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Science and society, grade 12 (3 units)

A kicked ball, a fountain’s jet, a braking car and a national exam: two mathematical models that describe real phenomena.Subjects

The topics, in order:

  • quadratic processes: the table and graph of a process, its extremum – the greatest height or the least value, its zeros – when the ball lands, where it increases and decreases;
  • quadratic models in life: an area as a function of a side, a stopping distance, a bridge’s arch;
  • fitting a model: when a table fits a linear, a quadratic or an exponential relation;
  • the normal distribution: the bell curve, the mean and the standard deviation, the 34/13.5/2.5 rule;
  • the standard score, a value from a percent, and comparing two distributions.

There are two kinds of lessons:

  • Learning lessons – an explanation, solved examples and a few questions to try;
  • Practice lessons – questions only.

How the page works: Using a page.

  1. A quadratic process: the height of a ballLearningA ball kicked upwards: the table and graph of its height against time, the parabola, the formula y = ax² + bx + c, and the symmetry of the table; more quadratic processes – a fountain, an area, a stopping distance.
  2. Reading the graph of a quadratic processLearningThe starting height, the greatest height and when, when the ball lands, where it increases and decreases – and the part of the graph that has a meaning; a process with a least value: the temperature at night.
  3. The vertex: the greatest height and the least valueLearningThe axis of symmetry midway between two times with the same height, the formula x = −b/2a, and the greatest height of a fountain's jet; a least value: the speed at which a car saves the most fuel.
  4. The zeros: when the ball landsLearningWhen a ball's height is 0 – a quadratic equation and the quadratic formula, and only the solution that has a meaning; when the ball is at a given height – twice, or never; a rounded answer when the root is not whole.
  5. Area as a function of a side: the largest areaLearningA fence of fixed length round a rectangular pen: the area as a function of one side, its table and graph, the largest area – a square; a pen against a wall; the sides that give a given area.
  6. Stopping distance and the arch of a bridgeLearningThe braking distance grows as the square of the speed: twice the speed – 4 times the distance; the reaction distance and the stopping distance; the speed at which a driver can stop in time; a bridge's arch – its width, its height, and where a ship passes.
  7. Choosing a model: linear, quadratic or exponentialLearningHow a table tells which model fits it: equal differences – linear, differences growing in equal steps – quadratic, a fixed ratio – exponential; measured braking distances, and a formula through given points.
  8. Practice: quadratic processesPracticeA dive, a fountain's jet, a fenced playground, braking distances on a wet road and visitors to an exhibition: the greatest value, when it lands, a value reached twice, and which model fits a table.
  9. The normal distribution: the bell curveLearningThe heights of a thousand girls: a histogram that looks like a bell; the normal distribution – symmetric about the mean, its width set by the standard deviation; birth weights, IQ, and data that are not normally distributed.
  10. The 34/13.5/2.5 rule of the normal distributionLearning34%, 13.5% and 2.5%: how many of the data lie in each band of one standard deviation; 68% and 95% round the mean; the percent between two values and above a value, and how many people that is – in IQ, birth weights and heights.
  11. The standard score: the distance from the mean in standard deviationsLearningThe standard score z = (x − μ)/σ: comparing a mark in maths with a mark in English, and a long jump with a sprint; from a standard score to a value; a value from a percent – the mark that gets into the top 2.5%.
  12. Comparing two normal distributionsLearningTwo curves on one axis: the heights of girls and of boys – different means, and who stands out more in their own group; two classes with the same mean and different spreads – who is likely to score over 90; two versions of an exam, and two kinds of battery.
  13. Practice: the normal distributionPracticeLoaves of bread, journey times, marks in physics and chemistry, blood pressure and oranges from two groves: the 34/13.5/2.5 rule, how many items of all of them, the standard score, a value from a percent and comparing distributions.
  14. Review: the quadratic model and the normal distributionPracticeQuestions from the whole course: a firework – its greatest height and when it lands, a dolphin's jump on the graph board, a pen against a wall, fitting a model to a table, the 34/13.5/2.5 rule in an IQ test, the standard score and comparing two classes.
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