular Lines in Coordinates](https://lightmysky.com/learn/mathematics/parallel-and-perpendicular-lines-in-coordinates-mt_xNEmm139Ka): Use equal gradients for parallel lines and a gradient product of -1 for perpendicular ones, and apply both to find missing lines, midpoints and distances. - [The Equation of a Circle](https://lightmysky.com/learn/mathematics/the-equation-of-a-circle-mt_YARqmi_ioe): Read a circle's centre and radius from (x - a)² + (y - b)² = r², and recover them from an expanded equation by completing the square in both variables. - [Lines Meeting Circles: Tangents and Chords](https://lightmysky.com/learn/mathematics/lines-meeting-circles-tangents-and-chords-mt_ELA4IzIVdv): Find where a line cuts a circle by substitution, use the discriminant to tell a tangent from a chord or a miss, and use the right angle between a tangent and the radius. - [The Unit Circle and the Graphs of Sine, Cosine and Tangent](https://lightmysky.com/learn/mathematics/the-unit-circle-and-the-graphs-of-sine-cosine-and-tangent-mt_VJaimevi3-): Define sine and cosine as the coordinates of a point travelling round the unit circle, which extends them beyond a right angle and explains the period, symmetry and repeated values of their graphs. - [The Pythagorean and Quotient Identities](https://lightmysky.com/learn/mathematics/the-pythagorean-and-quotient-identities-mt_NXBq37vxQh): sin²θ + cos²θ = 1 is Pythagoras on the unit circle, and tan θ = sin θ / cos θ is a ratio of coordinates. Use both to simplify expressions and to prove further identities. - [Parametric Equations of Curves](https://lightmysky.com/learn/mathematics/parametric-equations-of-curves-mt_7NC7SCeU2P): Describe a curve by giving x and y separately in terms of a parameter, plot it from a table of parameter values, and convert to Cartesian form by eliminating the parameter. - [Solving Trigonometric Equations in a Given Interval](https://lightmysky.com/learn/mathematics/solving-trigonometric-equations-in-a-given-interval-mt_74_jiK2xXA): Find every solution inside a stated interval, using the graph or the circle to get the solutions the calculator does not show, and adjusting the interval first when the angle is 2x or x + 30°. - [Compound Angle Formulae](https://lightmysky.com/learn/mathematics/compound-angle-formulae-mt_ANDmX8exwg): Expand sin(A ± B), cos(A ± B) and tan(A ± B), and use them for exact values of angles built from known ones and for equations mixing two angles. - [Double Angle Formulae](https://lightmysky.com/learn/mathematics/double-angle-formulae-mt_AC9MmZ_fam): Set B = A in the compound angle formulae to get sin 2A, cos 2A and tan 2A, and pick between the three forms of cos 2A to remove either a squared term or a doubled angle. - [Combining a Sine and a Cosine into a Single Wave](https://lightmysky.com/learn/mathematics/combining-a-sine-and-a-cosine-into-a-single-wave-mt_IwQkNDRWyq): Write a sin x + b cos x as R sin(x + α), so two waves become one with a single amplitude and shift. Maximum and minimum values, and the x where they happen, then read off directly. - [Conic Sections: Ellipse, Parabola and Hyperbola](https://lightmysky.com/learn/mathematics/conic-sections-ellipse-parabola-and-hyperbola-mt_V4YSI3a_Eu): Meet the three curves a plane cuts from a cone, in both their locus definitions and their standard equations, and read foci and axes off the equation. - [Polar Coordinates and Polar Curves](https://lightmysky.com/learn/mathematics/polar-coordinates-and-polar-curves-mt_R5QMUJvA1v): Locate a point by distance and angle instead of two displacements, convert between the two systems, and sketch curves that are simpler in polar form. - [Small Angle Approximations](https://lightmysky.com/learn/mathematics/small-angle-approximations-mt_fIsZes5Os3): Near zero, and only in radians, the sine of an angle is the angle, the tangent of it is the angle too, and the cosine drops off as one minus half the square. These are the approximations that turn an equation nobody can solve into one anybody can, and knowing when they stop being safe is half the point. - [Proving Geometry Facts with Vectors](https://lightmysky.com/learn/mathematics/proving-geometry-facts-with-vectors-mt_A00MA7Ecgf): Show that one vector expression is a multiple of another, and use that to prove lines are parallel or that three points lie on a straight line. - [Trigonometry basics](https://lightmysky.com/learn/mathematics/trigonometry-basics-mt_KB_Czd7RQH): Use the trigonometric ratios sin, cos, and tan in right-angled triangles to find unknown sides and angles, including setting up the correct ratio for a given problem - [The Dot Product: Angles and Projections](https://lightmysky.com/learn/mathematics/the-dot-product-angles-and-projections-mt_yFBcyE4sFw): One product of two vectors returns a number that encodes the angle between them. It tests perpendicularity and splits one vector into parts along and across another. - [The Cross Product and Oriented Area](https://lightmysky.com/learn/mathematics/the-cross-product-and-oriented-area-mt_QEBM_Ms-SK): The other product of two vectors returns a vector perpendicular to both, whose length is the area of the parallelogram they span and whose direction records the orientation. - [Lines and Planes in Space](https://lightmysky.com/learn/mathematics/lines-and-planes-in-space-mt_kIX5ak1GRv): Write lines and planes in vector, parametric and Cartesian form, and answer distance and intersection questions with dot and cross products. - [Arc Length and Curvature of a Space Curve](https://lightmysky.com/learn/mathematics/arc-length-and-curvature-of-a-space-curve-mt_8mXYE8bVG1): Reparametrise a curve by arc length and measure how fast its direction turns, separating the path from the speed along it. ## Measurement 68 topics, ages 4 to 11. https://lightmysky.com/learn/mathematics/areas/measurement - [Comparing durations](https://lightmysky.com/learn/mathematics/comparing-durations-mt_TcG90kS8nu): Use comparative language for time: quicker, slower, earlier, later - [Measurable attributes of objects](https://lightmysky.com/learn/mathematics/measurable-attributes-of-objects-mt_NtJYlJdUe9): Describe and identify measurable attributes of objects such as length, height, weight, and capacity; use comparative language (longer, shorter, heavier, lighter, more, less) - [Comparing Capacity](https://lightmysky.com/learn/mathematics/comparing-capacity-mt_zkFbMLpu3U): Compare and describe capacity and volume using language such as full, empty, more than, less than, half full - [Comparing Lengths & Heights](https://lightmysky.com/learn/mathematics/comparing-lengths-and-heights-mt_uhuxX8sg9f): Compare two objects directly by length or height and describe the difference using language such as long, short, tall, longer, shorter, taller, double, half - [Comparing mass and weight](https://lightmysky.com/learn/mathematics/comparing-mass-and-weight-mt_-P1kdZhHbL): Compare two objects directly by mass or weight and describe the difference using language such as heavy, light, heavier than, lighter than - [Ordering Events in Time](https://lightmysky.com/learn/mathematics/ordering-events-in-time-mt_-vvVxpOHG2): Sequence events in chronological order using language such as before, after, next, first, today, yesterday, tomorrow, morning, afternoon, evening - [Capacity and volume](https://lightmysky.com/learn/mathematics/capacity-and-volume-mt_4lp_b5Pzik): Measure and begin to record capacity and volume using non-standard and standard units - [Days, Weeks, Months & Years](https://lightmysky.com/learn/mathematics/days-weeks-months-and-years-mt_bj1YCgNWUx): Recognise and use language relating to dates, including days of the week, weeks, months, and years - [Measuring length and height (age 5+)](https://lightmysky.com/learn/mathematics/measuring-length-and-height-age-5-mt_KaF0SQvaiu): Measure and begin to record lengths and heights using non-standard and standard units - [Measuring mass and weight](https://lightmysky.com/learn/mathematics/measuring-mass-and-weight-mt_zd0YkB3xNj): Measure and begin to record mass/weight using non-standard and standard units - [Measuring Time](https://lightmysky.com/learn/mathematics/measuring-time-mt__N55B7u7HD): Measure and begin to record time in hours, minutes, and seconds - [Choosing measurement units](https://lightmysky.com/learn/mathematics/choosing-measurement-units-mt_DOe893F6gN): Choose and use appropriate standard units to measure length (m/cm), mass (kg/g), and capacity (litres/ml) to the nearest appropriate unit - [Number of minutes in an hour](https://lightmysky.com/learn/mathematics/number-of-minutes-in-an-hour-mt_2XDGT5tei1): Know the number of minutes in an hour and the number of hours in a day - [Ordering and Comparing Lengths](https://lightmysky.com/learn/mathematics/ordering-and-comparing-lengths-mt_skYly2Qm01): Order three objects by length and compare the lengths of two objects indirectly using a third object - [Measuring length (age 6+)](https://lightmysky.com/learn/mathematics/measuring-length-age-6-mt_cqSf213hSa): Measure the length of an object using same-size length units laid end to end with no gaps or overlaps - [Sequence intervals of time](https://lightmysky.com/learn/mathematics/sequence-intervals-of-time-mt_6XCURuNwPw): Compare and sequence intervals of time - [Measuring length (age 7+)](https://lightmysky.com/learn/mathematics/measuring-length-age-7-mt_e4x3l2JeLI): Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, metre sticks, and measuring tapes - [Comparing lengths by measuring](https://lightmysky.com/learn/mathematics/comparing-lengths-by-measuring-mt_xWg0lI_gG4): Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit - [Estimating Lengths](https://lightmysky.com/learn/mathematics/estimating-lengths-mt_OzRZ89GrQW): Estimate lengths using units of inches, feet, centimetres, and metres - [Measuring with different units](https://lightmysky.com/learn/mathematics/measuring-with-different-units-mt_jHv4BgRK8B): Measure the length of an object using two different length units and describe how the measurements relate to the size of the unit chosen - [Time Units and Calendar Facts](https://lightmysky.com/learn/mathematics/time-units-and-calendar-facts-mt_EXlmTURK_o): Know the number of seconds in a minute and the number of days in each month, year, and leap year - [Understanding Area](https://lightmysky.com/learn/mathematics/understanding-area-mt_6xNmQLzuqm): Understand that a unit square has one square unit of area and that the area of a plane figure is the number of unit squares that cover it without gaps or overlaps - [Area (age 8+)](https://lightmysky.com/learn/mathematics/area-age-8-mt_y1n0Zwhoca): Measure areas by counting unit squares (square cm, square m, square in, square ft) - [Area by Tiling](https://lightmysky.com/learn/mathematics/area-by-tiling-mt_Jvvh5P06NV): Find the area of a rectangle by tiling it with unit squares and show that the result equals the product of the side lengths - [Rectangle area by multiplying](https://lightmysky.com/learn/mathematics/rectangle-area-by-multiplying-mt_GzcJEVkNRn): Multiply side lengths to find areas of rectangles and represent whole-number products as rectangular areas - [Area and the distributive property](https://lightmysky.com/learn/mathematics/area-and-the-distributive-property-mt_AQo4u7O4sM): Use tiling to demonstrate the distributive property: the area of a rectangle with sides a and (b+c) equals a×b + a×c; use area models to represent the distributive property - [Area of compound shapes](https://lightmysky.com/learn/mathematics/area-of-compound-shapes-mt_eMtV6tBSJm): Recognise area as additive; find areas of rectilinear figures by decomposing into non-overlapping rectangles and summing their areas - [Coin Values](https://lightmysky.com/learn/mathematics/coin-values-mt_cPZwlUk8Nd): Recognise and know the value of different coins and notes - [Telling Time: Hours and Half Hours](https://lightmysky.com/learn/mathematics/telling-time-hours-and-half-hours-mt_0XxyaQLRhn): Tell the time to the hour and half past the hour, and draw clock hands to show these times - [Pounds & Pence Notation](https://lightmysky.com/learn/mathematics/pounds-and-pence-notation-mt_pbuhUQJjtt): Recognise and use symbols for pounds (£) and pence (p) and combine amounts to make a particular value - [Making the Same Amount with Different Coins](https://lightmysky.com/learn/mathematics/making-the-same-amount-with-different-coins-mt_AF2BeFQwfX): Find different combinations of coins that equal the same amount of money - [Money Addition & Subtraction](https://lightmysky.com/learn/mathematics/money-addition-and-subtraction-mt_6J1wmCWf41): Solve simple money problems involving addition and subtraction, including giving change - [Telling Time: Minutes](https://lightmysky.com/learn/mathematics/telling-time-minutes-mt_HNOPGJYiRK): Tell and write the time to five minutes, including quarter past and quarter to, and draw clock hands to show these times - [Giving Change](https://lightmysky.com/learn/mathematics/giving-change-mt_6_O6THdEDK): Add and subtract amounts of money to give change, using both £ and p in practical contexts - [Telling time to five minutes with a.m. and p.m. (age 7+)](https://lightmysky.com/learn/mathematics/telling-time-to-five-minutes-with-a-m-and-p-m-age-7-mt_4m8BimI4G5): Tell and write time from analogue and digital clocks to the nearest five minutes, using a.m., p.m., and 12-hour and 24-hour notation - [Estimating and reading time](https://lightmysky.com/learn/mathematics/estimating-and-reading-time-mt_3XJkeIn6J6): Estimate and read time with increasing accuracy to the nearest minute; record and compare time in terms of seconds, minutes, and hours - [Comparing Time Durations](https://lightmysky.com/learn/mathematics/comparing-time-durations-mt_4emC463IyW): Compare durations of events and calculate the time taken by particular events or tasks - [12-hour and 24-hour time](https://lightmysky.com/learn/mathematics/12-hour-and-24-hour-time-mt_WfrE_4r-kY): Read, write, and convert time between analogue and digital 12-hour and 24-hour clocks - [Telling time to the minute (age 8+)](https://lightmysky.com/learn/mathematics/telling-time-to-the-minute-age-8-mt_UNzojLkNdm): Tell and write time to the nearest minute using analogue and digital clocks - [Elapsed time problems](https://lightmysky.com/learn/mathematics/elapsed-time-problems-mt_IL86kadLSS): Solve word problems involving elapsed time by adding and subtracting time intervals in minutes, including using a number line - [Length word problems within 100](https://lightmysky.com/learn/mathematics/length-word-problems-within-100-mt_DLcEzmmj2r): Solve word problems involving lengths within 100, using addition and subtraction with drawings and equations - [Money problems with coins and notes](https://lightmysky.com/learn/mathematics/money-problems-with-coins-and-notes-mt_3tz3Otap5j): Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately - [Comparing and ordering measurements](https://lightmysky.com/learn/mathematics/comparing-and-ordering-measurements-mt_BFJ-ch_8QU): Compare and order lengths, mass, and capacity and record results using >, <, and = - [Calculating with measurements](https://lightmysky.com/learn/mathematics/calculating-with-measurements-mt_6oxQPNLHNv): Measure, compare, add, and subtract lengths (m/cm/mm), mass (kg/g), and volume/capacity (l/ml) using standard units - [Measuring Perimeters](https://lightmysky.com/learn/mathematics/measuring-perimeters-mt_wE7-Gs9ENL): Measure the perimeter of simple 2-D shapes - [Converting measurement units](https://lightmysky.com/learn/mathematics/converting-measurement-units-mt_d8al9JcajP): Convert between different units of measure (e.g. kilometre to metre, hour to minute, minute to second, year to month, week to day) - [Estimating and comparing measures](https://lightmysky.com/learn/mathematics/estimating-and-comparing-measures-mt_mFJ-2ZF6Tk): Estimate, compare, and calculate different measures including money in pounds and pence - [Measuring liquids and masses](https://lightmysky.com/learn/mathematics/measuring-liquids-and-masses-mt_-af65bxfdp): Measure and estimate liquid volumes and masses of objects using grams, kilograms, and litres; solve one-step word problems involving mass or volume - [Perimeters of polygons](https://lightmysky.com/learn/mathematics/perimeters-of-polygons-mt_WtcFrxGOgw): Solve problems involving perimeters of polygons: find perimeter from side lengths, find an unknown side length, and explore rectangles with same perimeter but different areas (or vice versa) - [Area and perimeter of rectangles](https://lightmysky.com/learn/mathematics/area-and-perimeter-of-rectangles-mt_eiB3-6pu6a): Apply the area formula (l × w) and perimeter formula (2l + 2w) for rectangles including squares in real-world and mathematical problems; calculate and compare areas using standard units (cm², m²) and estimate areas of irregular shapes - [Converting time units](https://lightmysky.com/learn/mathematics/converting-time-units-mt_2yas4Unc8o): Solve problems involving converting between units of time (hours↔minutes, minutes↔seconds, years↔months, weeks↔days) - [Estimating volume](https://lightmysky.com/learn/mathematics/estimating-volume-mt_NSC3LT_-ch): Estimate volume of cuboids using 1 cm³ blocks; estimate capacity of containers using water - [Perimeter of Compound Shapes](https://lightmysky.com/learn/mathematics/perimeter-of-compound-shapes-mt_n0AlyLQwC9): Measure and calculate the perimeter of composite rectilinear shapes in centimetres and metres - [Perimeter (age 10+)](https://lightmysky.com/learn/mathematics/perimeter-age-10-mt_MJZA90uc6H): Recognise that shapes with the same area can have different perimeters and vice versa; explore this relationship systematically - [Volume with unit cubes](https://lightmysky.com/learn/mathematics/volume-with-unit-cubes-mt_IuHa5UI5od): Recognise volume as an attribute of solid figures; understand that a unit cube (side length 1 unit) has 'one cubic unit' of volume and can be used to measure volume; a solid packed with n unit cubes has volume n cubic units - [Counting Unit Cubes](https://lightmysky.com/learn/mathematics/counting-unit-cubes-mt_d7XktBQPxm): Measure volumes by counting unit cubes using cubic cm, cubic in, cubic ft, and other units - [Volume of cuboids](https://lightmysky.com/learn/mathematics/volume-of-cuboids-mt_nTL-owFJTF): Find the volume of right rectangular prisms by packing with unit cubes and show it equals l × w × h (or base area × height); apply V = l × w × h and V = B × h to solve real-world problems; calculate, estimate, and compare volumes of cubes and cuboids in standard units (cm³, m³) - [Converting measurement units (age 9+)](https://lightmysky.com/learn/mathematics/converting-measurement-units-age-9-mt_Zks8xyInSG): Know relative sizes of measurement units within one system (km/m/cm/mm, kg/g, l/ml, hr/min/sec); convert between different metric units and express measurements in terms of a smaller unit; record equivalents in conversion tables - [Measure word problems with fractions and decimals](https://lightmysky.com/learn/mathematics/measure-word-problems-with-fractions-and-decimals-mt_5HV4mbgSGH): Solve word problems involving distances, time intervals, liquid volumes, masses, and money using the four operations with fractions or decimals; represent with diagrams including number lines - [Metric & Imperial Conversion](https://lightmysky.com/learn/mathematics/metric-and-imperial-conversion-mt_p6MhZJYYPN): Understand and use approximate equivalences between metric units and common imperial units (inches, pounds, pints) - [Volume as additive](https://lightmysky.com/learn/mathematics/volume-as-additive-mt_5TBUFnCy5-): Recognise volume as additive; find volumes of composite solid figures made of two or more non-overlapping right rectangular prisms - [Converting units with decimal notation](https://lightmysky.com/learn/mathematics/converting-units-with-decimal-notation-mt_2VpdPjvewx): Convert among different-sized standard measurement units within a given system (e.g. 5 cm to 0.05 m) using decimal notation to up to three decimal places; convert between smaller and larger units of length, mass, volume, and time - [Measurement Conversions](https://lightmysky.com/learn/mathematics/measurement-conversions-mt_SqhXQhAEUf): Solve problems involving the calculation and conversion of units of measure, using decimal notation and multi-step reasoning in real-world contexts - [Miles & Kilometres](https://lightmysky.com/learn/mathematics/miles-and-kilometres-mt_3tPI0HqqcN): Convert between miles and kilometres using the approximate relationship (5 miles ≈ 8 km) - [Area of Triangles & Parallelograms](https://lightmysky.com/learn/mathematics/area-of-triangles-and-parallelograms-mt_ML5t7n2-U8): Calculate the area of parallelograms and triangles using formulae (A = b × h for parallelograms, A = ½ × b × h for triangles) - [Measuring & Plotting Lengths](https://lightmysky.com/learn/mathematics/measuring-and-plotting-lengths-mt_mayItsxMUu): Generate measurement data by measuring lengths to the nearest whole unit and display the data on a line plot - [Measuring to the nearest half and quarter inch](https://lightmysky.com/learn/mathematics/measuring-to-the-nearest-half-and-quarter-inch-mt_ghK1mnEstc): Generate measurement data by measuring lengths to the nearest half and quarter inch; display the data on a line plot with a scale marked in whole numbers, halves, and quarters - [Measurement Line Plots](https://lightmysky.com/learn/mathematics/measurement-line-plots-mt_42QD6nYjiZ): Make a line plot to display measurement data in fractions of a unit (1/2, 1/4, 1/8); solve problems involving addition and subtraction of fractions using line plot data ## Multiplication & Division 57 topics, ages 4 to 13. https://lightmysky.com/learn/mathematics/areas/multiplication-and-division - [Division as equal sharing](https://lightmysky.com/learn/mathematics/division-as-equal-sharing-mt_LpSuPgL31x): Understand division as sharing equally into groups or as grouping (how many groups of a given size can be made) - [Multiplication as repeated addition](https://lightmysky.com/learn/mathematics/multiplication-as-repeated-addition-mt_PZ909yPrEC): Understand multiplication as repeated addition and grouping equal sets - [Arrays for multiplication](https://lightmysky.com/learn/mathematics/arrays-for-multiplication-mt_GRWwTDZ3wD): Use arrays to represent multiplication and division situations - [Commutative Multiplication](https://lightmysky.com/learn/mathematics/commutative-multiplication-mt_C9ZfT-4cgn): Understand and apply the commutative property of multiplication and recognise that division is not commutative - [Odd and even numbers](https://lightmysky.com/learn/mathematics/odd-and-even-numbers-mt_0u4KLbvBa1): Recognise