NSW Syllabuses

# Mathematics K–10 - Stage 5.1 - Measurement and Geometry Numbers of Any Magnitude

## Outcomes

#### A student:

• MA5.1-1WM

uses appropriate terminology, diagrams and symbols in mathematical contexts

• MA5.1-2WM

selects and uses appropriate strategies to solve problems

• MA5.1-3WM

provides reasoning to support conclusions that are appropriate to the context

• MA5.1-9MG

interprets very small and very large units of measurement, uses scientific notation, and rounds to significant figures

## Content

• Students:
• Investigate very small and very large time scales and intervals (ACMMG219)
• use the language of estimation appropriately, including 'rounding', 'approximate' and 'level of accuracy'
• identify significant figures
• round numbers to a specified number of significant figures
• determine the effect that truncating or rounding during calculations has on the accuracy of the results
• interpret the meaning of common prefixes, such as 'milli', 'centi' and 'kilo'
• interpret the meaning of prefixes for very small and very large units of measurement, such as 'nano', 'micro', 'mega', 'giga' and 'tera'
• record measurements of digital information using correct abbreviations, eg kilobytes (kB)
• investigate and recognise that some digital devices may use different notations to record measurements of digital information, eg 40 kB may appear as 40 K or 40 k or 40 KB (Communicating)
• convert between units of measurement of digital information, eg gigabytes to terabytes, megabytes to kilobytes
• use appropriate units of time to measure very small or very large time intervals
• describe the limits of accuracy of measuring instruments (±0.5 unit of measurement)
• explain why measurements are never exact (Communicating, Reasoning)
• recognise the importance of the number of significant figures in a given measurement (Reasoning)
• choose appropriate units of measurement based on the required degree of accuracy (Communicating, Reasoning)
• consider the degree of accuracy needed when making measurements in practical situations or when writing the results of calculations (Problem Solving, Reasoning)
• recognise the need for a notation to express very large or very small numbers
• express numbers in scientific notation
• explain the difference between numerical expressions such as $$2 × 10^4$$ and $$2^4$$ (Communicating, Reasoning)
• enter and read scientific notation on a calculator
• use index laws to make order of magnitude checks for numbers in scientific notation,
eg $$(3.12 \, \times \, 10^4) \, \times \, (4.2 \, \times \, 10^6) \approx 12 \, \times \, 10^{10} = 1.2 \, \times \, 10^{11}$$
• convert numbers expressed in scientific notation to decimal form
• order numbers expressed in scientific notation
• solve problems involving scientific notation
• communicate and interpret technical information using scientific notation (Communicating)

### Background Information

The metric prefixes 'milli', 'centi' and 'deci' for units smaller than the base International System of Units (SI, from the French Système international d'unités) unit derive from the Latin words mille, meaning 'thousand', centum, meaning 'hundred', and decimus, meaning 'tenth'. The metric prefixes 'kilo', 'hecto' and 'deca' for units larger than the base SI unit derive from the Greek words khilioi, meaning 'thousand', hekaton, meaning 'hundred', and deka, meaning 'ten'.

Scientific notation is also known as 'standard notation'.

#### Purpose/Relevance of Substrand

Measurements always have some degree of uncertainty and are therefore always approximations. Higher-precision instruments can only give better approximations of measurements. In applying knowledge, skills and understanding in measurement, it is necessary to be able to make reasonable estimates for quantities and to have a strong awareness of the levels of accuracy that are appropriate to particular situations. Appropriate approximations are also important for numbers that cannot be expressed exactly in decimal form, eg many fractions, such as $$\frac{1}{3}$$ and $$\frac{2}{7}$$ , and all irrational numbers, such as $$\sqrt{2}$$, $$\sqrt{3}$$, $$\sqrt{5}$$, etc, as well as $$\pi$$. The ability to round numbers appropriately is fundamental to the use of approximations in our everyday and working lives. For very large and very small numbers and measurements, it is necessary to be able to write the decimal forms (which generally contain a very high number of zeros as place-holders) in a much more compact form, for practicality and in order to appreciate their relative size, and for much greater facility in associated calculations. 'Scientific notation' is used wherever the writing and use of very large or very small numbers is needed. For example, to write the mass of the Sun in kilograms as a decimal would require a '2' followed by 30 zeros. In scientific notation, this (approximate) measurement is written simply as $$2 \times 10^{30}$$ kg. Even writing the equivalent approximation in megatonnes would require a '2' followed by 21 zeros in decimal form, but in scientific notation it is written simply as $$2 \times10^{21}$$ Mt. The ability to convert between metric units is very important in order to express very large and very small measurements in appropriate units.

### National Numeracy Learning Progression links to this Mathematics outcome

When working towards the outcome MA5.1-9MG the sub-elements (and levels) of Quantifying numbers (QuN12) 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.