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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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Slovak Lower Secondary School Mathematics Standards



Natural numbers
  • Reading and writing;
  • Representing on the number axis;
  • Comparing and rounding;
  • Adding, subtracting, multiplying and dividing
    (also with the remainder);
  • Solving the word oriented tasks;
  • Attributes of divisibility;
  • Prime number, composed number, factoring;
  • Divisor, common divisor, the greatest common divisor;
  • Multiple, the common multiple, the lowest common multiple;
  • Algorithmization of the calculation of the greatest common divisor and the lowest common multiple.


  • Writing in shortened and expanded forms;
  • Assigning the mirror of a number on the number axis;
  • Solving 11 types of simple word oriented tasks;
  • Solving composed word oriented tasks with at most three computing operations;
  • Dividing with remainder, making the check of the solution;
  • Calculating a multiple of a given number, common multiples of two numbers, the lowest common multiple of two numbers;
  • Using criteria of divisibility by numbers 2, 3, 4, 5, 10;
  • Calculating divisors of a given number, common divisors of two numbers, greatest common divisor of two numbers.

Decimal numbers

  • Concept of a decimal number, order of decimal numbers;
  • Comparing and rounding;
  • Adding and subtracting;
  • Multiplying and dividing;
  • Word oriented tasks;
  • Tasks on continual proportion and arithmetic average.


  • Reading and writing decimal numbers;
  • Representing decimal numbers on the number axis;
  • Comparing and rounding off decimal numbers;
  • Simple adding and subtracting of decimal numbers by rote;
  • Multiplying a decimal number by at most a three-digit number;
  • Dividing: a smaller natural number by a bigger natural number (by two-digit number at most), a decimal number by a natural number, a natural number by a decimal number, a decimal number by a decimal number;
  • Solving 11 types of simple word oriented tasks within the range of decimal numbers.
Integer numbers
  • Positive and negative numbers;
  • Opposite numbers;
  • Number axis, interpretation of an absolute value of a number;
  • Organizing integer numbers;
  • Negative decimal numbers;
  • Operations with integer numbers;
  • Problem solving.
  • Recognizing integer numbers;
  • Reading and writing down integer numbers
  • Giving pairs of opposite integer numbers and knowing their basic characteristics;
  • Comparing integer numbers;
  • Adding and subtracting integer numbers readily, multiplying and dividing by rote and in writing;
  • Solving word oriented tasks with integer numbers.
Fractions, rational numbers
  • Concept of a fraction;
  • Equation of fractions;
  • Fraction abbreviation and extension;
  • Representing fractions as decimal numbers;
  • Rational numbers;
  • Organizing fractions according to their value;
  • Mathematical operations with fractions.
  • Understanding a fraction correctly;
  • Reading and writing down a fraction;
  • Expressing a part of a whole in form of fraction;
  • Comparing fractions and marking the result of comparing with predicate symbols <, >, = ;
  • Expanding and abbreviating fractions;
  • Converting a fraction to its basic form;
  • Writing down fractions in a form of decimal numbers and vice versa;
  • Understanding the notion of  rational numbers correctly;
  • Representing a rational number on the number axis;
  • Comparing rational numbers using <, >, = ;
  • Understanding the notion of a mixed number correctly;
  • Writing down a mixed number in the form of a fraction and vice versa;
  • Adding and subtracting fractions and mixed numbers;
  • Calculating fractional part of a whole;
  • Multiplying fraction by another fraction;
  • Dividing integer number by a fraction and vice versa;
  • Dividing a fraction by another fraction;
  • Solving word oriented tasks based on operations with rational numbers.
  • Concept of a percent; ground;
  • Strength of percentage;
  • Part appertaining to the strength of percentage;
  • Bar and circular charts;
  • Solving word oriented tasks on percentage.
  • Determining 1 percent;
  • Distinguishing, naming and calculating the ground, strength of percentage and value appertaining to the strength of percentage;
  • Estimating and determining the strength of percentage by means of circular or bar charts;
  • Calculating the interest, the principal (or the capital) and the interest rate by yielding a simple interest;
  • Solving word oriented tasks on percentage from the area of finance.

