South 고등학교 3학년 Practical Mathematics
Chapters: 3
1. Patterns
Patterns in phenomena · Formulas in daily life · Similarity in daily life · Congruence in daily life · Problems with similarity and congruence
- 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.
- 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.
- Congruence: Same Shape, Same Size – Two figures are congruent (≅) if one can be moved exactly onto the other by rigid motions: translations (slides), reflections (flips) and rotations (turns), in any order. Rigid motions keep every length and every angle, so congruent figures have equal corresponding sides and equal corresponding angles. A dilation (enlargement) changes size, so it does not keep congruence; it gives similar figures. To prove two figures congruent, describe a sequence of rigid motions that maps one onto the other, and match vertices in order (A↔A′, B↔B′…). For triangles, SSS, SAS and ASA are shortcuts that follow from rigid motions.
2. Space
Shapes of plane and solid figures · Shapes in art · Sketches and nets · Using nets · Designs with figures
- 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).
- Maths in the Arts: Music, Pictures, Tiles, Poems and Film – Art is full of maths. Musical notes are ratios of string length (half length = octave, 2/3 length = fifth). Painters use perspective and the golden ratio. Tiles fit when their corner angles add up to 360°. Poems repeat sound patterns. A film is 24 pictures per second.
- What Is Design? Fields, Form and Function, and Design for Everyone – Design is planning something on purpose so that it does a job for people. It has many fields: product, graphic, fashion and textile, interior, landscape, architecture and digital (UX) design. Good design joins form (shape, colour, material) and function (the job it does): "form follows function". Designers use elements and principles, fit objects to the human body (ergonomics), and try to make things that work for everyone (inclusive design) and do less harm to the planet (sustainable design).
3. Data
Collecting and organising data · Real-life data and graphs · Drawing conclusions · Purposeful data projects
- 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.
- Data Analysis – Data analysis means turning raw data into answers. It follows a cycle: ask a question, collect data, clean it (remove errors, repeats and blanks), organise and transform it, analyse it with summaries such as mean, median, range and patterns, show it with a good chart, and draw a careful conclusion. Watch for outliers, small samples and bias, and remember that a correlation between two things does not prove that one causes the other. Data must also be stored safely and used with permission.
- The Statistical Inquiry Cycle – A statistical inquiry answers a question with data in five steps: Problem (ask a clear question), Plan (decide who to ask, which variable, how), Data (collect and record), Analysis (tables, graphs, averages) and Conclusion (answer the question, state limits, ask new questions). Then the cycle can start again.