TOLC-B structure and syllabus

The Basic Mathematics module is intended to test the student's basic preparation, which is required for all degree courses in the science area, even those that make relatively less use of mathematics. The same module can be found in both the TOLC-S and TOLC-B tests, as it provides important information for all the above-mentioned degree courses. However, it is worth remembering that the mathematical knowledge and skills that are tested in this module are only a part of those that it is good to have at the entrance to degree courses such as those in Mathematics or Physics. It is therefore suggested that those interested in such courses read the Reference Framework for Mathematics, which can be found on the Orientazione project website and which provides a more complete indication of the preparation that may be useful.

In order to answer the questions in this module, it is necessary to understand the text of the questions and answers and to reason about the information provided, using mathematical knowledge that is included in the first three or four years of the curricula of all secondary schools. The specific knowledge required is grouped in six nuclei. The general skills of understanding, representing, reasoning, modelling and problem-solving are described in three further cross nuclei. The questions are constructed in such a way that a calculator is not required, and its use is not permitted during the test. It should be noted that in a single question, there may be concepts that are indicated in the syllabus in several nuclei. The terms and symbols that are used vary from those most frequently used in school and in early university lessons.

For each core, there is a description of skills and abilities to work with the concepts that are useful for answering the questions, linking them appropriately. In the Reference Framework for Mathematics developed by the Orientazione Project, there is a more detailed description of the topics and skills indicated in the syllabus, set in a broader framework and accompanied by several examples of questions and exercises that may be useful for students to assess and possibly improve their preparation. The Reference Framework of Logic, Reasoning and Problems may also be useful for these purposes.

Numbers
In order to answer the questions involving the topics of this nucleus, it is necessary to work with numbers, with confidence and flexibility, using different representations of the numbers themselves and choosing those that are most useful depending on the situation and goals. The use of a calculator is not permitted during the test, as it is never necessary to answer the questions; instead, it is useful to be able to do simple calculations by mind, possibly helping the working memory with paper and pencil to perform elementary algorithms and to note down partial results. Among various equivalent procedures for performing calculations, it is important to choose the most efficient and simplest ones; to this end, the meaning and properties of operations and ordering must be known and coordinated. The ability to make estimates, besides being useful in itself, allows one to quickly assess the plausibility of the result of calculations and thus provides a useful control tool.

• addition and multiplication operations between integers, fractions, decimal numbers. Sorting. Properties of operations and ordering. Subtraction and division. Concept of rational number. Representations of numbers on a line. Real numbers
• integer division with remainder. Divisors and multiples of an integer; greatest common divisor (GCD) and least common multiple (LCM) of two or more positive integers
• power with integer exponent of a number and properties of powers. Positive integer root of a positive number. Power with rational exponent of a positive number
• estimates and approximations. Percentages. Calculation and transformation of simple expressions.

Algebra
In order to answer the questions involving the topics of this core, it is necessary to work with literal expressions and transform them appropriately according to the goals. Given equations must also be transformed in such a way as to obtain equivalent equations that can be solved more easily or from which the relevant information on solutions can be obtained. The same consideration applies to inequalities and systems. Knowledge of simple procedures and algorithms is important, but it is very important to recognise some simple structures that expressions can have, to see their properties and consequently to choose an effective operations strategy. A student is also expected to be able to use algebra as a tool to represent and process ideas or information, and thus to model and tackle problems in different contexts. In particular, in order to answer certain types of questions, it is necessary to translate the problem and the information provided by the text into an equation (or inequality or system); it is then necessary to transform and possibly solve the equation; finally, it is necessary to interpret in terms of the initial problem the meaning of the algebraic operations that have been performed.

• literal expressions: manipulation and evaluation
• concept of solution and 'set of solutions' of an equation, an inequality, a system of equations and/or inequalities. Algebraic equations and inequalities of first and second degree or related to them. Linear systems.

Geometry
In order to answer the questions involving the topics of this core, it is necessary to understand and use descriptions and representations of elementary geometric figures and their simple combinations, and to apply on them appropriate reasoned transformations, decompositions and recompositions. To analyse the properties of a certain geometric configuration, it is often useful to use different representations and both synthetic and analytical point of view, and to put together the information that can be obtained from the different approaches. Skills that are very useful in guiding representation and reasoning are the mental visualisation of geometric objects and the ability to imagine, when looking at a figure, even features that are not explicitly drawn but that complete it and allow one to understand its meaning.

