Week 9 Lesson 2 Learning Challenge:To begin to explore Pythagoras’ Theorem

• To learn about a famous Greek philosopher.

• To discover what maths from the Ancient Greeks is still used today.

• To learn about Pythagoras’ theorem.

Greek Week

at

STOC 2020

Introducing… Pythagoras• Pythagoras of Samos was a Greek

philosopher who lived from about 580 BC to about 500 BC. He made important developments in mathematics, astronomy, and the theory of music.

• Pythagoras spent much of his life studying mathematics and formed a special school where members followed strict rules, such as never eating meat. Pythagoras believed that everything in the world could be explained by numbers and his school worked hard to try to learn enough about numbers to be able to understand the universe.

It was a Pythagorean School of thought that numbers had meaning.

• Some of the things Ancient Greeks believed about numbers would seem odd to us now. For example, numbers were thought to have their own special nature.

• Pythagoreans thought numbers were male or female, ugly or beautiful, or had a special meaning. They believed:

– 1 is the number of reason.

– 2 is the first even or female number, the number of opinion.

– 3 is the first true male number, the number of harmony.

– 4 is the number of justice or retribution.

– 5 means marriage.

– 6 is creation.

Relevance of this maths today…

• Other ideas they worked on are things we still learn about in school and that mathematicians still use. Some of these are:

• Odd numbers, like 1, 3, 5, 7, 9, 11

• Even numbers, like 2, 4, 6, 8, 10, 12

• Triangular numbers, like 1, 3, 6, 10, 15

• Square numbers, like 1, 4, 9, 16, 25

One important Ancient Greek Theory…

It can be theorised that:‘The sum of the angles of a triangle is equal to two right angles.’

• This means that if you take any triangle, tear off the corners and fit them together like a puzzle, you will make a straight line (that is the same as two right-angles or 180°).

• Have a go! Draw a triangle on a piece of paper, cut it out, cut off each vertex and arrange them to show that the three angles in a triangle total 180°:

• This is how we know that the angles in any triangle will total 180° they make a straight line.

• Prove it: Draw some of your own triangles of differing sizes to test this theory out!

The famous theory…Pythagoras’ Theorem

• Perhaps the most famous theorem in the world is known as Pythagoras' theorem. It is thought that it was known to the Babylonians 1000 years earlier but Pythagoras may have been the first to prove it.

•The theorem is: for a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides.

• Let’s explore what this really means…

a² + b² = c²

Going Deeper: Mathematical Vocabulary

'Hypotenuse' is the name given to the side that is opposite the right angle. It is also the

longest side of a right angled triangle.

• Side a is 3 cm long, so the area of the square on that side is 3 x 3 = 9.

• Side b is 4 cm long, so the area of the square on that side is 4 x 4 = 16.

• Side c is 5 cm long, so the area of the square on that side is 5 x 5 = 25.

• Pythagoras' theorem says that if we add the smaller squares together we should get the big square: 9 + 16 = 25

• This can be written as can be written as a2 + b2 = c2

Pythagoras Theorem

• Side a is 8 cm long, so the area of the square on that side is 8 x 8 = ____

• Side b is 6 cm long, so the area of the square on that side is 6 x 6 = ___.

• Side c is 10 cm long, so the area of the square on that side is 10 x 10 = ___.

• Pythagoras' theorem says that if we add the two smaller squares together we should get the big square: __ + __ = __

• This can be written as can be written as a2 + b2 = c2

Pythagoras Theorem

8 cm

6 cm

10 cm

This time, we are going to visualise the squares.

Pythagoras Theorem

Greek Week Maths L.C. To begin to explore Pythagoras Theorem

Use the Greek Week Lesson 2 Sheet in your pack to complete this learning.

On the back of this sheet, there are 7 different triangles. Measure sides a and b and the hypotenuse c, of each triangle carefully to complete the table below. Remember, pay attention to the squares to ensure your measurements are accurate.

Remember, when you square a number you multiply it by itself:3² = 3 x 3 = 9So, use a calculator to square the decimal values that you may measure. You can use the x² button on your calculator or simply times the value by itself. (Round the product to 2d.p.)E.g.: 3.4cm x 3.4cm = 11.56cm

Check the accuracy of your measurements by using Pythagoras’ Theorem:a² + b² = c²Are there any measurements you need to retake to make sure you have been accurate?

Learning Reflection

Self-Mark your answers and reflect on the mathematical skills you have developed during this learning.

This lesson I have learnt…

I noticed…

The skills I have developed are…