NSW Syllabuses

# Mathematics K–10 - Stage 4 - Measurement and Geometry Length

## Outcomes

#### A student:

• MA4-1WM

communicates and connects mathematical ideas using appropriate terminology, diagrams and symbols

• MA4-2WM

applies appropriate mathematical techniques to solve problems

• MA4-12MG

calculates the perimeters of plane shapes and the circumferences of circles

Related Life Skills outcomes: MALS-25MG, MALS-26MG

## Content

• find the perimeters of a range of plane shapes, including parallelograms, trapeziums, rhombuses, kites and simple composite figures
• compare perimeters of rectangles with the same area (Problem Solving)
• solve problems involving the perimeters of plane shapes, eg find the dimensions of a rectangle, given its perimeter and the length of one side
• demonstrate by practical means that the ratio of the circumference to the diameter of a circle is constant, eg measure and compare the diameters and circumferences of various cylinders or use dynamic geometry software to measure circumferences and diameters
• define the number $$\pi$$ as the ratio of the circumference to the diameter of any circle
• compare the various approximations for $$\pi$$ used throughout the ages and investigate the concept of irrational numbers (Communicating)
• recognise that the symbol $$\pi$$ is used to represent a constant numerical value (Communicating)
• Investigate the relationship between features of circles, such as the circumference, radius and diameter; use formulas to solve problems involving circumference (ACMMG197)
• identify and name parts of a circle and related lines, including arc, tangent, chord, sector and segment
• develop and use the formulas to find the circumferences of circles in terms of the diameter d or radius r
$$\text{Circumference of circle}=\pi d$$
$$\text{Circumference of circle} = 2\pi r$$
• use mental strategies to estimate the circumferences of circles, using an approximate value of $$\pi$$ such as 3 (Problem Solving)
• find the diameter and/or radius of a circle, given its circumference (Problem Solving)
• find the perimeters of quadrants and semicircles
• find the perimeters of simple composite figures consisting of two shapes, including quadrants and semicircles
• find arc lengths and the perimeters of sectors
• solve a variety of practical problems involving circles and parts of circles, giving an exact answer in terms of $$\pi$$ and an approximate answer using a calculator's approximation for $$\pi$$

### Background Information

Students should develop a sense of the levels of accuracy that are appropriate to a particular situation, eg the length of a bridge may be measured in metres to estimate a quantity of paint needed, but would need to be measured much more accurately for engineering work.

The number $$\pi$$ is known to be irrational (not a fraction) and also transcendental (not the solution of any polynomial equation with integer coefficients). In Stage 4, students only need to know that the digits in its decimal expansion do not repeat (all this means is that it is not a fraction) and in fact have no known pattern.

#### Purpose/Relevance of Substrand

This substrand focuses on the 'perimeter' (or length of the boundary) of shapes (including the 'circumference' of a circle). The ability to determine the perimeters of two-dimensional shapes is of fundamental importance in many everyday situations, such as framing a picture, furnishing a room, fencing a garden or a yard, and measuring land for farming or building construction.

### Language

Pi $$(\pi)$$ is the Greek letter equivalent to 'p' and is the first letter of the Greek word perimetron, meaning 'perimeter'. The symbol for pi was first used to represent the ratio of the circumference to the diameter of a circle in the early eighteenth century.

The names for some parts of the circle (centre, radius, diameter, circumference, sector, semicircle and quadrant) are introduced in Stage 3. The terms 'arc', 'tangent', 'chord' and 'segment' are introduced in Stage 4.

Some students may find the use of the terms 'length/long', 'breadth/broad', 'width/wide' and 'height/high' difficult. Teachers should model the use of these terms in sentences, both verbally and in written form, when describing diagrams. Students should be encouraged to speak about, listen to, read about and write about the dimensions of given shapes using various combinations of these words, eg 'The length of this rectangle is 7 metres and the width is 4 metres', 'The rectangle is 7 metres long and 4 metres wide'. Students may also benefit from drawing and labelling a shape, given a description of its features in words, eg 'The base of an isosceles triangle is 6 metres long and its perimeter is 20 metres. Draw the triangle and mark on it the lengths of the three sides'.

In Stage 3, students were introduced to the term 'dimensions' to describe the length and width of a rectangle. However, some students may need to be reminded of this.