• most common plane and space figures (segments, straight lines, planes, angles, triangles, quadrilaterals, circumferences, parallelepipeds, prisms, pyramids, cylinders, cones, spheres), with their characterising properties and graphic representations
• pythagorean Theorem
• properties of similar triangles
• elementary language of geometric transformations (symmetries, rotations, translations, similarities). Effects of these transformations on geometric figures
• sine, cosine and tangent of an angle, obtained from the relationship between the sides of a right-angled triangle
• perimeter and area of the most common plane figures. Volume of elementary solids. Calculation of area and volume by sum and difference of figures
• cartesian coordinates in the plane and description of simple subsets of the plane using coordinates. Slope of a segment and equation of a straight line. Equations of straight lines that are parallel or perpendicular to a given line. Intersection of straight lines and representation of the solutions of a system of first degree equations
• distance between two points and equation of a circumference given its centre and radius.

Functions and graphs
In order to answer the questions involving the topics of this core, it is necessary to relate the information obtained from different representations of the same function; for example, thanks to the information that can be read on the graph of a function f, to determine the solutions of an inequality of the type f(x) > 0 or to establish that the value f(x) cannot be expressed by a formula of a given type. It is necessary to be aware of how the behaviour varies and how the graph of functions of a certain family changes as the parameters that define them vary. To answer the questions and to connect the properties of the different families of elementary functions, it is very useful to quickly visualise the graph of the functions x ↦ af(x), x ↦ f(x - a), x ↦ f(x) + g(x) from the graph of the functions x ↦ f(x) and x ↦ g(x). Finally, it is important to use the language of functions to express relationships between quantities describing a natural phenomenon and their variations, in different contexts.

• language and notations for functions. Graph of a function. Composition of functions. Existence and uniqueness of solutions of equations of the type f(x) = a, invertible functions and inverse function
• relative maxima and minima, monotonic intervals of a function
• characteristic properties, graph and behaviour of the following families of functions of a real variable: power functions and root functions; polynomial functions of first and second degree; functions of the type x ↦ 1/(ax+b) with a and b assigned constants; absolute value function; exponential functions and logarithmic functions in different bases; sine function and cosine function
• definition of logarithm and elementary algebraic properties of the exponential and logarithm functions based on the properties of power elevation
• equations and inequalities expressed by functions, for example f(x) = g(x), f(x) >a.

Combinatorics and probability
In order to count the elements of a set, it is necessary to represent them in some suitable way and to have suitable systematic listing and counting strategies.

The calculation of the probability of an event is only required in the case of random phenomena for which the possible events are finite in number. In such a situation, it is necessary to find an appropriate representation of the set of events and, in this representation, an appropriate description and decomposition of the event of interest in terms of simpler events whose probability is known.

• representation and counting of finite sets
• disjoint events. Independent events. Probability of the union event of disjoint events. Probability of the intersection event of independent events
• description of events in simple paradigmatic situations: tossing a coin, rolling a die, drawing from an urn. Tree diagrams
• conditional probability.

Means and variability
In order to answer the questions involving the topics of this core, it is necessary that in simple situations a student is able to read, interpret and relate different representations of a set of data, which refer to characteristics of a given population, identifying some essential features.

• qualitative and quantitative variables (discrete and continuous). Absolute and relative frequency
• representations of distributions (tables, bar graphs, pie charts, histograms, ...). Mean and median.

Understanding and representing

• understand texts that use, also contextually, languages and representations of different types
• depending on the situation and goal, use different representations of the same mathematical object
• understand and use elementary notations of set language and terms such as: element, belongs, subset, union, intersection.

Reasoning

• in a certain situation and given certain premises, determine whether a statement is true or false.
• denying a given statement
• understand and know how to use terms and phrases such as: for each, all, none, some, at least one, if... then..., necessary condition, sufficient condition, necessary and sufficient condition.

Modelling, problem solving

• formulate a situation or problem in mathematical terms
• solve a problem by adopting simple strategies, combining different knowledge and skills, making logical deductions and simple calculations.