odd and even numbers - [Times tables](https://lightmysky.com/learn/mathematics/times-tables-mt_HhuSDxwDNM): Recall and use multiplication and division facts for the 2, 5, and 10 multiplication tables - [Solving multiplication and division problems](https://lightmysky.com/learn/mathematics/solving-multiplication-and-division-problems-mt_wh3UqnWsa7): Solve problems involving multiplication and division using arrays, repeated addition, mental methods, and known facts - [Arrays for multiplication (age 7+)](https://lightmysky.com/learn/mathematics/arrays-for-multiplication-age-7-mt_B1zj1RwQ3a): Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and 5 columns; write an equation to express the total as a sum of equal addends - [Rows & Columns in Rectangles](https://lightmysky.com/learn/mathematics/rows-and-columns-in-rectangles-mt_UTnDKQkVX5): Partition a rectangle into rows and columns of same-size squares and count to find the total number of them - [Times tables (age 7+)](https://lightmysky.com/learn/mathematics/times-tables-age-7-mt_wQ89AEXhz3): Recall and use multiplication and division facts for the 3, 4, and 8 multiplication tables - [All times tables to 12×12](https://lightmysky.com/learn/mathematics/all-times-tables-to-12-12-mt_K5jM7vlVhA): Recall multiplication and division facts for multiplication tables up to 12 × 12 - [What Multiplication Means](https://lightmysky.com/learn/mathematics/what-multiplication-means-mt_gtTl3R5buH): Interpret products of whole numbers (e.g. 5 × 7 as the total number of objects in 5 groups of 7) - [Properties of Operations](https://lightmysky.com/learn/mathematics/properties-of-operations-mt_Lb2ZnMdkYR): Apply properties of operations (commutative, associative, distributive) as strategies to multiply and divide - [Factor Pairs & Commutativity](https://lightmysky.com/learn/mathematics/factor-pairs-and-commutativity-mt_nZkL5-XjRX): Recognise and use factor pairs and commutativity in mental calculations - [Factors, multiples, and primes](https://lightmysky.com/learn/mathematics/factors-multiples-and-primes-mt_FHIAv6dfhU): Find all factor pairs for numbers to 100, list common factors and multiples, and tell whether a number is prime - [Multiplicative Comparison as an Equation](https://lightmysky.com/learn/mathematics/multiplicative-comparison-as-an-equation-mt_U4cIBXVug4): Interpret a multiplication equation as a comparison (e.g. 35 = 5 × 7 means 35 is 5 times as many as 7); represent verbal statements of multiplicative comparisons as equations - [Square and cube numbers](https://lightmysky.com/learn/mathematics/square-and-cube-numbers-mt_gxCIASSezX): Recognise and use square numbers and cube numbers, and the notation for squared (²) and cubed (³) - [Factors, multiples, squares and cubes (age 9+)](https://lightmysky.com/learn/mathematics/factors-multiples-squares-and-cubes-age-9-mt_0Wg5F97osg): Solve problems involving multiplication and division using knowledge of factors, multiples, squares, and cubes - [Reading ×, ÷, and = Symbols](https://lightmysky.com/learn/mathematics/reading-and-symbols-mt_zOWwLxa77y): Read, write, and interpret the symbols ×, ÷, and = in multiplication and division number sentences - [What Division Means](https://lightmysky.com/learn/mathematics/what-division-means-mt_iNdrM2-oJf): Interpret whole-number quotients (e.g. 56 ÷ 8 as the number of objects in each share or the number of equal groups) - [Division as Unknown Factor](https://lightmysky.com/learn/mathematics/division-as-unknown-factor-mt_GDG9_SZmsO): Understand division as an unknown-factor problem (e.g. find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8) - [Fluent multiplication and division facts](https://lightmysky.com/learn/mathematics/fluent-multiplication-and-division-facts-mt_WX30dzi4dt): Fluently multiply and divide within 100 using strategies such as the relationship between multiplication and division - [Multiplication and Division Word Problems](https://lightmysky.com/learn/mathematics/multiplication-and-division-word-problems-mt_Zdv-b-iW5K): Use multiplication and division within 100 to solve word problems involving equal groups, arrays, and measurement quantities - [Patterns in addition and multiplication tables](https://lightmysky.com/learn/mathematics/patterns-in-addition-and-multiplication-tables-mt_w6MxaaoMXZ): Identify arithmetic patterns (including patterns in the addition table or multiplication table) and explain them using properties of operations - [Unknown in Multiplication & Division](https://lightmysky.com/learn/mathematics/unknown-in-multiplication-and-division-mt_xZvDCYA5Ae): Determine the unknown whole number in a multiplication or division equation relating three whole numbers (e.g. 8 × ? = 48, ? × 6 = 42) - [Generating a pattern from a rule](https://lightmysky.com/learn/mathematics/generating-a-pattern-from-a-rule-mt_A-FyLLLLzy): Generate a number or shape pattern that follows a given rule; identify apparent features of the pattern not explicit in the rule and explain informally why they occur - [Multiplicative Comparison Word Problems](https://lightmysky.com/learn/mathematics/multiplicative-comparison-word-problems-mt_6-j1NO2ZUH): Solve word problems involving multiplicative comparison using drawings and equations with a symbol for the unknown number - [Scaling by fractions and simple rates](https://lightmysky.com/learn/mathematics/scaling-by-fractions-and-simple-rates-mt_ZOJ6EbdPOb): Solve problems involving scaling by simple fractions and problems involving simple rates - [Short division](https://lightmysky.com/learn/mathematics/short-division-mt_MCu_SNg_OW): Divide numbers up to four digits by a one-digit number using short division (and place-value/array strategies); interpret remainders appropriately for the context - [Prime numbers](https://lightmysky.com/learn/mathematics/prime-numbers-mt_y1XCVsIelg): Know and use the vocabulary of prime numbers, prime factors, and composite numbers; establish whether a number up to 100 is prime; recall prime numbers up to 19 - [Factors, multiples, and primes (age 11+)](https://lightmysky.com/learn/mathematics/factors-multiples-and-primes-age-11-mt_xhoOWnhtHq): Use the concepts and vocabulary of prime numbers, factors, multiples, common factors, common multiples, highest common factor (HCF), lowest common multiple (LCM), and prime factorisation including product notation and the unique factorisation property - [Written Multiplication & Division](https://lightmysky.com/learn/mathematics/written-multiplication-and-division-mt_DyGBW3ZHh3): Write and calculate mathematical statements for multiplication and division using known tables, including two-digit × one-digit, using mental and progressing to formal written methods - [Multi-Step Multiply & Divide](https://lightmysky.com/learn/mathematics/multi-step-multiply-and-divide-mt_EmR5n58jZt): Solve problems involving multiplication and division, including scaling problems and correspondence problems where n objects are connected to m objects - [Multiplying by Tens](https://lightmysky.com/learn/mathematics/multiplying-by-tens-mt_isojCL9yy-): Multiply one-digit whole numbers by multiples of 10 in the range 10–90 using strategies based on place value and properties of operations - [Written Multiplication](https://lightmysky.com/learn/mathematics/written-multiplication-mt_18fK9sQdIz): Multiply two-digit and three-digit numbers by a one-digit number using formal written layout - [Multiply & Add Problems](https://lightmysky.com/learn/mathematics/multiply-and-add-problems-mt_AR-K72OIIO): Solve problems involving multiplying and adding, including using the distributive law, integer scaling problems, and harder correspondence problems - [Long multiplication](https://lightmysky.com/learn/mathematics/long-multiplication-mt_q9EaJc2FP8): Multiply a whole number of up to four digits by a one-digit number, and multiply two two-digit numbers, using formal written methods including long multiplication; illustrate with area models - [Multi-step word problems with remainders](https://lightmysky.com/learn/mathematics/multi-step-word-problems-with-remainders-mt_p-nbe0w_lf): Solve multi-step word problems using the four operations with whole numbers, including interpreting remainders in context; represent with equations using a letter for the unknown; check with estimation - [Long multiplication (age 10+)](https://lightmysky.com/learn/mathematics/long-multiplication-age-10-mt_XLP1IM3Qbb): Fluently multiply multi-digit whole numbers (up to 4 digits by 2 digits) using the formal written method of long multiplication - [Brackets in Expressions](https://lightmysky.com/learn/mathematics/brackets-in-expressions-mt_8jFSnXxqQD): Use parentheses, brackets, or braces in numerical expressions and evaluate expressions containing these grouping symbols - [Division with remainders (age 10+)](https://lightmysky.com/learn/mathematics/division-with-remainders-age-10-mt_WsM4EmdOLe): Divide numbers up to 4 digits by a two-digit divisor using formal written long division, interpreting remainders as whole numbers, fractions, or by rounding as appropriate - [Dividing by two-digit numbers](https://lightmysky.com/learn/mathematics/dividing-by-two-digit-numbers-mt_FvEq4heNBx): Divide numbers up to 4 digits by a two-digit number using formal written short division where appropriate, interpreting remainders according to context - [Order of operations](https://lightmysky.com/learn/mathematics/order-of-operations-mt_jHgRQ4hR0g): Understand and apply the conventional order of operations (PEMDAS/BODMAS) to carry out calculations involving the four operations - [Multi-step problems: choosing operations](https://lightmysky.com/learn/mathematics/multi-step-problems-choosing-operations-mt_RXnyhCRYXA): Solve problems using addition, subtraction, multiplication, and division. Choose the operation and method for each step, then solve multi-step problems in context. - [Writing Number Sentences](https://lightmysky.com/learn/mathematics/writing-number-sentences-mt_4WaKWECpcv): Write simple numerical expressions that record calculations with numbers, and interpret numerical expressions without evaluating them (e.g. recognise that 3 × (18932 + 921) is three times as large as 18932 + 921) - [Using inverse operations](https://lightmysky.com/learn/mathematics/using-inverse-operations-mt_VUQNveSYjQ): Recognise inverse relationships between operations and use them to check answers and simplify calculations. - [Mental multiplication and division](https://lightmysky.com/learn/mathematics/mental-multiplication-and-division-mt_pjfmCMMPjO): Use place value, known and derived facts to multiply and divide mentally, including multiplying by 0 and 1, dividing by 1, and multiplying together three numbers - [Mental multiplication and division (age 9+)](https://lightmysky.com/learn/mathematics/mental-multiplication-and-division-age-9-mt_8A3pZNOp7Z): Multiply and divide numbers mentally drawing upon known facts, including related facts and place-value adjustments - [Multiplying and dividing by 10, 100, and 1000](https://lightmysky.com/learn/mathematics/multiplying-and-dividing-by-10-100-and-1000-mt_LlMl2PbaZe): Multiply and divide whole numbers and those involving decimals by 10, 100, and 1000 - [Mental calculation strategies](https://lightmysky.com/learn/mathematics/mental-calculation-strategies-mt_vmW2cb5c7A): Perform mental calculations including with mixed operations and large numbers, using strategies such as partitioning, compensation, and derived facts - [Order of operations with powers and roots](https://lightmysky.com/learn/mathematics/order-of-operations-with-powers-and-roots-mt_X5cypSGoGU): Use conventional notation for the priority of operations including brackets, powers, roots, and reciprocals; apply BIDMAS/BODMAS consistently to evaluate complex numerical expressions - [Sign Rules for Multiplication and Division](https://lightmysky.com/learn/mathematics/sign-rules-for-multiplication-and-division-mt_rxInpOQ74w): Multiply and divide with positive and negative integers and rational numbers, understanding the rules for the sign of the product or quotient - [Multiplying and dividing (age 10+)](https://lightmysky.com/learn/mathematics/multiplying-and-dividing-age-10-mt_wPgpMJ0-PA): Multiply and divide numbers by 10, 100, and 1000 giving answers up to three decimal places, understanding that digits shift position in the place-value chart - [Division with Decimals](https://lightmysky.com/learn/mathematics/division-with-decimals-mt_lU-2aTRB9f): Use written division methods in cases where the answer has up to two decimal places; divide decimals to hundredths by whole numbers - [Multiplying decimals by whole numbers](https://lightmysky.com/learn/mathematics/multiplying-decimals-by-whole-numbers-mt_SoDP1fSQEB): Multiply one-digit numbers with up to two decimal places by whole numbers (e.g. 3.47 × 6) - [Estimation to check answers to calculations](https://lightmysky.com/learn/mathematics/estimation-to-check-answers-to-calculations-mt_ilPrU0cbtT): Use estimation to check answers to calculations and determine, in the context of a problem, an appropriate degree of accuracy - [Rounding Answers](https://lightmysky.com/learn/mathematics/rounding-answers-mt_QU5R7Aajy9): Solve problems which require answers to be rounded to specified degrees of accuracy ## Fractions 67 topics, ages 5 to 13. https://lightmysky.com/learn/mathematics/areas/fractions - [What Is a Half?](https://lightmysky.com/learn/mathematics/what-is-a-half-mt_-hTTat0mBR): Recognise, find, and name a half as one of two equal parts of an object, shape, or quantity - [Finding quarters (age 5+)](https://lightmysky.com/learn/mathematics/finding-quarters-age-5-mt_g3W0mdADVu): Recognise, find, and name a quarter as one of four equal parts of an object, shape, or quantity - [Fraction notation](https://lightmysky.com/learn/mathematics/fraction-notation-mt_vKcxX6iNOA): Read and write fraction notation. Name the numerator and denominator and explain equal parts. Use the names unit fraction, non-unit fraction, proper fraction, improper fraction, mixed number, equivalent fraction and simplest form. - [Fractions of amounts](https://lightmysky.com/learn/mathematics/fractions-of-amounts-mt_cFltwUQi-d): Recognise, find, name, and write fractions 1/3, 1/4, 2/4, and 3/4 of a length, shape, set of objects, or quantity - [Halves & Quarters of Shapes](https://lightmysky.com/learn/mathematics/halves-and-quarters-of-shapes-mt_xACS3rWWDp): Partition circles and rectangles into two and four equal shares and describe them using the words halves, fourths, and quarters - [Decomposing a shape into more equal shares](https://lightmysky.com/learn/mathematics/decomposing-a-shape-into-more-equal-shares-mt_hyvHv2BCwb): Understand that decomposing a shape into more equal shares creates smaller shares - [Fractions of a group of objects](https://lightmysky.com/learn/mathematics/fractions-of-a-group-of-objects-mt_k2WE0-22-4): Recognise, find, and write fractions of a discrete set of objects: unit fractions and non-unit fractions with small denominators - [Splitting shapes into equal parts (age 7+)](https://lightmysky.com/learn/mathematics/splitting-shapes-into-equal-parts-age-7-mt_Xp-rj46S2w): Partition circles and rectangles into two, three, or four equal shares; describe shares as halves, thirds, and fourths; recognise that equal shares of identical wholes need not have the same shape - [Tenths](https://lightmysky.com/learn/mathematics/tenths-mt_YzM5goBctT): Count up and down in tenths; recognise that tenths arise from dividing an object into 10 equal parts and from dividing one-digit numbers or quantities by 10 - [Fractions on a number line](https://lightmysky.com/learn/mathematics/fractions-on-a-number-line-mt_Kr3IyA6m-O): Recognise and use fractions as numbers: place unit fractions and non-unit fractions with small denominators on a number line - [Adding and subtracting fractions with the same bottom number within one whole](https://lightmysky.com/learn/mathematics/adding-and-subtracting-fractions-with-the-same-bottom-number-within-one-whole-mt_a1FdAsRKOF): Add and subtract fractions with the same denominator within one whole (e.g. 5/7 + 1/7 = 6/7) - [Comparing fractions](https://lightmysky.com/learn/mathematics/comparing-fractions-mt_IfEgu0X449): Compare and order unit fractions, and fractions with the same denominator - [Solving problems with fractions (age 7+)](https://lightmysky.com/learn/mathematics/solving-problems-with-fractions-age-7-mt_SsLWS_APM7): Solve problems involving counting in tenths, fractions of quantities, equivalence, fraction addition/subtraction, and fraction comparison - [Adding fractions (same denominator)](https://lightmysky.com/learn/mathematics/adding-fractions-same-denominator-mt_CBHwluE6Lp): Add and subtract fractions with the same denominator, including results greater than one whole (e.g. 5/8 + 6/8 = 11/8) - [Decimal & Percent Notation](https://lightmysky.com/learn/mathematics/decimal-and-percent-notation-mt_W17Kbwm0-u): Read, write, and use decimal and percentage notation correctly — decimal, decimal point, tenths, hundredths, thousandths, percentage, per cent, % symbol, convert, terminating decimal — and understand the relationships between fractions, decimals, and percentages as three ways of expressing the same value - [Fractions of a whole](https://lightmysky.com/learn/mathematics/fractions-of-a-whole-mt_ndGqFPWyen): Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a/b as a parts of size 1/b - [Comparing fractions (age 8+)](https://lightmysky.com/learn/mathematics/comparing-fractions-age-8-mt_HnKbuCliNS): Compare two fractions with the same numerator or the same denominator by reasoning about size; record comparisons with >, =, or < symbols - [Fractions as parts of shapes](https://lightmysky.com/learn/mathematics/fractions-as-parts-of-shapes-mt_idbKDrf9qZ): Partition shapes into parts with equal areas and express the area of each part as a unit fraction of the whole - [Fractions of amounts (harder)](https://lightmysky.com/learn/mathematics/fractions-of-amounts-harder-mt_3y7xKP9MjU): Solve problems involving increasingly harder fractions to calculate quantities, including non-unit fractions where the answer is a whole number - [Fractions on a number line (age 8+)](https://lightmysky.com/learn/mathematics/fractions-on-a-number-line-age-8-mt_NoB20kVa4w): Represent fractions on a number line: partition the interval 0 to 1 into b equal parts to locate 1/b, then mark off a lengths of 1/b from 0 to locate a/b - [Hundredths (age 8+)](https://lightmysky.com/learn/mathematics/hundredths-age-8-mt_doX1BhmFgk): Count up and down in hundredths; recognise that hundredths arise when dividing an object by 100 or dividing tenths by 10 - [Decimal equivalents of tenths and hundredths](https://lightmysky.com/learn/mathematics/decimal-equivalents-of-tenths-and-hundredths-mt_TqDq6jyOmL): Recognise and write decimal equivalents of any number of tenths or hundredths (e.g. 3/10 = 0.3, 27/100 = 0.27) - [Comparing two-decimal numbers](https://lightmysky.com/learn/mathematics/comparing-two-decimal-numbers-mt_IegHBHERVa): Compare numbers with the same number of decimal places up to two decimal places - [Decimals and fractions](https://lightmysky.com/learn/mathematics/decimals-and-fractions-mt_wB-GBDkoNr): Solve simple measure and money problems involving fractions and decimals to two decimal places - [Dividing by 10 and 100](https://lightmysky.com/learn/mathematics/dividing-by-10-and-100-mt_kDMKJ5Ztt6): Find the effect of dividing a one- or two-digit number by 10 and 100, identifying the value of the digits as ones, tenths, and hundredths - [Rounding decimals to whole numbers](https://lightmysky.com/learn/mathematics/rounding-decimals-to-whole-numbers-mt_Fl7b8q9pI1): Round decimals with one decimal place to the nearest whole number - [Whole numbers as fractions](https://lightmysky.com/learn/mathematics/whole-numbers-as-fractions-mt_AYzE1EAvI0): Express whole numbers as fractions (e.g. 3 = 3/1) and recognise fractions equivalent to whole numbers (e.g. 4/4 = 1, 6/1 = 6) - [Rounding decimals (age 9+)](https://lightmysky.com/learn/mathematics/rounding-decimals-age-9-mt_Ii1hV4V5ql): Round decimals with two decimal places to the nearest whole number and to one decimal place - [Understanding fractions (age 9+)](https://lightmysky.com/learn/mathematics/understanding-fractions-age-9-mt_09sySPqM9Z): Understand a fraction a/b with a > 1 as a sum of fractions 1/b (e.g. 3/5 = 1/5 + 1/5 + 1/5) - [Fraction addition concepts](https://lightmysky.com/learn/mathematics/fraction-addition-concepts-mt_VgOePicFYK): Understand addition and subtraction of fractions as joining and separating parts; decompose a fraction into a sum of fractions with the same denominator in more than one way - [Mixed numbers and improper fractions](https://lightmysky.com/learn/mathematics/mixed-numbers-and-improper-fractions-mt_o_p-3tCxiM): Recognise mixed numbers and improper fractions; convert from one form to the other (e.g. 2/5 + 4/5 = 6/5 = 1 1/5) - [Adding and subtracting mixed numbers](https://lightmysky.com/learn/mathematics/adding-and-subtracting-mixed-numbers-mt_70qDTI14td): Add and subtract mixed numbers with like denominators, including by converting to improper fractions or using properties of operations - [Multiplying fractions](https://lightmysky.com/learn/mathematics/multiplying-fractions-mt_TgHxujL81r): Understand a/b as a multiple of 1/b; multiply proper fractions and mixed numbers by whole numbers, supported by visual models (e.g. 3 × 2/5 = 6/5 = 1 1/5) - [Fractions of a whole (age 9+)](https://lightmysky.com/learn/mathematics/fractions-of-a-whole-age-9-mt_HJA2Oz-Zh1): Solve word problems involving multiplication of a fraction by a whole number using visual models and equations - [Word problems adding and subtracting fractions](https://lightmysky.com/learn/mathematics/word-problems-adding-and-subtracting-fractions-mt_ZLqYE7la4Z): Solve word problems involving addition and subtraction of fractions with like denominators, using visual models and equations - [Writing simple fractions](https://lightmysky.com/learn/mathematics/writing-simple-fractions-mt_MyGblah2yY): Write simple fractions (e.g. 1/2 of 6 = 3) and recognise the equivalence of 2/4 and 1/2 - [Equivalent fractions](https://lightmysky.com/learn/mathematics/equivalent-fractions-mt_FbDKeLfBCo): Recognise and show, using diagrams, equivalent fractions with small denominators - [Equivalent fractions on a number line](https://lightmysky.com/learn/mathematics/equivalent-fractions-on-a-number-line-mt_Ep7TDFuYUa): Understand two fractions as equivalent if they are the same size or the same point on a number line; recognise and show families of common equivalent fractions using diagrams - [Equivalent fractions (age 