Pythagorean triples inquiry
The prompt
Mathematical inquiry processes: Identify and extend patterns; make connections; generalise and prove. Conceptual field of inquiry: Pythagoras' Theorem; quadratic and cubic sequences; substitution.
Zack Miodownik, a secondary school teacher of mathematics, devised the prompt for his year 10 class. The students had already learnt about Pythagoras' theorem earlier in their schooling. Zack hoped the prompt would encourage students to make mathematical connections between the theorem and algebraic expressions for quadratic sequences.
Zack invited the students to question, notice and wonder. This is how they responded to the prompt:
There are consecutive odd numbers at the base, although the sequence does not start with one.
Pythagoras' Theorem works for all the triangles so they are all right-angled triangles.
The length of the hypotenuse is always one more than the length of the longest short side.
The areas of the triangles are 6, 30, 84, and 180 square units.
The length of the base in the next triangle in the sequence is 11. What are the lengths of the other two sides?
The square of the length of the base equals the sum of the lengths of the other two sides.
Label the sides a, b, and c (where a is the base and c the hypotenuse). If the length of b is x, then the length of c is x + 1 and the length of a equals the square root of (x + 1)2 - x2.
Does it work if there were even numbers at the base?
The class decided to work collaboratively to find the next cases in the sequence. The students used different approaches:
Square the length of the base, halve the number, then add and subtract a half to find the length of the other two sides.
Extend the linear sequence (2, 5, 7, 9, ...) and the quadratic sequences (4, 12, 24, 40, ... and 5, 13, 25, 41, ...).
Students then debated what triangles come before those in the prompt and what orientation they might have .
Collaborative inquiry
Zack reports that students collaborated throughout the inquiry as they followed different lines of inquiry (see below): "The students and I really enjoyed exploring the prompt. The whole class came together to collaborate on the inquiry. The energy in the room contributed to a brilliant sense of community."
May 2024
Lines of inquiry
In the second half of the lesson the class explored different lines of inquiry.
One group of students derived algebraic expressions from the sequences on each side of the triangle. They then used Pythagoras' theorem to prove that the expressions always give right-angled triangles.
Another group used the expression for the nth term of the 'base sequence' (2n + 1) to generate algebraic expressions for the other two sides.
They attempted to prove the expressions always give right-angled triangles by using Pythagoras' theorem - this time by finding the length of the hypotenuse (c).
A third group used Euclid's formula, which Zack introduced into the inquiry, to find Pythagorean triples that did not follow the rules of those in the prompt. They asked if the new triples follow their own rules.
As they explored systematically and recorded their results in a table, the students noticed that some triples were multiples of others. Zack told them about the concept of primitive triples to help them categorise the results.
Misconception
Zack cautions teachers who use the prompt about sowing a misconception. Students might generalise from the prompt that for all right-angled triangles the square of the length of the base equals the sum of the lengths of the other two sides. The teacher must address this idea whether it arises explicitly or not.
The best way is through the presentation and discussion of counter-examples, such as those illustrated. (The fact that, in the second counter-example, the square of the length of the base is twice the sum of the lengths of the other two sides can lead to further exploration.)
