YEAR 7 MATHS FOCUS

Students should gain an understanding of Pythagoras’ theorem, rather than just being able to recite the formula. By dissecting squares and rearranging the dissected parts, they will appreciate that the theorem is a statement of a relationship among the areas of squares.

In Stage 5, Pythagoras’ theorem becomes the formula for the circle on the Cartesian plane. These links can be developed later in the context of circle geometry and the trigonometry of the general angle.

Pythagoras’ theorem is named for the Greek philosopher and mathematician Pythagoras (c580–c500 BC), who is credited with its discovery. However, it is probable that the theorem was known to the Babylonians 1000 years earlier.

In the 1990s, the British mathematician Andrew Wiles (b 1953) finally proved a famous conjecture made by the French lawyer and mathematician Pierre de Fermat (1601–1665), known as ‘Fermat’s last theorem’, which states that if  $n$

“>n

 is an integer greater than 2, then  $a^n+b^n=c^n$

“>an+bn=cn

 has no positive-integer solutions.

Students need to understand the difference between an ‘exact’ answer and an ‘approximate’ answer.

They may find some of the terminology encountered in word problems involving Pythagoras’ theorem difficult to interpret, eg ‘foot of a ladder’, ‘inclined’, ‘guy wire’, ‘wire stay’, ‘vertical’, ‘horizontal’. Teachers should provide students with a variety of word problems and explain such terms explicitly.

establish the relationship between the lengths of the sides of a right-angled triangle in practical ways, including with the use of digital technologies

– describe the relationship between the sides of a right-angled triangle (Communicating)

use Pythagoras’ theorem to find the length of an unknown side in a right-angled triangle

– explain why the negative solution of the relevant quadratic equation is not feasible when solving problems involving Pythagoras’ theorem (Communicating, Reasoning)

write answers to a specified or sensible level of accuracy, using an ‘approximately equals’ sign, ie ≑ or $\approx$

“>≈

solve a variety of practical problems involving Pythagoras’ theorem, approximating the answer as a decimal

– apply Pythagoras’ theorem to solve problems involving the perimeters and areas of plane shapes (Problem Solving)

use technology to explore decimal approximations of surds

– A surd is a numerical expression involving one or more irrational roots of numbers. Examples of surds include 2–√$\sqrt{2}$, 5–√3$\sqrt[3]{5}$ and 43–√+76–√3$4\sqrt{3}+7\sqrt[3]{6}$.

– recognise that surds can be represented by decimals that are neither terminating nor have a repeating pattern (Communicating)

– solve a variety of practical problems involving Pythagoras’ theorem, giving exact answers (ie as surds where appropriate), eg $\sqrt{5}$

“>5–√