Maths curriculum in Polish

lower-secondary school

G I M N A Z J U M I M . A N N Y W A Z Ó W N Y , G O L U B - D O B R Z Y Ń

Class 1 ( 13-14 years old) Class 2 ( 14-15 years old) Class 3 ( 15-16 years old)

Arithmetic 1. Rational numbers - repetition from the

primary school a) natural numbers (multiples, factors,

common factors, highest common factors, lowest common multiples and primes)

b) integers c) fraction and decimals d) rounding off

2. Percentages a) expressing percentage as a fraction or

decimal b) expressing one quantity as a

percentage of another c) increasing/decreasing a quantity by a

given percentage reverse percentages d) problems involving percentages of

quantities and percentage increases or decreases;

e) problems involving, e.g., mobile phone tariffs, currency transactions, shopping, VAT, discount, simple interest

Arithmetic 1. Indices and roots

a) positive, negative and zero indices and the index laws for multiplication and division of positive integer powers,

b) laws of indices a) standard index form,

2. Roots a) square and cube roots b) simplifying roots, adding and

subtracting square roots, multiplying roots, dividing by square/cube roots,

c) basic rule of radicals: roots/surds - irrational numbers

d) rationalisation of denominators ( type

cb

a,

cb

da

Arithmetic 1. Real numbers - operations and their

properties - repetition 2. Operations on real numbers - repetition

Algebra 1. Algebraic expressions

a) monomials, binomials and polynomials

b) translation of simple real-world situations into algebraic expressions

c) collecting like terms, simplifying expressions, substituting,

Algebra 1. Algebraic expressions

a) expanding the product of two linear expressions including squaring a linear expression

b) multiply expressions of the form e.g.: (ax + b)(cx + d) (ax + b)( cx2+ dx + e)

Algebra 1. Functions

a) the intuitive concepts of functions, domains and co-domains, range, independent and dependent variables

b) the notation of functions and use tabular, algebraic and graphical methods to represent functions

d) simplifying polynomial expressions by adding, subtracting, and multiplying

e) multiplying a single term over a bracket,

f) taking out single term common factors,

g) rearrange formulae 2. Linear equations and inequalities

a) linear equations in one unknown, with integer or fractional coefficients

b) formulate a linear equation in one unknown to solve problems

c) solve simple linear inequalities in one variable, and represent the solution set on a number line

d) types of intervals 3. Direct and inverse proportion

c) multiplication of polynomials d) use of special products:

(a ± b)2 = a2 ± 2ab + b2 a2 − b2 = (a + b)(a − b)

f) rationalisation of denominators

( types cb

a

,

ed

cba

)

2. Linear simultaneous equations - algebraic methods a) the substitution method b) the elimination method. c) find the exact solution of two

simultaneous equations in two unknowns by eliminating a variable, and interpret the equations as lines and their common solution as the point of intersection

d) formulate a pair of linear equations in two unknowns to solve problems

a) simple linear functions and plot the corresponding graphs arising from real-life problems;

b) zeros of functions, c) determine y-intercept and x-intercept d) plot graphs of simple quadratic

functions y = ax2 + b, y =x

a

2. Linear functions a) Linear simultaneous equations -

graphical method b) interpreting and finding the equation

of a straight line graph in the form y = mx + c

c) condition for two lines to be parallel or perpendicuar

d) finding the gradient of a straight line given the coordinates of two points on it

e) determine the equation of a line, given its graph, the zero and y-intercept, or two points on the line.

3. Finding the exact solution of two simultaneous equations in two unknowns using determinants

4. System of three linear equations in three variables

5. Absolute value and distance

Statistics and Probability 1. Finding, collecting and organising data 2. Representing data graphically and

numerically 3. Analysing, interpreting and drawing

conclusions from data 4. Mean, median, mode 5. Outcomes of simple random processes

a) Finding the probability of equally likely outcomes - examples using coins, dice, urns with different coloured objects, playing cards,

Geometry 1. Basic figures - plane geometry

a) segments and lines b) properties of angles at a point, angles

on a straight line (including right angles), perpendicular lines, and opposite angles at a vertex - vertically opposite angles

c) alternate angles and corresponding angles

2. Properties of triangles a) types of triangles b) segments in triangles (altitudes and

medians) c) angle properties of equilateral,

isosceles and right-angled triangles d) the interior angles and exterior angle

of a triangle e) similarity of triangles (SSS, SAS,

ASA)and of other plane figures

3. Quadrilaterals a) properties of special types of

quadrilateral, including square,

Geometry 1. Circle

a) centre, radius, chord, diameter, circumference, tangent, arc, sector and segment, tangent

b) circumferences of circles and areas enclosed by circles

c) arc length and sector area d) area of a segment e) symmetry properties of circles the perpendicular bisector of a

chord passes through the centre tangents from an external point

are equal in length the line joining an external point

to the centre of the circle bisects the angle between the tangents

f) angle properties of circles 2. Right triangles

a) Theorem of Pythagoras b) determining whether a triangle is

right-angled given the lengths of three sides

a) equilateral triangle and its properties

Geometry 1. Similar figures 2. Similarity of triangles 3. Properties of similar polygons corresponding angles are equal corresponding sides are proportional

4. Ratio of areas of similar plane figures 5. Make and use scale drawings and

interpret maps 6. Ratio of volumes of similar solids 7. Map scales (distance and area) 8. Theorem of Thales 9. Pyramids

a) surface area of pyramids b) draw simple nets of solids e.g. regular

tetrahedron, square based pyramid, ect.

10. Cylinder, cone and sphere a) volume and surface area of cylinder,

cone and sphere 11. Trigonometric functions in right

triangles a) use of trigonometric ratios (sine,

cosine and tangent) of acute

rectangle, parallelogram, trapezium and rhombus;

b) classification of quadrilaterals by their geometric properties

4. Areas of triangles and quadrilaterals 5. Angle sum of interior and exterior angles

of any convex polygon 6. Cartesian coordinates in two dimensions

a) finding the length of a line segment given the coordinates of its end points

b) midpoint of line segment c) finding the area of simple rectilinear

figure given its vertices 7. Central symmetry and axial symmetry of

plane figures 8. Properties of perpendicular bisectors of

line segments and angle bisectors 9. Construction of simple geometrical

figures from given data using compasses, ruler, set squares and protractors, where appropriate

(the formulas of the height and the area of an equilateral triangle)

c) special right triangles - properties of sides in right triangles of 30°, 60°, 90° and of 45°, 45°, 90°

3. Polygons and circles b) use a straight edge and compasses to

construct: – the midpoint and perpendicular bisector of a line segment – the bisector of an angle

c) properties of perpendicular bisectors of line segments and angle bisectors

d) constructions: inscribed and circumscribed circles of a triangle, and a tangent line to a circle from a point outside a circle

e) inscribed and circumscribed circles of a triangle

f) the radius of the inscribed circle and circumscribed circle in an equilateral triangle/a right triangle

g) inscribed and circumscribed quadrilaterals

h) inscribed and circumscribed polygons

i) regular polygons j) calculating the interior or exterior

angle of any regular polygon k) inscribed and circumscribed

regular polygons 4. Polyhedra - prisms

a) surface area of cuboids and (rigth)prisms

angles to calculate unknown sides and angles in right-angled triangles

b) simple trigonometrical problems in two and three dimensions including angle between a line and a plane

b) volumes cuboids and (rigth)prisms c) draw simple nets of solids, e.g.

cuboid, triangular prism etc.

The topics in red are out of the curriculum but I am usually able to introduce these topics to my students because I prepare them to

different Maths contests.