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Project Details


112318-CP-1-2003-1 -IT-COMENIUS-C21


Project span

Project Coordinator

CAFRE Centro di Ateneo di Formazione e Ricerca Educativa
Università di Pisa

Contact person
Prof. Franco FAVILLI


Project Partners
(CZ) Univerzita Karlova v Praze

(DK) Skårup Seminarium

(FR) Institut Universitaire de Formation des Maîtres de Créteil

(IT) Università degli Studi di Firenze

(IT) Università degli Studi di Siena

(SK) Univerzita Mateja Bela

National Standards
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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.

Number systems

  • Real numbers;
  • Absolute value;
  • Rational numbers;
  • Fractions;
  • Operations with rational numbers;
  • Powers;
  • Roots.


  • 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

  • Percent;
  • Ratio;
  • 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.


  • Expressions;
  • Polynomials;
  • 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,    x -  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;
  • Functions;
  • 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.

Basic 2D-notions

  • Point, straight line, plane;
  • Half-line, line segment, half-plane, angle;
  • Sets of a given property;
  • Circle, circumference of a circle;
  • Polygon;
  • Triangle;
  • Quadrilateral;


  • 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.
Geometric mappings
  • Congruence of geometric figures;
  • Similarity of geometric figures;
  • Isometries;
  • Point symmetry;
  • Line reflection;
  • Translation.


  • 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.




  • Polyhedron;
  • Cube;
  • Cuboid;
  • Prism;
  • Pyramid.


  • 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[1];
  • 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.

[1] In our schools, special triangles are used for these constructions - isosceles right-angled triangles with the height to the base.


Franco Favilli and Giuseppe Fiorentino, eds.

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