8+)](https://lightmysky.com/learn/mathematics/equivalent-fractions-age-8-mt_FP-mjXaq3B): Generate simple equivalent fractions and explain why they are equivalent using visual fraction models - [Fraction-Decimal Equivalents](https://lightmysky.com/learn/mathematics/fraction-decimal-equivalents-mt_DRlbMok2lT): Recognise and write decimal equivalents of 1/4, 1/2, and 3/4 - [Equivalent fractions (age 9+)](https://lightmysky.com/learn/mathematics/equivalent-fractions-age-9-mt_ebPelt-qAl): Explain why a fraction a/b is equivalent to (n×a)/(n×b) using visual models; use this principle to recognise and generate equivalent fractions, including tenths and hundredths - [Adding fractions (different denominators)](https://lightmysky.com/learn/mathematics/adding-fractions-different-denominators-mt_-V7EnqU7gG): Add and subtract fractions when one denominator is a multiple of the other by rewriting with a common denominator. - [Comparing fractions (age 9+)](https://lightmysky.com/learn/mathematics/comparing-fractions-age-9-mt_SXbZ3bC9z7): Compare and order fractions with different numerators and denominators by creating common denominators/numerators or comparing to a benchmark such as 1/2; justify conclusions with visual models - [Converting tenths to hundredths](https://lightmysky.com/learn/mathematics/converting-tenths-to-hundredths-mt_NaqEP8xDhZ): Express a fraction with denominator 10 as an equivalent fraction with denominator 100 and use this to add fractions with denominators 10 and 100 (e.g. 3/10 + 4/100 = 34/100) - [Decimals for Tenths & Hundredths](https://lightmysky.com/learn/mathematics/decimals-for-tenths-and-hundredths-mt_Vi4Vo5xs_g): Use decimal notation for fractions with denominators 10 or 100; read and write decimal numbers as fractions (e.g. 0.62 = 62/100, 0.71 = 71/100) - [Understanding Percentages](https://lightmysky.com/learn/mathematics/understanding-percentages-mt_ZM9mhHsyYZ): Understand the per cent symbol (%); know that per cent means ‘number of parts per hundred’; write percentages as a fraction with denominator 100 and as a decimal - [Percentage and decimal equivalents](https://lightmysky.com/learn/mathematics/percentage-and-decimal-equivalents-mt_4ubP_RMg9o): Solve problems requiring knowledge of percentage and decimal equivalents of 1/2, 1/4, 1/5, 2/5, 4/5 and fractions with denominators that are multiples of 10 or 25 - [Simplifying Fractions](https://lightmysky.com/learn/mathematics/simplifying-fractions-mt_b7T-CjOYUR): Use common factors to simplify fractions to their simplest form; use common multiples to express fractions with a common denominator - [Adding fractions (unlike denominators)](https://lightmysky.com/learn/mathematics/adding-fractions-unlike-denominators-mt_14T5yPXUq_): Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions to produce a common denominator - [Comparing fractions (age 10+)](https://lightmysky.com/learn/mathematics/comparing-fractions-age-10-mt_PL9VkDwXfh): Compare and order fractions including fractions greater than 1, by converting to common denominators or using benchmarks - [Fraction word problems](https://lightmysky.com/learn/mathematics/fraction-word-problems-mt_e7filQgayF): Solve word problems involving addition and subtraction of fractions with unlike denominators, using visual models and benchmark fractions to estimate and assess reasonableness - [Multiplying fractions (age 10+)](https://lightmysky.com/learn/mathematics/multiplying-fractions-age-10-mt_AabJisinfi): Multiply a fraction or whole number by a fraction, including proper fractions by proper fractions; interpret (a/b) × q as a parts of q partitioned into b equal parts; write answers in simplest form - [Area with Fractions](https://lightmysky.com/learn/mathematics/area-with-fractions-mt_gx6KQK5-Kx): Find the area of a rectangle with fractional side lengths by tiling with unit-fraction squares; show that the area equals the product of the side lengths - [Multiplication as scaling](https://lightmysky.com/learn/mathematics/multiplication-as-scaling-mt_VKW8lOcFaw): Interpret multiplication as scaling (resizing): compare the size of a product to a factor based on the size of the other factor without computing; explain the effect of multiplying by fractions greater than, equal to, or less than 1 - [Real-world fraction multiplication](https://lightmysky.com/learn/mathematics/real-world-fraction-multiplication-mt_lvaSGHwvQ5): Solve real-world problems involving multiplication of fractions and mixed numbers, using visual fraction models or equations - [Fractions of a whole (age 10+)](https://lightmysky.com/learn/mathematics/fractions-of-a-whole-age-10-mt_4Km38F4L-6): Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b); solve word problems involving division of whole numbers leading to fractional or mixed-number answers - [Decimals and fractions (age 10+)](https://lightmysky.com/learn/mathematics/decimals-and-fractions-age-10-mt_4K1dr204Hi): Associate a fraction with division and calculate decimal fraction equivalents for simple fractions (e.g. 3/8 = 0.375); recall and use equivalences between simple fractions, decimals, and percentages in different contexts - [Dividing fractions (unit fractions)](https://lightmysky.com/learn/mathematics/dividing-fractions-unit-fractions-mt_ifPDOYvUqm): Interpret and compute division of a unit fraction by a non-zero whole number (e.g. 1/3 ÷ 4 = 1/12); use visual models and the relationship between multiplication and division to explain the result - [Dividing by Fractions](https://lightmysky.com/learn/mathematics/dividing-by-fractions-mt_1PAWhRhpdg): Interpret and compute division of a positive whole number by a positive unit fraction below 1; use visual models and the relationship between multiplication and division to explain why the quotient is larger than the dividend. - [Dividing unit fractions and whole numbers](https://lightmysky.com/learn/mathematics/dividing-unit-fractions-and-whole-numbers-mt_rCMdwG-YOE): Solve real-world problems involving division of unit fractions by whole numbers and whole numbers by unit fractions, using visual models and equations - [Calculations with fractions, decimals and negatives](https://lightmysky.com/learn/mathematics/calculations-with-fractions-decimals-and-negatives-mt_gNUE4B3vuk): Use the four operations with formal written methods applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative - [Decimals and fractions (age 11+)](https://lightmysky.com/learn/mathematics/decimals-and-fractions-age-11-mt_SBkTGjiZjZ): Work interchangeably with terminating decimals and their corresponding fractions (such as 3.5 and 7/2 or 0.375 and 3/8); convert fluently between the two forms - [Thousandths (age 9+)](https://lightmysky.com/learn/mathematics/thousandths-age-9-mt_kdWoAel3Zl): Recognise and use thousandths; relate them to tenths, hundredths, and their decimal equivalents (e.g. 1/1000 = 0.001, 35/1000 = 0.035) - [Comparing Decimals](https://lightmysky.com/learn/mathematics/comparing-decimals-mt_933BohS9BH): Compare two decimals to hundredths (or up to three decimal places) by reasoning about size using place-value understanding; record with >, =, < - [Decimals to three places](https://lightmysky.com/learn/mathematics/decimals-to-three-places-mt_sA2OvTiech): Solve problems involving numbers with up to three decimal places - [Fractions and percentages of amounts](https://lightmysky.com/learn/mathematics/fractions-and-percentages-of-amounts-mt_J339bO7qLe): Interpret fractions and percentages as operators — find a fraction or percentage of an amount by multiplying, understanding that 'of' means multiply (e.g., 3/4 of 200 = 3/4 × 200 = 150) - [Dividing fractions](https://lightmysky.com/learn/mathematics/dividing-fractions-mt_9Y96vxG_LH): Divide a fraction by a fraction using the 'keep-change-flip' method and visual models; interpret and solve word problems involving division of fractions by fractions ## Number Representation & Place Value 59 topics, ages 5 to 16. https://lightmysky.com/learn/mathematics/areas/number-representation-and-place-value - [Reading and writing numbers to 20](https://lightmysky.com/learn/mathematics/reading-and-writing-numbers-to-20-mt_fR0UtsSREU): Read and write numerals from 0 to 20 - [Number Words to Twenty](https://lightmysky.com/learn/mathematics/number-words-to-twenty-mt_9IzhGUZ30z): Read and write number words from one to twenty - [Reading and writing numbers to 100](https://lightmysky.com/learn/mathematics/reading-and-writing-numbers-to-100-mt_x8TshvbbQT): Read and write numerals from 0 to 100 - [The teen numbers](https://lightmysky.com/learn/mathematics/the-teen-numbers-mt_xmmgAzxe5j): Understand that the teen numbers (11–19) are composed of ten ones and some further ones (early place value) - [A Ten Is Ten Ones](https://lightmysky.com/learn/mathematics/a-ten-is-ten-ones-mt_r0VXbfAmsH): Understand that 10 can be thought of as a bundle of ten ones, called a 'ten' - [Number Words to 100](https://lightmysky.com/learn/mathematics/number-words-to-100-mt_76SPWvdI7r): Read and write numbers to at least 100 in words - [Reading and writing numbers to 120](https://lightmysky.com/learn/mathematics/reading-and-writing-numbers-to-120-mt_XV0B4kWwqL): Count to 120 starting at any number less than 120; read and write numerals to 120 - [The multiples of 10](https://lightmysky.com/learn/mathematics/the-multiples-of-10-mt_zfy1gOEewd): Understand that the multiples of 10 (10, 20, 30 … 90) represent one to nine tens and 0 ones - [The two digits of a two-digit number](https://lightmysky.com/learn/mathematics/the-two-digits-of-a-two-digit-number-mt_THl9GLxwoL): Understand that the two digits of a two-digit number represent amounts of tens and ones - [10 More or 10 Less](https://lightmysky.com/learn/mathematics/10-more-or-10-less-mt_kw7xmp68rU): Mentally find 10 more or 10 less than a given two-digit number without counting - [Comparing and ordering numbers](https://lightmysky.com/learn/mathematics/comparing-and-ordering-numbers-mt_U0waNfD8PB): Compare and order two-digit numbers using the symbols >, =, and <, based on place value understanding - [Place value understanding and number facts](https://lightmysky.com/learn/mathematics/place-value-understanding-and-number-facts-mt_kJGCjnuelW): Use place value understanding and number facts to solve problems - [A Hundred Is Ten Tens](https://lightmysky.com/learn/mathematics/a-hundred-is-ten-tens-mt_8gy7uxRlF6): Understand that 100 can be thought of as a bundle of ten tens, called a 'hundred' - [Odd or Even](https://lightmysky.com/learn/mathematics/odd-or-even-mt_hniI4E-OCE): Determine whether a group of objects (up to 20) has an odd or even number of members - [The multiples of 100](https://lightmysky.com/learn/mathematics/the-multiples-of-100-mt_R2ccrI-nKD): Understand that the multiples of 100 (100–900) each represent a number of hundreds with 0 tens and 0 ones - [The three digits of a three-digit number](https://lightmysky.com/learn/mathematics/the-three-digits-of-a-three-digit-number-mt_aPBzD28_mT): Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones - [Representing Numbers](https://lightmysky.com/learn/mathematics/representing-numbers-mt_9L3NQqgqRd): Identify, represent, and estimate numbers using different representations including the number line - [10 or 100 More or Less](https://lightmysky.com/learn/mathematics/10-or-100-more-or-less-mt_izien3ZX51): Find 10 or 100 more or less than a given number up to 1000 - [Ordering Numbers to 1000](https://lightmysky.com/learn/mathematics/ordering-numbers-to-1000-mt_AQcVRBddko): Compare and order numbers up to 1000 using >, =, and < symbols, based on place-value understanding - [Place Value to 1000](https://lightmysky.com/learn/mathematics/place-value-to-1000-mt_98c2qwEF7Q): Solve number problems and practical problems involving place value of numbers up to 1000 - [Reading and writing numbers to 1000](https://lightmysky.com/learn/mathematics/reading-and-writing-numbers-to-1000-mt_VFsuftfvYM): Read and write numbers to 1000 in numerals, number names, and expanded form - [Negative Numbers](https://lightmysky.com/learn/mathematics/negative-numbers-mt_vXRzMbiPff): Count backwards through zero to include negative numbers - [Place value of each digit](https://lightmysky.com/learn/mathematics/place-value-of-each-digit-mt_jY7uf0Cb7o): Recognise the place value of each digit in a four-digit number (thousands, hundreds, tens, and ones) - [1000 More or Less](https://lightmysky.com/learn/mathematics/1000-more-or-less-mt_9XVFje6Tyr): Find 1000 more or less than a given number - [Numbers to 10,000](https://lightmysky.com/learn/mathematics/numbers-to-10-000-mt_lC_Q5mSL_I): Identify, represent, and estimate numbers up to 10,000 using different representations - [Comparing Large Numbers](https://lightmysky.com/learn/mathematics/comparing-large-numbers-mt_MHaiUd2FLA): Order and compare numbers beyond 1000 - [Roman numerals to 100](https://lightmysky.com/learn/mathematics/roman-numerals-to-100-mt_MewIRdzpzz): Read Roman numerals to 100 (I to C) and understand that the numeral system changed over time to include zero and place value - [Rounding to 10, 100, 1000](https://lightmysky.com/learn/mathematics/rounding-to-10-100-1000-mt_NLSfvB9vUl): Round any number to the nearest 10, 100, or 1000 - [Place Value Problem-Solving](https://lightmysky.com/learn/mathematics/place-value-problem-solving-mt__sMrmOv3bx): Solve number and practical problems involving place value with increasingly large positive numbers - [Place Value × 10 Pattern](https://lightmysky.com/learn/mathematics/place-value-10-pattern-mt_QqG6IdmTSE): Recognise that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right (e.g. 700 ÷ 70 = 10) - [Counting forwards and backwards (age 9+)](https://lightmysky.com/learn/mathematics/counting-forwards-and-backwards-age-9-mt_tAtMET4EIU): Count forwards and backwards in steps of powers of 10 (10, 100, 1000, 10,000, 100,000) for any given number up to 1,000,000 - [Negative numbers in context](https://lightmysky.com/learn/mathematics/negative-numbers-in-context-mt_1KkvzwYxbR): Interpret negative numbers in context (temperature, sea level, bank balance); count forwards and backwards with positive and negative whole numbers, including through zero - [Reading and writing numbers (age 9+)](https://lightmysky.com/learn/mathematics/reading-and-writing-numbers-age-9-mt_JwP9QFv6gQ): Read, write, order, and compare whole numbers up to at least 1,000,000 using base-ten numerals, number names, expanded form, and place-value understanding - [Roman numerals to 1000](https://lightmysky.com/learn/mathematics/roman-numerals-to-1000-mt_To9HdLy8vq): Read Roman numerals to 1000 (M) and recognise years written in Roman numerals - [Rounding Large Numbers](https://lightmysky.com/learn/mathematics/rounding-large-numbers-mt_5NwqN6pf_A): Round any whole number up to 1,000,000 to the nearest 10, 100, 1,000, 10,000, or 100,000 using place-value understanding - [Working with Large Numbers](https://lightmysky.com/learn/mathematics/working-with-large-numbers-mt_xMt1TLTs--): Solve number and practical problems involving reading, writing, ordering, comparing, and rounding whole numbers up to 1,000,000 - [Negative Numbers Across Zero](https://lightmysky.com/learn/mathematics/negative-numbers-across-zero-mt_RVK655t391): Use negative numbers in context (temperature, finance, sea level); calculate intervals across zero - [Integer powers and roots](https://lightmysky.com/learn/mathematics/integer-powers-and-roots-mt_hCVPYlF-7Y): Use integer powers and associated real roots (square, cube, and higher); recognise powers of 2, 3, 4, and 5; distinguish between exact representations of roots and their decimal approximations - [Number Sets & Infinity](https://lightmysky.com/learn/mathematics/number-sets-and-infinity-mt_b4lbTOJYwI): Appreciate the infinite nature of the sets of integers, real numbers, and rational numbers; position integers on a number line and distinguish between rational and irrational numbers - [Recurring Decimals as Exact Fractions](https://lightmysky.com/learn/mathematics/recurring-decimals-as-exact-fractions-mt_Mb-sXIZDWd): A decimal that repeats forever is still an exact fraction, and multiplying by a power of ten before subtracting is what pins it down. This is the step that makes the claim about rational numbers repeating into something a learner can act on rather than believe. - [Place Value × 10 and ÷ 10](https://lightmysky.com/learn/mathematics/place-value-10-and-10-mt_EDgw64OmfA): Recognise that in a multi-digit number, a digit in one place represents 10 times as much as in the place to its right and 1/10 of what it represents in the place to its left - [Patterns with Powers of Ten](https://lightmysky.com/learn/mathematics/patterns-with-powers-of-ten-mt_HLUqHJ9Y7n): Explain patterns in zeros when multiplying by powers of 10 and in decimal-point placement when multiplying/dividing by a power of 10; use whole-number exponents to denote powers of 10 (e.g. 10³ = 1000) - [Reading and writing numbers (age 10+)](https://lightmysky.com/learn/mathematics/reading-and-writing-numbers-age-10-mt_Ac7oMWhyPw): Read, write, and compare decimals to thousandths using base-ten numerals, number names, and expanded form; compare using >, =, < based on place-value meaning - [Reading and writing numbers to 10,000,000](https://lightmysky.com/learn/mathematics/reading-and-writing-numbers-to-10-000-000-mt_Gag_h98jWP): Read, write, order, and compare numbers up to 10,000,000 and determine the value of each digit - [Reading Decimal Places](https://lightmysky.com/learn/mathematics/reading-decimal-places-mt_lNGpnILM5C): Identify the value of each digit in numbers given to three decimal places (e.g. in 4.378, the 7 represents 7 hundredths) - [Rounding to a required degree of accuracy](https://lightmysky.com/learn/mathematics/rounding-to-a-required-degree-of-accuracy-mt__casygEB85): Round decimals to any place using place-value understanding; round any whole number to a required degree of accuracy - [Numbers to ten million](https://lightmysky.com/learn/mathematics/numbers-to-ten-million-mt_XNmGwNggdU): Solve number and practical problems involving reading, writing, ordering, comparing, rounding, and negative numbers up to 10,000,000 - [Absolute value as distance from zero](https://lightmysky.com/learn/mathematics/absolute-value-as-distance-from-zero-mt_RWUY7_IXvw): Understand the absolute value of a rational number as its distance from zero on the number line; interpret absolute value as magnitude in real-world contexts; distinguish absolute value comparisons from ordering statements - [Place value for decimals, measures and integers](https://lightmysky.com/learn/mathematics/place-value-for-decimals-measures-and-integers-mt_PsylzZ9lHW): Understand and use place value for decimals, measures, and integers of any size; extend the number system to include all positive and negative integers, decimals, and fractions on a single number line - [Order integers, decimals and fractions on a number line](https://lightmysky.com/learn/mathematics/order-integers-decimals-and-fractions-on-a-number-line-mt_uDJY0X0hgo): Order positive and negative integers, decimals, and fractions on a number line; use the symbols =, ≠, <, >, ≤, ≥ to compare values including negative numbers and mixed representations - [Rounding to decimal places and significant figures](https://lightmysky.com/learn/mathematics/rounding-to-decimal-places-and-significant-figures-mt_U9sme87C32): Round numbers and measures to an appropriate degree of accuracy, including a specified number of decimal places or significant figures. - [Powers of Ten Notation](https://lightmysky.com/learn/mathematics/powers-of-ten-notation-mt_bO-njVOige): Interpret and compare positive numbers in standard form A × 10ⁿ, where 1 ≤ A < 10 and n is an integer; convert between ordinary notation and standard form - [Zero and Negative Indices](https://lightmysky.com/learn/mathematics/zero-and-negative-indices-mt_jAtXBSVqkj): Extend the index laws so that any non-zero number to the power 0 is 1 and a negative index means the reciprocal. - [Fractional Indices and Roots](https://lightmysky.com/learn/mathematics/fractional-indices-and-roots-mt_fdK-nbQcBu): Read x^(1/n) as the nth root and x^(m/n) as a root and a power together, and move between root form and index form. - [Standard Form](https://lightmysky.com/learn/mathematics/standard-form-mt_GbZZ2xjBTt): Write very large and very small numbers as a number between 1 and 10 times a power of ten, and calculate with them by hand and on a calculator. - [Surds: Simplifying and Rationalising](https://lightmysky.com/learn/mathematics/surds-simplifying-and-rationalising-mt_cqr0crUUXS): Simplify surds by taking out square factors, add and multiply them, and clear a surd from the bottom of a fraction. - [Bounds and error in rounded values](https://lightmysky.com/learn/mathematics/bounds-and-error-in-rounded-values-mt_VVn1IXjkzn): Use approximation through rounding to estimate answers and calculate possible resulting errors expressed using inequality notation a < x ≤ b; understand upper and lower bounds of rounded values - [Upper and Lower Bounds](https://lightmysky.com/learn/mathematics/upper-and-lower-bounds-mt_HaoOqujpRU): Give the upper and lower bounds of a rounded measurement and write the error interval, using the half-unit rule for the accuracy stated. - [Calculating with Bounds in Compound Measures](https://lightmysky.com/learn/mathematics/calculating-with-bounds-in-compound-measures-mt_zfMz7x8dlf): Combine bounds in a calculation by picking the values that give the greatest and least results, including for speed, density and other compound measures. ## Data & Statistics 49 topics, ages 5 to 22. https://lightmysky.com/learn/mathematics/areas/data-and-statistics - [Sorting into categories](https://lightmysky.com/learn/mathematics/sorting-into-categories-mt_xppl18avyY): Classify objects into given categories, count the number in each category, and sort the categories by count - [Data collection and display vocabulary](https://lightmysky.com/learn/mathematics/data-collection-and-display-vocabulary-mt_u5HkSxZECM): Read, write, and use the vocabulary of data collection and display — data, tally, tally chart, frequency, frequency table, survey, pictogram, bar chart, axis/axes, scale, label, category, discrete data, continuous data, line graph, pie chart — and apply these terms when collecting, organising, and presenting data - [Sorting data into categories](https://lightmysky.com/learn/mathematics/sorting-data-into-categories-mt_ylXdiVRAYv): Organise and represent data with up to three categories by counting objects in each category and sorting categories by quantity - [Pictograms and tally charts](https://lightmysky.com/learn/mathematics/pictograms-and-tally-charts-mt_c29FaCTNsx): Interpret and