# Essential Mathematics

### Rationale

Mathematics is the study of order, relation and pattern. From its origins in counting and measuring, it has evolved in highly sophisticated and elegant ways to become the language used to describe much of the physical world.

### Links to Foundation to Year 10

For all content areas of Essential Mathematics, the proficiency strands of Understanding, Fluency, Problem solving and Reasoning from the F–10 curriculum are still very much applicable and should be inherent in students’ learning of the subject.

### Representation of General capabilities

The seven general capabilities of Literacy, Numeracy, Information and Communication Technology (ICT) capability, Critical and creative thinking, Personal and social capability, Ethical understanding, and Intercultural understanding are identified where they offer opportunities to add depth and richness to student learning.

### Structure of Essential Mathematics

Essential Mathematics has four units each of which contains a number of topics. It is intended that the topics be taught in a context relevant to students’ needs and interests. In Essential Mathematics, students use their knowledge and skills to investigate realistic problems of interest which involve the application of mathematical relationships and concepts.

## Unit 3

### Unit 3 Description

This unit provides students with the mathematical skills and understanding to solve problems related to measurement, scales, plans and models, drawing and interpreting graphs, and data collection. Teachers are encouraged to apply the content of the four topics in this unit – ‘Measurement’, ‘Scales, plans and models’, ‘Graphs’ and ‘Data collection’ – in a context which is meaningful and of interest to the students. A variety of approaches can be used to achieve this purpose. Two possible contexts which may be used in this unit are Mathematics and design and Mathematics and medicine. However, as these contexts may not be relevant to all students, teachers are encouraged to find suitable contexts relevant to their particular student cohort.

It is assumed that an extensive range of technological applications and techniques will be used in teaching this unit. The ability to choose when and when not to use some form of technology, and the ability to work flexibly with technology, are important skills.

### Unit 3 Learning Outcomes

By the end of this unit, students:

• understand the concepts and techniques used in measurement, scales, plans and models, graphs, and data collection
• apply reasoning skills and solve practical problems in measurement, scales, plans and models, graphs, and data collection
• communicate their arguments and strategies when solving mathematical and statistical problems using appropriate mathematical or statistical language
• interpret mathematical and statistical information and ascertain the reasonableness of their solutions to problems.

### Unit 3 Content Descriptions

#### Topic 1: Measurement

Examples in context
• calculating and interpreting dosages for children and adults from dosage panels on medicines, given age or weight
• calculating and interpreting dosages for children from adults’ medication using various formulas (Fried, Young, Clark) in milligrams or millilitres
• calculating surface areas of various buildings to compare costs of external painting.

#### Linear measure:

review metric units of length, their abbreviations, conversions between them, estimation of lengths, and appropriate choices of units

(ACMEM090)

calculate perimeters of familiar shapes, including triangles, squares, rectangles, polygons, circles, arc lengths, and composites of these.

(ACMEM091)

find the area of irregular figures by decomposition into regular shapes (ACMEM094)

#### Area measure:

review metric units of area, their abbreviations, and conversions between them (ACMEM092)

use formulas to calculate areas of regular shapes, including triangles, squares, rectangles, parallelograms, trapeziums, circles and sectors (ACMEM093)

find the surface area of familiar solids, including cubes, rectangular and triangular prisms, spheres and cylinders (ACMEM095)

find the surface area of pyramids, such as rectangular- and triangular-based pyramids (ACMEM096)

use addition of the area of the faces of solids to find the surface area of irregular solids. (ACMEM097)

#### Mass:

review metric units of mass (and weight), their abbreviations, conversions between them, and appropriate choices of units (ACMEM098)

recognise the need for milligrams (ACMEM099)

convert between grams and milligrams. (ACMEM100)

#### Volume and capacity:

review metric units of volume, their abbreviations, conversions between them, and appropriate choices of units (ACMEM101)

recognise relations between volume and capacity, recognising that $$1\mathrm c\mathrm m^3=1\mathrm m\mathrm L$$ and $$1\mathrm m^3=1\mathrm k\mathrm L$$ (ACMEM102)

use formulas to find the volume and capacity of regular objects such as cubes, rectangular and triangular prisms and cylinders (ACMEM103)

use formulas to find the volume of pyramids and spheres. (ACMEM104)

#### Topic 2: Scales, plans and models

Examples in context
• drawing scale diagrams of everyday two-dimensional shapes
• interpreting common symbols and abbreviations used on house plans
• using the scale on a plan to calculate actual external or internal dimensions, the lengths of the house and the dimensions of? particular rooms
• using technology to translate two-dimensional house plans into three-dimensional buildings
• creating landscape designs using technology.

