Module 1
N, Z, Q and inclusions
Prerequisites: Integers, fractions and number sets
What you will learn: Classify a number in the smallest of N, Z and Q and understand inclusions between sets.
Learning paths · Mathematics — first biennium
Complete first-biennium micro-skill path: numbers, sets, algebraic expressions, monomials and polynomials, equations and inequalities, systems, radicals and algebraic fractions, Euclidean and analytic geometry, functions, data and probability. Flex and school-profile-dependent nodes remain identifiable in the catalog.
The common core excludes genuinely profile-specific modules. The Scientific / Applied Sciences track adds quadratics, vectors, matrices, advanced analytic geometry and trigonometry where required.
Skills marked as «often anticipated to year one in scientific tracks» remain in the common biennium: the timing changes, not their role in the learning path.
It is ready and unlocks 5 more skills in the learning path.
Start module →The map shows prerequisites, mastered skills and the next step in the selected learning track.
Prerequisites: Integers, fractions and number sets
What you will learn: Classify a number in the smallest of N, Z and Q and understand inclusions between sets.
Prerequisites: Operations with natural numbers
What you will learn: Recognize and apply commutative, associative and distributive properties and identity elements.
Prerequisites: Operations with natural numbers
What you will learn: Use divisibility tests and factor a number into primes.
Prerequisites: Divisibility and prime numbers
What you will learn: Compute GCD and LCM while understanding Euclid's successive-division method.
Prerequisites: Operations with fractions
What you will learn: Move between fractions and decimals and correctly compare rational numbers.
Prerequisites: Laws of exponents and fraction operations
What you will learn: Apply exponent rules deliberately with integer and rational bases.
Prerequisites: No specific prerequisite
What you will learn: Represent a finite set and correctly use the membership symbols ∈ and ∉.
Prerequisites: Set, element and membership
What you will learn: Distinguish an element from a subset and deliberately use ∈, ⊂ and ⊆.
Prerequisites: Subsets and inclusion
What you will learn: Compute union, intersection and difference of finite sets while interpreting the symbols correctly.
Prerequisites: Properties of operations
What you will learn: Interpret letters as quantities and distinguish variables, constants and algebraic expressions.
Prerequisites: Variable, constant and algebraic expression
What you will learn: Translate verbal relationships into symbolic expressions while preserving structure and parentheses.
Prerequisites: Algebraic expressions and rational numbers
What you will learn: Substitute values for variables and correctly evaluate an expression.
Prerequisites: Algebraic expressions and operation precedence
What you will learn: Identify the operation to perform first and the subexpressions that make up an algebraic expression.
Prerequisites: Algebraic expressions and the meaning of powers
What you will learn: Recognize a monomial in the chosen school domain and distinguish it from non-examples.
Prerequisites: What is a monomial and laws of exponents
What you will learn: Separate coefficient and literal part and normalize a monomial.
Prerequisites: Coefficient, literal part and normal form
What you will learn: Determine the degree with respect to one variable and the total degree of a non-zero monomial.
Prerequisites: Normal form and degree of a monomial
What you will learn: Classify two monomials by structurally comparing their literal parts.
Prerequisites: Similar monomials and integer addition/subtraction
What you will learn: Reduce algebraic sums of similar monomials by combining coefficients and preserving the literal part.
Prerequisites: Normal form, laws of exponents and sign rules
What you will learn: Multiply monomials by combining coefficients and adding exponents of powers with the same base.
Prerequisites: Multiplying monomials and laws of exponents
What you will learn: Raise a monomial to a power by applying the exponent to the coefficient and every factor in the literal part.
Prerequisites: Multiplying monomials, fractions, sign rules and quotient of powers
What you will learn: Divide monomials with rational coefficients, subtract equal-base exponents and recognize when the quotient leaves the monomial domain.
Prerequisites: Monomial operations and expression structure
What you will learn: Plan and simplify mixed monomial expressions while respecting precedence, parentheses and normal form.
Prerequisites: Like monomials and their addition/subtraction
What you will learn: Recognize a polynomial and reduce it by combining only like terms.
Prerequisites: Polynomial reduction and monomial degree
What you will learn: Determine degree, ordered form and completeness of a polynomial in one variable.
Prerequisites: Polynomial reduction and like terms
What you will learn: Add and subtract polynomials while correctly handling parentheses, signs and like terms.
Prerequisites: Monomial multiplication and distributive property
What you will learn: Apply the distributive property by multiplying a monomial by every term of a polynomial.
Prerequisites: Monomial times polynomial, addition and reduction
What you will learn: Multiply every term of one polynomial by every term of the other and correctly combine like terms.
Prerequisites: Multiplying polynomials
What you will learn: Understand and apply (a+b)(a-b)=a²-b² by recognizing cancellation of the mixed products.