List of topics

1. Biological molecules – The importance of water in biology. Knowledge of the chemical composition of organisms: carbohydrates, lipids, proteins and nucleic acids. Polymers and monomers. Structure and function of macromolecules
   • Water and its characteristics
   • Hydrophilic and hydrophobic substances
   • Chemical composition, structure and function of the main biological molecules: carbohydrates, lipids, amino acids and proteins, nucleotides and nucleic acids.

2. The organization of cells– The fundamental differences between prokaryotic and eukaryotic cells; the structure and basic functions of the plasma membrane and the main organelles of the eukaryotic cell; identification through schematic drawings. The fundamental differences between animal cells and plant cells. Theories explaining the origin of the eukaryotic cell, with particular reference to endosymbiotic theory for mitochondria and chloroplasts
   • Organization of the prokaryotic cells
   • Organization of the eukaryotic cells
   • Difference between animal cells and plant cells
   • Structure and function of: plasma membrane, cell wall, nucleus, cytoplasm, mitochondria, chloroplasts, ribosomes, endoplasmic reticulum, Golgi apparat, lysosomes, vacuoles, cytoskeleton
   • Evolution of the eukaryotic cells.

3. Fundamentals of genetics – Transmission and expression of hereditary characters in prokaryotic and eukaryotic cells, in individuals and populations. The structure of the genetic material and its levels of organization in microbial, plant and animal systems, including humans. Regulation of gene expression and mechanisms of mutagenesis
   • Chromosomes
   • Mendelian genetics
   • Conservation of genetic information and its expression
   • Genetic code
   • DNA and genes
   • Transcription and translation.

4. Cell basis of reproduction and heredity. Reproduction and development. Vital cycles.
   – Cell division in prokaryotes. Cell division in unicellular and multicellular eukaryotic organisms. Mitosis and cell duplication. Meiosis and sexual reproduction. Gametes and zygote formation. The main stages of embryonic development. Differences in the life cycle of animals (diplontic) and plants (hapodiplontic)
   • Cell division. Mitosis and meiosis. Cytodieresis
   • Gametes, fertilization and some knowledge on embryonic development
   • Reproduction and life cycles in animals
   • Reproduction and life cycles in plants.

5. Anatomy and physiology of animals and humans – Hierarchical organization of multicellular organisms: cells, tissues, organs and systems
   • Structure and functions of the main tissues. Structure of body systems and their fundamental functions in animals and humans. Main structure and functions of animal tissues (epithelial, connective, muscular and nervous)
   • General organization of human digestive, respiratory, circulatory, skeletal muscle, excretory, reproductive, immune, nervous and endocrine systems. The sense organs
   • Vital functions in animals and humans. Nutrition and digestion. Respiration. Circulation. Excretion. Nervous and chemical communication. Protection, support and movement. Immunity. Reproduction.

6. Plant anatomy and physiology– Basic knowledge of the structure and function of the main plant tissues and organs. Basic knowledge of photosynthesis, to convert light energy into chemical energy for the production of organic molecules. The importance of plant organisms in ecosystems for the nutrition of other organisms as well as for the production of oxygen and the consumption of carbon dioxide that occur during the photosynthetic process. The importance of roots for land plants, their help in anchoring them to the ground and means to absorb water and mineral nutrients
   • Structure and function of plant tissues and organs: leaf, root, stem, flower. Fruits and seeds
   • Growth
   • Photosynthesis
   • Mineral nutrition
   • Water absorption and transpiration.

7. Biodiversity, classification, evolution – A general outline of the evolution of living beings and their classification into domains and kingdoms. Recognizing biodiversity: general characteristics of bacteria, protists, fungi, plants, animals. Viruses. Classification of biodiversity: general concepts of classification and phylogenesis, homology and analogy. The mechanisms of evolution: genetic variability, natural selection, adaptation, speciation and extinction
   • Bacteria
   • Viruses
   • Protists
   • Fungi
   • General characteristics of the main plant phyla (mosses, ferns, gymnosperms, angiosperms)
   • General characteristics of the main animal phyla (Porifera, Cnidaria, Platyhelminthes, Nematoda, Mollusca, Annelida, Arthropoda, Echinodermata, Chordata)
   • Classification and phylogenesis, homology and analogy
   • Evolution: genetic variability, natural selection, adaptation, speciation, extinction.

