Year level: 8

Strand: Measurement / Number

Lesson length: 60 mins

This lesson is one in a series of lessons on Pythagoras theorem. The purpose of this lesson is to have students undertake a mathematical exploration to find Pythagorean triples, that is, sets of positive integers {a, b, c} such that a2 + b2 = c2. Students apply trial and error to identify sets of positive integers {a, b, c} that are Pythagorean triples using a spreadsheet. They apply scaling to develop new triples from a known triple, and use a formula (Euclid’s formula) to generate Pythagorean triples. They then take a systematic approach to explore if there are any emerging patterns. This lesson is an example of an exploration within mathematics itself rather than an application of mathematics to contexts involving practical problems or other learning areas.

### Achievement standard

By the end of Year 8, students recognise irrational numbers and terminating or recurring decimals. They use Pythagoras’ theorem to solve measurement problems involving unknown lengths of right-angle triangles.

### Content description

• Students recognise irrational numbers in applied contexts, including square roots and π. AC9M8N01
• Students use Pythagoras’ theorem to solve problems involving the side lengths of right-angled triangles. AC9M8M06

### General capabilities

Numeracy

• Understanding geometric properties (Level 7)
• Understanding units of measurement (Level 10)
• Multiplicative strategies (Level 9)

Critical and creative thinking

• Interpret concepts and problems (Level 5)
• consider alternatives (Level 5)

Digital literacy

• Select and operate tools (Level 5)

The following assessment opportunity is suggested in the exit ticket found below and in the teacher’s slides on slide 9.

• Question 1: which one of the following is a Pythagorean triple: {4, 6, 8}, {6, 8, 10}, {8, 10, 12}?
• Question 2: what is the value of b for {12, b, 37} to be a Pythagorean triple?

Answers: {6, 8, 10}, b = 35

• Some students lack knowledge of square numbers. This can be addressed by having students create a reference table of squares on numbers from 1 to 20, and practising calculating squares of larger numbers using a calculator or another tool such as a spreadsheet.
• Some students may have difficulty interpreting and applying algebraic expressions. This can be addressed by writing out a list of written instructions for applying a formula and practising these. Students will have previously been exposed to and worked with applications of Pythagoras’ theorem to calculate lengths, distances, as well as lengths in the coordinate plane. This will have developed the ability to interpret and apply formulas involving squaring, addition and subtraction (sums and differences).

Prior to this lesson, it is expected that students have:

• knowledge of Pythagoras’ theorem and the formula a2 + b2 = c2
• knowledge of the number terms and notations for integer, square and square root
• familiarity with squares of small integers and the concept of a perfect square
• familiarity with the use of a scientific calculator.

## What you need:

• Lesson plan (Word)

• Teacher’s slides (PowerPoint)

• Testing triples and Euclid's formula spreadsheet (Excel)