Pythagoras Rods

Task 97 ... Years 4 - 10

Summary

Pythagoras proved that the defining property of a right angled triangle is:

The area of the square drawn on the hypotenuse is equal to the sum of the areas of squares drawn on the other two sides.

A triangle is only right angled if this space/measurement property is true. Given the side lengths of any triangle these squares can easily be constructed and their areas calculated. The task makes this a tactile, colourful situation for some examples by using material that allows the areas to be created and reorganised to discover what is equal to what.

Materials

Multiple Cuisenaire rods of lengths 3, 4, 5, 6, 8, 10 centimetres

Content

classification of triangles
mathematical language such as hypotenuse, right angle and adjacent
concept of square numbers and cube numbers
calculation of square numbers and cube numbers
relationship between length and area
spatial perception
Pythagoras Theorem

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

In the first example, the area of the square built on the hypotenuse is:

5 x 5 = 5² = 25 sq.cm The areas of the other squares are 16 and 9, which total to 25. The numbers work, but using the rods from the two smaller squares to cover the square built from the hypotenuse lifts Pythagoras Theorem well beyond the number-crunching level. It's even more exciting that the 'equality' can be demonstrated in three ways:

The card offers another example and similar constructions apply. But the critical thing about Pythagoras is that it only works for right angle triangles. Students are asked to build another triangle where the numbers don't 'add up' and the rods can't be made to cover the square on the longer side. Explaining what does and doesn't work requires use of language such as right angle, hypotenuse and the sides adjacent to the right angle.

Extensions

(3, 4, 5) and (6, 8, 10) are called Pythagorean Triples or Triads. Students will see a relationship between these two Triples. Encourage them to extend the relationship to discover whether, for example, a triangle with sides (9, 12, 15) would be right angled.
Cut out these three triangles from graph paper and stack them on top of each other. What else can be discovered about these triangles?
Are there Pythagoreans Triples which are not in this family of triangles? How many can the students find?
If a square can be built on the side of a triangle, then so can a cube. Is it possible to find side lengths for a triangle so that:
The volume of the cube built on the longest side is equal to the sum of the volumes of cubes built on the other two sides.
Research Fermat's Last Theorem.

Whole Class Investigation
Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.
To convert this task to a whole class lesson you only need a bag of Cuisenaire Rods for each student. Many schools have this equipment in their storeroom because it is also very useful for developing fractions content. If you don't have access you can contact the Distribution Manager and ask about having sets made up.
At this stage, Pythagoras Rods does not have a matching lesson on Maths300. However, a related lesson is Lesson 157, Pythagoras & Other Polygons which asks a different What happens if ...? question. For more ideas and discussion about Pythagoras & Other Polygons, open a new browser tab (or page) and visit Maths300.

Is it in Maths With Attitude?
Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.
The Pythagoras Rods task is an integral part of:

MWA Chance & Measurement Years 9 & 10

The Pythagoras & Other Polygons lesson is an integral part of:

MWA Chance & Measurement Years 9 & 10

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