# Framework curricula for secondary schools

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Arithmetic and algebra

 DEVELOPMENTAL TASKS, ACTIVITIES CONTENTS PREREQUISITES OF MOVING AHEAD The expansion of a concept for a specific purpose. The magnitude order of numbers. Powers with 0 and negative integral exponent; identities in connection with powers; absolute value of numbers, normal form of numbers. Knowing and applying identities. Special identical equations: commutativity, associativity, distributivity; (ab)2, a2-b2 expressed as a product; (ab)3, a3-b3 expressed as a product. Absolute value, normal form of numbers. Applying identical equations of the second degree Operations with numbers and algebraic expressions, using mathematical terms. Applying these identities in operations with simple algebraic fractions. The four basic operation with algebraic fractions. Expressing certain variables in formulae used in physics and chemistry. Review of solving simple equations. Algorithmic thinking and modelling practical problems, knowledgeable reading. Solution of linear equation systems of two variables. Problems leading to equation systems, calculation of percentage, calculation of interest. Examples to equation systems of several variables. Practice in solving simple equation systems. The application of percentage calculations in practice. Improving ordination skills. Equations with absolute values. Strengthening interest in mathematics through some basic problems of elementary number theory with respect to the history of mathematics. Relative primes. Divisibility problems, the number of prime numbers, examples to number systems. Familiarity with divisibility by 3 and 9. Factorising numbers into a product of primes.

Functions and sequences

 DEVELOPMENTAL TASKS, ACTIVITIES CONTENTS PREREQUISITES OF MOVING AHEAD Developing a function based attitude: interpreting assignments as rules. Finding a suitable model. The concept of function, and its basic features; linear functions, absolute value functions, quadratic functions, square root function, practical examples to further functions (integral part function, fractional part function, sign function), inverse proportionality. Knowledge of the characteristics of basic functions. Represent functions expressed in formulae with the help of a table of values. Using tools for specific purposes. Functional transformations. Transformation of basic functions with one step. Solving equation systems of two variables with the graphical method.

Geometry

 DEVELOPMENTAL TASKS, ACTIVITIES CONTENTS PREREQUISITES OF MOVING AHEAD Familiarity with the characteristics of planar configurations discussed at primary schools. Basic geometrical concepts, expanding and organising knowledge about triangles, rectangles, polygons. Knowing the characteristics of special triangles, rectangles and regular polygons. Formulating guesses, working out new relationships, developing a for proofs. Special lines in triangles, the inscribed circle and circumscribed circle of a triangle Familiarity with the remarkable lines, the inscribed circle and circumscribed circle of a triangle. The theorem of Thales, the circle and tangents to circles, the concept of escribed polygons. Familiarity with concepts related to circles and the properties of tangents. Transformations as the interpretation of functions, establishing relationships between various fields of mathematics. Reflection about a line and reflection about a point, the review and organisation of knowledge about translation, examples to further congruent transformations (rotation about a point). Using the properties of discussed transformations in simple, concrete cases. The study of planes, planning, developing construction and analysing skills and a for discussion. All-round demonstration, learning about construction software. The concept of rotation angle, radian measure, the central angle of a circle, the length of a circular arc, the circumference and area of a sector. Simple construction exercises.

Probability, statistics

 DEVELOPMENTAL TASKS, ACTIVITIES CONTENTS PREREQUISITES OF MOVING AHEAD The correct interpretation of statistical data. Statistical data and their representation (circular chart, bar diagram, etc.), arithmetic mean, median, modal value, measuring dispersion of data. Calculating the arithmetic mean of an assembly of numbers, familiarity with median and modal value. Interpreting data of circular charts and bar diagrams.

Year 10
Number of teaching hours per year: 111
Methods of thinking

 DEVELOPMENTAL TASKS, ACTIVITIES CONTENTS PREREQUISITES OF MOVING AHEAD Differences between everyday thinking and mathematical thinking. Further strengthening the demand for proofs. Theorems and their inverses. Methods of verification, typical methods of reasoning (indirect method, pigeon-hole principle). Making a distinction between stated relationships and stated and verified relationships. A range of combinatorial problems. Simple sorting and selection tasks with a concrete number of elements.

Arithmetic and algebra

 DEVELOPMENTAL TASKS, ACTIVITIES CONTENTS PREREQUISITES OF MOVING AHEAD The principle of permanence in the expansion of the concept of the number. A demonstrative concept of real numbers, their relation with the datum line, the decimal form of real numbers, some examples to irrational numbers. In-depth knowledge of the set of real numbers, the decimal form of rational and irrational numbers, familiarity with special irrational numbers. The identities of square-roots, the concept of n degree root. The application of the identities of square roots in simple cases. Finding a solution in various ways, students’ own findings, looking for personal procedures. Improving algorithmic thought. Solving equations of the second degree, solution formula, the root-factor form, the relationship between roots and coefficients, the relationship between the arithmetic mean and geometric mean of two positive numbers. Established knowledge and application of the solution formula. The concept the arithmetic mean and geometric mean of two positive numbers. Using mathematics as a tool to solve practical and scientific problems. Problems leading to equations of the second degree. Solving various types of simple problems. Demand for discussion in solving algebraic problems. Equivalent and non-equivalent steps of equation transformations. Simple equations with square roots. Solving simple equations with square roots. Checking solutions. Using algebraic and graphical methods together to solve problems. Finding solutions for inequalities of the second degree.

