Italian Lower Secondary School Mathematics Standards
Knowledge 
Abilities 
The number
 The number sets and the properties of the operations.
 Multiples and divisors of a number; prime numbers, least common multiple, greatest common divisor.
 Quotients, percentage and proportions.
 Rational numbers; decimal representation of the rational numbers; comparison between relative rational numbers.
 Decimal and periodic numbers; examples of irrational numbers.
 The square root as the inverse operation of the second power.
 Order of size, approximation, error, aware use of the computational tools.
 The formal writing of the properties and the use of the letters as a generalization (from the number to the symbol).
 Basic elements of the algebraic calculus.
 Simple first grade equations.

 To solve problems and calculate easy expressions with rational numbers.
 To decompose a natural number into its prime divisors.
 To read and write numbers in decimal base using the polynomial and the scientific representations.
 To recognize equivalent fractions; to compare rational numbers and represent them on the number line.
 To recognize the different number sets with their formal properties and to operate within them.
 To make easy sequences of approximated computations.
 To use letters to denote the main properties of operations.
 To explore modelizable situations with easy equations; solution of equations in easy cases.

Geometry
 Plane figures; significant elements and characteristic properties of triangles and quadrilaterals.
 The sum of the angles of a triangle and of a polygon.
 Equivalent decomposition of simple polygonal figures.
 The theorem of Pythagoras.
 Intuitive notion of geometrical transformation: translations, rotations, symmetries.
 Similarity.
 The length of the circumference and the area of the circle.
 The meaning of π and related historical remarks.
 The concept of reference system: the Cartesian coordinates, the Cartesian plane.
 The solids; computation of the volume of the main solids (cube, parallelepiped, pyramid, cone, cylinder, sphere).

 To know the properties of solid and plane figures; their classification according to different types of properties.
 To build up isometric figures with given properties.
 To use the transformations to observe, classify and discuss the properties of the figures.
 To compute the perimeter and the area of plane figures; to calculate the length of circumferences and the area of circles.
 To represent points, segments and figures on the Cartesian plane.
 To solve problems by the use of geometrical properties of the figures, also making use of physical models, easy deductions and suitable representations or instruments (rule, square, compass and, in case, geometric software).
 To compute the surfaces and the volume of the main solid figures.
 To visualize 3D objects from a 2D representation and, conversely, to give a plane representation of a solid figure.

The measure
 The international measure system.

 To express measures by means of the measure international system, by using the powers of 10 and significant figures.
 To calculate and express a measure in direct and indirect way.
 To recognize proportional measures in different contexts; to scale off.
 To evaluate the significance of the figures of the result of a given measure.

Relations
 Intuition of the set notion and introduction of the elementary operations between sets.
 Some significant relations (to be equal to, to be a multiple of, to be greater than, to be parallel or perpendicular to,…).
 Functions: tables and diagrams.
 Functions of type y=ax, y=a/x, y=ax^{2} and their graphs.
 Simple models of experiments and of mathematical laws.

 In different contexts, to identify, denote or build up meaningful relations; to recognize analogies and differences.
 To use letters to denote in general form simple properties (numerical, geometrical, physical,… properties).
 To recognize relations among measures in real contexts.
 To use the Cartesian plane, tables and diagrams to represent relations and functions.

Data and estimates
 Data collection of continuous quantities: construction of intervals of equal or different amplitude.
 Histograms.
 Relative or cumulated frequencies and percentages.
 Official data sources and their use.
 Adequate understanding of the different notions of probability: classic, frequentist and subjective.

 Construction of histograms and their interpretation.
 To identify a problem affordable with a statistical survey, to identify the population and the statistical units related to the problem, to prepare a questionnaire, to collect data, to organize them in frequency tables.
 To represent by a diagram and to analyse the indexes adequate to the characteristics (the mode if qualitatively incoherent, also the mean if they may be ordered, the arithmetic average if quantitative).
 To estimate, in easy contexts, the possible results of an experience and their probabilities.
 To recognize wrong diagrams and to correct them, if possible.
 To get information from data collections and graphics from different sources.
 To use IT tools to organize and represent data.
 To compute relative frequencies and percentages and understand their meaning.
 To use relative or cumulated frequencies and percentages to compare data collections.
 To understand when and how to use the different probability measures.

Rational thinking (to be coordinated to all other studies and educational subjects)
 From the natural to the formal language; to understand and to use suitable words for different contexts; the propositions and the introduction of the logical connections non, et, vel.
 To understand the role of definitions.
 To use different logic processes: induction and generalization, deduction. The meaning and use of examples and counterexamples.
 To conjecture about observations in different contexts.
 To properly motivate enunciations, recognizing statements induced by the observation, guessed and conjectured, discussed and proved.
 To recognize problems, data and goals.
 To depict and schematize in different ways a problem, in order to detect a possible way to solve it.
 To prove the processes chosen and implemented in the problems solutions.
 To express clearly the solution, the necessary steps and their connections.
 To evaluate critically the different strategies to solve a problem.

Italian Input Situation Summary of mathematics taught at the primary level (5 years)
Arithmetic
 Natural numbers and their notation in the decimal system;
 Their presentation on a number line;
 Comparison of natural numbers;
 Operations with natural numbers including division by one  and more  digit divisors and powers of natural numbers;
 Solving word problems resulting in one or more operations with natural numbers;
 Decimal numbers, their addition and subtraction;
 Using decimal numbers in real contexts (money, weight, length);
 Fraction as a part of a whole, representation of fractions.
Geometry
 Recognition of 2Dfigures (triangle, rectangle, square, quadrilateral, circle);
 Measuring lengths of line segments  units of length: mm, cm, dm, m, km and their conversions;
 Perimeter of a figure;
 Construction of simple regular geometrical figures;
 Symmetries and simple geometrical transformations of geometrical figures;
 Units of area mm^{2}, cm^{2}, dm^{2}, m^{2};
 Determination of the area of simple polygons (from the triangle to the regular hexagon) using formulas;
 Solving word problems resulting in simple calculations of perimeters and areas of simple polygons.
The measure
 Measures by means of the measure international system, converting them into each other in real simple cases;
 Determination in simple cases of lengths, areas and volumes.
Data and estimates
 Data collection of quantities;
 Histograms.
Rational thinking
 To use the suitable mathematics words formerly introduced in different contexts;
 To use examples to verify hypotheses and conjectures;
 In a word problem, to be able to detect data and their connections and to plan a solution to solve it.

