Master Mathematics with Sangakoo

Learning platform designed to make mathematics accessible, engaging, and comprehensive for learners at (almost) every level.

• Angles - Learn to identify the different types of angles and the basic operations on them by using different units of measure: degrees and radians.
• Areas and volumes of geometric bodies - Learn how to calculate the area and volume of three-dimensional geometric bodies such as the tetrahedron,the cube, the cuboid, the prism, the pyramid, the cylinder, the cone, the sphere, and others.
• Areas of plane figures - Learn how to calculate areas and perimeters of various plane shapes: the triangle, quadrilaterals, regular polygons and the circle.
• Arithmetical progressions - Learn how to identify arithmetic progressions to then calculate their general term or add their first terms.
• Circumference - Learn the basic elements of a circle, how to calculate its area and perimeter and also its positions in relation with a point, a line and another circle.
• Combinatorics - Learn the notion of a factorial and a combinatorial number, as well as the principle of addition and multiplication, i.e. combinatorial concepts focused on solving counting problems such as variations, permutations and combinations.
• Complex numbers - Learn the concept of complex numbers and moreover expressed in different forms: binomial, polar, trigonometric and exponential. Learn how to operate with them as well as mechanisms to graph them in the complex plane.
• Conic classification - Learn how to classify a conic defined by an equation or by its associated matrix and Euclidean invariants.
• Conics - Learn about the main properties and different equations of conics, such as the circle, the ellipse, the hyperbola and the parabola.
• Continuity - Learn the concept of continuity of a function and the different types of discontinuities that can be found. Also discover results on continuous functions as the Weierstrass theorem, Bolzano's theorem or the Darboux property (also known as the Intermediate Value Theorem).
• Derivatives - Learn the concept of a derivative function, studying applications (mathematical and physical) for better understanding and practice with the calculation of derivatives of all types of functions.
• Determinants - Learn how to calculate determinants of any order, about some of their most important properties and how to find the adjugate matrix.
• Differential equations - Learn the general concepts of Ordinary Differential Equations (ODE) and methods to solve different types of ODE, as well as different systems of differential equations.
• Diophantine equations - Learn to solve linear Diophantine equations, after previously having reviewed the Euclid algorithm to calculate the greatest common divisor of two numbers and the Bézout coefficients. There is also a very specific type of Diophantine equations: the quadratic equation.
• Distributions - Learn the concept of the random variable and then work with the probability function, distribution function as well as the mean, the variance and the standard deviation of random variables. Also learn the most used and known distributions such as the Binomial (or Bernoulli) and Normal distribution.
• Divisibility - Learn the concepts of the divisor and multiple of a number, prime and composite numbers, and of the decomposition of a number into prime factors. Mastering all this allows you to calculate the greatest common divisor and the least common multiple of a set of numbers.
• Exponential equations - Learn to solve equations where the unknown appears in the exponent. You will discover various methods such as using logarithms or the reduction to the same base.
• Finance Mathematics - Learn the algorithms to calculate simple and compound interest, to then solve banking problems.
• First degree equations - Learn how to find unknown values (unknown variables) with first degree equations, the most simple. But be careful, there are also ones with n unknown variables. It's best to practice by solving problems.
• Fractions - Learn the concept of fractions, how to operate with them and everything around them, as the irreducible fraction or mixed numbers. At a more advanced level you will learn how to write rational numbers.
• Functions - Learn the concept of a function and work with all kinds of them: related, rational, irrational, piece-wise... Study the general characteristics of the functions and operations between them.
• Geometric progressions - Learn how to identify geometric progressions to then calculate their general term, add or multiply their first terms and, where possible, to add their infinite elements.
• Geometry in space - Learn to recognize and classify the main geometric shapes in space.
• Geometry in the space - Learn concepts of affine geometry in space: the relative positions and distances between points, lines and planes.
• Graphic representation of functions - Learn how to represent functions studying symmetry, cut-off, asymptotes, maximums/minimums and points of inflection.
• Inequations - Learn how to work with inequalities and inequations of different types: of first and second degree, with one variable or two and inequality systems.
• Integration - Enter the world of integration with the indefinite and definite integral, and learn the methods to solve them, such as by parts or by substitution of the variable. The level also increases with the integrals of two variables, introducing integrals on curves and surfaces, and the theorems of Green, Gauss and Stokes. And let's not forget the two basic applications of integrals, the calculation of areas and volumes.
• Interpolation - Learn the concept of interpolation and two methods to calculate the polynomial interpolation of a network of points: the Hermite polynomial and the Taylor polynomial.
• Limits - Learn the concept of the limit of a function at finite values and infinity, and the calculation of both finite and infinite limits, including the case of uncertainties.
