Poland Szkoła podstawowa, klasa VIII Mathematics
Chapters: 8
1. Properties of plane figures
Angles and lines · Triangles · Pythagoras and simple proofs
- Lines and Angles: Linear Pair, Vertically Opposite and Parallel Lines – An angle is the turn between two rays that start from the same point. Angles on a straight line add up to 180° (linear pair). When two lines cross, the opposite angles are equal. When a transversal cuts two parallel lines, corresponding angles are equal, alternate interior angles are equal, and interior angles on the same side add up to 180°. The reverse is also true, and we can prove such facts by contradiction: assume the opposite and show it leads to something impossible.
- Congruence of Triangles – Two triangles are congruent when one can be placed exactly on top of the other: all three sides and all three angles match. You do not need to check all six parts. Any one of SAS, SSS, ASA, AAS or RHS is enough. SSA is not enough. In an isosceles triangle the angles opposite the equal sides are equal, and the converse is also true.
- Pythagoras Theorem – In any right-angled triangle, the square on the longest side (the hypotenuse) equals the sum of the squares on the other two sides: a² + b² = c².
2. Polygons
Polygons and area
- Area and Perimeter: Heron's Formula, Circles and Sectors – Perimeter is the length of the edge. Area is the space inside. For a triangle with three known sides, Heron's formula gives the area without any height. For a four-sided shape whose corners sit on a circle, Brahmagupta's formula does the same. For circles, the edge is π times the diameter, and a slice (sector) is just a fraction of the whole circle.
3. Number line and coordinate plane
Number line · Coordinate plane
- Inequalities: Rules, Intervals and Solving Them – An inequality says one amount is bigger or smaller than another, using <, >, ≤ or ≥. On a number line, the smaller number is on the left. You may add or subtract the same number on both sides, and multiply or divide by the same positive number, and the sign stays. If you multiply or divide by a negative number, the sign flips. The answer is usually a whole set of numbers, written as an interval such as (−∞, 4]. A quadratic inequality is solved from its roots and the shape of its graph. |x| < a means −a < x < a. Some inequalities are true for every number, like x² ≥ 0 and the AM–GM inequality.
- The Cartesian Plane: Coordinates, Distance and Midpoint – Two number lines that cross at right angles turn a flat surface into a map where every point has an address (x, y). The axes meet at the origin O(0, 0) and cut the plane into four quadrants. Distance between two points comes from Pythagoras: square the x-gap and the y-gap, add, take the square root. The midpoint is the average of the x values and the average of the y values. Three points are collinear when the two short distances add up to the long one, and they make a right angle when the squares of the two short sides add up to the square of the long side.
4. Solid geometry
Prisms and pyramids
- Surface Area and Volume of Cuboid, Cylinder, Cone, Pyramid and Sphere – Surface area is the total area of the outside skin of a solid, like the paper needed to wrap it. Volume is the space inside, like the water it can hold. A cuboid's volume is the number of 1 cm cubes that fit in it. A cone holds one third of a cylinder with the same base and height, and a pyramid holds one third of the matching prism.
5. Introduction to combinatorics and probability
Counting and chance
- Introduction to Probability: Scale, Experiments, Sample Spaces and Trees – Probability is a number from 0 to 1 that tells how likely something is. 0 means it can never happen, 1 means it will surely happen. We can find it by doing an experiment many times (empirical probability), or by listing every possible result (the sample space) and counting the ones we want. Tree diagrams and tables help us list results when two things happen together.
6. Reading data and descriptive statistics
Data and averages
- Statistics: Asking Questions, Averages and Stacked Bar Graphs – Statistics starts with a question that has many possible answers. We collect data, organise it, show it in a graph, and find one number that sums it up. The mean is the fair share, the median is the middle value, and the mode is the most common value. A weighted average gives more importance to some values. Stacked and 100% stacked bar graphs show how a whole is split into parts.
7. Circumference and area of a circle
Circle measures
- Area of Sector and Segment of a Circle – A sector is a pizza slice of a circle: its area is θ/360 × πr². Its crust is an arc of length θ/360 × 2πr. A segment is the slice minus the triangle inside it: segment = sector − triangle.
8. Symmetries
Bisectors and symmetry
- Symmetry: Mirror Lines, Centres and Bisectors – A figure is symmetric when one half is a perfect copy of the other. In line (axial) symmetry the copy is a mirror image across a line. In point (central) symmetry the copy is the figure turned half a circle (180°) about a centre. The perpendicular bisector of a segment and the bisector of an angle are lines of symmetry, so every point on them is the same distance from two things.