Czech Lower Secondary School Mathematics Standards
- Multiples and divisors of a number;
- Prime numbers;
- Least common multiple, greatest common divisor;
- Powers of natural numbers;
- Integer numbers.
- To decide whether a given number is/is not a multiple/divisor of a certain number.
- To find all divisors of a given number.
- To decide without dividing if the number is divisible by 2, 3, 4, 5, 6, 8, 9, 10;.
- To decompose a natural numbers into its prime divisors.
- To find the LCM and GCD of a group of numbers.
- Real numbers;
- Absolute value;
- Rational numbers;
- Operations with rational numbers;
- To recognize a natural, integer, rational, irrational number.
- To write numbers in the decimal system
- To represent all rational and some irrational numbers on the number line.
- To understand the notions positive, negative, non-positive, non-negative numbers.
- Find the absolute value of a real number.
- To define a rational number.
- To compare rational numbers.
- To write a fraction as a decimal or periodic number.
- To understand that one rational number might be expressed as infinitely many fractions.
- To reduce and raise a fraction, simplify a fraction; to find irreducible fractions.
- To make calculations with mixed numbers.
- To make calculations with rational numbers using different methods.
- To calculate second and third powers of a number.
- To make calculations with powers.
- To calculate square roots of some numbers, to find second and cube roots using tables and calculators.
Percents, ratio, proportion
- Proportion, rule of three
- To calculate the percent as 1/100 of a whole
- To make calculations about percents.
- To apply percents - simple interest.
- To express the ratio in its simplest form by cancellation.
- To understand the progressive ratio as a short form of simple ratios.
- To divide a number in a given progressive ratio.
- To understand proportion as the equality of two ratios.
- To calculate the unknown term of a proportion.
- To apply the rule of three when solving problems.
- Rational expressions.
- To recognize expressions.
- To construct expression.
- To find the value of an expression for a given value of variable.
- To apply the notions member, coefficient, degree, value of a polynomial.
- To add, subtract, multiply polynomials, to divide a polynomial by a linear binomial.
- To factorize polynomials.
- To use formulas for (A + B)2, A2 - B2, A3 - B3, A3 + B3.
- To distinguish between polynomials and rational expressions.
- To reduce and raise rational expressions by a number, monomial and linear binomial.
- To make calculations with rational expressions.
- Equations and their rearranging;
- Linear equations;
- Quadratic equations;
- Systems of linear equations.
- To recognize equations.
- To verify if a given object is a root of the given equation.
- To define a root of an equation.
- To add the same expression to both sides of an equation, to multiply both sides of an equation by the same non-zero expression.
- To determine whether a given equation is linear.
- To solve linear equations.
- To classify the number of roots of a linear equation.
- To define a root of a linear equation as the intersection on the graph of the corresponding linear function and x-axis.
- To transform an equation with the unknown in the denominator into a linear equation.
- To apply linear equation s in problem solving.
- To solve quadratic equations ax2 + b = 0, x2 - m = 0, m > 0.
- To recognize systems of linear equations.
- To understand that the solution of a system of two linear equations with two unknowns is an ordered number pair.
- To have the idea about the number of solutions of a system of two linear equations with two unknowns.
- To solve a system of two linear equations with two unknowns.
- To find the solution of two linear equations with two unknowns as coordinates of the intersection of graphs of two linear functions.
- Coordinate system;
- Direct proportion;
- Indirect proportion;
- Linear function;
- Quadratic function;
- Trigonometric functions.
- To choose an appropriate coordinate system in a plane.
- To represent a point in a given coordinate system.
- To determine coordinates of a point represented in a given coordinate system.
- To recognize a function given by a table, graph and formula.
- To decide if a number belongs to the range of a function.
- To find the range of a function from its graph, formula, table.
- To find the function value for a given element from its range.
- To decide if a given set of points is a graph of a function.
- To decide from the graph if the function is increasing/decreasing.
- To select functions which are direct/indirect proportions.
- To find the coefficient of a given direct/indirect proportion.
- To construct a graph of a direct/indirect proportion.
- To select functions which are linear.
- To construct a graph of a linear function.
- To select functions which are quadratic of the form y = ax2, a > 0.
- To construct a graph of a quadratic function.
- To define tangent, sine, cosine of an acute number.
- To grasp tangent, sine, cosine as dependencies, to sketch their graphs.
- To find values of trigonometric functions using table and calculators.
- To find α when knowing sin α, cos α, tg α.
- To use trigonometric functions in problem solving.
