Outcomes
A student:

 MA41WM
communicates and connects mathematical ideas using appropriate terminology, diagrams and symbols

 MA42WM
applies appropriate mathematical techniques to solve problems

 MA43WM
recognises and explains mathematical relationships using reasoning

 MA47NA
operates with ratios and rates, and explores their graphical representation
Related Life Skills outcome: MALS19NA
 Students:
 Recognise and solve problems involving simple ratios (ACMNA173)
 use ratios to compare quantities measured in the same units
 write ratios using the \(:\) symbol, eg \(4\!:\!7\)
 express one part of a ratio as a fraction of the whole, eg in the ratio \(4\!:\!7\), the first part is \( \frac{4}{11} \) of the whole (Communicating)
 simplify ratios, eg \(4\!:\!6 = 2\!:\!3\), \(\quad \frac{1}{2}\! :\! 2 = 1\!:\!4 \), \(\quad 0.3\!:\!1 = 3\!:\!10 \)
 apply the unitary method to ratio problems
 divide a quantity in a given ratio
 Solve a range of problems involving ratios and rates, with and without the use of digital technologies (ACMNA188)
 interpret and calculate ratios that involve more than two numbers
 solve a variety of reallife problems involving ratios, eg scales on maps, mixes for fuels or concrete
 use rates to compare quantities measured in different units
 distinguish between ratios, where the comparison is of quantities measured in the same units, and rates, where the comparison is of quantities measured in different units
 convert given information into a simplified rate, eg 150 kilometres travelled in 2 hours = 75 km/h
 solve a variety of reallife problems involving rates, including problems involving rate of travel (speed)
 Investigate, interpret and analyse graphs from authentic data (ACMNA180)
 interpret distance/time graphs (travel graphs) made up of straightline segments
 write or tell a story that matches a given distance/time graph (Communicating)
 match a distance/time graph to a description of a particular journey and explain the reasons for the choice (Communicating, Reasoning)
 compare distance/time graphs of the same situation, decide which one is the most appropriate, and explain why (Communicating, Reasoning)
 recognise concepts such as change of speed and direction in distance/time graphs
 describe the meaning of straightline segments with different gradients in the graph of a particular journey (Communicating)
 calculate speeds for straightline segments of given distance/time graphs (Problem Solving)
 recognise the significance of horizontal line segments in distance/time graphs
 determine which variable should be placed on the horizontal axis in distance/time graphs
 draw distance/time graphs made up of straightline segments
 sketch informal graphs to model familiar events, eg noise level during a lesson
 record the distance of a moving object from a fixed point at equal time intervals and draw a graph to represent the situation, eg move along a measuring tape for 30 seconds using a variety of activities that involve a constant rate, such as walking forwards or backwards slowly, and walking or stopping for 10second increments (Problem Solving)
 use the relative positions of two points on a line graph, rather than a detailed scale, to interpret information
Background Information
Work with ratios may be linked to the 'golden rectangle'. Many windows are golden rectangles, as are rectangles used in some of the ancient buildings in Athens, such as the Parthenon.
The relationship between the ratios involving the dimensions of the golden rectangle was known to the Greeks in the sixth century BC: \(\frac{\mbox{length}}{\mbox{width}} = \frac{\mbox{length + width}}{\mbox{length}} \).
In Stage 4, the focus is on examining situations where the data yields a constant rate of change. It is possible that some practical situations may yield a variable rate of change. This is the focus in Ratios and Rates in Stage 5.3.
It is the usual practice in mathematics to place the independent variable on the horizontal axis and the dependent variable on the vertical axis. This is not always the case in other subjects, such as economics.
Purpose/Relevance of Substrand
As we often need to compare two numbers, amounts or quantities in our daily lives, ratios and rates are important aspects of our study of mathematics. Ratios are used to compare two (or more) numbers, amounts or quantities of the same kind (eg objects, people, weights, heights) and can be expressed as '\(a\) to \(b\)' or \(a\!:\!b\). In simple terms, a ratio represents that for a given number or amount of one thing, there is a certain number or amount of another thing (eg 'I have 4 ties for every shirt, so the ratio of ties:shirts is 4:1 and the ratio of shirts:ties is 1:4'). A rate is a particular type of ratio that is used to compare two measurements of different kinds. Speed is a rate in which the distance travelled (by a person, car, etc) is compared to the time taken (to cover the distance travelled). It follows that speed is measured (or expressed) in units such as metres per second and kilometres per hour. Other examples of rates that are important in our everyday lives include interest rates, exchange rates, heart rates, birth rates and death rates.
Language
The use of the word 'per', meaning 'for every', in rates should be made explicit to students.
When solving ratio and rate problems, students should be encouraged to write a few key words on the lefthand side of the equals sign to identify what is being found in each step of their working, and to conclude with a statement in words.
When describing distance/time graphs (travel graphs), supply a modelled story and graph first, or jointly construct a story with students before independent work is required. When constructing stories and interpreting distance/time graphs, students can use present tense, 'The man travels …', or past tense, 'The man travelled …'.
Students should be aware that 'gradient' may be referred to as 'slope' in some contexts.
National Numeracy Learning Progression links to this Mathematics outcome
When working towards the outcome MA4‑7NA the subelements (and levels) of Comparing units (CoU2CoU3) and Positioning and locating (PoL5) describe observable behaviours that can aid teachers in making evidencebased decisions about student development and future learning.
The progression subelements and indicators can be viewed by accessing the National Numeracy Learning Progression.