construct simple pictograms, tally charts, block diagrams, and simple tables - [Picture & bar graphs](https://lightmysky.com/learn/mathematics/picture-and-bar-graphs-mt_VhBH8wrFC6): Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories; solve put-together, take-apart, and compare problems using information presented in a bar graph - [Scaled picture graphs and bar graphs](https://lightmysky.com/learn/mathematics/scaled-picture-graphs-and-bar-graphs-mt_r8XnXwRA6g): Draw a scaled picture graph and a scaled bar graph to represent a data set; solve one- and two-step comparison, sum, and difference problems using bar charts, pictograms, and tables - [Bar charts and time graphs](https://lightmysky.com/learn/mathematics/bar-charts-and-time-graphs-mt_ChjMU2GDJa): Interpret and present discrete and continuous data using appropriate graphical methods, including bar charts and time graphs - [Reading and comparing bar and line graphs](https://lightmysky.com/learn/mathematics/reading-and-comparing-bar-and-line-graphs-mt_H6LlpWgEYS): Solve comparison, sum, and difference problems using information presented in a bar graph - [Reading tables](https://lightmysky.com/learn/mathematics/reading-tables-mt_Cqm8iy48UI): Complete, read, and interpret information in tables, including timetables - [Statistical analysis vocabulary](https://lightmysky.com/learn/mathematics/statistical-analysis-vocabulary-mt_fmm-P17Vka): Read, write, and use the vocabulary of statistical analysis (mean, median, mode, range, frequency, data, sample, average, chart, table, graph, pie chart, scatter graph, correlation) with an understanding of what each term describes - [Statistical Questions](https://lightmysky.com/learn/mathematics/statistical-questions-mt_4Ax4JPXJWD): Tell a statistical question from one with a single fixed answer: a statistical question anticipates variability ('How tall are the students in my class?' rather than 'How tall am I?') and is answered by collecting many values - [Scatter Graphs](https://lightmysky.com/learn/mathematics/scatter-graphs-mt_BdAeZJUOir): Plot bivariate data on a scatter graph with correctly labelled axes and appropriate scales; describe the correlation (positive, negative, none) and draw an estimated line of best fit where appropriate - [Populations, Samples and Bias](https://lightmysky.com/learn/mathematics/populations-samples-and-bias-mt_c7knRZy3Vt): Tell a population from a sample, judge whether a way of collecting data is fair, and say how a biased sample bends the conclusion. - [Calculating the mean](https://lightmysky.com/learn/mathematics/calculating-the-mean-mt_Cg8VPguS_V): Calculate and interpret the mean as an average of a data set - [Two-Way Tables](https://lightmysky.com/learn/mathematics/two-way-tables-mt_oD1Rgg52n3): Construct and read two-way tables that sort the same people by two categories at once (e.g., year group against preferred sport); use row and column totals and relative frequencies to judge whether the two categories are associated - [Interpreting categorical data](https://lightmysky.com/learn/mathematics/interpreting-categorical-data-mt_1VSfm9yiLn): Interpret categorical data by asking and answering questions about totals, how many in each category, and how many more or less one category has than another - [Line plots with fractions of a unit](https://lightmysky.com/learn/mathematics/line-plots-with-fractions-of-a-unit-mt_IHipFGTFEY): Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8); use operations on fractions to solve problems involving data in line plots (e.g. redistribute total equally) - [Pie charts and line graphs](https://lightmysky.com/learn/mathematics/pie-charts-and-line-graphs-mt_bESTSBB0wK): Interpret and construct pie charts and line graphs; use these to solve problems - [Stratified Sampling](https://lightmysky.com/learn/mathematics/stratified-sampling-mt_CWzOOk0609): Work out how many to take from each group so that a sample matches the proportions in the population, and round the results sensibly. - [Sampling Methods and Their Trade-offs](https://lightmysky.com/learn/mathematics/sampling-methods-and-their-trade-offs-mt_aSsRQvnKa6): Compare simple random, systematic, stratified, quota and opportunity sampling by what each costs to run and what bias each risks, and describe the sampling frame a method needs. - [Frequency tables, bar charts and pie charts](https://lightmysky.com/learn/mathematics/frequency-tables-bar-charts-and-pie-charts-mt_eKJG-0eC6D): Construct and interpret frequency tables, bar charts, pie charts, pictograms, and vertical line charts for both categorical and grouped numerical data, choosing appropriate representations for the data type - [Comparing data sets with averages and range](https://lightmysky.com/learn/mathematics/comparing-data-sets-with-averages-and-range-mt_XSXnTQoQ4l): Describe, interpret, and compare distributions of a single variable using appropriate measures of central tendency (mean, median, mode) and spread (range), including the effect of outliers - [Scatter Graphs & Correlation](https://lightmysky.com/learn/mathematics/scatter-graphs-and-correlation-mt_8atyuvPUZc): Describe simple mathematical relationships between two variables using scatter graphs, identify positive, negative, or no correlation, and use a line of best fit to make predictions - [Averages from frequency tables](https://lightmysky.com/learn/mathematics/averages-from-frequency-tables-mt_5WF_Rhdb1X): Find the mean, median, mode and range from a frequency table by working with the counts instead of listing every value. - [Estimating the mean from grouped data](https://lightmysky.com/learn/mathematics/estimating-the-mean-from-grouped-data-mt_3PHP7NLZAw): Estimate a mean from grouped data using class midpoints, say why the answer is only an estimate, and identify the modal class and the class holding the median. - [Lines of Best Fit: Prediction and Its Limits](https://lightmysky.com/learn/mathematics/lines-of-best-fit-prediction-and-its-limits-mt_CnYCx1a0j5): Use a line of best fit to predict inside the range of the data, say why predicting far outside it is unsafe, and separate correlation from cause. - [Quartiles and the Interquartile Range](https://lightmysky.com/learn/mathematics/quartiles-and-the-interquartile-range-mt_vN10d1I9dA): Find the median, the quartiles and the interquartile range of a data set, and use the interquartile range to compare how spread out two sets are. - [Box Plots and Comparing Distributions](https://lightmysky.com/learn/mathematics/box-plots-and-comparing-distributions-mt_G_SnfEllzQ): Draw a box plot from a five-figure summary, and compare two distributions by their medians and their spread in words about the situation. - [Cumulative Frequency Curves](https://lightmysky.com/learn/mathematics/cumulative-frequency-curves-mt_5U70qgbnST): Build a cumulative frequency table from grouped data, plot the curve at the upper bound of each class, and read off the median, the quartiles and how many lie above a value. - [Histograms and Frequency Density](https://lightmysky.com/learn/mathematics/histograms-and-frequency-density-mt_0QAOgzD-Qw): Draw and read histograms where the classes have different widths, using frequency density so that area, not height, shows the frequency. - [Standard deviation and variance](https://lightmysky.com/learn/mathematics/standard-deviation-and-variance-mt_eNpow5tIX2): Measure spread by how far the values sit from the mean on average, using squared deviations so that distances on both sides count. Compare data sets by mean and standard deviation together. - [Sampling Distributions and the Central Limit Theorem](https://lightmysky.com/learn/mathematics/sampling-distributions-and-the-central-limit-theorem-mt_nugaEeUSii): For independent observations from one population with finite mean and finite positive variance, sample means centre on the population mean and have standard error σ / √n. Their standardised distribution approaches a standard normal one as n grows. - [Setting Up a Hypothesis Test](https://lightmysky.com/learn/mathematics/setting-up-a-hypothesis-test-mt_JmpLNHK0Nf): State a null and an alternative hypothesis about a population, fix a significance level before looking at the data, and decide in advance what result would count as surprising enough to reject the null. - [Testing a Binomial Proportion](https://lightmysky.com/learn/mathematics/testing-a-binomial-proportion-mt_ZJ8MTLYYKE): Test a claim about a proportion by finding the probability of a result at least as extreme under the null model, comparing it with the significance level, and writing the conclusion in the words of the situation. - [Correlation and the Product-Moment Coefficient](https://lightmysky.com/learn/mathematics/correlation-and-the-product-moment-coefficient-mt_Rs_Z0P7ELJ): Summarise the strength and direction of a linear relationship in one number between -1 and 1, judge it against a critical value for the sample size, and keep it apart from any claim about cause. - [The Regression Line and Making Predictions](https://lightmysky.com/learn/mathematics/the-regression-line-and-making-predictions-mt_tsjctRmkyE): Fit the least-squares line through the data, read its gradient and intercept in the units of the problem, and predict only inside the range the data covers. - [The Law of Large Numbers and the Central Limit Theorem](https://lightmysky.com/learn/mathematics/the-law-of-large-numbers-and-the-central-limit-theorem-mt_NknG3L5rQp): Averages settle on the population mean as samples grow, and their distribution approaches a normal one whatever the population looked like. The two results are what makes inference possible at all. - [Point Estimation: Bias, Variance and Consistency](https://lightmysky.com/learn/mathematics/point-estimation-bias-variance-and-consistency-mt_I374r2UU_i): An estimator is a random variable computed from a sample. Judge it by whether it is centred on the truth, how much it varies, and whether it settles down as the sample grows. - [Maximum Likelihood Estimation](https://lightmysky.com/learn/mathematics/maximum-likelihood-estimation-mt_ZRTrcy7-KV): Write the probability of the observed data as a function of the unknown parameter and choose the parameter that makes the data most likely. Logs turn the product into a sum you can differentiate. - [Confidence Intervals for Means and Proportions](https://lightmysky.com/learn/mathematics/confidence-intervals-for-means-and-proportions-mt_Pg2fPDW1sn): Turn the sampling distribution around to produce an interval and a stated coverage. The t-distribution replaces the normal when the population standard deviation is estimated. - [Hypothesis Tests for a Mean with the t-Distribution](https://lightmysky.com/learn/mathematics/hypothesis-tests-for-a-mean-with-the-t-distribution-mt_ig2o39hAwh): Test a population mean using a one-sample t statistic when the population standard deviation is unknown. Check the sampling model, use the planned tails and degrees of freedom, and interpret the p-value assuming the null hypothesis. - [Errors, Power and the Design of a Test](https://lightmysky.com/learn/mathematics/errors-power-and-the-design-of-a-test-mt_D7ZXbvSD0l): A Type I error rejects a true null hypothesis. A Type II error fails to reject a false null. Power is the probability of rejecting the null for a specified real effect, and depends on the test, significance level, variability and sample size. - [Comparing Two Populations](https://lightmysky.com/learn/mathematics/comparing-two-populations-mt_v34rZkMfqr): Two means or two proportions, with the paired design treated separately because pairing removes a source of variation rather than adding a group. - [Chi-Square Tests for Categorical Data](https://lightmysky.com/learn/mathematics/chi-square-tests-for-categorical-data-mt_yxc87pzaXX): Compare observed counts with the counts a model predicts. The same statistic serves goodness of fit, independence and homogeneity, with the degrees of freedom telling them apart. - [One-Way Analysis of Variance](https://lightmysky.com/learn/mathematics/one-way-analysis-of-variance-mt_WYGK3Ktnqz): Compare several group means at once by splitting the total variation into a between-groups and a within-groups part. The F ratio asks whether the split is more lopsided than chance allows. - [Simple Linear Regression and Inference on the Slope](https://lightmysky.com/learn/mathematics/simple-linear-regression-and-inference-on-the-slope-mt_W2huNTKsMj): State regression as a model with an error term, estimate its slope by least squares, then test and bound that slope. Residual plots are the check on whether the model was reasonable. - [Multiple Regression and Model Diagnostics](https://lightmysky.com/learn/mathematics/multiple-regression-and-model-diagnostics-mt_TUx1tDF0jv): Several predictors at once, where a coefficient reads as the effect of one variable with the others held fixed. Correlated predictors, influential points and overfitting are the standing hazards. - [How a Chart Can Mislead](https://lightmysky.com/learn/mathematics/how-a-chart-can-mislead-mt_z7yMoRWipf): The same numbers can be drawn to argue opposite things. A vertical axis that does not start at zero, a broken scale, unequal category widths, a 3-D effect, or a picture whose area grows faster than its height all change what a reader sees without changing a single value. - [Who Was Asked: Samples and the People Left Out](https://lightmysky.com/learn/mathematics/who-was-asked-samples-and-the-people-left-out-mt_NQju6n-CKb): A statistic is about the people who answered, not the people you care about. Covers the difference between a population and a sample, why the method of asking decides who is missing, and how a poll of a self-selected group can be precisely wrong. ## Mathematical Thinking 65 topics, ages 5 to 24. https://lightmysky.com/learn/mathematics/areas/mathematical-thinking - [Early Maths Vocabulary](https://lightmysky.com/learn/mathematics/early-maths-vocabulary-mt_lMz9nAs7VO): Use mathematical words carefully when counting, comparing, and describing shapes and positions - [Finding efficient methods](https://lightmysky.com/learn/mathematics/finding-efficient-methods-mt_RFeVlw0QvX): Notice when a calculation or pattern repeats and use this to count more efficiently or predict results - [Hands-On Problem Solving](https://lightmysky.com/learn/mathematics/hands-on-problem-solving-mt_20WfHhnL39): Select and use familiar tools (concrete objects, fingers, ten frames) to help solve a mathematical problem - [Making Sense of Problems](https://lightmysky.com/learn/mathematics/making-sense-of-problems-mt_WkKkb7W9Qd): Make sense of a problem by identifying what is being asked, choosing concrete objects or pictures to represent the situation, and explaining a pathway to the solution - [Real-World to Maths Connections](https://lightmysky.com/learn/mathematics/real-world-to-maths-connections-mt_IM1G_7QzTa): Move between a real-world situation and a mathematical representation using concrete objects, drawings, diagrams, tables, number sentences, or bar models - [Showing Your Working](https://lightmysky.com/learn/mathematics/showing-your-working-mt_nyK25mNOeR): Show and tell how a mathematical answer was found using objects, drawings, and spoken words - [Spotting mathematical patterns](https://lightmysky.com/learn/mathematics/spotting-mathematical-patterns-mt_uG2mjHFOlO): Notice simple patterns and structures: spot that changing order doesn't change the total, and recognise how numbers relate to each other - [Using objects to model real problems](https://lightmysky.com/learn/mathematics/using-objects-to-model-real-problems-mt_fLOkq-HfPB): Use objects, drawings, or simple number sentences to represent a real-world situation (early mathematical modelling) - [Choosing a tool or representation](https://lightmysky.com/learn/mathematics/choosing-a-tool-or-representation-mt_zMEvtigoM3): Select and use appropriate tools and representations (number lines, hundred squares, rulers, part-whole models) to support problem-solving - [Connecting Representations](https://lightmysky.com/learn/mathematics/connecting-representations-mt_EBYGhd8X3x): Move between real-world situations, drawings, and number sentences, explaining how each representation connects to the others (quantitative reasoning) - [Explaining Mathematical Reasoning](https://lightmysky.com/learn/mathematics/explaining-mathematical-reasoning-mt_kJ5wYzO8qC): With teacher prompting, explain and justify mathematical reasoning using drawings, number sentences, or words - [Generalising Patterns](https://lightmysky.com/learn/mathematics/generalising-patterns-mt_fZTn0W_iZR): Recognise and use repeated reasoning to generalise: spot calculation patterns, describe rules for sequences, and predict results using known mathematical facts - [Precise Maths Communication](https://lightmysky.com/learn/mathematics/precise-maths-communication-mt_TMoHjMhRS2): Communicate with mathematical precision: use correct vocabulary, specify units, and use symbols accurately - [Using structure to calculate efficiently](https://lightmysky.com/learn/mathematics/using-structure-to-calculate-efficiently-mt_DW2D1c0fKx): Look for and use mathematical structure: apply properties of operations, place-value patterns, and relationships between shapes to solve problems efficiently - [Choosing the right strategy](https://lightmysky.com/learn/mathematics/choosing-the-right-strategy-mt_mQcWGh02no): Choose the best tool for each task, such as mental maths, written working, a number line or a ruler, and explain why it fits. - [Extending Table Patterns](https://lightmysky.com/learn/mathematics/extending-table-patterns-mt_2jbUekyTu4): Recognise and use repeated reasoning to generalise: extend multiplication table patterns, derive unknown facts from known ones, and describe rules for sequences - [Moving between situations, diagrams and equations](https://lightmysky.com/learn/mathematics/moving-between-situations-diagrams-and-equations-mt_dXq9VWm31W): Move fluently between real-world situations, diagrams, and symbolic equations involving three-digit numbers and fractions, explaining what each part represents - [Precision in mathematical language and notation](https://lightmysky.com/learn/mathematics/precision-in-mathematical-language-and-notation-mt_HY8Yycu_rz): Communicate with mathematical precision: use correct place-value and fraction vocabulary, specify units in measurement answers, and use notation accurately - [Using structure in numbers and shapes](https://lightmysky.com/learn/mathematics/using-structure-in-numbers-and-shapes-mt_FAXjFkgG6X): Look for and use mathematical structure: apply place-value patterns to three-digit operations, use multiplication/division relationships, and exploit shape properties to classify - [Choosing mathematical tools](https://lightmysky.com/learn/mathematics/choosing-mathematical-tools-mt_d-WZC2OyMB): Select and use appropriate tools and representations strategically: choose between mental, written, and diagrammatic methods; use calculators for checking; select fraction models suited to the task - [Generalising from repeated reasoning](https://lightmysky.com/learn/mathematics/generalising-from-repeated-reasoning-mt_aivrWs6jrS): Recognise and use repeated reasoning to generalise: extend patterns in times tables and equivalent fractions, derive unknown facts from known facts efficiently, describe general rules - [Mathematical Precision](https://lightmysky.com/learn/mathematics/mathematical-precision-mt_mywsN77hGZ): Communicate with mathematical precision: use correct fraction/decimal vocabulary, name angle types accurately, specify units in measurement and money, and use notation (=, <, >, ÷, ×) correctly - [Moving between diagrams and equations with fractions](https://lightmysky.com/learn/mathematics/moving-between-diagrams-and-equations-with-fractions-mt_3VmBdlAeOZ): Move fluently between real-world situations, diagrams, number lines, and symbolic equations involving multiplication, fractions, and decimals, explaining what each representation shows - [Using Mathematical Structure](https://lightmysky.com/learn/mathematics/using-mathematical-structure-mt__Itf4aQZUj): Look for and use mathematical structure: exploit place-value patterns for ×10/×100, use the distributive property to break apart multiplications, apply fraction equivalence to compare and compute, use shape properties to classify quadrilaterals - [Choosing representations strategically](https://lightmysky.com/learn/mathematics/choosing-representations-strategically-mt_IwEOCN6bL1): Select and use tools and representations strategically: choose between mental methods, formal written methods, protractors, fraction strips, and diagrams based on the demands of the problem - [Fractions, Decimals & Percentages](https://lightmysky.com/learn/mathematics/fractions-decimals-and-percentages-mt_hlGKg5M7qJ): Look for and use mathematical structure: exploit the relationship between fractions, decimals, and percentages; use factor pairs to simplify multiplication; apply angle facts to find unknowns; use properties of regular polygons systematically - [Moving between bar models, number lines and equations](https://lightmysky.com/learn/mathematics/moving-between-bar-models-number-lines-and-equations-mt_CUmjcE7W6c): Move fluently between real-world situations, diagrams, number lines, bar models, and symbolic equations involving multi-digit multiplication, fractions, decimals, and percentages, explaining connections between representations - [Precise Maths Vocabulary](https://lightmysky.com/learn/mathematics/precise-maths-vocabulary-mt_xfwv0M83mJ): Communicate with mathematical precision: use correct vocabulary for primes, factors, multiples, angle types, and polygon regularity; specify units including cm², m³, °; use notation for squares/cubes and percentages accurately - [Reasoning with Equivalences](https://lightmysky.com/learn/mathematics/reasoning-with-equivalences-mt_jCy07DyBNU): Recognise and use repeated reasoning to generalise: extend patterns in equivalent fractions and percentage conversions, derive unknown facts from known facts, describe general rules for sequences and predict terms - [Choosing Maths Tools](https://lightmysky.com/learn/mathematics/choosing-maths-tools-mt_ZJC7JnnPCu): Select and use tools and representations strategically: choose between mental methods, formal written methods, algebraic approaches, coordinate grids, and technology based on the demands of the problem - [Using structure in shapes and expressions](https://lightmysky.com/learn/mathematics/using-structure-in-shapes-and-expressions-mt_hImiKNiaNh): Look for and use mathematical structure: exploit the hierarchy of 2-D shapes to deduce properties; use order of operations and algebraic structure to simplify expressions; connect