This unit is an introduction to Pythagoras’ Theorem. It includes history, proofs, and practise in the application of the theorem.
This unit looks at using Pythagoras’ Theorem to find unknown sides of right-angled triangles. It also includes three proofs of the Theorem.
The learning opportunities in this unit can be differentiated by providing or removing support to students, and by varying the task requirements. Ways to differentiate include:
The focus of this unit on investigating Pythagoras’ Theorem can be applied to a variety of engaging, real-life contexts, such as the slope of hills, the height of buildings, physical construction problems, navigation, and construction. Session 5 asks students to think of these, however, you could make reference to these contexts throughout the unit.
You can increase the relevance of the learning in this unit by providing ample opportunities for students to create their own problems, create their own representations of a task, and participate in productive learning conversations.
Te reo Māori kupu such as ture (formula, rule), tāroa (hypotenuse), ture a Pythagoras (theorem of Pythagoras), and koki hāngai (right angle) could be introduced in this unit and used throughout other mathematical learning.
In this session we look at the history of Pythagoras, introduce the theory, and look at the 3,4,5 triangle using Cuisenaire rods.
Pythagoras of Samos lived from about 569 BC to about 475 BC. He was a Greek philosopher who made important developments in mathematics, astronomy, and the theory of music. He spent most of his life in the Greek colonies in Sicily and southern Italy. He never married, and he had a group of followers who taught other people what he had taught them. The Pythagoreans were known for their pure lives. They wore their hair long, wore only simple clothing, and went barefoot.
Pythagoreans were interested in philosophy, but especially in music and mathematics, two ways they saw as making order out of chaos. Music as noise that has meaning, and mathematics as the whole world reduced to rules.
Pythagoras himself is best known for proving that the Pythagorean Theorem was true. The Sumerians, two thousand years earlier, already knew that it was generally true, and they used it in their measurements, but Pythagoras proved that it would always be true.
Pythagoras’ Theorem "For any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.” a2 + b2 = c2 |
This may require some revision of knowledge of geometry. Ensure your students understand what is meant by right-angled triangle (a triangle with 900 as one of its interior angles) and hypotenuse (the longest side of a right-angled triangle, which is always the one opposite the right angle). Also ensure that the class understands what the square of a number is, and what the sum of two numbers means. Explain that the theorem is often stated as a2+b2=c2, where a and b are the short sides and c is the hypotenuse.
A Pythagorean Triple is a triple of natural numbers (a,b,c) such that a2 + b2 = c2 The following is a list of all Pythagorean Triples where a, b, c are all no larger than 100; (3,4,5), (5,12,13), (6,8,10), (7,24,25), (8,15,17), (9,12,15), (9,40,41), (10,24,26), (11,60,61), (12,16,20), (12,35,37), (13,84,85), (14,48,50), (15,20,25), (15,36,39), (16,30,34), (16,63,65), (18,24,30), (18,80,82), (20,21,29), (20,48,52), (21,28,35), (21,72,75), (24,32,40), (24,45,51), (24,70,74), (25,60,65), (27,36,45), (28,45,53), (28,96,100), (30,40,50), (30,72,78), (32,60,68), (33,44,55), (33,56,65), (35,84,91), (36,48,60), (36,77,85), (39,52,65), (39,80,89), (40,42,58), (40,75,85), (42,56,70), (45,60,75), (48,55,73), (48,64,80), (51,68,85), (54,72,90), (57,76,95), (60,63,87), (60,80,100), (65,72,97). |
In this session we explore of proofs of Pythagoras’ Theorem. Depending on your class you might want to use all, some, one or none of these proofs. They could be used in a station approach, as a series of teacher directed explorations, or as a demonstration to the class.
This session uses Pythagoras’ Theorem to find the hypotenuse of right-angled triangles.
This session uses Pythagoras’ Theorem to find side lengths other than the hypotenuse.
In this concluding session we work on applications of Pythagoras’ Theorem.
Dear families and whānau,
Recently we have been investigating the history, proofs, and applications of Pythagoras’ Theorem. Ask your child to share their learning with you.
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/pythagoras-theorem at 8:43pm on the 26th February 2024