National Year 9 Mathematics
Chapters: 6
1. Number
Place value, ordering and integers · Factors, primes and powers · Calculation · Fractions, decimals, percentages · Measures and estimation
- Place Value, Ordering and Negative Numbers – In our number system every place is worth 10 times the place on its right. A digit's value = digit × the value of its place. To compare numbers, line up the places and compare from the left. Negative numbers lie left of zero; the further left, the smaller.
- Fundamental Theorem of Arithmetic – Every whole number bigger than 1 is either a prime or can be written as a product of primes in exactly one way (only the order can change). This is the Fundamental Theorem of Arithmetic. Using these prime "building blocks": HCF = product of the smallest powers of the common primes; LCM = product of the greatest powers of all primes. For two numbers, HCF × LCM = a × b.
- Fractions: The Four Operations Made Simple – A fraction is part of a whole cut into equal pieces. The bottom number (denominator) says how many pieces make the whole; the top number (numerator) says how many we take. Equivalent fractions name the same amount (3/4 = 6/8). To add or subtract, first make the denominators the same. To multiply, multiply top by top and bottom by bottom. To divide, keep the first fraction, change ÷ to ×, and flip the second. The same four operations work for decimals and negative numbers, and BIDMAS tells us which operation to do first. An inverse operation (the opposite one) lets us check any answer.
- Percentages – Per cent means "out of 100", so x% = x/100. To find x% of an amount, multiply by x/100. Percentage change = (change ÷ original) × 100. To increase by r%, multiply by (1 + r/100); to decrease by r%, multiply by (1 − r/100). Successive changes multiply their multipliers. For reverse percentage, divide the new value by the multiplier.
- Measurement and Units: How We Measure Anything – To measure something is to compare it with a fixed amount called a unit. Every measurement has a number and a unit. Scientists everywhere use the SI system, with seven base units such as the metre, kilogram and second. Prefixes like kilo (×1000), centi (÷100) and milli (÷1000) make units bigger or smaller. A good measurement starts at zero, is read with the eye straight above the mark, and is only as accurate as the smallest division (least count). Rounded values hide a small range, given by upper and lower bounds.
2. Algebra
Expressions and formulae · Equations · Graphs · Sequences
- Exploring Algebraic Identities – An identity is an equation that is true for every value of the letters. Pictures prove them: a square of side (a + b) splits into a², ab, ab and b², so (a + b)² = a² + 2ab + b². In the same way we get (a − b)², a² − b² = (a + b)(a − b), (x + a)(x + b), (a + b + c)² and (a + b)³. Read backwards, identities help us factorise, calculate fast and simplify rational expressions.
- Linear Equations in One Variable – A linear equation in one variable has one unknown (like x) with power 1, for example 2x + 3 = 11. An equation is like a balance: both sides are equal. To solve it, do the same thing to both sides (add, subtract, multiply or divide by the same non-zero number) until x is alone. Always check by putting the answer back in.
- Linear Graphs: Straight Lines and y = mx + c – A linear graph is a straight line. Its equation is y = mx + c. m is the gradient (how steep: rise ÷ run). c is the y-intercept (where the line cuts the y-axis). Lines with the same m are parallel. If m₁ × m₂ = −1 the lines are perpendicular. Where two lines cross, both equations are true, so the crossing point solves them together. In real-life graphs the gradient is a rate (like speed) and the area under the graph can be a total (like distance).
- Sequences and Progressions – A sequence is a list of numbers in a fixed order. We can describe it with a recursive rule (how to get the next term from the last one) or an explicit rule (a formula for term n). In an arithmetic progression (AP) we add the same number every time. In a geometric progression (GP) we multiply by the same number every time. Fractals and the Tower of Hanoi are fun patterns that hide these rules.
3. Ratio, proportion and rates of change
Ratio and scale · Proportion and rates
- Ratio and Proportion – A ratio a : b compares two amounts of the same kind. Divide both parts by the same number to simplify it. To share an amount in a ratio, add the parts, find one part, then multiply. Two quantities are in direct proportion when y = k·x (double one, double the other) and in inverse proportion when x·y = k (double one, halve the other).
- Compound Units: Speed, Density, Unit Price and Other Rates – A compound unit joins two units, like metres per second (m/s) or grams per cubic centimetre (g/cm³). "Per" means "in each one", so a rate tells you how much of one amount fits into ONE unit of another. Speed = distance ÷ time, density = mass ÷ volume, unit price = cost ÷ amount. To convert a compound unit, convert the top unit and the bottom unit separately.
4. Geometry and measures
Perimeter, area and volume · Constructions and properties · Angles · Transformations and similarity · Pythagoras and trigonometry · 3D shapes
- 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.
- Geometric Constructions: Drawing Exactly with Compass and Straightedge – A geometric construction draws a figure exactly using only a compass and a straightedge (a ruler used for straight lines). Key constructions: the perpendicular bisector of a segment, the bisector of an angle, a perpendicular from a point to a line, angles of 60°, 30°, 90° and 45°, a triangle from three sides (SSS), two sides and the included angle (SAS) or two angles and a side (ASA), and regular polygons such as the hexagon. Each works because equal compass arcs make equal lengths, which give congruent triangles.
- 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.
- Transformations: Slide, Flip, Turn and Resize a Shape – A transformation moves a shape to a new place. Translation slides it, reflection flips it in a mirror line, rotation turns it about a centre, enlargement changes its size from a centre. The first three keep size and shape (congruent image). Enlargement keeps shape but changes size (similar image). Each has a simple coordinate rule.
- 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.
- 3D Shapes: Faces, Edges, Vertices, Nets and Views – A 3D shape (solid) has length, width and height, so it takes up space. We describe it by counting its faces (flat or curved surfaces), edges (lines where two faces meet) and vertices (corners). Prisms have two equal ends joined by rectangles; pyramids have one base and triangles that meet at a top point; cylinders, cones and spheres have curved surfaces. A net is a flat pattern that folds into the solid. Drawing a solid from the top, front and side gives its three views. For every solid with flat faces, Faces + Vertices − Edges = 2 (Euler's rule).
5. Probability
Probability
- 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. Statistics
Statistics
- 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.