Lesson 25 Triangle ratios

Episode 2

Reasoning Resources: Triangle measurements from Episode 1, Worksheet 1 (A3/A2
size) or on OHT, plain paper (A4 and A3) Worksheet 2, Protractors, Calculators
Whole class preparation
Understanding that the tangent is a particular relationship between the opposite and adjacent sides of a right-angled triangle.

Finding experimental values for the tangent of the given angle. Noticing that the tangent ratio is independent of the size of the triangle.

Pairing and teachers’ support should help weaker pupils in measuring lengths by ruler, dividing by calculator and rounding, so that these do not prevent them from engaging with thinking about relationships.

Working out how to find the missing side, using ideas of ratio. Formulating different ways of thinking about the missing side. Deciding which way works best for themselves.

Explain that the tangent ratio is the relationship between the opposite and the adjacent sides. It is found by dividing the opposite by the adjacent. How many times will the adjacent fit into the opposite? If the opposite is smaller it will be a number less than one. Using the measurements from Episode 1, calculate the tangent of 30° for each triangle. Establish that the ratio is the same no matter which size of triangle is used.

Can you work out the tangents of different angles by drawing triangles?
Pair and small group work

Collating found values of tangents on a simple list allows recognition of pattern as well as cross checking.

Linking what they know about isosceles right-angled triangles to the work on tangent.
Assign each pair of pupils one of the angles 10°, 20°, 30°, ...,80°. Explain that they are going to draw a right-angled triangle with that angle (display the diagram on Worksheet 1) and with adjacent side 5 cm long. They then need to measure the opposite side and work out the tangent (to 1 d.p.) by dividing the opposite side by the adjacent side. You could organise the class so that at least two pairs of pupils work on each angle as a check. The larger angles are best worked out on A3 paper.
In pairs pupils work on their given angle. Accuracy in measuring angles and lengths are not emphasised.

Record the ratio for each angle in the table on Worksheet 1. Ask pupils to repeat for an adjacent side of length 10 cm, and encourage them to interpret the ratios, e.g.’a bit less than one, so the opposite side is only a bit shorter than the adjacent side’ and ‘nearly half” Record the ratios and repeat for an adjacent side 20 cm long. Check that pupils find the same ratio when the size of the triangle is doubled.
Whole class sharing and discussion

Recognising intuitively and putting in their own words how the range of values for the tangent ratio starts from 0, increases slowly, passes 1, then increases much more quickly.
Discuss the patterns in the tangent ratio table. Then hand out Worksheet 2 and ask the pupils to think about the questions, working in pairs. Encourage pupils to share their ideas:

  • What can we say about the tangent ratio as the angles get bigger?
  • 45° is not in the table - can you estimate what the tangent of 45° will be? Think about a right-angled triangle with an angle of 45° - what else can you tell me about it? How does this confirm that the tangent of 45° is 1?

    For most classes you could ask - What do you expect when the angle is 5°? 85°? 1°? 89°?

    Ask pupils to sketch such extreme angles in the largest triangle, to get a rough idea of how the tangent ratio behaves with angles nearer 0° and 90°.
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