fraction–decimal–percentage equivalences; use ratio structure to solve proportion problems efficiently - [Generalising with repeated reasoning](https://lightmysky.com/learn/mathematics/generalising-with-repeated-reasoning-mt_iyovKgZC1q): Recognise and use repeated reasoning to generalise: describe algebraic rules for nth terms, use properties of operations to simplify, and verify generalisations with specific cases - [Connecting maths to real life](https://lightmysky.com/learn/mathematics/connecting-maths-to-real-life-mt_f67qGDhyfi): Represent real-world problems with number sentences, bar models, or diagrams, and interpret the mathematical result back in context - [Guided Multi-Step Problem Solving](https://lightmysky.com/learn/mathematics/guided-multi-step-problem-solving-mt_O_UOTiMvT_): With teacher guidance, make sense of multi-step and more complex problems by planning a pathway to the solution, identifying relevant information, and choosing appropriate operations - [Justifying mathematical reasoning](https://lightmysky.com/learn/mathematics/justifying-mathematical-reasoning-mt_pyMD_SIiYO): Construct and follow multi-step mathematical arguments; identify errors in reasoning and explain why a method works or does not work - [Multi-Step Problem Solving](https://lightmysky.com/learn/mathematics/multi-step-problem-solving-mt_RKeheOL9uo): With teacher support, make sense of multi-step problems involving larger numbers or mixed operations by breaking them into parts, choosing strategies, and checking answers for reasonableness — children at this stage are developing the habit with guidance; independent strategy evaluation comes later - [Solving money measurement and time problems](https://lightmysky.com/learn/mathematics/solving-money-measurement-and-time-problems-mt_iQYPw8bMfN): Model real-world problems involving measurement, money, and time by choosing appropriate representations and interpreting results in context - [Justifying mathematical reasoning (age 8+)](https://lightmysky.com/learn/mathematics/justifying-mathematical-reasoning-age-8-mt_a3dov8CZkq): Construct and present multi-step mathematical arguments; critique the reasoning of others and explain clearly why a method works or fails - [Modelling with multiplication and fractions](https://lightmysky.com/learn/mathematics/modelling-with-multiplication-and-fractions-mt_K3R0yaHVcx): Model real-world problems involving multiplication, area, fractions, and unit conversion by choosing appropriate representations and interpreting mathematical results in context - [Real-World Maths Modelling](https://lightmysky.com/learn/mathematics/real-world-maths-modelling-mt_8xVHooT4aI): Model real-world problems involving scaling, unit conversion, area/perimeter, and percentage by selecting appropriate mathematical representations and interpreting results in context - [Advanced Maths Vocabulary](https://lightmysky.com/learn/mathematics/advanced-maths-vocabulary-mt_3rTIyJDw7-): Communicate with mathematical precision: use correct vocabulary for ratio, proportion, algebra, volume, coordinate geometry, and circle parts; specify units including cm³, m³, and miles/km; use notation for algebraic expressions and order of operations accurately - [Real-World Mathematical Modelling](https://lightmysky.com/learn/mathematics/real-world-mathematical-modelling-mt_yqZiX6vS7D): Model real-world problems involving ratio, scale, volume, unit conversion, and proportional reasoning with appropriate tools, diagrams, or equations - [Moving between representations of a relationship](https://lightmysky.com/learn/mathematics/moving-between-representations-of-a-relationship-mt_9KmVWCuh5_): Move fluently between real-world situations, diagrams, coordinate grids, algebraic expressions, tables, and symbolic equations involving fractions, ratio, and algebra, explaining connections between representations - [Choosing a Model and Naming What It Ignores](https://lightmysky.com/learn/mathematics/choosing-a-model-and-naming-what-it-ignores-mt_UtK9aauJ6e): Set up a mathematical model for a real situation, state the assumptions it rests on, and say which of them would break the answer if they were wrong. - [Algebraic Proof with Odd, Even and Consecutive Numbers](https://lightmysky.com/learn/mathematics/algebraic-proof-with-odd-even-and-consecutive-numbers-mt_p9bajOC17q): Prove a general claim about integers by naming them algebraically, expanding, and showing the required factor is always there. - [Multi-Step Problem Solving (age 8+)](https://lightmysky.com/learn/mathematics/multi-step-problem-solving-age-8-mt_SmghasIvbT): Make sense of multi-step problems involving four operations, fractions, and area/volume by identifying sub-steps, choosing a strategy, and monitoring progress - [Building and critiquing a mathematical argument](https://lightmysky.com/learn/mathematics/building-and-critiquing-a-mathematical-argument-mt_2ESZh70NyS): Construct and present logical mathematical arguments involving multiple steps; critique others' reasoning about fractions, angles, or calculations and clearly explain errors or alternative methods - [Complex Multi-Step Problems](https://lightmysky.com/learn/mathematics/complex-multi-step-problems-mt_GpltXPoaoc): Make sense of complex multi-step problems involving large numbers, fractions, decimals, and percentages by analysing what is known and unknown, planning multi-step strategies, and evaluating reasonableness through estimation and inverse operations - [Advanced Multi-Step Problems](https://lightmysky.com/learn/mathematics/advanced-multi-step-problems-mt_FieL-vVTI_): Make sense of complex multi-step problems involving ratio, proportion, algebra, negative numbers, and all four operations with fractions and decimals by analysing given and unknown quantities, planning solution strategies, and evaluating reasonableness using estimation and inverse operations - [Constructing mathematical arguments](https://lightmysky.com/learn/mathematics/constructing-mathematical-arguments-mt_j5YqQnN6xe): Construct and present logical mathematical arguments involving multiple steps and formal reasoning; critique others' reasoning about fractions, algebra, ratio, or geometry and clearly explain errors or alternative approaches - [Using a Calculator Well](https://lightmysky.com/learn/mathematics/using-a-calculator-well-mt_FZE80D2jS_): Use a calculator to get accurate answers and interpret them sensibly: key in multi-step calculations respecting the order of operations, use brackets and memory, round the display appropriately, and sanity-check results by estimating first - [From Examples to a General Argument](https://lightmysky.com/learn/mathematics/from-examples-to-a-general-argument-mt_vm7cigeRtg): Move from checking a pattern on a few cases to an argument that covers every case, using a letter to stand for any number. - [Finding a Counterexample](https://lightmysky.com/learn/mathematics/finding-a-counterexample-mt_EoLyw6eP87): Disprove a claim by producing one case where it fails, and see why a single case is enough to sink a general statement. - [Proof by Deduction and Exhaustion](https://lightmysky.com/learn/mathematics/proof-by-deduction-and-exhaustion-mt_QNWvbkg04f): Argue from what is given to what is claimed with every step justified, and settle a claim about a small finite set by checking every case in it. - [Proof by Contradiction and Counterexample](https://lightmysky.com/learn/mathematics/proof-by-contradiction-and-counterexample-mt_pbFTv4v6k8): Kill a general claim with a single counterexample, and prove one by assuming the opposite and following it until something impossible appears. - [Statements, Quantifiers and Negation](https://lightmysky.com/learn/mathematics/statements-quantifiers-and-negation-mt_S5NvyPxEZ_): Read mathematical statements as objects: connectives, the two quantifiers, and the order they appear in. Negating a quantified statement correctly is the skill everything else rests on. - [Sets and Functions in the Language of Proof](https://lightmysky.com/learn/mathematics/sets-and-functions-in-the-language-of-proof-mt_XAcHX_3DVz): Set operations, images and preimages, and what injective, surjective and bijective actually require. Proofs at this level are mostly element-chasing arguments in this language. - [Proof by Induction and Strong Induction](https://lightmysky.com/learn/mathematics/proof-by-induction-and-strong-induction-mt_7lvC02JBOC): Prove a statement for every natural number by settling a base case and showing each case forces the next. Strong induction assumes all earlier cases when one predecessor is not enough. - [Equivalence Relations and Partitions](https://lightmysky.com/learn/mathematics/equivalence-relations-and-partitions-mt_wgEupYwEUF): A reflexive, symmetric and transitive relation divides a set into classes with no overlap. Modular arithmetic, quotient constructions and much of later algebra use this same idea. - [Countable and Uncountable Sets](https://lightmysky.com/learn/mathematics/countable-and-uncountable-sets-mt_VTHPloNBbJ): Compare infinite sets by pairing their elements. The rationals pair with the naturals; the reals do not, and the diagonal argument shows why. - [Reading a Mathematics Paper](https://lightmysky.com/learn/mathematics/reading-a-mathematics-paper-mt_pTF-HzLWE6): Identify the main theorem, its hypotheses and its conclusion. Use examples to interpret the claim, locate needed definitions, and trace how lemmas support the proof. - [Writing a Proof Someone Else Can Check](https://lightmysky.com/learn/mathematics/writing-a-proof-someone-else-can-check-mt_nZ-zMzj18N): A written proof is a claim about what a reader can verify, not a record of how it was found. Announced structure, named hypotheses and explicit quantifiers are what make it checkable. - [Definitions as Design Decisions](https://lightmysky.com/learn/mathematics/definitions-as-design-decisions-mt_b6ip8eP0-d): A definition specifies which objects a term includes. Comparing candidate definitions shows how their clauses affect equality, edge cases and the conclusions that follow. - [Counterexamples and Testing a Conjecture](https://lightmysky.com/learn/mathematics/counterexamples-and-testing-a-conjecture-mt_SjA4jqNkaM): Before proving a statement, spend real effort trying to break it. A counterexample settles the question outright, and a failed attempt usually shows which hypothesis the proof will need. - [Open Problems as a Genre](https://lightmysky.com/learn/mathematics/open-problems-as-a-genre-mt_qBWcaUWG9J): An open problem is a question stated precisely enough to be attacked and known to have resisted attack. Reading a few shows what makes a question worth asking and what a partial result looks like. ## Ratio & Proportion 24 topics, ages 9 to 16. https://lightmysky.com/learn/mathematics/areas/ratio-and-proportion - [Ratio and proportion vocabulary](https://lightmysky.com/learn/mathematics/ratio-and-proportion-vocabulary-mt_ESgc4YBw-a): Know and use the vocabulary of ratio and proportion — ratio, proportion, percentage, scale, equivalent, unequal, relative size, part-to-part, part-to-whole, and out of — and understand the difference between ratio (comparing parts to parts) and proportion (comparing a part to the whole) - [Bar Models for Ratios](https://lightmysky.com/learn/mathematics/bar-models-for-ratios-mt_ePXg_XyCKU): Represent ratio and proportion problems using bar models (rectangular strips divided into equal parts labelled with quantities) and tape diagrams (segmented strips showing part-to-part and part-to-whole relationships); use these visual models to set up and solve unequal sharing, scaling, and percentage problems — drawing the diagram first, then reading off the answer - [Calculating percentages](https://lightmysky.com/learn/mathematics/calculating-percentages-mt_PNSyfH56eQ): Solve problems involving the calculation of percentages of amounts (e.g. 15% of 360) and the use of percentages for comparison - [Ratio Problems](https://lightmysky.com/learn/mathematics/ratio-problems-mt_h0gJcSuwdL): Solve problems involving the relative sizes of two quantities where missing values can be found by using integer multiplication and division facts - [Scale and similar shapes](https://lightmysky.com/learn/mathematics/scale-and-similar-shapes-mt_2OtRUM_0zW): Solve problems involving similar shapes where the scale factor is known or can be found - [Unequal sharing and grouping](https://lightmysky.com/learn/mathematics/unequal-sharing-and-grouping-mt_FnUJMXPUZX): Solve problems involving unequal sharing and grouping using knowledge of fractions and multiples - [Ratio Notation](https://lightmysky.com/learn/mathematics/ratio-notation-mt_XWSGuFW7It): Use ratio notation to describe the relationship between two or more quantities, simplify ratios to their simplest form, and convert between ratio and fraction representations - [Proportion Graphs](https://lightmysky.com/learn/mathematics/proportion-graphs-mt_FspV_imUGK): Represent proportional relationships using double number lines (two parallel number lines aligned at 0) and ratio tables; recognise that equivalent ratios generate straight lines through the origin when graphed - [Proportional Reasoning Vocabulary](https://lightmysky.com/learn/mathematics/proportional-reasoning-vocabulary-mt_ATYLKt0je-): Know and use advanced vocabulary of multiplicative reasoning — direct proportion, inverse proportion, ratio, rate, unit rate, compound unit, scale factor — accurately in problem-solving contexts - [Dividing Quantities by Ratio](https://lightmysky.com/learn/mathematics/dividing-quantities-by-ratio-mt_nOCPx5qw0Z): Divide a given quantity into two parts in a given part:part or part:whole ratio, and express the division as a fraction of the whole - [One Quantity as a Fraction](https://lightmysky.com/learn/mathematics/one-quantity-as-a-fraction-mt_xAf2bu9wYK): Express one quantity as a fraction of another where the result may be less than 1 or greater than 1, and interpret the meaning in context - [Unit Conversions](https://lightmysky.com/learn/mathematics/unit-conversions-mt_tpT9brpI6D): Convert freely between related standard units of measurement (time, length, area, volume/capacity, mass) using decimal notation to up to three decimal places where appropriate - [Scale drawings and maps](https://lightmysky.com/learn/mathematics/scale-drawings-and-maps-mt_1badik7iKJ): Use scale factors to interpret and create scale diagrams and maps, calculating real-life distances from map measurements and vice versa - [Ratio Notation and Relationships](https://lightmysky.com/learn/mathematics/ratio-notation-and-relationships-mt_yK51ZnKA8m): Understand that a multiplicative relationship between two quantities can be expressed as a ratio; use ratio notation; simplify ratios - [Proportion](https://lightmysky.com/learn/mathematics/proportion-mt_5mIcmKRCgA): Recognise and solve problems involving direct proportion (as one quantity increases, the other increases at a constant rate) and inverse proportion (as one increases, the other decreases), including graphical and algebraic representations - [Percentages as fractions](https://lightmysky.com/learn/mathematics/percentages-as-fractions-mt_ALUrJpY0cZ): Define percentage as 'number of parts per hundred'; interpret percentages and percentage changes as a fraction or a decimal; express one quantity as a percentage of another; compare quantities using percentages; work with percentages greater than 100% - [Percentages (age 12+)](https://lightmysky.com/learn/mathematics/percentages-age-12-mt_iEXqN48w3x): Solve problems involving percentage increase, percentage decrease, finding the original value after a percentage change, and calculating simple interest - [Compound Interest and Repeated Percentage Change](https://lightmysky.com/learn/mathematics/compound-interest-and-repeated-percentage-change-mt_2vJz9VKle_): Applying a percentage change again and again multiplies rather than adds, so the multiplier raised to a power gives the answer in one step. The same calculation covers savings growing, a debt growing, and a value depreciating. - [Percentage Change with Multipliers](https://lightmysky.com/learn/mathematics/percentage-change-with-multipliers-mt_uqclfF4OBR): Use one multiplier for an increase or a decrease, chain multipliers for repeated changes, and divide by the multiplier to find the original amount. - [Compound Growth and Decay](https://lightmysky.com/learn/mathematics/compound-growth-and-decay-mt_xf5FQ2sugf): Apply a repeated percentage change as a multiplier raised to a power, for savings, depreciation and populations, and find how many steps it takes to pass a target. - [Compound Units](https://lightmysky.com/learn/mathematics/compound-units-mt_kLQOzZYrd5): Use compound units such as speed (distance ÷ time), unit pricing (cost ÷ quantity), and density (mass ÷ volume); solve problems involving compound units - [Ratio Problems: Sharing, Changing and Combining](https://lightmysky.com/learn/mathematics/ratio-problems-sharing-changing-and-combining-mt_vJelayH4uv): Solve ratio problems where a total is shared, one part is known, a ratio changes, or two ratios have to be joined through the quantity they share. - [Direct Proportion as a Formula](https://lightmysky.com/learn/mathematics/direct-proportion-as-a-formula-mt_nhSf9I7Bjz): Write y = kx for two quantities in direct proportion, find k from one pair of values, and use the formula to predict others. - [Inverse Proportion](https://lightmysky.com/learn/mathematics/inverse-proportion-mt_ZbbN2x6tcL): Use y = k/x when one quantity falls as the other rises, find k from a given pair, and recognise the curve it produces. ## Algebra 65 topics, ages 10 to 18. 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. ## Mechanics 14 topics, ages 16 to 18. https://lightmysky.com/learn/mathematics/areas/mechanics - [Modelling Motion in a Straight Line](https://lightmysky.com/learn/mathematics/modelling-motion-in-a-straight-line-mt_no78_6w1rS): Set up a motion problem before calculating anything: choose which direction counts as positive, treat the object as a particle, and keep displacement, velocity and acceleration apart from distance, speed and everyday words like slowing down. - [Motion Graphs: Gradients and Areas](https://lightmysky.com/learn/mathematics/motion-graphs-gradients-and-areas-mt_KhJeCMYBSL): Read the gradient of a displacement-time graph as velocity and the area under a velocity-time graph as displacement, and describe a whole journey from the shape of the graph alone. - [The Constant-Acceleration Equations](https://lightmysky.com/learn/mathematics/the-constant-acceleration-equations-mt_Nyc3BcikQ9): Derive the five constant-acceleration equations from a straight-line velocity-time graph, then choose the one that uses the three quantities you know and gives the one you want. - [Vertical Motion Under Gravity](https://lightmysky.com/learn/mathematics/vertical-motion-under-gravity-mt_IgNj9bn14O): Apply the constant-acceleration equations to an object thrown or dropped, taking the acceleration as g downwards and keeping one sign convention for the whole solution. - [Kinematics with Calculus](https://lightmysky.com/learn/mathematics/kinematics-with-calculus-mt_2kNspOOoDw): When the acceleration is not constant, differentiate displacement to get velocity and velocity to get acceleration, and integrate back the other way, using an initial condition to fix each constant. - [Newton's Second Law: Force, Mass and Acceleration](https://lightmysky.com/learn/mathematics/newtons-second-law-force-mass-and-acceleration-mt_SKO85AcKdb): A resultant force gives an acceleration in its own direction, with F = ma. Weight is mass times g, which is why objects of different mass fall together. - [Forces as Vectors: Resolving and Resultants](https://lightmysky.com/learn/mathematics/forces-as-vectors-resolving-and-resultants-mt_-qDRgd2g8v): Draw every force acting on a body as an arrow, add them as vectors to get the resultant, and split any force into components along two perpendicular directions of your choosing. - [Equilibrium and Newton's First Law](https://lightmysky.com/learn/mathematics/equilibrium-and-newtons-first-law-mt_e6RFRQkRPo): A body with no resultant force stays still or keeps a constant velocity. Setting the components in two perpendicular directions to zero turns that into equations that find unknown forces. - [Friction and the Coefficient of Friction](https://lightmysky.com/learn/mathematics/friction-and-the-coefficient-of-friction-mt_EwFUd9N24W): Use the static friction limit to decide whether a body can remain at rest, and use a separate kinetic coefficient when surfaces slide. - [Moments and the Equilibrium of a Rigid Body](https://lightmysky.com/learn/mathematics/moments-and-the-equilibrium-of-a-rigid-body-mt_FX5egGL4LI): Measure the turning effect of a force as force times perpendicular distance from a point, and solve a beam problem by setting both the total force and the total moment to zero. - [Newton's Second Law in Two Dimensions](https://lightmysky.com/learn/mathematics/newtons-second-law-in-two-dimensions-mt_yNUFF3lxlj): Resolve the forces along the direction of motion and perpendicular to it, then apply F = ma along that direction while the perpendicular components balance. This is what makes a slope problem solvable. - [Connected Particles and Pulleys](https://lightmysky.com/learn/mathematics/connected-particles-and-pulleys-mt_pZ6zV0RFdC): Two bodies joined by a string share one acceleration and pull on each other equally and oppositely. Write F = ma for each body, then solve the two equations together for the acceleration and the tension. - [Motion on a Rough Inclined Plane](https://lightmysky.com/learn/mathematics/motion-on-a-rough-inclined-plane-mt_0rBy66HFNO): Resolve weight along and perpendicular to a slope, include friction in the correct direction, and find the acceleration or the angle at which sliding starts. - [Projectile Motion](https://lightmysky.com/learn/mathematics/projectile-motion-mt_O1C-QxCSJN): Split the launch velocity into horizontal and vertical components, then run constant velocity across and constant acceleration downwards. For a same-height flight with gravity alone and a stated g, range, greatest height and flight time follow. ## Linear Algebra 19 topics, ages 18 to 24. https://lightmysky.com/learn/mathematics/areas/linear-algebra - [Systems of linear equations and their solution sets](https://lightmysky.com/learn/mathematics/systems-of-linear-equations-and-their-solution-sets-mt_sQL6JfR7Te): A linear system has no solution, exactly one, or infinitely many, and nothing else. Reading which case you are in, and describing an infinite solution set, is the starting point for the whole subject. - [Gaussian elimination and row echelon