South 고등학교 2학년 Workplace Mathematics
Chapters: 4
1. Number and operations
Arithmetic at work · Estimating large numbers · Standard units
- Estimation and Order of Magnitude – An estimate is a quick, sensible answer made with rounded numbers. Count a small part, multiply up, and give the answer to a sensible precision. The order of magnitude is the nearest power of ten, such as 10² for about 100.
2. Change and relationships
Ratios at work · Percentages · Tables of correspondence · Growth, decline and cycles · Linear equations and inequalities at work
- 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.
- Functions: Composite, Inverse and Standard Graphs – A function is a rule that gives exactly one output for each allowed input. The allowed inputs are the domain; the outputs are the range. Two functions can be joined: g(f(x)) means do f first, then g. An inverse function f⁻¹ undoes f, and its graph is the mirror image of the graph of f in the line y = x. Only one-to-one functions have an inverse.
- 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.
- 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.
3. Shapes and measurement
Sketches and nets of solids · Views from top, front, side · Transformations, congruence, similarity · Perimeter and area · Surface area and volume
- 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).
- Similar Triangles – Two figures are similar when they have the same shape but maybe a different size. Two triangles are similar if their matching angles are equal and their matching sides are in the same ratio. Basic Proportionality Theorem (BPT): a line parallel to one side of a triangle cuts the other two sides in the same ratio; its converse is also true. You can prove similarity with AA, SSS or SAS.
- 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.
4. Data and chance
Counting at work · Likelihood as numbers · Tables and graphs · Reading tables and graphs · Choosing data displays
- Permutations and Combinations – Counting without listing is the heart of this chapter. The fundamental principle of counting says: if one job can be done in m ways and the next in n ways, both together can be done in m × n ways. n! (n factorial) is 1 × 2 × … × n, with 0! = 1. A permutation is an arrangement, where order matters: the number of ways to arrange r things out of n different things is ⁿPᵣ = n!/(n − r)!. A combination is a selection, where order does not matter: ⁿCᵣ = n!/(r!(n − r)!). Each selection of r things can be arranged in r! ways, so ⁿPᵣ = ⁿCᵣ × r!. Useful facts: ⁿCᵣ = ⁿCₙ₋ᵣ and ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ. When some objects repeat, divide by the factorial of each repeat count.
- Probability: Chances of a Single Event – Probability tells how likely something is, as a number from 0 to 1. When all outcomes are equally likely, P(event) = number of favourable outcomes ÷ total number of outcomes. An impossible event has probability 0 and a sure event has probability 1.
- Data Handling: Collect, Organise and Show Data – Data is a set of facts, such as answers, counts or measurements. Data can be qualitative (words) or quantitative (numbers); numbers are discrete (counted) or continuous (measured). We collect data by surveys, observation, experiments or from existing sources, then organise it with tally marks into a frequency table. We show it with the right graph: bar graphs to compare groups, pie charts to show parts of a whole, line graphs for change over time and scatter graphs for links between two variables. Computers store data as structured tables or unstructured text, images and sound.