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The plane and space, grade 11 (3 units)

Geometry of the world around you: maps and plans, heights no tape measure can reach, and the angles of ladders, ramps and roads.Subjects

The topics, in order:

  • scale: maps, a flat’s plan and models; area and scale;
  • similarity: enlarging a photo, the ratio of areas, a height by its shadow and with a mirror;
  • trigonometry: sine, cosine and tangent, finding a side and finding an angle;
  • trigonometry at work: angles of elevation and depression, a road’s gradient, the isosceles triangle and the rectangle.

There are two kinds of lessons:

  • learning lessons – an explanation, worked examples and a few questions to try;
  • practice lessons – questions only.

How the page works: Using a page.

  1. The scale of a mapLearningFrom the map to the real world and back, finding the scale, a scale bar, and a journey's time from a map.
  2. Plans and modelsLearningA flat's plan at 1:50 and 1:100, a model car, and the scale of an enlargement.
  3. Area and scaleLearningArea grows by the square of the scale: the area of a flat, a field and a lake, dunams and km², and the price of a floor from a plan.
  4. Practice: scalePracticeMaps, plans and models: lengths, the scale and areas.
  5. Similar shapesLearningEnlarging and reducing a photo, the scale factor, the area grows k² times, two pizzas, and A4 and A3 paper.
  6. Heights and distances by similarityLearningThe height of a tree and a building by its shadow, a height with a mirror, the width of a pond with a parallel segment, and a ramp with a support.
  7. Practice: similarityPracticeEnlarging photos, heights by a shadow and with a mirror, and the ratio of areas.
  8. Sine, cosine and tangentLearningThe side opposite the angle, the side next to it and the hypotenuse; ratios that depend only on the angle; values from a calculator in degrees, and rounding with the sign ≈.
  9. Finding a side of a right-angled triangleLearningChoosing the right ratio, a ladder against a wall, the length of a kite's string, a slide, and a mast with a cable.
  10. Finding an angle of a right-angled triangleLearningFrom the ratio to the angle: sin⁻¹, cos⁻¹ and tan⁻¹; the angle of a ramp, of a ladder and of the sun's rays.
  11. Angles of elevation and depressionLearningThe height of a building and a tower, a boat's distance from a lighthouse, a kite's height, a river's width and a plane's landing.
  12. The gradient of a road: percent and angleLearningA gradient in percent is the tangent of the angle times 100: a sign on a mountain road, an access ramp, and the height climbed along a road.
  13. The isosceles triangle and the rectangle in trigonometryLearningThe height splits an isosceles triangle into two right-angled triangles, and the diagonal splits a rectangle: a roof, a tent, a stepladder, a screen and a gate.
  14. Practice: trigonometryPracticeSides and angles of a right-angled triangle, angles of elevation and depression, a road's gradient and the front of a roof.
  15. Review: the plane and space, grade 11PracticeQuestions from the whole course: scale on maps and plans, area and scale, similar shapes and a height by its shadow, and trigonometry – a ladder, a kite and a ramp.
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