form](https://lightmysky.com/learn/mathematics/gaussian-elimination-and-row-echelon-form-mt_PZBce3UvRm): Reduce a system with row swaps, nonzero row scaling and row replacement to echelon or reduced echelon form, then read pivots, free variables and the full solution set. - [Matrix multiplication and what it represents](https://lightmysky.com/learn/mathematics/matrix-multiplication-and-what-it-represents-mt_0fJOMPlCxr): A matrix records a linear rule. Multiplying matrices composes the rules, which is why the product is defined by rows against columns and why order matters. - [The Inverse of a Matrix and When It Exists](https://lightmysky.com/learn/mathematics/the-inverse-of-a-matrix-and-when-it-exists-mt_1K6lcfMQ_b): An inverse undoes the rule a matrix applies. Only square matrices with full pivot count have one, and elimination both decides the question and computes the answer. - [Determinants and What They Measure](https://lightmysky.com/learn/mathematics/determinants-and-what-they-measure-mt_fWr2D_1BdM): The determinant is the factor by which a map scales area or volume, with a sign for orientation. It vanishes exactly when the map collapses space, which is exactly when no inverse exists. - [LU Factorisation, Pivoting and the Cost of a Solve](https://lightmysky.com/learn/mathematics/lu-factorisation-pivoting-and-the-cost-of-a-solve-mt_5Gg2v3Oxrr): Record Gaussian elimination as a product of two triangular matrices, add pivoting for stability, and count the operations each stage costs. - [Vector Spaces and Subspaces](https://lightmysky.com/learn/mathematics/vector-spaces-and-subspaces-mt_Zrddx-E6_n): Strip vectors down to the two operations and the axioms they satisfy, so that polynomials, matrices and functions count as vectors too. A subspace is a subset closed under both operations. - [Linear Independence, Span and Basis](https://lightmysky.com/learn/mathematics/linear-independence-span-and-basis-mt_5RGbip9yC9): Span is everything you can reach by combining vectors; independence says none of them is redundant. A basis is a set that is both, so every element has exactly one expression. - [Dimension and the Rank-Nullity Theorem](https://lightmysky.com/learn/mathematics/dimension-and-the-rank-nullity-theorem-mt_aMG8PcwPAR): Every basis of a space has the same size, which is its dimension. For a matrix, the rank and the nullity add to the number of columns, which explains the shape of every solution set met so far. - [Linear Maps and Their Matrices](https://lightmysky.com/learn/mathematics/linear-maps-and-their-matrices-mt_pzrfiQFdto): A linear map respects addition and scaling. Once a basis is fixed, every such map is a matrix, and the matrix columns are the images of the basis vectors. - [Change of Basis](https://lightmysky.com/learn/mathematics/change-of-basis-mt_HTGuJhX484): The same map has different matrices in different bases. A change-of-basis matrix converts coordinates, and similar matrices are the same map seen from two viewpoints. - [Eigenvalues and Eigenvectors](https://lightmysky.com/learn/mathematics/eigenvalues-and-eigenvectors-mt_TVqqaw11qa): Some directions survive a map unchanged except for scaling. Finding them means solving the characteristic equation, and they explain what the map does far better than its entries do. - [Diagonalisation and Powers of a Matrix](https://lightmysky.com/learn/mathematics/diagonalisation-and-powers-of-a-matrix-mt_w15uCVhdhX): A matrix with a basis of eigenvectors becomes diagonal in that basis. Its diagonal form lets us compute powers by raising the eigenvalues. - [Inner Products, Length and Orthogonality](https://lightmysky.com/learn/mathematics/inner-products-length-and-orthogonality-mt_X0HPRGto4W): An inner product gives a real vector space length and angle. Orthogonal sets of nonzero vectors are independent. Inner products with an orthonormal basis give a vector's coordinates. - [Orthogonal Projection and Least Squares](https://lightmysky.com/learn/mathematics/orthogonal-projection-and-least-squares-mt_kGvRrYJrat): Project a vector onto a subspace to get the closest point in it. Applied to an inconsistent system, this is the least squares solution, which is where regression comes from. - [The Gram-Schmidt Process](https://lightmysky.com/learn/mathematics/the-gram-schmidt-process-mt_nkp231EA_E): Turn any basis into an orthonormal one by subtracting off projections onto what has already been built, which gives the QR factorisation used by numerical solvers. - [Symmetric Matrices and the Spectral Theorem](https://lightmysky.com/learn/mathematics/symmetric-matrices-and-the-spectral-theorem-mt_sdQv4m7Nbk): A real symmetric matrix has real eigenvalues and an orthonormal eigenbasis. That single fact is behind quadratic forms, covariance matrices and principal axes. - [The Singular Value Decomposition](https://lightmysky.com/learn/mathematics/the-singular-value-decomposition-mt_UsUZbM9VRK): Every matrix, square or not, factors into a rotation, a scaling and another rotation. The singular values rank the directions by how much the map stretches them, which is how low-rank approximation works. - [Iterative Methods for Large Linear Systems](https://lightmysky.com/learn/mathematics/iterative-methods-for-large-linear-systems-mt_FKkxKYVGYJ): For a large sparse system, elimination costs too much and the solution is approached instead of computed. Convergence is decided by the spectrum of the iteration, and preconditioning is the act of changing that spectrum. ## Calculus & Analysis 83 topics, ages 16 to 24. https://lightmysky.com/learn/mathematics/areas/calculus-and-analysis - [The Power Rule for Differentiating Polynomials](https://lightmysky.com/learn/mathematics/the-power-rule-for-differentiating-polynomials-mt_l7f4j44bR3): Differentiate any sum of powers of x with the rule that xⁿ becomes nxⁿ⁻¹, including negative and fractional powers once the term is rewritten as a power. - [Integration as the Reverse of Differentiation](https://lightmysky.com/learn/mathematics/integration-as-the-reverse-of-differentiation-mt_edtk3ArxRk): Recover a function from its derivative by raising the power and dividing, and carry the constant of integration that differentiation destroyed. Use a known point to pin that constant down. - [The Gradient of a Curve as a Limit](https://lightmysky.com/learn/mathematics/the-gradient-of-a-curve-as-a-limit-mt_LAlhcwVjln): See that a curve has a different gradient at every point, and that the gradient of a chord settles on a single value as the second point slides in. That limiting value is the gradient of the tangent. - [Differentiation from First Principles](https://lightmysky.com/learn/mathematics/differentiation-from-first-principles-mt_pdfPztEjw_): Write the chord gradient as [f(x + h) - f(x)] / h, simplify it algebraically, and let h approach zero to get the derivative. This is the definition every rule later rests on. - [Locating Roots and Iterative Methods](https://lightmysky.com/learn/mathematics/locating-roots-and-iterative-methods-mt_2COO6TWerM): Trap a root between two values by a change of sign, then rearrange the equation into the form x = g(x) and iterate from a starting value until the digits stop moving. - [The Definite Integral and the Area Under a Curve](https://lightmysky.com/learn/mathematics/the-definite-integral-and-the-area-under-a-curve-mt_S2IO2S7kgd): Estimate the area under a curve with rectangles, then evaluate it exactly by substituting the limits into the antiderivative. Areas below the axis come out negative and have to be handled separately. - [Areas Between Two Curves](https://lightmysky.com/learn/mathematics/areas-between-two-curves-mt_lDBZmnLK1b): Find where two curves meet, then integrate the difference of the two functions between those limits to get the area enclosed. - [The Derivative as a Rate of Change](https://lightmysky.com/learn/mathematics/the-derivative-as-a-rate-of-change-mt_RqyLE3jrAW): Read dy/dx as how fast one quantity changes per unit of another, with the units to match, so a derivative answers questions about cost per item, litres per second or metres per second. - [Tangents and Normals to a Curve](https://lightmysky.com/learn/mathematics/tangents-and-normals-to-a-curve-mt_BzMw7IT6kp): Find the equation of the tangent at a point on a curve from the derivative there, and the normal from the perpendicular gradient. - [Increasing and Decreasing Functions](https://lightmysky.com/learn/mathematics/increasing-and-decreasing-functions-mt_KOrXxQvBrZ): Use the sign of the derivative to say where a curve rises and where it falls, which turns a question about shape into an inequality in x. - [Stationary Points and the Second Derivative](https://lightmysky.com/learn/mathematics/stationary-points-and-the-second-derivative-mt_VbGJEFFgfs): Solve f'(x) = 0 to locate the stationary points, then use the sign of f''(x) to tell a maximum from a minimum, and check the sign of f'(x) either side when f''(x) = 0. - [Optimisation with Calculus](https://lightmysky.com/learn/mathematics/optimisation-with-calculus-mt_9I0ql39Lzw): Turn a worded problem into one function of one variable, use a constraint to remove the second variable, then differentiate to find the best value and justify that it is a maximum or minimum. - [The Newton-Raphson Method](https://lightmysky.com/learn/mathematics/the-newton-raphson-method-mt_jkjJre5ETd): Follow the tangent at the current guess down to the x-axis to get the next guess, which is x - f(x)/f'(x). Note when it fails: a nearly flat tangent throws the next guess far away. - [The Trapezium Rule](https://lightmysky.com/learn/mathematics/the-trapezium-rule-mt_rAlJ1WIRmZ): Estimate a definite integral by slicing the region into trapezia of equal width, and use the curvature to say whether the estimate is above or below the true value. - [One-Sided Limits and When a Limit Fails to Exist](https://lightmysky.com/learn/mathematics/one-sided-limits-and-when-a-limit-fails-to-exist-mt__L0B_WQq-P): Compare one-sided behavior. Matching finite limits give a finite two-sided limit. Shared growth toward positive or negative infinity is called an infinite limit, not a finite limit. Disagreement or persistent failure to settle prevents a shared two-sided limit. - [The Limit Laws and Indeterminate Forms](https://lightmysky.com/learn/mathematics/the-limit-laws-and-indeterminate-forms-mt_rAJs-V4Jlx): Apply finite limit laws where their conditions hold. Investigate 0/0 forms with suitable algebra or bounds; the form alone guarantees neither a removable factor nor a finite limit. - [Continuity and the Intermediate Value Theorem](https://lightmysky.com/learn/mathematics/continuity-and-the-intermediate-value-theorem-mt_kaQR6cWjbV): Define continuity at a point as the limit agreeing with the value there, classify the ways that can fail, and use the intermediate value theorem to guarantee a root inside an interval. - [The Mean Value Theorem](https://lightmysky.com/learn/mathematics/the-mean-value-theorem-mt_09iApYVelx): Over an interval where a function is continuous and differentiable, some interior point has instantaneous rate equal to the average rate across the whole interval. Most facts linking a derivative to a function's behaviour rest on this. - [L'Hopital's Rule and Comparing Growth Rates](https://lightmysky.com/learn/mathematics/lhopitals-rule-and-comparing-growth-rates-mt_Ai-A2sPIoE): Resolve 0/0 and infinity over infinity limits by differentiating numerator and denominator separately, and use the result to rank how fast logarithms, powers and exponentials grow. - [Riemann Sums and the Definite Integral as a Limit](https://lightmysky.com/learn/mathematics/riemann-sums-and-the-definite-integral-as-a-limit-mt_pwa2SRU6P9): Build the definite integral as the limit of sums of rectangle areas, with sample points anywhere in each subinterval. This is the definition every later property of the integral is argued from. - [Double Integrals over General Regions](https://lightmysky.com/learn/mathematics/double-integrals-over-general-regions-mt_UKS9_AhAWw): Integrate over a plane region by iterating two single integrals, with the inner limits describing the region. Choosing the order of integration is often the whole problem. - [Sequences and Their Limits](https://lightmysky.com/learn/mathematics/sequences-and-their-limits-mt_PQl3Q6n5dc): Treat a sequence as a function on the positive integers and ask what its terms approach. Monotone bounded sequences converge, which is the first place completeness is doing real work. - [Infinite Series and the Geometric Series](https://lightmysky.com/learn/mathematics/infinite-series-and-the-geometric-series-mt_EPE-ERFFQt): Define the sum of a series as the limit of its partial sums, work out the geometric and telescoping cases exactly, and use the term test to rule out convergence. - [The Fundamental Theorem of Calculus](https://lightmysky.com/learn/mathematics/the-fundamental-theorem-of-calculus-mt_EsS_ovdz2A): Both halves: differentiating an accumulation function returns the integrand, and a definite integral can be evaluated from any antiderivative. This is why the two calculus operations undo each other. - [Double Integrals in Polar Coordinates](https://lightmysky.com/learn/mathematics/double-integrals-in-polar-coordinates-mt_93e-KSzQMT): Integrate over circular regions by switching to polar coordinates, where the area element carries an extra factor of r. - [The Completeness Axiom: Suprema and Infima](https://lightmysky.com/learn/mathematics/the-completeness-axiom-suprema-and-infima-mt_xjI-pIfh95): The rationals have gaps and the reals do not. Completeness is stated as every bounded set having a least upper bound, and it is the axiom every later theorem in analysis is traced back to. - [Conditioning, Stability and Floating-Point Error](https://lightmysky.com/learn/mathematics/conditioning-stability-and-floating-point-error-mt_q8IoCElXIV): Conditioning says how much the answer moves when the data moves, and stability says how much an algorithm adds on top of that. Rounding error is tiny, and those two amplifiers are what make it visible. - [The Epsilon-Delta Definition of a Limit](https://lightmysky.com/learn/mathematics/the-epsilon-delta-definition-of-a-limit-mt_jFzRYQzCmk): State what a limit claims with no appeal to motion: for every tolerance around the limiting value there is a distance around the point that keeps outputs inside it. Prove a simple limit by producing that distance. - [The Product and Quotient Rules](https://lightmysky.com/learn/mathematics/the-product-and-quotient-rules-mt_NJ7INJ0jfv): Apply the product and quotient rules to differentiable functions, keeping both product contributions and the quotient's subtraction order. Use the quotient rule only where its denominator is nonzero. - [The Chain Rule](https://lightmysky.com/learn/mathematics/the-chain-rule-mt_kdhl4dmwJn): Differentiate a composition by multiplying the rate of the outer function by the rate of the inner one. The rule is what makes substitution, related rates and implicit differentiation possible. - [Derivatives of Exponential, Logarithmic and Trigonometric Functions](https://lightmysky.com/learn/mathematics/derivatives-of-exponential-logarithmic-and-trigonometric-functions-mt_695WPZKADT): Establish the derivatives of the standard non-polynomial functions, including why e is the base that makes the exponential its own derivative. - [Implicit Differentiation and Derivatives of Inverse Functions](https://lightmysky.com/learn/mathematics/implicit-differentiation-and-derivatives-of-inverse-functions-mt_P4ABBGsKFT): Differentiate a relation that is not solved for y, and use the same move to get the derivative of an inverse function from the derivative of the original. - [Related Rates](https://lightmysky.com/learn/mathematics/related-rates-mt_3XTIPt8Vmf): Link two changing quantities by an equation, differentiate the equation with respect to time, and solve for the rate you cannot measure directly. - [Integration by Substitution](https://lightmysky.com/learn/mathematics/integration-by-substitution-mt_jH-j_XsdC7): Use a change of variables to reverse a chain-rule pattern when the differential matches. For definite integrals, either convert the bounds or return to the original variable before evaluating. Distinguish a valid substitution from a shortcut that does not simplify the integral. - [Integration by Parts](https://lightmysky.com/learn/mathematics/integration-by-parts-mt_v6jZVs2_tf): Reverse the product rule to trade one integral for another that is easier, and choose the parts so the trade is an improvement. - [Trigonometric Integrals and Trigonometric Substitution](https://lightmysky.com/learn/mathematics/trigonometric-integrals-and-trigonometric-substitution-mt_JqysS41psg): Handle powers of sine and cosine with identities, and clear expressions such as √(a² - x²) by substituting a trigonometric function for x. - [Partial Fractions for Rational Integrands](https://lightmysky.com/learn/mathematics/partial-fractions-for-rational-integrands-mt_PA8qxs424c): Split a rational function into simpler fractions whose integrals are logarithms and arctangents, after checking the degree of the numerator first. - [Improper Integrals and Their Convergence](https://lightmysky.com/learn/mathematics/improper-integrals-and-their-convergence-mt_gvtZPFRA6Z): Give meaning to an integral with an infinite limit or an unbounded integrand by evaluating a proper integral and taking a limit, then decide whether the result is finite. - [Convergence Tests for Series of Positive Terms](https://lightmysky.com/learn/mathematics/convergence-tests-for-series-of-positive-terms-mt_8N49Q4BlO8): Decide convergence with the integral, comparison, limit comparison, ratio and root tests, and choose the test that matches the shape of the terms. - [Alternating Series and Absolute Convergence](https://lightmysky.com/learn/mathematics/alternating-series-and-absolute-convergence-mt_sk6F4n1USH): Handle series whose terms change sign: the alternating test, the error bound from the first omitted term, and the difference between converging absolutely and converging only conditionally. - [Power Series and the Radius of Convergence](https://lightmysky.com/learn/mathematics/power-series-and-the-radius-of-convergence-mt_RnYb0JLKbD): Read a power series as a function defined where it converges, find its radius with the ratio test, and check the endpoints separately. - [Taylor and Maclaurin Series](https://lightmysky.com/learn/mathematics/taylor-and-maclaurin-series-mt_SgI9Pn9RIO): Build the power series of a function from its derivatives at a point, use the remainder to say how good a truncation is, and reuse the standard expansions instead of starting over. - [Vector-Valued Functions and Motion Along a Curve](https://lightmysky.com/learn/mathematics/vector-valued-functions-and-motion-along-a-curve-mt_Xcjt6exMy3): Describe a path in space by a vector that depends on one parameter. Differentiating gives velocity and acceleration, and the arc length is an integral of speed. - [Functions of Several Variables and Level Curves](https://lightmysky.com/learn/mathematics/functions-of-several-variables-and-level-curves-mt_u04fZ_XiJ8): Read a function of two inputs as a surface, and read the surface through its level curves. A contour map is the working picture for the rest of the spine. - [Partial Derivatives](https://lightmysky.com/learn/mathematics/partial-derivatives-mt_tu11fd9Xk9): Differentiate with respect to one variable while holding the others fixed. Interpret coordinate slopes and compare mixed second partials, which agree when both mixed derivatives are continuous near the point. - [Tangent Planes and Linear Approximation](https://lightmysky.com/learn/mathematics/tangent-planes-and-linear-approximation-mt_EYMcjChX_m): Replace a surface near a point by the plane its partials define, and use that plane to estimate values and propagate small errors. - [The Multivariable Chain Rule](https://lightmysky.com/learn/mathematics/the-multivariable-chain-rule-mt_CHSHdNxPmJ): Differentiate a composition when several inputs each depend on other variables, by summing one contribution per path through the dependency diagram. - [Directional Derivatives and the Gradient](https://lightmysky.com/learn/mathematics/directional-derivatives-and-the-gradient-mt_LW_KckY5Ad): The rate of change in an arbitrary direction is the dot product of the gradient with a unit vector, so the gradient points the steepest way uphill and sits perpendicular to level curves. - [Critical Points and Optimisation in Two Variables](https://lightmysky.com/learn/mathematics/critical-points-and-optimisation-in-two-variables-mt_4qB5pAGeuR): Find points where the gradient vanishes and classify them with the second derivative test, including the saddle point, which has no one-variable analogue. - [Lagrange Multipliers](https://lightmysky.com/learn/mathematics/lagrange-multipliers-mt_0RPOBWZ1gs): Optimise subject to a constraint by setting the gradient of the objective parallel to the gradient of the constraint, which is the moment the level curves touch. - [Volumes of Revolution: Disks, Washers and Shells](https://lightmysky.com/learn/mathematics/volumes-of-revolution-disks-washers-and-shells-mt_mzZ22MkWD4): Set up a volume as an integral of cross-sections, and choose between slicing perpendicular to the axis and using cylindrical shells based on which integral is easier. - [Arc Length and the Area of a Surface of Revolution](https://lightmysky.com/learn/mathematics/arc-length-and-the-area-of-a-surface-of-revolution-mt_W4uujT9Z1t): Set up the integrals that measure the length of a plane curve and the area swept when it is spun about an axis. - [Convergence of Sequences, Rigorously](https://lightmysky.com/learn/mathematics/convergence-of-sequences-rigorously-mt_yB5M-DCH8V): Prove convergence from the definition: for every tolerance there is a point in the sequence past which every term stays inside it. Limit laws become theorems with proofs rather than rules to apply. - [Cauchy Sequences and the Bolzano-Weierstrass Theorem](https://lightmysky.com/learn/mathematics/cauchy-sequences-and-the-bolzano-weierstrass-theorem-mt_nreDylVkSU): Terms that eventually crowd together converge, in the reals but not in the rationals. Every bounded sequence has a convergent subsequence, which is the workhorse behind existence proofs. - [Continuity and Uniform Continuity](https://lightmysky.com/learn/mathematics/continuity-and-uniform-continuity-mt_plWc9raOwz): Continuity as a sequence or epsilon-delta condition, then the stronger version where one delta works across the whole domain. Heine-Cantor says a continuous function on a closed bounded interval is uniformly continuous, and the extreme value theorem says it attains its bounds. - [Differentiability and the Theorems Behind