8. Bioenergetics – The main metabolic processes through which cells convert, store, use and exchange energy. A general outline of photosynthesis, aerobic and anaerobic respiration, glycolysis and fermentation. Understanding of the differences between catabolism and anabolism. Definitions of autotrophic and heterotrophic metabolism. Basic knowledge on enzymatic catalysis. Basic knowledge on human nutrition – nutrients and other substances in foods that play a role in human nourishment, growth, reproduction, and health
   • Energy flow and biological significance of photosynthesis, aerobic and anaerobic respiration, glycolysis, fermentation
   • Catabolism and anabolism
   • Autotrophic and heterotrophic metabolism
   • Enzymatic catalysis
   • Some knowledge of nutrition in humans.

9. Ecology – Basic knowledge of a) interactions between organisms and between organisms and the environment, at different levels of biological hierarchy: individuals, populations (groups of organisms of the same species that colonize a given territory), communities (structured sets of populations) and ecosystems (communities with their physical and chemical environment), (b) energy flows and material cycles that allow ecosystem functions to be maintained, (c) the factors determining the abundance and distribution of organisms and biodiversity
   • Individuals, populations, communities, and ecosystems
   • Trophic chains
   • Habitat and ecological niche
   • Biotic interactions.

10. Biotechnology– Basic knowledge of techniques used for the production of goods and services, whose applications range from the pharmaceutical to the food industry and may also have important applications in the medical field.
    • Genetic engineering, GMO
    • Animal and plant biotechnology
    • Microbial biotechnology.

Introduction
The syllabus of the Physics module is deliberately limited to what can be found in high school texts and the topics listed do not require additional study. On the other hand, those relating to the modelling of physical phenomena are considered as indispensable mathematical skills and, in particular:

• The use of graphical representations and functional models related at least to direct and inverse proportionality, linear dependence, increasing and decreasing quadratic proportionality, sinusoidal, exponential and logarithmic dependence
• Recognising proportional relationships between the quantities used in a law, both in numerical and symbolic exercises

It is also essential to know how to use: the units of measurement of the International System, including prefixes, and the practical units most commonly used in science, scientific notation, the concept of order of magnitude, vector calculation limited to the composition and decomposition of vectors, scalar and vector product.

List of topics

1. Kinematics and dynamics of the material point– Description of motion: velocity and acceleration, graph of time law, angular and peripheral velocity, angular acceleration, simple harmonic motion. Rectilinear motion, acceleration of gravity, free fall of bodies. Two-dimensional curvilinear motion, for example motion of a projectile and uniform circular motion, acceleration and centripetal force. Galilean principle of relativity and fictitious force: velocity and acceleration in reference frames in uniform or accelerated relative motion. The three laws of dynamics. Equilibrium of an extended rigid body (resulting from forces and moments of forces) with applications: inclined plane, lever, pulley, winch. Hooke’s law. Forces of friction. Barycentre motion of a rigid body. Momentum and impulse, Newton’s second law as a change in momentum. Work. Power. Kinetic energy. Conservative forces. Gravitational potential energy, elastic potential energy. Conservation laws. Elastic and inelastic collisions (special cases: central collision, collision against a rigid wall). Universal gravitation, gravitational potential energy and force, acceleration of gravity on a planet, motion of satellites and planets
2. Fluid mechanics– Quantities: density, pressure (in liquids and gases), flux, flow rate. Fluid statics: Pascal’s principle, Stevin’s law, Archimedes’ principle. Continuity equation. Torricelli’s principle, Bernoulli’s equation
3. Kinetic theory of gases and thermodynamics –Perfect gas law. Perfect gas equation of state Pressure and internal energy of a perfect monatomic gas. Absolute temperature. Heat, specific heat and heat capacity. Changes of state and latent heat. First law of thermodynamics. Efficiency of a thermal machine (Carnot cycle), reversible and irreversible cycles
4. Electrostatica and electric currents–Electrical charge. Coulomb’s law and electric field. Electric field flow and Gauss’s theorem (e.g. point charge, sphere charge and uniformly charged plane). Motion of charged particles in uniform electric fields. Conductors and electrostatic induction. Electrostatic potential, equipotential surfaces, potential difference. Potential energy for a uniform field and two point charges. Charge distribution, field and potential for a conductor in electrostatic equilibrium. Condenser capacity, equivalent capacity of condensers in series and in parallel. Electrostatic energy of a uniform field. Electric current, motion of charges, Ohm’s laws, electrical resistance, equivalent resistance for resistors in series and in parallel. Electromotive force and generator internal resistance. Joule effect
5. Oscillations, waves and optics– Simple harmonic motion: period, pulse, amplitude. Waves: amplitude, frequency, wavelength, velocity. Superposition principle and interference of harmonic waves. Standing waves. Energy transport: energy density and intensity of a wave, attenuation with the distance from a point source for a spherical wave. Interference. Diffraction. Reflection and refraction, Snell’s law and refractive index, total internal reflection. Plane and spherical mirrors: image formation and conjugated points. Thin lenses: image formation and conjugated points. Chromatic dispersion
6. Magnetism –Magnetic dipole, permanent magnets. Lorentz force: motion of point charges in uniform magnetic fields. Ampère’s circuital law, Biot-Savart law. Magnetic field of a wire and in an undefined solenoid. Force exerted by a magnetic field on an electric current, forces between current carrying wires (straight and parallel)
7. Electromagnetic field –Faraday-Neumann-Lenz law. Electromagnetic waves. The electromagnetic spectrum and the nature of light
8. Modern physics –The structure of the atom and the nucleus, radioactive decay. Special relativity: constant of C, length contraction and time expansion, relativistic energy, conservation law. Photon, energy and frequency, photoelectric effect. Wave-particle dualism, Young’s double-slit experiment. Uncertainty principle.