#### Geometry:

recognise the properties of common two-dimensional geometric shapes and three-dimensional solids (ACMEM105)

interpret different forms of two-dimensional representations of three-dimensional objects, including nets and perspective diagrams (ACMEM106)

use symbols and conventions for the representation of geometric information; for example, point, line, ray, angle, diagonal, edge, curve, face and vertex. (ACMEM107)

#### Interpret scale drawings:

interpret commonly used symbols and abbreviations in scale drawings (ACMEM108)

find actual measurements from scale drawings, such as lengths, perimeters and areas (ACMEM109)

estimate and compare quantities, materials and costs using actual measurements from scale drawings; for example, using measurements for packaging, clothes, painting, bricklaying and landscaping. (ACMEM110)

#### Creating scale drawings:

understand and apply drawing conventions of scale drawings, such as scales in ratio, clear indications of dimensions, and clear labelling (ACMEM111)

construct scale drawings by hand and by using software packages. (ACMEM112)

#### Three dimensional objects:

interpret plans and elevation views of models (ACMEM113)

sketch elevation views of different models (ACMEM114)

interpret diagrams of three-dimensional objects. (ACMEM115)

#### Right-angled triangles:

apply Pythagoras’ theorem to solve problems (ACMEM116)

apply the tangent ratio to find unknown angles and sides in right-angled triangles (ACMEM117)

work with the concepts of angle of elevation and angle of depression (ACMEM118)

apply the cosine and sine ratios to find unknown angles and sides in right-angled triangles (ACMEM119)

solve problems involving bearings. (ACMEM120)

#### Topic 3: Graphs

Examples in context
• interpreting graphs showing growth ranges for children (height or weight or head circumference versus age)
• interpreting hourly hospital charts showing temperature and pulse
• interpreting graphs showing life expectancy with different variables.

#### Cartesian plane:

demonstrate familiarity with Cartesian coordinates in two dimensions by plotting points on the Cartesian plane (ACMEM121)

generate tables of values for linear functions, including for negative values of $$x$$ (ACMEM122)

graph linear functions for all values of $$x$$ with pencil and paper and with graphing software. (ACMEM123)

#### Using graphs:

interpret and use graphs in practical situations, including travel graphs and conversion graphs (ACMEM124)

draw graphs from given data to represent practical situations (ACMEM125)

interpret the point of intersection and other important features of given graphs of two linear functions drawn from practical contexts; for example, the ‘break-even’ point. (ACMEM126)

#### Topic 4: Data collection

Examples in context
• analysing data obtained from medical sources, including bivariate data.

#### Census:

investigate the procedure for conducting a census (ACMEM127)

#### Surveys:

understand the purpose of sampling to provide an estimate of population values when a census is not used (ACMEM129)

investigate the different kinds of samples; for example, systematic samples, self-selected samples, simple random samples (ACMEM130)

investigate the advantages and disadvantages of these kinds of samples; for example, comparing simple random samples with self-selected samples. (ACMEM131)

#### Simple survey procedure:

identify the target population to be surveyed (ACMEM132)

investigate questionnaire design principles; for example, simple language, unambiguous questions, consideration of number of choices, issues of privacy and ethics, and freedom from bias. (ACMEM133)

#### Sources of bias:

describe the faults in the collection of data process (ACMEM134)

describe sources of error in surveys; for example, sampling error and measurement error (ACMEM135)

investigate the possible misrepresentation of the results of a survey due to misunderstanding the procedure, or misunderstanding the reliability of generalising the survey findings to the entire population (ACMEM136)

investigate errors and misrepresentation in surveys, including examples of media misrepresentations of surveys. (ACMEM137)

#### Bivariate scatterplots:

describe the patterns and features of bivariate data (ACMEM138)

describe the association between two numerical variables in terms of direction (positive/negative), form (linear/non-linear) and strength (strong/moderate/weak). (ACMEM139)

#### Line of best fit:

identify the dependent and independent variable (ACMEM140)

find the line of best fit by eye (ACMEM141)

use technology to find the line of best fit (ACMEM142)

interpret relationships in terms of the variables (ACMEM143)

use technology to find the correlation coefficient (an indicator of the strength of linear association) (ACMEM144)

use the line of best fit to make predictions, both by interpolation and extrapolation (ACMEM145)

recognise the dangers of extrapolation (ACMEM146)

distinguish between causality and correlation through examples. (ACMEM147)