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 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. Instead they 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.
Lines of inquiry
(1) The size of other angles
Students might choose (or be directed) to find different angles in the scalene triangles. If the teacher splits the class into three groups, then the students could verify their calculations by sharing the results from each group. The inquiry could move away from a focus on whether the size of one angle increases or decreases to consider how and why the three angles are changing in relation to each other.
(2) A given included angle
Another line of inquiry is initiated by a new prompt that gives an included angle. The angles increase in a linear sequence while the lengths of the two given sides decrease. Students speculate about what happens to the length of the third side and go on to use the rearranged cosine rule to check their conjectures. The teacher might use the prompt to introduce the sine rule to find the size of the other angles.
(3) Algebraic expressions
The cosine of the angle can be expressed algebraically by using the nth terms for the sequences (see mathematical notes). Students can then use the algebraic expression to calculate the size of each angle as n increases before going on to create their own examples.
(4) Area of the triangle
The teacher might take the opportunity to introduce the trigonometric formula for the area of a triangle, especially if a student has posed a question or made a conjecture about the areas of the triangles in the prompt at the start of the inquiry. Students could use numerical and algebraic approaches (see mathematical notes).