Prerequisites: Multiplying polynomials
What you will learn: Understand a binomial square as a product and obtain the first square, double product and second square.
Prerequisites: Addition/subtraction, polynomial multiplication and notable products
What you will learn: Choose the correct strategy, expand the priority block and reduce an expression involving polynomials and notable products.
Prerequisites: Algebraic expressions and the core polynomial path
What you will learn: Distinguish numerical equality, identity and equation by correctly interpreting the role of the = sign.
Prerequisites: Meaning of an equation
What you will learn: Verify a value by substituting it into both members and comparing the results.
Prerequisites: Solution verification and algebraic sums
What you will learn: Apply the same addition or subtraction to both members while keeping explicit the operation that preserves balance.
Prerequisites: Solution verification and fraction operations
What you will learn: Multiply or divide both members by the same non-zero number and simplify correctly.
Prerequisites: First and second equivalence principles
What you will learn: Solve equations ax+b=c using a reasoned sequence of transformations and a final check.
Prerequisites: Simple linear equations, polynomials, distribution and fractions
What you will learn: Simplify parentheses and rational coefficients before applying equivalence principles and solving the equation.
Prerequisites: Linear equations with complete simplification
What you will learn: Interpret the terminal form of an equation and distinguish one solution, no solution and infinitely many solution values.
Prerequisites: Meaning of equations and reading algebraic expressions
What you will learn: Choose the unknown and translate a verbal relationship into an equation without jumping directly to the solution.
Prerequisites: Modeling and solving linear equations
What you will learn: Complete the cycle text → unknown → equation → solution → interpretation in context.
Prerequisites: No specific prerequisite
What you will learn: Distinguish the role of primitive notions, definitions, postulates, theorems and proofs.
Prerequisites: Geometry language
What you will learn: Correctly recognize and name the main basic geometric objects.
Prerequisites: Point, line, ray and segment
What you will learn: Recognize angle relations and calculate measures using complementary, supplementary and vertically opposite angles.
Prerequisites: Geometric objects and angles
What you will learn: Classify a triangle by sides and by angles while keeping the two classifications distinct.
Prerequisites: Proof language and triangle classification
What you will learn: Recognize when data are sufficient for congruence and correctly apply SSS, SAS and ASA.
Prerequisites: Triangle congruence criteria
What you will learn: Use the isosceles property and its converse and complete simple proof steps through congruence.
Prerequisites: Integers: sign, order and number line
What you will learn: Read and write coordinates (x,y) and identify quadrants.
Prerequisites: Sets and ordered pairs; relations recommended
What you will learn: Recognize a function from input-output pairs and identify domain and image.
Prerequisites: Functions, coordinates and substitution
What you will learn: Move between formulas, tables and graph points for simple functions.
Prerequisites: Ratios and function representations
What you will learn: Distinguish direct and inverse proportionality and determine the constant k from data.
Prerequisites: Counting; percentages recommended
What you will learn: Distinguish qualitative, quantitative discrete and continuous data and calculate absolute and relative frequencies.
Prerequisites: Data types and frequencies
What you will learn: Choose a suitable graphical representation and correctly read labels and scale.
Prerequisites: Data types, frequencies and fraction operations
What you will learn: Calculate mean, median and mode and deliberately choose a simple measure of centre.
Prerequisites: Rational numbers and number line
What you will learn: Distinguish rational and irrational numbers and interpret approximation on the number line.
Prerequisites: Sets and membership
What you will learn: Build A×B and understand ordered pairs.
Prerequisites: Sets; true and false
What you will learn: Use negation, and, or and implication in elementary cases.
Prerequisites: Cartesian product
What you will learn: Read a relation as a set of ordered pairs and move between simple representations.
Prerequisites: Square of a binomial
What you will learn: Derive and apply the cube of a binomial while checking coefficients and signs.
Prerequisites: Polynomials and monomial division
What you will learn: Extract a common factor and verify factorization using the distributive law.
Prerequisites: Notable products and common factors
What you will learn: Recognize differences of squares and perfect-square trinomials.
Prerequisites: Common factors and polynomial reduction
What you will learn: Group terms strategically until a common binomial factor appears.
Prerequisites: Notable products and factoring by grouping
What you will learn: Factor simple monic trinomials by finding two numbers with the required sum and product.
Prerequisites: Ordered polynomials and monomial division
What you will learn: Perform simple polynomial divisions and check quotient and remainder.
Prerequisites: Equations and integer order
What you will learn: Interpret an inequality as a solution set and represent it as an interval.
Prerequisites: Intervals and equivalence principles
What you will learn: Solve linear inequalities, reversing the sign when multiplying or dividing by a negative number.
Prerequisites: Angles and fundamental relations
What you will learn: Use angle relations created by parallel lines and a transversal.
