Subject Maths Key stage 4
Topic Area Number and algebra: Pythagoras’ theorem Programme of study links
Objectives

To understand Pythagoras' theorem and how it can be used to calculate the hypotenuse

1.3b, 2.2, 3.2c
Lesson plan context This lesson uses large-scale demonstrations and a dynamic real-life context to engage students and reinforce the relevance of the theory behind the theorem of Pythagoras. It begins with a review of the terminology of right-angled triangles and clear visual explanations of how Pythagoras’ theorem can be expressed geometrically and algebraically. The formula is made relevant by its application to a real context and its use in calculating the length of the hypotenuse is demonstrated. The new concepts are reinforced with practice questions and reviewed using the ‘Tick or trash’ feature.
Teaching context Class and group work. The lesson can be carried out using video clips and a projector/whiteboard. Two worksheets are provided to help consolidate the students’ understanding of the key mathematical concepts covered in this lesson plan. These worksheets can be used alongside course materials on this topic.
Notes on timing This lesson will cover 1-2 hours depending on the ability level of the students. Lower-ability students may need more time to develop their understanding of the principles and application of Pythagoras’ theorem. If time is limited, the worksheet in the Main section could be carried out as an alternative homework activity and the plenary activity could be used as a starter in a follow-up lesson. For higher-ability students who take less time to grasp the key ideas, the lesson could be extended by incorporating the extension activities in the classroom.
Key question How can Pythagoras’ theorem be useful?
Starter

Start the lesson with the clip Pythagoras’ theorem (1): introduction, which provides an overview of right-angled triangles and two large-scale demonstrations of the theorem of Pythagoras. The prompt questions can be used to focus the students’ attention and then stimulate discussion to establish what they already know about this topic.

Students could be presented with triangles in a variety of orientations and discuss how they can identify the hypotenuse. They could also try identifying right-angled triangles within other common figures, such as diagonals in rectangles and heights in isosceles and other triangles.

The formula for Pythagoras’ theorem is given as c² = a² + b², which could be compared with versions of the formula given in classroom texts. Students could be challenged to express the theorem algebraically for triangles labelled in a different order, or with a different set of letters, or using vertices to names the sides.

In preparation for the activities in the Main section, an extra question is given to get students thinking about how Pythagoras’ theorem could be useful.

Main Activity

Start the main section of the lesson with the clip Pythagoras’ theorem (2): find the hypotenuse in which Ben tackles an aerial ropeway in order to demonstrate how Pythagoras’ theorem can be applied to the real-life context. He explains how to identify the right-angled triangle and how to use Pythagoras’ theorem to calculate the length of the hypotenuse.

It is suggested that the clip is watched in two sections to give an opportunity to review the first part of the calculation and to introduce the use of the square root. Students could find the symbol on their calculators and establish how to use the function. Both sections are supported by accompanying questions, which will enable students to work through Ben’s calculation. They could then generate more examples in the same context. For example, what if the tower was 12 metres high? What if the field was bigger or smaller?

The worksheet Pythagoras’ theorem: Practice questions gives students the opportunity to practise this type of calculation:

  • Questions 1 and 2 revise calculating the squares and square roots.
  • Questions 3-6 all involve using Pythagoras’ theorem to calculate the hypotenuse and follow an increasing level of difficulty.
  • Question 3 is a multiple-choice problem based on simple right-angled triangles. These are similar triangles based on the Pythagorean triple 3, 4, 5, so the hypotenuse calculations could be carried out without use of a calculator. Diagrams are provided.
  • Question 4 involves identifying a diagonal in a rectangle as the hypotenuse of a right-angled triangle. Diagrams are provided.
  • Question 5 is based on an aerial ropeway such as that featured in the clip Pythagoras’ theorem (2): find the hypotenuse. A diagram is provided.
  • Question 6 is based on the route taken by a ship sailing due north and then east and recognising that the distance of the ship from its starting point is the hypotenuse of a right-angled triangle. A diagram is not provided and so students have to interpret the information given in the question to draw their own right-angle triangle.