Functions and sequences

 DEVELOPMENTAL TASKS, ACTIVITIES CONTENTS PREREQUISITES OF MOVING AHEAD Learning new function properties, further application of transformations. Using four-digit tables of functions for specific purposes. Interpreting the trigonometric functions of the rotation angle, relationships between them. The properties of trigonometric functions (domain of definition, monotony, zeros, extreme values, periodicity, range), function representation . Knowing the definition of trigonometric functions, the representation and properties of functions xsinx and xcosx.

Geometry

 DEVELOPMENTAL TASKS, ACTIVITIES CONTENTS PREREQUISITES OF MOVING AHEAD Improving transformation skills. Similarity transformation, the similarity of planar configurations. Knowing the graphic content of similarity, the application of magnification and reduction about a point in simple practical exercises. Creative problem solving. The application of geometrical knowledge, good arithmetic skills, using a calculator for specific purposes. The basic instances of similarity of triangles. Various applications of similarity: the medians and the centroid of a triangle, theorems of proportionality in a right triangle. The ratio of the area of two similar planar configurations, the ratio of the volume of two similar bodies. The application of the theorem of Pythagor. Using trigonometric functions to calculate the missing data of a right triangle, practical exercises. Two and three dimensional calculations, calculating the trigonometric function values of special angles. Familiarity with the basic cases. Knowing the listed theorems, and applying them in calculations with one or two steps. Further application of vectors. Multiplying vectors by numbers, splitting vectors into components in the plane.

Probability, statistics

 DEVELOPMENTAL TASKS, ACTIVITIES CONTENTS PREREQUISITES OF MOVING AHEAD Interpreting, understanding and evaluating realistic situations. Further experiments in probability, estimating and calculating probability in simple cases. The illustrated concept of probability, and its calculation in concrete cases. Solving simple problems on the basis of the classical model of probability.

Year 11
Number of teaching hours per year: 111

Methods of thinking

 DEVELOPMENTAL TASKS, ACTIVITIES CONTENTS PREREQUISITES OF MOVING AHEAD Improving combinatorial skills. Finding alternative ways leading to the solution of a problem. Getting used to a priori estimates, and comparing estimates with calculations. Permutation, variation, combination. Binomial coefficients. Mixed combinatorial exercises. Solving simple combinatorial problems. Using graphs as models. The basic concepts of graph theory and their application. Using graphs to solve problems. The graphic concept of a graph. Simple graph applications.

Arithmetic and algebra

 DEVELOPMENTAL TASKS, ACTIVITIES CONTENTS PREREQUISITES OF MOVING AHEAD Equations and equation systems which can be converted to second degree equations and equation systems. An expedient extension of the concept of mathematics, applying the principle of permanence to the generalisation of concepts. Extending raising to a power to powers with a positive base and rational exponent. Identities in connection with powers. The definition of raising to a power, operations and identities in connection with powers with integral exponents. Strengthening the demand for proofs. Linking material to some aspects of the history of mathematics (Internet use and work in the library). Explaining logarithm. Logarithm as the inverse of raising to a power. Identities in connection with logarithm. Familiarity with the concept of logarithm, using identities in connection with logarithms in simple cases. Improving abstraction and synthesising skills. Strengthening the demand for self-checks. Exponential and logarithmic equations. The simple application of the definitions and identities of exponential, logarithmic and trigonometric equations.

Functions and sequences

 DEVELOPMENTAL TASKS, ACTIVITIES CONTENTS PREREQUISITES OF MOVING AHEAD Developing the concept of function even further. Recognising links between the various fields of mathematics. Strengthening the pursuit of verification. The investigation of functions 2x, 10x, and the exponential function. Exponential processes in nature. The logarithmic function as the inverse of the exponential function. Using a PC for the analysis of functions and transformations. Review of previous knowledge about trigonometric functions. The properties of functions learned earlier (domain of definition, range, zeros, extreme values, monotony, periodicity, parity). The transformations of trigonometric functions: f(x) + c; f (x+c); c f (x); f (c x). The graphs of the basic functions and the investigation of their key properties (domain of definition, range, zeros, extreme values).

Geometry, measurement

 DEVELOPMENTAL TASKS, ACTIVITIES CONTENTS PREREQUISITES OF MOVING AHEAD Improving the way of looking at space. Efforts to create accurate ideation. Further strengthening the demand for proofs. Demonstrating the productive nature of the relationship between mathematics and physics. Review of vectors. The properties of vector operations. The scalar product of vectors. Listing the properties of scalar product. Vector operations and their properties (addition, difference, scalar product). Vector applications. Learning to work according to a plan. Improving aesthetic skills. Law of sines, law of cosines. Simple trigonometric equations necessary for their application. Applying the law of sines and the law of cosines for the solution of simple problems (determining the missing data of a triangle). Using mathematics in practice. Calculator and computer use. Judging how the realistic and accurate results are. Determining distance, height and angle in practical situations and in physics. Finding solutions to geometrical problems using algebraic tools. Position vector. Operations with vectors in the co-ordinate plane. Using vector co-ordinates with certainty. Improving verification skills. The point of division of a segment. The centroid of a triangle. Determining the co-ordinates of the point of division of a segment. The distance between two points, the length of a segment. Equations in connection with circles. Knowing the equation of the centre of the circle. Alternative approaches to a given problem. The concept of the direction vector and the angular coefficient. The equation of the straight line. The condition of the parallelism and perpendicularity of two lines. The intersection point of two lines. The correlate position of a circle and a straight line. The tangents to a circle. The parabola as a point set. Knowing one of the equations of the straight line according to free choice. Determining the intersection point of two lines. The investigation of the correlate position of a circle and a straight line.