the Rules](https://lightmysky.com/learn/mathematics/differentiability-and-the-theorems-behind-the-rules-mt_1xLEeoA69z): Define the derivative as a limit, prove differentiability implies continuity, state and prove Rolle's theorem using the extreme value theorem, and derive the mean value theorem from Rolle's theorem. - [Polynomial Interpolation and Its Error Term](https://lightmysky.com/learn/mathematics/polynomial-interpolation-and-its-error-term-mt_NSnzY9Xodd): Fit the unique polynomial through given points, build it in Lagrange and Newton form, and bound how far it strays between the points. - [Numerical Quadrature: Newton-Cotes and Adaptive Rules](https://lightmysky.com/learn/mathematics/numerical-quadrature-newton-cotes-and-adaptive-rules-mt_fT5kRCUiD5): Derive quadrature rules by integrating an interpolating polynomial, compare their error orders, and refine only where the integrand misbehaves. - [The Riemann Integral and When a Function Is Integrable](https://lightmysky.com/learn/mathematics/the-riemann-integral-and-when-a-function-is-integrable-mt_Ahl2MYIA4N): Define the integral by upper and lower sums and call a function integrable when the two meet. Continuous functions qualify; a function that is discontinuous everywhere need not. - [Pointwise and Uniform Convergence](https://lightmysky.com/learn/mathematics/pointwise-and-uniform-convergence-mt_iAGk1ML5MA): A sequence of functions can converge at every point and still lose continuity in the limit. Uniform convergence is the stronger condition that preserves continuity and permits term-by-term integration. - [Triple Integrals and Coordinates for Solids](https://lightmysky.com/learn/mathematics/triple-integrals-and-coordinates-for-solids-mt_RFeK8LD_jT): Integrate over a solid region, and choose cylindrical or spherical coordinates when the solid has an axis or a centre of symmetry. - [Change of Variables and the Jacobian](https://lightmysky.com/learn/mathematics/change-of-variables-and-the-jacobian-mt_1_LRE2cJft): Change coordinates in a multiple integral, and see the Jacobian determinant as the local factor by which area or volume is stretched. - [Vector Fields and Line Integrals](https://lightmysky.com/learn/mathematics/vector-fields-and-line-integrals-mt_VZxKEmsVgO): Attach a vector to every point of a region, then integrate along a curve to accumulate work done by the field or mass along a wire. - [Conservative Fields and Path Independence](https://lightmysky.com/learn/mathematics/conservative-fields-and-path-independence-mt_5djKpI5F33): Some fields are gradients of a potential function. For those, the line integral depends only on the endpoints, and there is a test to recognise them. - [Green's Theorem in the Plane](https://lightmysky.com/learn/mathematics/greens-theorem-in-the-plane-mt_UDgivmPVEJ): Trade a line integral around a closed plane curve for a double integral of a derivative expression over the region inside it. The first of three theorems with the same shape. - [Divergence and Curl](https://lightmysky.com/learn/mathematics/divergence-and-curl-mt_JbjW8WKOPE): Two derivative operations on a field: divergence measures net outflow at a point, curl measures local rotation. Both are the language of the theorems that follow. - [Surface Integrals and Flux](https://lightmysky.com/learn/mathematics/surface-integrals-and-flux-mt_b3FIxOVaTO): Parameterise a surface, build its area element from a cross product, and integrate a field through it to measure flux. - [Stokes' Theorem and the Divergence Theorem](https://lightmysky.com/learn/mathematics/stokes-theorem-and-the-divergence-theorem-mt_sRojZcrw-q): Both theorems say that integrating a derivative over a region equals integrating the original object over its boundary. With Green's theorem and the fundamental theorem, they are one statement in four settings. - [Sigma-Algebras and Measurable Sets](https://lightmysky.com/learn/mathematics/sigma-algebras-and-measurable-sets-mt_cpvegazufk): A translation-invariant, countably additive length that gives each interval its usual length cannot be assigned to every subset of the line, so first choose a family of sets closed under complement and countable union. - [Outer Measure and the Construction of Lebesgue Measure](https://lightmysky.com/learn/mathematics/outer-measure-and-the-construction-of-lebesgue-measure-mt_6u5_Um71wE): Cover a set by intervals, take the infimum of cover lengths, and obtain an outer measure for every subset. The Caratheodory condition selects a sigma-algebra on which outer measure becomes countably additive. - [Measurable Functions and Approximation by Simple Functions](https://lightmysky.com/learn/mathematics/measurable-functions-and-approximation-by-simple-functions-mt_VwNRjXA7ro): A function is measurable when the preimage of every interval is a measurable set. Every non-negative measurable function is an increasing limit of simple functions, and that staircase is what makes an integral definable in stages. - [The Lebesgue Integral and What It Repairs](https://lightmysky.com/learn/mathematics/the-lebesgue-integral-and-what-it-repairs-mt_KO92nLc5YU): Define the integral first for simple functions, then as a supremum over the simple functions underneath. Functions the Riemann theory cannot touch become integrable, and the value agrees with the Riemann integral wherever that one exists. - [Monotone Convergence, Fatou and Dominated Convergence](https://lightmysky.com/learn/mathematics/monotone-convergence-fatou-and-dominated-convergence-mt_7n6ZDkHeBe): Three theorems that say when a limit may be moved inside an integral, each with its own price. They are the reason this integral, and not the earlier one, is the one used in probability and in partial differential equations. - [Product Measure and Fubini's Theorem](https://lightmysky.com/learn/mathematics/product-measure-and-fubinis-theorem-mt_1VzzVmMjTB): Two measure spaces combine into one, and an integral over the product can be done one variable at a time. The hypotheses of the theorem say exactly when the order of integration is free. - [Lp Spaces and the Inequalities They Rest On](https://lightmysky.com/learn/mathematics/lp-spaces-and-the-inequalities-they-rest-on-mt_Qd8jU26qgE): Functions with integrable p-th power form a normed space once functions that agree almost everywhere are treated as equal. Holder's and Minkowski's inequalities are what make the norm behave like a length. - [Normed Spaces, Completeness and Banach Spaces](https://lightmysky.com/learn/mathematics/normed-spaces-completeness-and-banach-spaces-mt_y_jueKnovX): A vector space with a length function, plus the question of whether every Cauchy sequence in it converges. Completeness is what lets an existence proof finish by taking a limit and knowing the limit is still in the space. - [Bounded Linear Operators and the Operator Norm](https://lightmysky.com/learn/mathematics/bounded-linear-operators-and-the-operator-norm-mt_Qi_mbSNMMx): For a linear map between normed spaces, continuity and boundedness turn out to be the same condition. The operator norm records the worst stretching the map can perform on a unit vector. - [Baire Category and the Uniform Boundedness Principle](https://lightmysky.com/learn/mathematics/baire-category-and-the-uniform-boundedness-principle-mt_11Rd73trAX): In a nonempty complete metric space, countably many nowhere dense sets cannot cover the space. Baire's theorem yields uniform boundedness for pointwise-bounded families of bounded linear operators on Banach spaces and proves the open mapping theorem for surjective bounded linear operators between Banach spaces. - [The Hahn-Banach Theorem and the Dual Space](https://lightmysky.com/learn/mathematics/the-hahn-banach-theorem-and-the-dual-space-mt_pJuHFxkvjS): On a real or complex normed space, a bounded linear functional on a subspace has an extension to the whole space with the same norm. The continuous dual collects bounded linear functionals, which can detect nonzero vectors and separate points from closed subspaces. - [The Fourier Transform on the Line](https://lightmysky.com/learn/mathematics/the-fourier-transform-on-the-line-mt_BMrt5FPe3U): Extend Fourier series from an interval to the whole line, where the sum over discrete frequencies becomes an integral over a continuum. - [Hilbert Spaces and Orthogonal Projection in Infinite Dimensions](https://lightmysky.com/learn/mathematics/hilbert-spaces-and-orthogonal-projection-in-infinite-dimensions-mt_xmKq8SAgvy): A complete inner-product space keeps the geometry of angles and projections when the dimension is infinite. Closest points to a closed subspace still exist and are unique, and every bounded functional is an inner product with a single fixed vector. - [Compact Self-Adjoint Operators and the Spectral Theorem](https://lightmysky.com/learn/mathematics/compact-self-adjoint-operators-and-the-spectral-theorem-mt_P_Jr9LWFQj): The spectrum contains all eigenvalues and can contain other values. A compact self-adjoint Hilbert-space operator has an orthonormal eigenbasis with real eigenvalues. Nonzero eigenvalues have finite multiplicity; if there are infinitely many, they tend to zero. - [Convolution, Plancherel and Transforming a Derivative](https://lightmysky.com/learn/mathematics/convolution-plancherel-and-transforming-a-derivative-mt_zSwMJrQIMt): Establish the three properties that make the transform useful: convolution becomes multiplication, energy is preserved, and differentiation becomes multiplication by the frequency. ## Complex Analysis 12 topics, ages 19 to 22. https://lightmysky.com/learn/mathematics/areas/complex-analysis - [Complex Functions and the Complex Plane as a Domain](https://lightmysky.com/learn/mathematics/complex-functions-and-the-complex-plane-as-a-domain-mt_hMo851VTEs): Treat a function of a complex variable as a map of the plane to itself, and set up limits and continuity in that setting. - [Complex Differentiability and the Cauchy-Riemann Equations](https://lightmysky.com/learn/mathematics/complex-differentiability-and-the-cauchy-riemann-equations-mt_nZNmZcl9Ss): Define the complex derivative as a limit over all approach directions, derive the Cauchy-Riemann equations, and state the continuity condition that makes them sufficient. - [Harmonic Functions and What Analyticity Forces](https://lightmysky.com/learn/mathematics/harmonic-functions-and-what-analyticity-forces-mt_jqajP39XI4): Show the real and imaginary parts of an analytic function each satisfy Laplace's equation, and recover one from the other. - [Contour Integrals Along Parametrised Paths](https://lightmysky.com/learn/mathematics/contour-integrals-along-parametrised-paths-mt_CaJ7ruAv8P): Integrate a complex function along a path by parametrising it, and bound the result by length times maximum modulus. - [Cauchy's Theorem and Deforming a Contour](https://lightmysky.com/learn/mathematics/cauchys-theorem-and-deforming-a-contour-mt_aacYFVN-HH): Prove Cauchy's theorem from Green's theorem plus the Cauchy-Riemann equations under explicit hypotheses, and use valid contour deformation with correct boundary orientations. - [Cauchy's Integral Formula and Derivatives of Every Order](https://lightmysky.com/learn/mathematics/cauchys-integral-formula-and-derivatives-of-every-order-mt_mViFwrFNWQ): Recover a function's value inside a contour from its values on the contour, then differentiate under the integral to get every derivative at once. - [Power Series and Analyticity in the Complex Plane](https://lightmysky.com/learn/mathematics/power-series-and-analyticity-in-the-complex-plane-mt_Mwxapy_pU8): Expand an analytic function as a power series on a disc, and read the radius of convergence off the nearest singularity. - [Laurent Series and Classifying Isolated Singularities](https://lightmysky.com/learn/mathematics/laurent-series-and-classifying-isolated-singularities-mt_WoQV0gD1O-): Allow negative powers to expand a function on an annulus, and sort singularities by how the negative part behaves. - [The Residue Theorem](https://lightmysky.com/learn/mathematics/the-residue-theorem-mt_BN2biZ_zos): Reduce an integral around a closed contour to a sum of residues, one number per singularity enclosed. - [Evaluating Real Integrals by Residues](https://lightmysky.com/learn/mathematics/evaluating-real-integrals-by-residues-mt_Lj_HXtjzAh): Close a real integral into a contour in the plane, discard the added arc with an estimate, and read the answer off the residues. - [The Argument Principle and Rouche's Theorem](https://lightmysky.com/learn/mathematics/the-argument-principle-and-rouches-theorem-mt_J1dC428M84): Count zeros and poles inside a contour by watching how the argument turns, and transfer a count to a nearby function. - [Conformal Maps and Mobius Transformations](https://lightmysky.com/learn/mathematics/conformal-maps-and-mobius-transformations-mt_WtHCbAT4GI): See that an analytic map with nonzero derivative preserves angles, and use Mobius maps to move standard regions onto one another. ## Differential Equations 22 topics, ages 19 to 24. https://lightmysky.com/learn/mathematics/areas/differential-equations - [What a Differential Equation Says and What a Solution Is](https://lightmysky.com/learn/mathematics/what-a-differential-equation-says-and-what-a-solution-is-mt_tHlHg2nK3o): An equation relating a function to its own derivatives constrains a whole family of functions. A general solution carries arbitrary constants; an initial condition picks one member out of the family. - [Direction Fields and Euler's Method](https://lightmysky.com/learn/mathematics/direction-fields-and-eulers-method-mt_Ak-TZOX3GR): Draw the slope the equation prescribes at each point and the solution curves appear without any formula. Stepping along those slopes numerically is Euler's method, with error you can reason about. - [Separable Equations](https://lightmysky.com/learn/mathematics/separable-equations-mt_itSTsLTkwQ): When the equation factors into a part in y and a part in x, separate the variables and integrate both sides. The constant of integration is where the initial condition enters. - [First-Order Linear Equations and the Integrating Factor](https://lightmysky.com/learn/mathematics/first-order-linear-equations-and-the-integrating-factor-mt_BXmt2pbWp8): Multiply through by a factor chosen to turn the left side into the derivative of a product, then integrate once. The factor is built from the coefficient of y. - [The Logistic Equation and Saturating Growth](https://lightmysky.com/learn/mathematics/the-logistic-equation-and-saturating-growth-mt_7Y_48Ga33L): Growth proportional to the population and to the room left produces an S-shaped curve with a carrying capacity. The model is where equilibrium solutions first appear. - [Second-Order Linear Equations with Constant Coefficients](https://lightmysky.com/learn/mathematics/second-order-linear-equations-with-constant-coefficients-mt_G7sMq1EHwQ): Try an exponential and the equation becomes a quadratic in the exponent. Real, repeated and complex roots give three shapes of general solution. - [Nonhomogeneous Equations and Particular Solutions](https://lightmysky.com/learn/mathematics/nonhomogeneous-equations-and-particular-solutions-mt_gbkQZte_JZ): The full solution is the homogeneous solution plus any particular one. Undetermined coefficients guesses the shape of the forcing; variation of parameters handles the rest. - [Damped and Driven Oscillations](https://lightmysky.com/learn/mathematics/damped-and-driven-oscillations-mt_Y8K1njb1PH): The same equation governs a mass on a spring and a series circuit. Damping decides how the free motion dies away, and driving near the natural frequency produces resonance. - [Fourier Series](https://lightmysky.com/learn/mathematics/fourier-series-mt_DJ3iwfj7NK): Any reasonable periodic function is a sum of sines and cosines. The coefficients are recovered by integrating against each one, which is what lets a system driven by a complicated periodic input be solved one frequency at a time. - [Series Solutions About an Ordinary Point](https://lightmysky.com/learn/mathematics/series-solutions-about-an-ordinary-point-mt_E6NiSd7MED): When the coefficients are not constant, assume a power series, match coefficients and generate a recurrence. Named functions arise this way, such as the Airy functions, which solve y'' = xy. - [The Laplace Transform](https://lightmysky.com/learn/mathematics/the-laplace-transform-mt_8j8GXzITN0): An improper integral converts a function of time into a function of a new variable, turning differentiation into multiplication. Its usefulness rests on the transform being reversible. - [Solving Initial Value Problems with Laplace Transforms](https://lightmysky.com/learn/mathematics/solving-initial-value-problems-with-laplace-transforms-mt_0BhjPVNUwl): Transform the equation, solve the resulting algebra for the transform of the unknown, then invert. Initial conditions enter at the start rather than at the end, which is what makes the method suit discontinuous forcing. - [Sturm-Liouville Problems and Eigenfunction Expansions](https://lightmysky.com/learn/mathematics/sturm-liouville-problems-and-eigenfunction-expansions-mt_oqadAPaSsW): Putting a second-order boundary value problem in self-adjoint form makes its eigenfunctions orthogonal, which is why a solution can be written as a series in them at all. Fourier series is then one case rather than a trick that happened to work. - [Green's Functions for Boundary Value Problems](https://lightmysky.com/learn/mathematics/greens-functions-for-boundary-value-problems-mt_2ctFrfKIgW): Solve the problem once for a point source and the solution for any source is an integral against that answer. The Green's function carries the boundary conditions, so it is built from two homogeneous solutions chosen to die at the right end. - [Systems of Differential Equations and Eigenvalue Solutions](https://lightmysky.com/learn/mathematics/systems-of-differential-equations-and-eigenvalue-solutions-mt_BbOqJ4S1ur): Several coupled unknowns become one vector equation. The eigenvalues of the coefficient matrix give the growth rates and the eigenvectors give the directions along which the system decouples. - [Phase Portraits, Equilibria and Stability](https://lightmysky.com/learn/mathematics/phase-portraits-equilibria-and-stability-mt_ULuZk4lmGr): Draw the trajectories of a system in the plane of its unknowns. Equilibria are classified by the eigenvalues, and the same classification says whether nearby solutions return or leave. - [Classifying Second-Order PDEs and What the Type Decides](https://lightmysky.com/learn/mathematics/classifying-second-order-pdes-and-what-the-type-decides-mt_hlQWbd1kzR): The sign pattern of the second-order coefficients sorts equations into elliptic, parabolic and hyperbolic. The type decides which data may be prescribed and whether a solution smooths its data out or carries it along. - [Separation of Variables and the Heat Equation](https://lightmysky.com/learn/mathematics/separation-of-variables-and-the-heat-equation-mt_XnPDL0X1TL): Assume the solution is a product of one-variable functions and the equation splits into ordinary ones. The boundary conditions pick out which modes are allowed, and the initial profile is matched by a series in those modes. - [The Wave Equation and Its Characteristics](https://lightmysky.com/learn/mathematics/the-wave-equation-and-its-characteristics-mt_-_QCiBWg1w): Along two families of lines the wave equation reduces to an ordinary derivative, and the solution reads as two fixed shapes travelling in opposite directions. A discontinuity in the data is carried along, not smoothed away. - [Laplace's Equation and the Maximum Principle](https://lightmysky.com/learn/mathematics/laplaces-equation-and-the-maximum-principle-mt_jSEkaTav1V): A solution of Laplace's equation equals its own average over any small sphere, so it can have no interior peak. Uniqueness and stability for the boundary value problem follow from that one observation. - [Weak Solutions and Test Functions](https://lightmysky.com/learn/mathematics/weak-solutions-and-test-functions-mt_AKlC3Ol70p): Multiply the equation by a smooth test function, integrate by parts, and the derivative moves off the unknown. A function with a corner can then solve an equation that in the classical sense it cannot. - [Numerical Schemes for Evolution Equations](https://lightmysky.com/learn/mathematics/numerical-schemes-for-evolution-equations-mt_RuDbOapexB): Replace the derivatives by differences and a differential equation becomes a recursion. Consistency and stability together give convergence, and the stability condition is what caps the time step. ## Probability 37 topics, ages 9 to 24. https://lightmysky.com/learn/mathematics/areas/probability - [Likelihood Language](https://lightmysky.com/learn/mathematics/likelihood-language-mt_iFFKZd-Vgv): Use probability language to describe and compare the likelihood of everyday events using words such as certain, likely, even chance, unlikely, impossible - [Ordering Likelihoods](https://lightmysky.com/learn/mathematics/ordering-likelihoods-mt_UcGn2hjhYU): Compare the likelihood of different events and order them from least to most likely, including situations with unequal outcomes such as a bag with more of one colour than another or a spinner with sections of different sizes. Explain reasoning using informal language. - [Equally Likely Outcomes](https://lightmysky.com/learn/mathematics/equally-likely-outcomes-mt_xSgAgg9Ej_): Understand that equally likely means every outcome has exactly the same chance; identify equal and unequal chances in fair coins, fair dice, equal-section spinners, unequal-section spinners, and bags with different colour counts. - [Simple Chance Experiments](https://lightmysky.com/learn/mathematics/simple-chance-experiments-mt_J4j7d3iAfg): Conduct simple probability experiments — flipping a coin, rolling a die, pulling coloured counters from a bag — record results, and compare experimental outcomes with expected theoretical outcomes - [Probability as a Fraction](https://lightmysky.com/learn/mathematics/probability-as-a-fraction-mt_-c4Ca_nBzX): Describe the probability of simple equally likely outcomes with fractions, including unit fractions such as 1/6, 1/2, and 1/3, and place these probabilities on a 0-to-1 scale. - [The 0-to-1 Probability Scale](https://lightmysky.com/learn/mathematics/the-0-to-1-probability-scale-mt_t0g2SlP404): Understand probability as a measure expressed as a number between 0 (impossible) and 1 (certain); place events on the probability scale; express probabilities as fractions, decimals, and percentages - [Calculating Simple Probability](https://lightmysky.com/learn/mathematics/calculating-simple-probability-mt_Zt30Gxi-qp): Calculate the probability of a simple