Essential Mathematics (Version 8.4)

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Unit 1
Unit 2
Unit 3
Unit 4

Understand how Essential Mathematics works

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.

Glossary

Achievement Standards

Unit 2

Unit 2 Description

This unit provides students with the mathematical skills and understanding to solve problems related to representing and comparing data, percentages, rates and ratios, the mathematics of finance, and time and motion. Teachers are encouraged to apply the content of the four topics in this unit – ‘Representing and comparing data’, ‘Percentages’, ‘Rates and ratios’ and ‘Time and motion’ – in a context which is meaningful and of interest to their students. A variety of approaches can be used to achieve this purpose. Two possible contexts which may be used are Mathematics and cars and Mathematics and independent living. 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 2 Learning Outcomes

By the end of this unit, students:

understand the concepts and techniques used in representing and comparing data, percentages, rates and ratios, and time and motion
apply reasoning skills and solve practical problems in representing and comparing data, percentages, rates and ratios, and time and motion
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 2 Content Descriptions

Topic 1: Representing and comparing data

Examples in context

analysing and interpreting a range of statistical information related to car theft, car accidents and driver behaviour
using statistics and graphs to find the number of people in each blood type, given the population percentages of blood types in different countries
using blood usage statistics to predict the amount of blood needed at different times of the year
using blood donation statistics to predict how much blood will be needed and when.

Classifying data:

identify examples of categorical data (ACMEM043)

identify examples of numerical data. (ACMEM044)

Data presentation and interpretation:

display categorical data in tables and column graphs (ACMEM045)

display numerical data as frequency distributions, dot plots, stem and leaf plots, and histograms (ACMEM046)

recognise and identify outliers (ACMEM047)

compare the suitability of different methods of data presentation in real-world contexts. (ACMEM048)

Summarising and interpreting data:

identify the mode (ACMEM049)

calculate measures of central tendency, the arithmetic mean and the median (ACMEM050)

investigate the suitability of measures of central tendency in various real-world contexts (ACMEM051)

investigate the effect of outliers on the mean and the median (ACMEM052)

calculate and interpret quartiles, deciles and percentiles (ACMEM053)

use informal ways of describing spread, such as spread out/dispersed, tightly packed, clusters, gaps, more/less dense regions, outliers (ACMEM054)

calculate and interpret statistical measures of spread, such as the range, interquartile range and standard deviation (ACMEM055)

investigate real-world examples from the media illustrating inappropriate uses, or misuses, of measures of central tendency and spread. (ACMEM056)

Comparing data sets:

compare back-to-back stem plots for different data-sets (ACMEM057)

complete a five number summary for different datasets (ACMEM058)

construct box plots using a five number summary (ACMEM059)

compare the characteristics of the shape of histograms using symmetry, skewness and bimodality. (ACMEM060)

Topic 2: Percentages

Examples in context

calculating stamp duty costs involved in buying a car, using percentages and tables
calculating depreciation of a vehicle over time
using statistics and graphs to find the number of people in each blood type, given the population percentages of blood types in different countries.

Percentage calculations:

review calculating a percentage of a given amount (ACMEM061)

review one amount expressed as a percentage of another. (ACMEM062)

Applications of percentages:

determine the overall change in a quantity following repeated percentage changes; for example, an increase of 10% followed by a decrease of 10% (ACMEM063)

calculate simple interest for different rates and periods. (ACMEM064)

Topic 3: Rates and ratios

Examples in context

Rates – for example:

using rates to find fuel consumption for different vehicles under different driving conditions
calculating food, clothing, transport costs per day, week or month using tables, spreadsheets, and estimation
calculating clothing costs per week or month using tables, spreadsheets, and estimation.

Ratios – for example:

discussing various ratios used in bicycle gears
comparing ratios such as people per household.

Ratios:

demonstrate an understanding of the elementary ideas and notation of ratio (ACMEM065)

understand the relationship between fractions and ratio (ACMEM066)

express a ratio in simplest form (ACMEM067)

find the ratio of two quantities (ACMEM068)

divide a quantity in a given ratio (ACMEM069)

use ratio to describe simple scales. (ACMEM070)

Rates:

review identifying common usage of rates such as km/h (ACMEM071)

convert between units for rates; for example, km/h to m/s, mL/min to L/h (ACMEM072)

complete calculations with rates, including solving problems involving direct proportion in terms of rate (ACMEM073)

use rates to make comparisons (ACMEM074)

use rates to determine costs; for example, calculating the cost of a tradesman using rates per hour, call-out fees. (ACMEM075)

Topic 4: Time and motion

Examples in context

Time – for example:

calculating reaction times through experiments.

Distance – for example:

calculating distances travelled to school and the time taken, considering different average speeds.

Speed – for example:

calculating stopping distances for different speeds by using formulas for different conditions such as road type, tyre conditions and vehicle type.

Time:

use units of time, conversions between units, fractional, digital and decimal representations (ACMEM076)

represent time using 12-hour and 24-hour clocks (ACMEM077)

calculate time intervals, such as time between, time ahead, time behind (ACMEM078)

interpret timetables, such as bus, train and ferry timetables (ACMEM079)

use several timetables and electronic technologies to plan the most time-efficient routes (ACMEM080)

interpret complex timetables, such as tide charts, sunrise charts and moon phases (ACMEM081)

compare the time taken to travel a specific distance with various modes of transport (ACMEM082)

Distance:

use scales to find distances, such as on maps; for example, road maps, street maps, bushwalking maps, online maps and cadastral maps (ACMEM083)

optimise distances through trial-and-error and systematic methods; for example, shortest path, routes to visit all towns, and routes to use all roads. (ACMEM084)

Speed:

identify the appropriate units for different activities, such as walking, running, swimming and flying (ACMEM085)

calculate speed, distance or time using the formula speed = distance/time (ACMEM086)

calculate the time or costs for a journey from distances estimated from maps (ACMEM087)

interpret distance-versus-time graphs (ACMEM088)

calculate and interpret average speed; for example, a 4-hour trip covering 250 km. (ACMEM089)