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Students are guided through the discovery of Pythagoras’ Theorem using Pythagorean Triples. They use Pythagoras’ Theorem to find the hypotenuse and shorter side of any right-angled triangle. As learning progresses they begin to find lengths in 3D shapes.

Pythagoras’ Theorem takes place during Year 10 Term and Year 9 Term 6 for Higher. This is the prerequisite for Trigonometry at both Foundation and Higher levels.

**Prerequisite Knowledge**

- derive and apply the properties and definitions of: special types of quadrilaterals, including square, rectangle, parallelogram, trapezium, kite and rhombus; and triangles and other plane figures using appropriate language
- calculate the perimeters of 2D shapes, including composite shapes;
- use the standard conventions for labelling and referring to the sides and angles of triangles; draw diagrams from written description

**Success Criteria**

- know the formulae for: Pythagoras’ theorem, a
^{2}+ b^{2}= c^{2} - apply angle facts, triangle congruence, similarity and properties of quadrilaterals to conjecture and derive results about angles and sides, including Pythagoras’ Theorem and the fact that the
- base angles of an isosceles triangle are equal, and use known results to obtain simple proofs

**Key Concepts**

- Pythagoras’ Theorem identifies how the three sides of a right angled triangle are connected by the areas of shapes on each edge. To fully engage with this concept students could construct the theorem using a 3,4,5 triangle to measure the hypotenuse and calculate the area of each square. Their hypothesis can then by tested on a 5, 12, 13 triangle.
- Pythagoras’ Theorem can be applied to a wide variety of geometrical and real world problems. Students need to practise identifying when the theorem can be applied by recognising triangular components.

**Common Misconceptions**

- Students often believe that the areas of the shapes on the edges have to be squares in order for a
^{2}+ b^{2}= c^{2}to apply. In fact, the formula applies for all shapes as long as the dimensions are in proportion to the edges of the triangle. - Confusion often lies in identifying the Hypotenuse side of a right-angled triangle since it is not always apparent which side is longest. Encourage students to identify the hypotenuse as opposite the right angle.
- There is often difficulty when trying to calculate a shorter side of a triangle since students tend to memorise the formula with the hypotenuse as the subject.

March 6, 2021

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