- Point, straight line, plane;
- Half-line, line segment, half-plane, angle;
- Sets of a given property;
- Circle, circumference of a circle;
- To know what defines a straight line in a plane.
- To characterize the mutual position of 3 points/2 lines in a plane.
- To find out if two lines are parallel, perpendicular.
- To understand properties of half-lines, line segments, angles.
- To measure an angle, classify angles by their size.
- To know properties of an angle bisector.
- To know and determine properties of adjacent angles, vertically opposite angles, alternate angles, corresponding angles on parallel lines.
- To find simple sets of points of a given property.
- To know and apply Theorem of Thales.
- To define a circle, circumference of a circle.
- To know the notion of an arc, to determine its length.
- To characterize the mutual position of a circle and line.
- To know properties of a tangent and of a secant to a circle.
- To know formulas for the perimeter and area of a circle.
- To recognize a polygon.
- To decide if a given polygon is convex or concave.
- To characterize a regular polygon.
- To know and be able to use properties of triangles.
- To classify triangles.
- To know properties of angles, lines joining the mid-points of 2 sides, heights, medians of a triangle.
- To characterize the circumscribed circle, inscribed circle.
- To calculate the height and area of a triangle.
- To describe an isosceles and equilateral triangle.
- To know the Theorem of Pythagoras and to apply it when solving problems.
- To recognize a quadrilateral.
- To calculate the area of a quadrilateral by dividing it into two triangles.
- To know the sum of inner angles of a quadrilateral.
- To classify quadrilaterals.
- To describe trapezium and its special types, parallelogram, rhombus, rectangle, square.
- To calculate and to apply formulas for the area of a trapezium, parallelogram, rhombus, rectangle, square.
- Congruence of geometric figures;
- Similarity of geometric figures;
- Point symmetry;
- Line reflection;
- To understand intuitively the concept of the congruence/similarity of figures.
- To have an active command of theorems conditions for the congruence/similarity of triangles.
- To understand the notions object, image, to know that the image of congruent figures in an isometry are congruent figures, to understand the notion of a fixed point.
- To find the image of a point, line, triangle and circle in an isometry.
- To define a point symmetry/line reflection/translation, to know their properties.
- To recognize figures which are isometric,
- To apply symmetries when solving problems.
- Basic geometric constructions;
- Constructions of an angle;
- Construction of a triangle;
- Construction of a quadrilateral;
- Construction of a regular polygon;
- Construction of a circle.
- To recognize polyhedra.
- To describe their vertices, edges, faces on a model.
- To define and recognize a cube/cuboid, to sketch it in an oblique projection.
- To know its properties.
- To mark its face and body diagonals.
- To know and apply formulas for a cube/cuboid volume and surface area.
- To know the properties and be able to sketch a prism/a pyramid.
- To describe its basis and lateral surface.
- To use Theorem of Pythagoras and trigonometric functions for determining the height and lengths of edges.
- To sketch nets.
Czech Input Situation
Summary of mathematics taught at the primary level
- Natural numbers and their notation in the decimal system;
- Representation on a number line;
- The number zero;
- Comparison of natural numbers;
- Rounding natural numbers;
- Operations with natural numbers including division by one- and two-digit divisors;
- Solving word problems resulting in one or two operations with natural numbers;
- Decimal numbers with tenths and hundreds, their addition and subtraction;
- Using decimal numbers in concrete models (money, weight, length);
- Fraction as a part of a whole, representation of fractions;
- Simple cases of addition of fractions with the same denominator.
- Recognition of 2D-figures (triangle, rectangle, square, quadrilateral, circle);
- Measuring lengths of line segments - units of length mm, cm, dm, m, km and their conversions;
- Construction of line segments of given lengths;
- Perimeter of a figure;
- Construction of lines parallel to a given line, perpendicular to a given line;
- Construction of a circle with a given center and radius;
- Construction of a rectangle, square and right-angles triangle;
- Determination of the area of a rectangle using the square grid;
- Calculation of the area of a circle, rectangle;
- Units of area mm2, cm2, dm2, m2, ha;
- Recognition of 3D-figures (cube, cuboid, prism, cylinder, pyramid, cone, sphere);
- Nets of cuboids and cubes;
- Calculation of the surface area of a cuboid and cube by adding the area of their bases and faces;
- Solving word problems resulting in simple calculations of perimeters and areas of rectangles and squares.
 In our schools, special triangles are used for these constructions - isosceles right-angled triangles with the height to the base.