Four scalene triangles inquiry

The prompt

Mathematical inquiry processes: Generate more examples; conjecture; create and test particular cases. Conceptual field of inquiry: Sine and cosine rules; sequences; algebraic expressions for the nth term of a sequence.

The prompt is designed to follow on from the right-angled triangles inquiry. It requires a knowledge of the cosine rule to find an angle. The sine rule could also be used to find a second angle. Students have asked the following questions and made the following observations when asked to compare and contrast the diagram in the prompt with the original one:

  • The triangles are not right-angled, but scalene.

  • The lengths of all the sides are given, not just two.

  • The length of the sides increase in a linear sequence like the others. What would happen if the sequences decreased?

  • Do the angles increase or decrease in a linear sequence?

  • How do you work out the size of an angle? Could you use the sine, cosine or tangent ratio?

The prompt highlights the difference between inquiry and discovery learning. During the inquiry, the teacher will, perhaps in response to a question like the one above about finding the size of an angle, explain how to use the cosine and sine rules. Students are not expected to deduce the rules for themselves, but use them in a mathematical exploration of non-right-angled triangles whose side lengths form a sequence.

One cautionary point about the use of this prompt relates to the lengths of the sides of the triangles. Students can unwittingly create impossible triangles by following their sequences (even when all three are ascending). For example, triangles with sides of lengths (5, 6, 9) (7, 8, 14) gives (9, 10, 19) on the third diagram. The sum of the lengths of the two shorter sides must be greater than the length of the longest side and the sum of the increases of the two shorter sides must be greater than the increase of the longest side.

Exploring the prompt

A year 10 class at Haverstock School (Camden, UK) explored the prompt. They conjectured that the size of the angle marked with a green dot (above) would increase because the sequences are increasing. The conjecture turned out to be false, which can explained by considering the ratio of (the length of side b):(the sum of the lengths of sides a and c).

As the ratio approaches 1:1, at which point the angle disappears, the angle is getting smaller. The class then went on to explore their own sequences, aiming to find triangles in which the angle stayed the same and increased in size.

The examples on this page come form Mona, Kacey and Abdul Rahim.