Prerequisites: Parallelism and triangles
What you will learn: Classify quadrilaterals using an inclusive hierarchy of properties.
Prerequisites: Right triangles and powers
What you will learn: Apply the Pythagorean theorem by correctly identifying hypotenuse and legs.
Prerequisites: Functions, coordinates and equations
What you will learn: Read slope and intercept and construct points on a line.
Prerequisites: Mean, median and mode
What you will learn: Understand range and compare datasets with the same centre but different spread.
Prerequisites: Fractions and sets
What you will learn: Build an equally likely sample space and calculate classical probability.
Prerequisites: Real numbers, irrationals and powers
What you will learn: Simplify radicals by extracting perfect-square factors.
Prerequisites: Simplifying radicals
What you will learn: Operate with like radicals and recognize valid reductions.
Prerequisites: Complete factorization and monomial division
What you will learn: Find domain restrictions and simplify algebraic fractions using appropriate factorization.
Prerequisites: Algebraic-fraction domain and numerical fractions
What you will learn: Use common denominators while preserving domain restrictions.
Prerequisites: Algebraic-fraction operations and linear equations
What you will learn: State domain restrictions, clear denominators and verify that the solution is admissible.
Prerequisites: Equations and linear functions
What you will learn: Interpret a system as the intersection of two lines and classify its solutions.
Prerequisites: Meaning of systems and linear equations
What you will learn: Solve 2×2 systems by substitution or elimination and verify the pair.
Prerequisites: Linear inequalities
What you will learn: Intersect solution sets of several linear inequalities.
Prerequisites: Linear inequalities and factorization
What you will learn: Build a sign chart from the critical points of linear factors.
Prerequisites: Product/quotient signs and algebraic-fraction domain
What you will learn: Solve simple rational inequalities by distinguishing numerator zeros from excluded denominator values.
Prerequisites: Functions and representations
What you will learn: Calculate f(g(x)) by distinguishing inner and outer functions and understand that composition order matters.
Prerequisites: Function concept and linear equations
What you will learn: Interpret the inverse as recovering the input and recognize the role of one-to-one behaviour in simple cases.
Prerequisites: Tables, formulas and graphs
What you will learn: Read vertex, symmetry and opening direction of a translated absolute-value function.
Prerequisites: Inverse proportionality and fraction domain
What you will learn: Interpret domain, asymptotes and translations of a simple reciprocal function.
Prerequisites: Coordinates and Pythagoras
What you will learn: Calculate midpoint and distance between two points in the Cartesian plane.
Prerequisites: Linear functions and coordinates
What you will learn: Find and interpret slope, intercept and points of a line.
Prerequisites: Line equations and geometric parallelism
What you will learn: Compare lines through slopes and recognize their relations.
Prerequisites: Distance, coordinates and line equation
What you will learn: Derive a line from two points and move between slope, point-slope form and slope-intercept form.
Prerequisites: Parallel lines and proportions
What you will learn: Use parallelism to establish proportions between corresponding segments.
Prerequisites: Thales and triangles
What you will learn: Recognize similarity criteria and use a consistent scale factor.
Prerequisites: Coordinates and quadrilaterals
What you will learn: Recognize translations, reflections and rotations and apply coordinate rules.
Prerequisites: Similarity and Pythagorean theorem
What you will learn: Apply both Euclid theorems by linking legs, altitude, hypotenuse and projections.
Prerequisites: Geometric objects and angles
What you will learn: Recognize centre, radius, diameter, chord, arc, secant and tangent and use their fundamental relations.
Prerequisites: Circle elements and angles
What you will learn: Relate central and inscribed angles intercepting the same arc.
Prerequisites: Circle geometry and Pythagoras
What you will learn: Use the perpendicular from the centre to a chord and relate equal chords to equal arcs.
Prerequisites: Angles, segments and congruence criteria
What you will learn: Construct a perpendicular bisector and angle bisector with compass and straightedge and understand their locus properties.
Prerequisites: Introductory variability
What you will learn: Compare centre and spread of simple distributions.
Prerequisites: Comparing distributions
What you will learn: Calculate deviations, variance and standard deviation and interpret them as measures of spread around the mean.
Prerequisites: Measures of centre and spread
What you will learn: Critically choose measures of centre and spread in the presence of extreme values and compare the graph with the numbers.
Prerequisites: Classical probability and sets
What you will learn: Build compound sample spaces and use complementary events.
Prerequisites: Relative frequencies and classical probability
What you will learn: Compare observed relative frequency and theoretical probability without confusing a finite sample with the model.
Prerequisites: Compound events and set operations
What you will learn: Use union, intersection and complement to describe events and calculate their probabilities in simple equally likely spaces.
Progress is stored in a protected technical cookie in your browser.