NSW Syllabuses

# Mathematics K–10 - Stage 4 - Number and Algebra Ratios and Rates

## 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

• MA4-3WM

recognises and explains mathematical relationships using reasoning

• MA4-7NA

operates with ratios and rates, and explores their graphical representation

Related Life Skills outcome: MALS-19NA

• 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 real-life 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 real-life 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 straight-line 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 straight-line segments with different gradients in the graph of a particular journey (Communicating)
• calculate speeds for straight-line 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 straight-line 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 10-second 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 left-hand 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 sub-elements (and levels) of Comparing units (CoU2-CoU3) and Positioning and locating (PoL5) describe observable behaviours that can aid teachers in making evidence-based decisions about student development and future learning.

The progression sub-elements and indicators can be viewed by accessing the National Numeracy Learning Progression.