YEAR 7 MATHS FOCUS
NUMBER AND ALGEBRA
MEASUREMENT AND GEOMETRY
RIGHT ANGLED TRIANGLES
- applies Pythagoras’ theorem to calculate side lengths in right-angled triangles, and solves related problems
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
is an integer greater than 2, then
has no positive-integer solutions.
|RELEVANCE||Pythagoras’ theorem is of great importance in the mathematics learned in secondary school and is also important in many areas of further mathematics and science study. The theorem is part of the foundations of trigonometry, an important area of mathematics introduced in Stage 5, which allows the user to determine unknown sides and angles in both right-angled and non-right-angled triangles. Pythagoras’ theorem has many everyday applications, such as building construction (to ensure that buildings, rooms, additions, etc are square) and determining the length of any diagonal (eg the screen size of a television, the shortest distance between two points).|
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.
EXPECTATIONS OF ATTAINMENT
|Investigate pythagoras-theorem and its application to solving simple problems involving right-angled triangles||identify the hypotenuse as the longest side in any right-angled triangle and also as the side opposite the right angle|
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
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)
|identify a Pythagorean triad as a set of three numbers such that the sum of the squares of the first two equals the square of the third|
|use the converse of Pythagoras’ theorem to establish whether a triangle has a right angle|
|Investigate the concept of irrational numbers|
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, and .
– solve a variety of practical problems involving Pythagoras’ theorem, giving exact answers (ie as surds where appropriate), eg