At a Glance...

E2.1 Represent very large (mega, giga, tera) and very small (micro, nano, pico) metric units using models, base ten relationships, and exponential notation.

Skill: Representing Very Large (Mega, Giga, Tera) Metric Units Using Models

Using a scale model, students can represent very large objects and units. This can be used to compare, for example, the diameter of the Earth and the Moon. On a diagram, one centimetre could be one megametre.

Examples

Very Large Metric Units

• kilo-unit: 1 thousand (1 ×10³) units
  • kilogram: mass of an object
  • kilometre: distances travelled on land and in the air
• mega-unit: 1 million (1 ×10⁶) units
  • megahertz: frequency of electromagnetic radiation for broadcasting stations
  • megapixels: picture resolution
• giga-unit: 1 billion (1 ×10⁹) units
  • gigametre: distance of planets from the Sun
  • gigabit: bandwidth of a network link
• tera-unit: 1,000 trillion (1 ×10¹²) units
  • terabyte: data storage
  • terasecond: approximately 32 000 years

Source: The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.

Skill: Representing Very Small (Micro, Nano, Pico) Metric Units

Using a scale model, students can represent very small objects and units. This can be used to compare, for example, the thickness of a hair and the length of a microscopic object such as a bacterium. On a diagram, one centimetre could correspond to 10 micrometres.

Examples

Very Small Metric Units

• milli-unit: 1 thousandth (1 ×10^-3) of a unit
  • millimetre: thickness of a card
  • millilitre: volume in cooking and baking
• micro-unit: 1 millionth (1 ×10^-6) of a unit
  • micrometre: measure of microscopic objects
  • microsecond: duration of a high-speed strobe flash
• nano-unit: 1 billionth (1 ×10^-9) of a unit
  • nanosecond: time for light to travel
  • nanometre: length a fingernail grows in 1 s
• pico-unit: 1 trillionth (1 ×10^-12) of a unit
  • picometre: measure of an atom
  • picosecond: speed of lasers

Source: The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.

Knowledge: Base Ten Relationships

All metric units are based on a system of tens, and the metric prefixes describe the relative size of a unit. Whereas units from kilo- to milli- are scaled by powers of 10, units beyond these are scaled by powers of 1000. Exponents are helpful for representing these relationships.

Inverse Relationship

The number of units required to determine the measure of an attribute is inversely proportional to the size of the metric unit used. In other words:

• the smaller the metric unit used, the greater the number of units required to determine the measurement of the attribute;
• the larger the metric unit used, the smaller the number of units required to determine the measurement of the attribute.

Relationships Between Standard Metric Units

When students have a good grasp of the concept of an inverse relationship between the number of units required to determine a measurement and the size of that unit, they can more easily understand and establish relationships between some of the standard metric units.

In order for students to develop an understanding of these relationships, teachers must provide learning situations that allow them to make sense of standard metric units as well as to explore different strategies for converting from one unit to another. These strategies are based on the recognition that any metric unit can be expressed:

• as a multiple of a smaller unit (for example, one metre equals 1000 millimetres, one minute equals 60 seconds);
• as a fraction of a larger unit (for example, one metre is \(\frac{1}{1000}\) of a kilometre, one minute is \(\frac{1}{60} \) of an hour).

Use of a Conversion Table as a Teaching Aid

In order to detect difficulties in students' understanding of metric units, teachers may wish to invite them to explore the table below. While students can name common metric units, it is not always obvious to them that several relationships exist between these units; for example, there is a relationship between the prefixes of units and powers of 10.

Ask students to look at the table and discuss what they notice and what they wonder about. The discussion helps to identify students' misconceptions and is a way to get them to recognize vocabulary they already know.

The Relationship Between the Place Value and the Metric System

Prefixes are also used with other metric units:

• in computer science (byte)
• power output (watt)
• pressure (pascal)
• frequency (hertz)

Note: The intent is not to ask students to memorize vocabulary or to use this table to make a multitude of unit conversions, but rather to invite them to notice relationships and become familiar with the order of magnitude of some of the units already known.

Source: translated from Guide d’enseignement efficace des mathématiques de la 7^e à la 10^e année, Mesure et géométrie, Fascicule 3, p. 46-48.

Knowledge: Exponential Notation

Units from kilo-to milli-are converted using a conversion factor of 10, and other units are converted using a conversion factor of 1000. Exponents help represent these conversion factors.

Source: The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.