--- title: "What a 15-year-old learns" description: "34 topics are written for age 15, across 2 subjects." canonical: https://lightmysky.com/learn/age/15 source: https://lightmysky.com/learn/age/15.md retrieved: 2026-09-02 --- > **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`. # What a 15-year-old learns 34 topics are written for age 15, across 2 subjects. Topics are placed by what they need, not by school year, so this is a starting point rather than a syllabus. Page: https://lightmysky.com/learn/age/15 ## Mathematics - [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 - [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. - [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. - [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. - [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. - [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. - [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. - [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. - [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. - [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. - [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): Recognise and sketch the shapes of y = x³, y = 1/x and y = kˣ, and match each shape to the kind of situation it describes. - [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. - [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. - [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. - [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. ## Science - [Inside the Atom: Protons, Neutrons and Electrons](https://lightmysky.com/learn/science/inside-the-atom-protons-neutrons-and-electrons-mt_WqDXAr2IkA): Place protons, neutrons and electrons in the atom with their relative charges and masses. Read the atomic number and mass number off the periodic table and use them to count every particle in an atom. - [Isotopes and Relative Atomic Mass](https://lightmysky.com/learn/science/isotopes-and-relative-atomic-mass-mt_NFoRZd6UN_): Isotopes of an element differ in neutron count but behave the same chemically. Relative atomic mass is the weighted mean of an element's isotopes, which is why most table values are not whole numbers. - [Electron Shells and the Shape of the Periodic Table](https://lightmysky.com/learn/science/electron-shells-and-the-shape-of-the-periodic-table-mt__Sdcw4RnZF): Fill electron shells for the first twenty elements and use the result to explain the table itself: the group number gives the outer-shell electrons, the period number gives how many shells are occupied. - [Group 1, Group 7 and Group 0: Trends You Can Predict](https://lightmysky.com/learn/science/group-1-group-7-and-group-0-trends-you-can-predict-mt_cy1OrVAjIo): Use outer-shell electrons to predict how alkali metals, halogens and noble gases behave, including why group 1 gets more reactive down the group while group 7 gets less. - [Ionic Bonding: Electron Transfer and Charged Lattices](https://lightmysky.com/learn/science/ionic-bonding-electron-transfer-and-charged-lattices-mt_sf24C4YLtN): Metal atoms hand their outer electrons to non-metal atoms, and the ions that result pull on each other in a giant lattice. Work out ionic charges and formulas from the group numbers. - [Covalent Bonding: Shared Pairs and Simple Molecules](https://lightmysky.com/learn/science/covalent-bonding-shared-pairs-and-simple-molecules-mt_fn5Dq6qUjP): Two non-metal atoms fill their outer shells by sharing electron pairs. The shared bond is strong, but the pull between separate molecules is weak, which sets the properties of simple molecular substances. - [Giant Covalent Structures: Diamond, Graphite and Graphene](https://lightmysky.com/learn/science/giant-covalent-structures-diamond-graphite-and-graphene-mt_YTEj9srwPG): Some covalent substances never stop bonding. Explain diamond, graphite, graphene and silicon dioxide from how many bonds each atom makes and what is left over. - [Metallic Bonding and Alloys](https://lightmysky.com/learn/science/metallic-bonding-and-alloys-mt_yXOGOwHmZ4): Metal ions sit in a regular lattice inside a sea of delocalised electrons. That one picture explains conduction, malleability, and why an alloy is harder than the pure metal.