List of topics

1. Macroscopic properties of matter – The macroscopic properties of matter are the observable properties of matter itself. Understanding the behaviour of materials is useful for interpreting situations you may encounter in everyday life. It is also important to understand the difference between physical and chemical changes in materials
   
   • State of matter and physical changes
   • Particle model of matter on a macroscopic scale
   • Macroscopic properties of gases, liquids and solids (kinetic theory, fixed points, phase transitions)
   • Homogeneous and heterogeneous mixtures (suspensions, colloids, dispersions)
   • Separation of mixtures
   • Chemical transformations
   • Fundamental laws in chemistry (Lavoisier, Proust, Gay-Lussac, Avogadro).

2. Microscopic properties of matter and composition of substances –Understanding the particle model of matter is important to explain the properties of materials, their interactions and their uses. The structure of the matter can be defined through particles called atoms composed of protons, neutrons and electrons. The study of the atomic structure, the electron configuration and the theories of binding allows a better understanding of the properties of metals, ionic substances, covalent solid compounds and covalent molecular structures
   
   • Particle model of matter on a macroscopic scale
   • Simple substances, compounds and ions
   • The structure of the atom. Atomic mass and relative atomic mass (Ar), relative molecular mass (Mr)
   • Types of chemical bond: ionic, covalent and metallic
   • Lewis structures (electron dot structures)
   • Intermolecular forces and hydrogen bonds
   • Polarity of the chemical bond
   • Oxidation number and atomic valence
   • Molecular geometry (VSEPR theory) and hybridization.

3. Chemical reactions and stoichiometry – It is very important to acquire the ability to read, write and interpret reaction patterns correctly, as well as being able to work with the necessary units of measurement to determine the quantities of substances used in a chemical process or transformation. Stoichiometry calculates the proportions between atoms in molecules and between reagents and the products in chemical reactions. This information is used to balance chemical reaction patterns. The study of how the fundamental laws of chemistry were formulated helps to understand and apply the particle model of matter on a microscopic scale
   
   • Balancing of chemical reactions
   • Definition of the concept of mole and the Avogadro constant
   • Units of measurement for concentration (mol dm-3, g dm-3, percentage composition) and their calculations
   • Mass to mole conversions
   • Concepts of limiting reagent and theoretical yield
   • Relationship between the number of moles (chemical amount) and mass in the reaction patterns.