event with equally likely outcomes using the formula: probability = number of favourable outcomes ÷ total number of possible outcomes; express the result as a fraction in its simplest form; apply to rolling dice, drawing from bags, and other simple chance situations - [Experimental vs Theoretical](https://lightmysky.com/learn/mathematics/experimental-vs-theoretical-mt_-bMnJcPJy8): Compare observed relative frequencies with a probability model. With independent trials and fixed outcome probabilities matching the model, larger samples usually give closer estimates. A particular longer run can still be farther away. - [Probabilities Sum to One](https://lightmysky.com/learn/mathematics/probabilities-sum-to-one-mt_3fwYu7imd4): Understand that when all possible outcomes of a trial are listed, their probabilities must add up to 1; use this to find the probability of an event NOT happening: P(not A) = 1 − P(A); apply this shortcut to avoid counting all unfavourable outcomes directly - [Venn Diagrams and Counting Outcomes](https://lightmysky.com/learn/mathematics/venn-diagrams-and-counting-outcomes-mt_eSv_w46u6H): Construct and interpret Venn diagrams with two or three sets to organise and count outcomes; use systematic listing and the product rule for counting to enumerate all possible outcomes of combined events - [Counting Outcomes with the Product Rule](https://lightmysky.com/learn/mathematics/counting-outcomes-with-the-product-rule-mt_74gSbYIG8p): Count the outcomes of several choices made in a row by multiplying the number of options at each stage, instead of listing them all. - [The Addition Rule and Mutually Exclusive Events](https://lightmysky.com/learn/mathematics/the-addition-rule-and-mutually-exclusive-events-mt_QsIusFefeE): Use P(A or B) = P(A) + P(B) - P(A and B), and see why the subtraction disappears when two events cannot happen together. - [Independent Events and the Multiplication Rule](https://lightmysky.com/learn/mathematics/independent-events-and-the-multiplication-rule-mt_MK8EPHEUdN): Use P(A and B) = P(A) × P(B) for independent events, and check whether learning one event happened changes the other's probability. - [Discrete Random Variables and Probability Distributions](https://lightmysky.com/learn/mathematics/discrete-random-variables-and-probability-distributions-mt_yKztNkvvq3): A random variable attaches a number to each outcome, and its distribution lists every value the variable can take with the probability of taking it. Those probabilities have to total 1. - [Probability Spaces: Sample Spaces, Events and Axioms](https://lightmysky.com/learn/mathematics/probability-spaces-sample-spaces-events-and-axioms-mt_4gsBeqJtIQ): Probability has a sample space, events as subsets, and a measure governed by three axioms: it is nonnegative, totals one and adds over disjoint events. Every rule met earlier follows from these axioms. - [The Probability Scale](https://lightmysky.com/learn/mathematics/the-probability-scale-mt_vmQJAtAFuy): Understand probability as a measure on a scale from 0 (impossible) to 1 (certain); use the language of probability including likely, unlikely, certain, and impossible - [Complementary events](https://lightmysky.com/learn/mathematics/complementary-events-mt_XfyqXLqzpx): Understand and apply the rule that probabilities of all mutually exclusive outcomes sum to one; use this to find the probability of a complementary event (P(not A) = 1 − P(A)) - [Experimental probability](https://lightmysky.com/learn/mathematics/experimental-probability-mt_sHJqh6UUya): Record, describe, and analyse the frequency of outcomes from probability experiments to develop an understanding of relative frequency as an estimate of probability - [Sets & Venn Diagrams](https://lightmysky.com/learn/mathematics/sets-and-venn-diagrams-mt_1YwOCMMwD8): Enumerate sets and their unions and intersections systematically using tables, grids, and Venn diagrams to organise and count outcomes - [Tree diagrams](https://lightmysky.com/learn/mathematics/tree-diagrams-mt_GDtFU5fyUv): Generate theoretical sample spaces for single and combined events using listing, tables, and tree diagrams, and calculate theoretical probabilities as the number of favourable outcomes divided by the total number of equally likely outcomes - [Relative Frequency and Expected Outcomes](https://lightmysky.com/learn/mathematics/relative-frequency-and-expected-outcomes-mt_OQCnzkcHUH): Estimate a probability from experimental results, and use a probability to predict how many times an outcome should happen in a set number of trials. - [Tree Diagrams Without Replacement](https://lightmysky.com/learn/mathematics/tree-diagrams-without-replacement-mt_NfiAY_wMwZ): Draw tree diagrams for two or three stages where the first outcome changes the second, multiply along a branch, and add the routes that fit the question. - [Conditional Probability from Tables and Venn Diagrams](https://lightmysky.com/learn/mathematics/conditional-probability-from-tables-and-venn-diagrams-mt_Vx88ugViYP): Work out the probability of one event given that another has happened, by restricting attention to the row, column or region the condition allows. - [Expected Value and Variance of a Discrete Random Variable](https://lightmysky.com/learn/mathematics/expected-value-and-variance-of-a-discrete-random-variable-mt_PFdo0Nm3FP): Weight each value by its probability to find the expected value. Calculate variance as E(X²) minus the square of E(X), and compare a game's stake with its expected payout to judge mathematical fairness. - [The Binomial Distribution](https://lightmysky.com/learn/mathematics/the-binomial-distribution-mt_X2kVGK67Md): Count successes in a fixed number of independent trials that each have the same probability. The number of arrangements times the probability of one arrangement gives each term. - [The Normal Distribution and Standardised Scores](https://lightmysky.com/learn/mathematics/the-normal-distribution-and-standardised-scores-mt_Cs32DT9KnA): A continuous symmetric model where most values sit near the mean. Converting a value to z = (x - μ) / σ says how many standard deviations from the mean it is, which makes different data sets comparable. - [Finding Probabilities and Values from a Normal Model](https://lightmysky.com/learn/mathematics/finding-probabilities-and-values-from-a-normal-model-mt_naXMRySvos): Get the probability of a range of values from a normal model, and go the other way from a stated probability back to the cut-off value, sketching the region every time. - [Conditional Probability, Independence and Bayes' Theorem](https://lightmysky.com/learn/mathematics/conditional-probability-independence-and-bayes-theorem-mt_wIi_XRzwwK): Conditioning restricts the sample space. Independence is the case where it changes nothing, and Bayes' theorem reverses the conditioning, which is how evidence updates a belief. - [Discrete Models Beyond the Binomial](https://lightmysky.com/learn/mathematics/discrete-models-beyond-the-binomial-mt_9zc5xYjDg_): Geometric, hypergeometric and Poisson models make different assumptions. Choosing a model means deciding which assumptions fit the story before using any formula. - [Continuous Random Variables and Density Functions](https://lightmysky.com/learn/mathematics/continuous-random-variables-and-density-functions-mt_jlf4zDTeNm): For a continuous variable, probability is area under a density, so any single value has probability zero. The cumulative function and the density are linked by calculus in both directions. - [Expectation and Variance by Integration](https://lightmysky.com/learn/mathematics/expectation-and-variance-by-integration-mt_kRSaKgp3bs): The mean and variance of a continuous variable are integrals against the density, and the same linearity rules from the discrete case still hold. - [Joint Distributions, Covariance and Independence](https://lightmysky.com/learn/mathematics/joint-distributions-covariance-and-independence-mt_YyTFTAqhSF): Two variables have a joint distribution, from which marginals follow. Covariance measures how they move together, and independence forces zero covariance without the converse holding. - [Conditional Expectation Given a Sigma-Algebra](https://lightmysky.com/learn/mathematics/conditional-expectation-given-a-sigma-algebra-mt_i2sxZc-ER8): Define conditioning on information rather than on an event, as the best prediction measurable with respect to what is known. - [Markov Chains and the Memoryless Assumption](https://lightmysky.com/learn/mathematics/markov-chains-and-the-memoryless-assumption-mt_2wDa_JxJ_o): The next state depends on the current state and on nothing before it. That single assumption turns a process into a matrix, and questions about the far future become questions about matrix powers. - [Stationary Distributions and Long-Run Behaviour](https://lightmysky.com/learn/mathematics/stationary-distributions-and-long-run-behaviour-mt_MGfHvUGkeh): A distribution left unchanged by one step is an eigenvector of the transition matrix for eigenvalue one. Irreducibility and aperiodicity are the conditions under which every starting distribution converges to it. - [Martingales and Optional Stopping](https://lightmysky.com/learn/mathematics/martingales-and-optional-stopping-mt_198uXX3jRJ): A martingale is a process whose expected next value, given everything so far, is its current value. Stopping it at a random time preserves that expectation, as long as the stopping rule is not allowed to look ahead. - [Brownian Motion and Its Defining Properties](https://lightmysky.com/learn/mathematics/brownian-motion-and-its-defining-properties-mt_T5KtWIk07Z): Independent normal increments, continuous paths, and no derivative anywhere. It is the scaling limit of a random walk, and it is the process most continuous-time models are written against. ## Discrete Mathematics 12 topics, ages 18 to 22. https://lightmysky.com/learn/mathematics/areas/discrete-mathematics - [Counting with the Product and Sum Rules](https://lightmysky.com/learn/mathematics/counting-with-the-product-and-sum-rules-mt_3NZC5F7RKP): Count a finite set by breaking the choice into independent stages to multiply, or into disjoint cases to add, and say which structure the problem has. - [Permutations and Combinations](https://lightmysky.com/learn/mathematics/permutations-and-combinations-mt_1L58kuu-w3): Count arrangements where order matters and selections where it does not, and see the second as the first divided by the arrangements you refuse to distinguish. - [Binomial Coefficients and Combinatorial Identities](https://lightmysky.com/learn/mathematics/binomial-coefficients-and-combinatorial-identities-mt_lHXFvudTh6): Read the binomial coefficient as a count and prove identities about it by counting one set two ways rather than by algebra. - [The Pigeonhole Principle](https://lightmysky.com/learn/mathematics/the-pigeonhole-principle-mt_wgfpmZqzd9): Use the observation that more objects than boxes forces a repeat to prove existence results no construction would give. - [Inclusion-Exclusion](https://lightmysky.com/learn/mathematics/inclusion-exclusion-mt_MgU-IMpVHA): Count a union by adding the parts, subtracting the pairwise overlaps, adding the triples back, and prove the alternating pattern is right. - [Recurrence Relations and Characteristic Roots](https://lightmysky.com/learn/mathematics/recurrence-relations-and-characteristic-roots-mt_q0b2bi0iAB): Turn a rule that defines each term from earlier ones into a closed formula by solving the characteristic equation, including the repeated-root case. - [Generating Functions](https://lightmysky.com/learn/mathematics/generating-functions-mt_WAX3O-HjRk): Store a whole sequence as the coefficients of a power series, then use algebra on the series to answer questions about the sequence. - [Graphs, Degrees and the Handshake Lemma](https://lightmysky.com/learn/mathematics/graphs-degrees-and-the-handshake-lemma-mt_1oqO_o3bTO): Define a graph as a set of vertices with a set of pairs, and prove the first theorem about it: the degrees sum to twice the number of edges. - [Trees and Spanning Trees](https://lightmysky.com/learn/mathematics/trees-and-spanning-trees-mt_kMWpsz44yJ): Characterise a tree in several equivalent ways, prove the edge count, and count the labelled trees on n vertices. - [Euler Trails and Hamilton Cycles](https://lightmysky.com/learn/mathematics/euler-trails-and-hamilton-cycles-mt_RU5P5Iqekt): Settle exactly when a graph has a walk using every edge once, and see why the analogous question about vertices has no such clean answer. - [Graph Colouring and Planarity](https://lightmysky.com/learn/mathematics/graph-colouring-and-planarity-mt_6xy6UPBppw): Colour vertices so that neighbours differ, bound the colours needed, and use Euler's formula to see what a graph drawn without crossings cannot contain. - [Matchings and Hall's Theorem](https://lightmysky.com/learn/mathematics/matchings-and-halls-theorem-mt_hkWslW5b5U): Decide when every element of one side can be paired off, and prove that the obvious obstruction is the only one. ## Abstract Algebra 15 topics, ages 21 to 24. https://lightmysky.com/learn/mathematics/areas/abstract-algebra - [Groups, Subgroups and Symmetry](https://lightmysky.com/learn/mathematics/groups-subgroups-and-symmetry-mt_HFAhcaGo2L): A group is a set with a closed, associative operation, an identity and inverses. Integers under addition, permutations under composition and square symmetries share these rules. The choice of set and operation matters. - [Homomorphisms, Cosets and Lagrange's Theorem](https://lightmysky.com/learn/mathematics/homomorphisms-cosets-and-lagranges-theorem-mt_yKiZ8sg7yN): Homomorphisms preserve a group's operation. Left cosets of a subgroup partition the group, as do right cosets separately. For a finite group, Lagrange's theorem says that each subgroup's order divides the group's order. Kernels connect these maps with quotient groups. - [Rings, Fields and Their First Properties](https://lightmysky.com/learn/mathematics/rings-fields-and-their-first-properties-mt_BmhIook13o): Two operations instead of one. Rings cover the integers and polynomials, fields are the rings where division works, and the difference explains why some equations are solvable and others are not. - [Ideals and Quotient Rings](https://lightmysky.com/learn/mathematics/ideals-and-quotient-rings-mt_uQBA9E1JIP): The ring counterpart of a normal subgroup is an ideal, and quotienting by it declares its elements to be zero. Whether an ideal is prime or maximal is read directly off the ring the quotient produces. - [Normal Subgroups and Quotient Groups](https://lightmysky.com/learn/mathematics/normal-subgroups-and-quotient-groups-mt_RU7A-t7wQ5): Cosets can be multiplied consistently exactly when the subgroup is normal. The quotient group that results turns the kernel of a homomorphism into a complete description of the map. - [Group Actions, Orbits and the Class Equation](https://lightmysky.com/learn/mathematics/group-actions-orbits-and-the-class-equation-mt_urut0TFTOR): A group acting on a set splits it into orbits, and each orbit has the size of an index of a stabiliser. Counting one set two ways with that relation proves results no direct argument reaches. - [Polynomial Rings, Irreducibility and Unique Factorisation](https://lightmysky.com/learn/mathematics/polynomial-rings-irreducibility-and-unique-factorisation-mt_m2G9PIHj6y): Polynomials over a field divide with remainder, and that single fact forces unique factorisation into irreducibles. Deciding whether a given polynomial is irreducible over the rationals needs tests rather than inspection. - [Field Extensions and Their Degrees](https://lightmysky.com/learn/mathematics/field-extensions-and-their-degrees-mt_CjOMNt5_i0): Adjoining a root to a field produces a larger field, which is a vector space over the smaller one. Its dimension is called the degree, and degrees multiply along a tower of extensions. - [Modules: Linear Algebra Over a Ring](https://lightmysky.com/learn/mathematics/modules-linear-algebra-over-a-ring-mt_0GMVp-s8wD): A module allows scalars from a ring and need not have a basis. The structure theorem classifies finitely generated modules over a principal ideal domain into a free part and cyclic torsion summands. Over Z this classifies finite abelian groups; over C[x], with a matrix defining the action of x, it gives Jordan form. - [Group Representations and the Group Algebra](https://lightmysky.com/learn/mathematics/group-representations-and-the-group-algebra-mt_y5JlvQGKYW): Realise an abstract group as matrices acting on a vector space, and recognise a representation as a module over the group algebra. - [Maschke's Theorem and Complete Reducibility](https://lightmysky.com/learn/mathematics/maschkes-theorem-and-complete-reducibility-mt_FVNCwUqeTx): Prove that over a field of the right characteristic every representation splits into irreducible pieces, and see exactly where the proof needs that hypothesis. - [The Galois Correspondence](https://lightmysky.com/learn/mathematics/the-galois-correspondence-mt_30kyliQR_L): The symmetries of a field extension form a group whose subgroups match the intermediate fields, in reverse order. Whether that group can be built from abelian pieces is what decides if the roots can be written with radicals. - [The Sylow Theorems and Groups of Small Order](https://lightmysky.com/learn/mathematics/the-sylow-theorems-and-groups-of-small-order-mt_vg2VYkLuz8): Subgroups of prime-power order exist, are all conjugate, and their number is pinned down by two arithmetic conditions. That is enough to rule out simple groups of many orders and to finish the classification of small ones. - [Characters and the Orthogonality Relations](https://lightmysky.com/learn/mathematics/characters-and-the-orthogonality-relations-mt_RleR7YGXjX): Reduce a representation to the trace of each group element, and prove the resulting functions are orthogonal for an inner product on class functions. - [Decomposing a Representation from Its Character](https://lightmysky.com/learn/mathematics/decomposing-a-representation-from-its-character-mt_ccEcWW9xAa): Build the character table of a small finite group over the complex numbers and use weighted character inner products to decompose finite-dimensional representations. ## Number Theory 8 topics, ages 18 to 22. https://lightmysky.com/learn/mathematics/areas/number-theory - [Divisibility and the Division Algorithm](https://lightmysky.com/learn/mathematics/divisibility-and-the-division-algorithm-mt_UkQNs-6-40): Fix what divides means, prove that division leaves a unique quotient and remainder, and use that uniqueness as the base of everything after it. - [Euclid's Algorithm and Bezout's Identity](https://lightmysky.com/learn/mathematics/euclids-algorithm-and-bezouts-identity-mt_26hM0EnJpq): Find a greatest common divisor by repeated division instead of by factorising, then run the algorithm backwards to write it as a combination of the two numbers. - [Modular Arithmetic and Congruence Classes](https://lightmysky.com/learn/mathematics/modular-arithmetic-and-congruence-classes-mt_Lt4_Aazx2j): Agree to identify integers with the same remainder, check the arithmetic survives that identification, and compute in the resulting finite system. - [Linear Congruences and the Chinese Remainder Theorem](https://lightmysky.com/learn/mathematics/linear-congruences-and-the-chinese-remainder-theorem-mt_J9_RMpwcRX): Solve a single linear congruence, say exactly when it is solvable, and combine congruences with coprime moduli into one. - [Fermat's Little Theorem and Euler's Theorem](https://lightmysky.com/learn/mathematics/fermats-little-theorem-and-eulers-theorem-mt_7wW4z4Eidx): Prove that raising a unit to the size of the unit group returns one, and use it to shortcut enormous modular powers. - [Primitive Roots and the Units Modulo n](https://lightmysky.com/learn/mathematics/primitive-roots-and-the-units-modulo-n-mt_KwTDckkMze): Ask when the units modulo n are generated by a single element, and settle which n admit one. - [Quadratic Residues and the Law of Reciprocity](https://lightmysky.com/learn/mathematics/quadratic-residues-and-the-law-of-reciprocity-mt_0KS9eTCY4p): Decide which residues are squares modulo a prime, compute the Legendre symbol, and use reciprocity to answer the question quickly. - [The Distribution of Primes and What Is Still Open](https://lightmysky.com/learn/mathematics/the-distribution-of-primes-and-what-is-still-open-mt_PJwm9kuQWQ): Prove the primes never run out, meet the estimate for how thinly they spread, and see where the honest boundary of current knowledge sits. ## Topology 7 topics, ages 20 to 24. https://lightmysky.com/learn/mathematics/areas/topology - [Metric Spaces: Distance as an Axiom](https://lightmysky.com/learn/mathematics/metric-spaces-distance-as-an-axiom-mt_A3TMb8wL6k): Keep only the properties distance must have, and check how much of the analysis of the real line survives on that alone. - [Open Sets, Closed Sets and Limit Points](https://lightmysky.com/learn/mathematics/open-sets-closed-sets-and-limit-points-mt_9bBdT38dbB): Rebuild openness, closure and limit points from balls alone, and restate continuity as preimages of open sets. - [Completeness and the Contraction Mapping Theorem](https://lightmysky.com/learn/mathematics/completeness-and-the-contraction-mapping-theorem-mt_DYsLzg-IaK): Say what a complete metric space is, and prove that a contraction on one has exactly one fixed point that iteration finds. - [Compactness in Metric Spaces](https://lightmysky.com/learn/mathematics/compactness-in-metric-spaces-mt_dBLXU0wSN9): Compare the open-cover and sequential definitions, prove they agree in a metric space, and see what compactness buys for continuous functions. - [Topological Spaces: Continuity Without Distance](https://lightmysky.com/learn/mathematics/topological-spaces-continuity-without-distance-mt_D2oPq0AVhI): Declare which sets count as open and continuity can be defined with no distance anywhere in sight. One set can carry several topologies, and the choice decides which functions are continuous. - [Compactness and Connectedness](https://lightmysky.com/learn/mathematics/compactness-and-connectedness-mt_ybV1lUedHs): Two properties that continuous maps preserve, which is why proofs lean on them so heavily. Compactness turns any open cover into a finite one, and connectedness rules out splitting a space in two. - [The Fundamental Group and Loops on a Circle](https://lightmysky.com/learn/mathematics/the-fundamental-group-and-loops-on-a-circle-mt_zH50_L0nu4): Loops based at a point, counted up to continuous deformation, form a group. Computing it for the circle is what proves a disc and an annulus are genuinely different spaces rather than merely drawn differently.