Students could work through these questions individually before discussing their progress in groups and providing peer-to-peer feedback. The workings and answers could then be reviewed as a whole class.

Worksheet

Pythagoras' theorem: Practice questions

Worksheet Answers

  • Question 1
    a) 7² = 7 x 7 = 49; b) 25² = 25 x 25 = 625; c) 1.2² = 1.2 x 1.2 = 625
  • Question 2
    a) √16 = 4; b) √1296 = 26; c) √2.89 = 1.7
  • Question 3
    a) x = 5; b) x = 8 These two right-angled triangles are similar triangles.
  • Question 4
    The statement is true.
    Length of hypotenuse of right-angled triangle
    = √ (19² + 22²) = √ (361 + 484) = √ 845 = 29.1 to 1 d.p.
    Length of diagonal of rectangle
    = √ (182 + 232) = √ (324 + 529) = √ 853 = 29.2 to 1 d.p.
  • Question 5
    Length of rope = √ (20² + 35²) = √ (400 + 1225) = √ 1625 = 40.3 m to 1 d.p.
  • Question 6
    Distance of ship from starting point
    = √ (24² + 47²) = √ (576 + 2209) = √ 2785 = 53 km to the nearest kilometre
Plenary

The plenary is based on the clip Tick or trash: Pythagoras' theorem in which the presenters both tackle a typical exam question involving the use of Pythagoras’ theorem to calculate the length of the hypotenuse. The task is to look carefully at their working out and decide who has the correct answer.

Instructions are given for watching this clip in three stages which facilitates an excellent group activity: discuss or attempt the question; vote on whose working to tick and whose to trash; and then discover the correct outcome. The peer assessment approach of this regular feature in Clipbank Maths allows students to review their own understanding and develop their analytical skills.

The worksheet Pythagoras’ theorem: Tick or trash is provided as a homework activity. It contains three sets of further ‘Tick or trash’ questions and answers based on using Pythagoras’ theorem to calculate the hypotenuse. These problems highlight a number of typical exam errors and students could write their own revision tips based on their analysis of these questions.

Worksheet

Pythagoras' theorem: Tick or trash


Worksheet Answers

  • Question 1
    Tick: Ben
    Trash: Katie – Instead of squaring 15 and 12, Katie multiplied them by 2 - the squared notation is often confused with multiplying by 2. Checking the answer to see if it was appropriate, which is always good practice, she should have seen that her answer was too small to be the length of the hypotenuse, which is always the longest side of a right-angled triangle.
  • Question 2
    Tick: Katie
    Trash: Ben – He correctly substituted the values into Pythagoras’ theorem but then forgot to square them – another common mistake with the squared notation. If it helps, students can convert "a²" to "(a x a)" in their working out.
  • Question 3
    Tick: Ben
    Trash: Katie – She incorrectly identified the sides of the right-angled triangle and so carried out the wrong calculation. The orientation of the triangle can contribute to confusion about which side is the hypotenuse – it is always opposite the right angle and is always the longest side.
Extension

Three extension activities are suggested.

  1. Students watch the clip Pythagoras’ theorem (3): find a shorter side in which Ben explains another use of Pythagoras’ theorem: in this case, to calculate the length of one of the shorter sides of a right-angled triangle. They have to use their calculators to work out the final answer of the problem covered and then repeat the calculation with a different set of input values.
  2. Students can demonstrate their understanding of Pythagoras’ theorem by designing a poster to illustrate and explain the demonstrations shown in the clip Pythagoras' theorem (1): introduction.
  3. Find out more about Pythagoras’ life and theorem from these two useful sites, which include many facts and anecdotes about one of the most famous ancient Greek mathematicians:
Notes on differentiation The students’ responses to the worksheet tasks will be differentiated by outcome. Using these as group activities facilitates feedback from peers, which can be used to support lower ability pupils.
Cross-curricular links / Functional skills English
Maths
ICT
Representing, analysing and interpreting.