Lesson 25 Triangle ratios

Episode 3

Reasoning Resources: Pupils’ triangles from Episode 2, tangent values from Episode 2, Worksheet 3, Calculators
Whole class preparation
Understanding the sine ratio as the relationship between different sides of a triangle. Depending on how the time is going, you may want to leave exploration of the sine ratio for lesson — in which case, go to the end of lesson reflection.

Explain that the sine is another ratio that can be found for triangles — it is the relationship between the opposite side and the hypotenuse.
Pair and small group work
Understanding what the range means in terms of the properties of triangles. Display and give out Worksheet 3 asking them first to write the class values for the tangent from Episode 2. Pupils find the sine ratio for one of their triangles (to 1 d.p.) and write it into Worksheet 3. The class table of values can be completed so that the full picture can be seen and discussed.

Ask the pupils to estimate the sine of 45° by considering the patterns in the table. Then ask them to draw and measure a triangle to check.
Whole class sharing and discussion
Pupils should accept that with advanced concepts, technical and foreign words sometimes are used, often to
avoid the mix-ups that can occur if ordinary words are used. That is why sine and tangent are not translatable.
Could they think of any translations for such ratios?
As a class consider the remaining questions on Worksheet 3. Take pupils’ suggestions and encourage them to try them out by drawing triangles where appropriate.

Ask pupils to consider the range for the tangents (zero to infinity), and sines (zero to one) of angles between 0° and 90°.

  • What can you say about a triangle where the sine is almost 1?
  • What can’t the sine ever equal exactly 1?
  • In the same triangle, what can you tell me about the tangent?
  • End of Lesson Reflection
    Comparing their experimental values with calculator values, with understanding of the many decimal places as accuracy that does not detract from the meaning of the sine as ratio. Transferring what they have done into real-life
  • How do the values (for tangent and/or sine ratios) we've found compare with the values on our calculators?
  • Right-angled triangles are useful. How could you use what we've done today to find the height of a tree? Draw a diagram to illustrate the similar triangles involved - show that the height could be found by using similar triangle properties or the tangent of the angle of sight.
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    Thinking Mathematics Lessons Copyright © by Michael Shayer and Mundher Adhami. All Rights Reserved.

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