• Linear programming - Learn how to write problems with constraints as two-variable inequalities and how to get the area of possible solutions. All this will be useful to you to find unknown values, and not just any, but the optimal.
• Logarithmic equations - Learn how to solve equations that involve logarithms. You will handle logarithmic equations of the first and second degree and systems with two equations and two unknowns.
• Logarithms - Learn the basic concepts and properties of logarithms and algorithms to deal with ease with algebraic expressions containing logarithms.
• Matrices - Learn about matrices, their elements and how to operate with them. You will also learn two methods to calculate the very important invertible matrix.
• Measurements - Learn to move between units of measurement of the metric system, the traditional measurement system and the English system.
• Movements in the plane - Learn the different movements in the plane, such as geometric transformations, translations, rotations and symmetries.
• Plane geometry - Learn about the different equations of lines in the plane, their relative positions, and how to calculate the angle and the distance between them and other elements.
• Polygons - Learn the basic concept of polygons and their elements, so you can distinguish and classify them.
• Polynomials - Enter the world of polynomials, starting with monomials, and learn to apply rules as the indispensable Ruffini rule, which will take you to the remainder and factor theorems. Finally, you will get to algebraic fractions and how to use polynomials.
• Powers and roots - Learn to master powers and radicals. This will enable you, for example, to calculate with rational exponents and rationalizing radicals.
• Probability - Enter the world of probability and start learning the basics to solve problems, perform operations between events and apply Laplace's rule. At a more advanced level discover conditional probability, the total probability theorem and Bayes' formula. To get there it is always a good idea to learn to work with contingency tables and tree diagrams.
• Proportionality - Learn the concepts of direct and inverse proportionality that allow you to solve problems with more complex statements, such as proportional distributions.
• Quadratic equations - Learn about the algorithms needed to solve a second degree equation, even if they are incomplete. You will also discover how to build a second degree equation knowing the value of the solutions.
• Quadric classification - Learn how to determine what an analytical quadric is by using an algorithm that is based on equations or Euclidean invariants.
• Sequences - Learn to handle an infinite list of real numbers, studying and classifying them, e.g. in monotone, bounded, and calculating the limit of the sequence or some families of sequences.
• Set theory - Learn the basic definitions of the set theory to work with the Algebra of sets, i.e. unions, intersections, complementations...
• Similarities - Learn the meaning of similarity between triangles and polygons in general. Discover what looks like a simple statement but actually is as important as the theorem of Thales.
• Statistics - Learn the basics of statistics, such as the arithmetic mean, the mode or the median, and others, more complex, as the mathematical expectation and dispersion measures.Also learn to construct frequency tables to make the various statistical graphs.
• Systems of equations - Learn how to solve all kinds of systems: using the famous methods of substitution and inspection, Cramer's rule, Gauss's law for systems with several equations and unknown variables, and last but not least with the Rouche-Frobenius theorem that lets you discuss the systems.
• The decimal numbers - Learn to identify sets of decimal numbers and the different types, to then learn the operations you can do with them, as the conversion of decimals to fractions.
• The integers - Learn to identify sets of whole numbers and the basic operations you can do with them. You will also see what you should do first when you have a combination of operations, combined operations on sets.
• The irrational numbers - Learn what irrational numbers are and how to represent them on the number-line.
• The natural numbers - Learn to identify sets of natural numbers and the properties of addition and subtraction of these numbers.
• The rational numbers - Learn to identify rational numbers by their construction. At a more advanced level, you will learn about algebraic structures with binary operations such as addition and multiplication of rational numbers.
• The real numbers - Learn to identify the different sets of real numbers and their operations. These operations include addition and multiplication, absolute value, the calculation of distances and interval operations.
• The sexagesimal system - Learn to operate with numbers expressed in the sexagesimal system and other important number systems such as the decimal, the binary and the hexadecimal.
• Triangles - Learn the definition and the elements of a triangle and discover its properties and classifications. Learn how to calculate its perimeter and area, for example by using Heron's formula.
• Trigonometric equations - Learn how to solve certain trigonometric equations and solve any type of triangle depending on the data given by the problem using trigonometric relations and the laws of sine and cosine.
• Trigonometry - Discover trigonometric functions like sine, cosine, tangent and reciprocal functions. Learn how to use them to solve problems involving right triangles.
• Vector analysis - Learn how to parameterize a curve or surface to calculate the gradient of a scalar field and the curl and divergence of a vector field.
• Vectors - Learn the main concepts on vectors, as the basic "module", and start operating with them, not only with addition and subtraction, but also with linear combinations, bases, scalar and vector product.