Ratio; Direct ratio and indirect ratio

  • Concept of a ratio;
  • Inverted ratio;
  • Sequential ratio;
  • Direct and indirect ratio;
  • Graph of direct and indirect ratio;
  • Simple and compound  rule of proportion;
  • Scale of plans and maps;
  • Solution of tasks.
  • Correct understanding of  the terms ratio, inverted ratio, sequential ratio and knowing how to use them in a solution of the tasks;
  • Dividing, multiplying and diminishing a given number in a given ratio;
  • Marking points in a rectangular system of coordinates in the plain;
  • Understanding the right meaning of the outright and indirect proportions and knowing how to represent it in the coordinate system in the plane;
  • Knowing how to read from the graphs of the outright and indirect proportions;
  • Solving tasks concerning the outright and indirect proportions;
  • Solving exercises using a to-scale map and plans.
Powers and roots
  • Square and cube powers;
  • Square and cube roots;
  • Powers with natural exponent;
  • Operations with powers, with natural exponent, to the 0th power; 
  • Writing down numbers of type a.10n, where 1≤ a < 10.
  • Writing down the products of the same factors into the shape of a power and conversely;
  • Reading the natural power of an arbitrary rational number and designating individual numbers in powers;
  • Reading square and cube roots of arbitrary positive rational numbers and naming particular numbers in powers/roots;
  • Calculating the value of the square and cube roots by rote and on a calculator;
  • Calculating the square of a product;
  • Calculating the product of square roots of non-negative numbers;
  • Writing numbers in the form of a.10n, where 1≤ a < 10;
  • Adding and subtracting powers with the same base and the same natural exponents;
  • Multiplying and dividing powers with the same base;
  • Raising products and quotients to a given power.


  • Numerical expressions;
  • Expressions with variables;
  • Arithmetic functions with numerical expressions; 
  • Arithmetic functions with whole expressions;
  • Fractional expressions;
  • Arithmetic functions with fractional expressions.
  • Understanding letters in the meaning of numbers;
  • Correctly writing algebraic expressions;
  • Adding, subtracting and multiplying algebraic expressions;
  • Modifying the algebraic expressions with the help of formulas (a + b)2,  a2 - b2;
  • Using fractional expressions and giving conditions for them to have a meaning;
  • Shortening and expanding of fractional expressions;
  • Adding and subtracting fractional expressions;
  • Dividing and multiplying fractional expressions by whole and fractional expressions;
  • Expressing the unknown from the equation.

Linear equations and their systems

  • Equality of two expressions;
  • Solving a linear equation with one unknown using equivalent operations;
  • Solving an equation with unknown in denominator;
  • Solving systems of two linear equations with two unknowns;
  • Expressing the unknown from the equation.
  • Deciding if two numerical and algebraic expressions are equal;
  • Equivalent operations with an equation and their use in solving a linear equation with one unknown
  • Solving a linear equation with unknown in the denominator;
  • Making the check for the solution of an equation;
  • Solving the system of two linear equations with two unknowns;
  • Solving word oriented tasks leading to a linear equation with one unknown or to a system of two linear equations with two unknowns.

Linear inequalities

  • Concept of a solution of a linear inequality;
  • Step-by-step substitution;
  • Equivalent operations with a linear inequality.
  • Solving a linear inequality;
  • Multiplying a linear inequality with a negative number;
  • Solving the word oriented tasks leading to a linear inequality.
Functions; Linear function
  • Concept of a function;
  • Dependent and independent variable;
  • Domain and codomain of a function;
  • Linear function, graph of a linear function;
  • Properties of a linear function.
  • Recognizing a linear functional dependence of two variables;
  • Constructing graphs of linear functions using a table for values of dependent and independent variables


Slovak Input Situation
Summary of mathematics taught at the primary level (4 years)



Natural numbers. Quantitative aspects of numbers. Notation of natural numbers in the decimal system. Representation on a number line. Sequence of natural numbers. Comparison of natural numbers. Reading and writing of big numbers. Rounding of natural numbers. First acquaintance with decimal numbers and with their use in concrete models (length, weight). Fraction as a part of a whole.



Addition and subtraction of natural numbers. Multiplication of natural numbers. Algorithm of division of a natural number by one digit divisor.



Variable. Equation. Inequality. Solving of inequalities of type x < a, x > b.



Simple word problems leading to the use of addition or subtraction or multiplication or division. Compound word problems leading to the use of expressions of types a + b + ca - b - ca - (b + c),  (a + b) - ca + (a + b),  a + (a - b),  a + a.ba + a:ba.b + ca.b + c.d.



Recognition of 2D- and 3D-figures: triangle, square, rectangle, circle, sphere, cube, cylinder. Simple lines and curves. Open curves, closed curves. Point, line segment, line. Units of length (mm, cm, dm, m, km), length of a line segment. Construction of line segments of given lengths. Conversions of units of length. Construction of a circle. Construction of lines parallel to a given line. Square grid. Determination of the area of a rectangle using the square grid. Calculation of the area of a rectangle. Units of area (mm2, cm2, dm2, m2, ha). Triangle, quadrilateral - notation of vertices and construction. Construction of a rectangle and a square.  Construction of perpendiculars. Addition and subtraction of lengths of line segments. Perimeter of a triangle, rectangle, square. Solving word problems leading to simple calculations of perimeters and areas of rectangles and squares.


Franco Favilli and Giuseppe Fiorentino, eds.

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