4. Periodic trends and atomic structure –Many properties of simple substances and atoms show a periodic trend. The electronic configuration of the atom of an element determines both its position in the periodic table and its reactivity to the other atoms of the table. Periodic trends can be used to predict atomic properties
   
   • Periods and groups
   • Atomic models
   • Quantum numbers
   • Electronic configuration of atoms: Aufbau principle and the Pauli exclusion principle.

5. Compounds, properties and nomenclature of compounds. Solutions and Solution Properties – Acquiring the correct terminology and knowing how to classify compounds and ions is essential to the understand and discuss chemistry. Despite this premise, this knowledge can be achieved step by step while learning the basic chemical principles and understanding the various chemical reactions
   
   • Formulas of substances and compounds
   • Nomenclature of substances and compounds (IUPAC and traditional)
   • Properties of the main inorganic compounds (carbonates, sulphates, oxides, hydroxides)
   • Chemical properties of metals
   • Electrolytes
   • Properties of solutions, solubility
   • Colligative properties of solutions.

6. Thermodynamics and kinetics –Particle motions explain the properties of gases. The motion of atoms and molecules, as well as kinetics, allows a connection with chemical balances. Relationship between matter and energy. In a chemical reaction the energy can be absorbed or released. The rate of a chemical reactions of atoms and molecules depends on the frequency with which they collide. The number of these impacts is a function of the concentration, temperature and pressure of the reactive species. Catalysts can be used to change the rate of a chemical reaction. Under certain conditions, a reaction can reach equilibrium. To define the properties of covalent substances it is important to have understood the concepts of intermolecular forces, hydrogen bond, dipole-dipole interaction and dispersion forces
   
   • Ideal gas laws (Boyle, Charles, Gay Lussac)
   • Partial pressure
   • Laws of thermodynamics: internal energy, enthalpy, entropy and Gibbs free energy
   • Exothermic and endothermic reactions
   • Dynamic chemical balance (equilibrium constant and reaction quotient).
   • Reaction rate: factors that affect the reaction rate
   • Activation energy and catalysis.

7. Acids and Bases –Acids and bases have particular characteristics and are chemical products that can easily be found in any home. Acid-base theory and the use of indicators can be used to understand the acidic and basic properties of saline solutions, the balances in solution, as well as providing useful links to practical applications
   
   • Definitions of acids and bases
   • Common acids and bases
   • Strength of acids and bases
   • pH calculation
   • Neutralization reactions and salt formation
   • Acid-base reactions and use of pH indicators
   • Buffer solutions

8. Oxidations and reductions – reduction oxidations (redox) are reactions in which the atoms change their oxidation state. These reactions are characterised by the transfer of electrons between chemical species. These reactions play an important role in many phenomena of everyday life
   
   • Redox reactions and interpretive models
   • Identification of the oxidant and the reducing agent in a simple chemical redox transformation or in a reaction pattern
   • Balancing simple redox reaction patterns
   • Galvanic and electrolytic cells
   • Scale of redox potentials.

9. Organic chemistry – Organic chemistry studies carbon compounds other than carbon monoxide, carbon dioxide, and carbonates. Hydrocarbons, compounds containing only carbon and hydrogen, undergo specific reactions such as substitution reaction, combustion and addition reaction. Many organic compounds are characterised by the presence of functional groups. The student should be able to identify these functional groups, to assign them the correct nomenclature and the type of reactivity
   
   • Origin and characteristics of hydrocarbons
   • Carbon hybridization
   • Organic compounds: structure and nomenclature. Isomerism, relationship between structure and property
   • Alkanes, alkenes, alkynes, cycloalkanes
   • Benzene and aromatic compounds
   • Alcohols, aldehydes, ketones and carboxylic acids
   • Nucleophiles and electrophiles: substitution and addition reactions
   • Combustion reactions
   • Oxidation and reduction reactions.

10. Applied chemistry –Scientific measurements and their reliability are essential in the study of chemical processes. Understanding chemical processes can be used to describe, explain and predict biological, environmental and industrial processes.
    • easurements and units of measurement
    • Uncertainties in experimental measures, average and errors
    • Chemical transformations in daily life
    • Correct reading of commercial product labels (beverages, food, chemicals)
    • Main environmental issues (acid rain, greenhouse effect, smog…
    • Safety regulations.

Depending on the result obtained in the test, the grid below shows the initial preparation level and how to improve your results, if necessary.