Year level: 8

Strand: Measurement / Number

Lesson length: 60 mins

This is an introductory lesson in a series of lessons on Pythagoras theorem. Students learn about Pythagoras’ theorem and its application in calculating lengths in right-angled triangles. They explore the statement of the theorem, some of its history, a simple geometric proof and visual demonstrations of the theorem. Students apply the theorem to the two types of problem calculating the length of the third side of a right-angled triangle, given the lengths of the other two sides.

### 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 opportunity for assessment is suggested below.

The exit ticket is found on slide 17 in the downloadable Teacher’s slides. Students work through the following questions in their exercise books for you to collect at the end of the lesson.

• Question 1: Can {7, 11, 14} be the side lengths of a right-angled triangle?
• Question 2: If {a, 12, 18} are the side lengths of a right-angled triangle, what is the value of a?
• Question 3: If {8, 8, c} are the side lengths of a right-angled triangle, what is the value of c?

Answers: No, as 49 + 121 ≠ 196; 13.42 (correct to two decimal places); 11.31 (correct to two decimal places).

• Some students may not yet realise that while the triangle inequality applies for all triangles, Pythagoras’ theorem only applies to right-angled triangles; the careful use of counterexamples of triangles with side lengths {a, b, c} that satisfy the inequality but not the equation for Pythagoras’ theorem will help illustrate the difference.
• Some students may have little experience dealing with square numbers and square roots and applying these operations on a calculator; completing a table of values of square numbers from 1 to 30 and calculating square roots of numbers like 87, 243 and 456 with a calculator and noting which numbers they lie between will be helpful for familiarisation.
• Some students may not yet understand that the square root of a sum/difference of two numbers is not the same as the sum/difference of the square roots of the two numbers; a selection of suitably chosen counterexamples will be of assistance, for example,
9 + 16 = 25 but √9 + √16 = 3 + 4 = 7.

Prior to this lesson, it is assumed that students:

• have knowledge of triangle inequality (that the sum of the lengths of any two sides must be greater than the length of the remaining side)
• are familiar with the spatial terms: point, line segment, side, right-angle, triangle and square and their representation and labelling in diagrams
• are familiar with the measurement terms: 90 degrees, unit, cm, square cm, length, area
• are familiar with the number terms and notations for integer, square and square root.

The following terms will need to be introduced in the context of the theorem: leg, opposite, hypotenuse.

## What you need:

• Lesson plan (Word)

• Teacher’s slides (PowerPoint)

• How old is the theorem worksheet (Word)