Grade by Grade
E. Spatial Sense · Spatial Sense: Angles, Tessellations, Transformations, Scale, Pythagoras and Composite Measurement

Very Large and Very Small Metric Units

E2.1 A1About 45 minutes
Learning goals
  • I can name the metric prefixes mega, giga and tera (very large) and micro, nano and pico (very small), with their symbols.
  • I can write what each prefix means as a power of 10, using exponential notation.
  • I can convert between units with these prefixes using base ten relationships.
  • I can choose a sensible unit for a very large or very small measurement.

Words to know

prefix
A word part at the start of a unit that tells its size, like kilo- in kilometre (1 000 metres).
mega (M)
One million: 1 000 000 = 10⁶.
giga (G)
One billion: 1 000 000 000 = 10⁹.
tera (T)
One trillion: 1 000 000 000 000 = 10¹².
micro (µ)
One millionth: 0.000 001 = 10⁻⁶. The symbol µ is the Greek letter mu.
nano (n)
One billionth: 0.000 000 001 = 10⁻⁹.
pico (p)
One trillionth: 0.000 000 000 001 = 10⁻¹².
negative exponent
A short way to write one over a power of 10: 10⁻³ means 1 ÷ 10³ = 1/1 000 = 0.001.

Remember from earlier grades

You already know kilo (1 000), centi (1/100) and milli (1/1 000): 1 km = 1 000 m and 1 m = 1 000 mm. In Grade 7 you learned to write powers of 10 with exponents, like 10³ = 1 000. In Grade 8 you'll meet prefixes for things that are much bigger and much smaller.

Let's start with a phone

A phone might store 128 GB of photos. A song might be 4 MB. The light from its screen has waves only a few hundred nanometres long, and its processor can do one step in less than a nanosecond. Gigabytes, megabytes and nanometres all use the same metric idea: a prefix in front of a unit tells you how many times bigger or smaller it is.

PrefixSymbolMeaningAs a numberAs a power of 10Example
teraTone trillion1 000 000 000 00010¹²1 TB (terabyte) of storage
gigaGone billion1 000 000 00010⁹1 GHz (gigahertz) computer speed
megaMone million1 000 00010⁶1 MW (megawatt) of power
kilokone thousand1 00010³1 km (kilometre)
(none)one110⁰1 m, 1 g, 1 s
millimone thousandth0.00110⁻³1 mm (millimetre)
microµone millionth0.000 00110⁻⁶1 µg (microgram) on a vitamin label
nanonone billionth0.000 000 00110⁻⁹1 nm (nanometre)
picopone trillionth0.000 000 000 00110⁻¹²1 ps (picosecond)
Each row is 1 000 times the row below it. (Other prefixes, like centi, fit in between, but these jump by 1 000 each time.)

What does a negative exponent mean?

Look at the pattern going down the table: 10³ = 1 000, then 10⁰ = 1, then 10⁻³ = 0.001. Each time the exponent goes down by 1, you divide by 10. So a negative exponent means a fraction:

  • 10⁻³ = 1/10³ = 1/1 000 = 0.001
  • 10⁻⁶ = 1/10⁶ = 1/1 000 000 = 0.000 001

The exponent also tells you how many places to move the decimal point: 10⁶ moves it 6 places right, 10⁻⁶ moves it 6 places left.

1 metre (m)1 millimetre (mm), zoomed in 1 000 timesCut it into 1 000 equal parts: each is 1 micrometre (µm).
A model of base ten relationships. Zoom in on a metre so that 1 mm looks as long as the metre bar. The thin lines cut that millimetre into 10 equal parts of 100 µm each. Cut each of those into 100 more equal parts and every tiny part is 1 µm. A micrometre is 1/1 000 of a millimetre and 1/1 000 000 of a metre. (Not to scale: no screen can show a real micrometre!)

Converting

In the table, each step is a factor of 1 000.

  • From a larger unit to a smaller unit: multiply (you need more of the small units). 3 GB = 3 × 1 000 MB = 3 000 MB.
  • From a smaller unit to a larger unit: divide. 2 500 nm = 2 500 ÷ 1 000 µm = 2.5 µm.
  • Using exponents: 1 GB = 10⁹ bytes and 1 MB = 10⁶ bytes, so 1 GB = 10⁹ ÷ 10⁶ = 10³ = 1 000 MB.

For very large or very small numbers, scientists often write a number between 1 and 10 times a power of 10, like 3.84 × 10⁸ m. This is called scientific notation.

teraT10¹²gigaG10⁹megaM10⁶kilok10³base110⁰millim10⁻³microµ10⁻⁶nanon10⁻⁹picop10⁻¹²to a BIGGER unit: divide by 1 000 for each stepto a SMALLER unit: multiply by 1 000 for each step3 GB → MB: 1 step right, 3 × 1 000 = 3 000 MB2 500 nm → µm: 1 step left, 2 500 ÷ 1 000 = 2.5 µm
The prefixes in order, each 1 000 times the next one. Count the steps between two prefixes: moving right (to a smaller unit) multiply by 1 000 per step, and moving left (to a bigger unit) divide by 1 000 per step.
Worked example

Example 1: The distance to the Moon

The Moon is, on average, about 384 000 km from Earth. Write this distance in megametres (Mm) and in metres using a power of 10.

  1. 1 Mm = 1 000 km. 384 000 ÷ 1 000 = 384 Mm.
  2. In metres: 384 000 km × 1 000 = 384 000 000 m.
  3. Move the decimal point 8 places to get a number between 1 and 10: 3.84 × 10⁸ m.

384 Mm, or 3.84 × 10⁸ m (384 000 000 m).

Worked example

Example 2: A wavelength of light

Green light has a wavelength of about 550 nm. Write this in micrometres and in metres.

  1. 1 µm = 1 000 nm, so 550 nm = 550 ÷ 1 000 = 0.55 µm.
  2. 1 nm = 10⁻⁹ m, so 550 nm = 550 × 10⁻⁹ m = 0.000 000 55 m.
  3. In scientific notation: 5.5 × 10⁻⁷ m.

0.55 µm, or 5.5 × 10⁻⁷ m.

Worked example

Example 3: Photos on a memory card

A memory card holds 64 GB. Each photo is about 5 MB. About how many photos fit?

  1. Change to the same unit: 64 GB = 64 × 1 000 MB = 64 000 MB.
  2. 64 000 ÷ 5 = 12 800.

About 12 800 photos. (Computers sometimes count 1 GB as 1 024 MB instead of 1 000 MB, so real numbers can differ a little.)

Think like a mathematician

  • Ask: is the new unit bigger or smaller? A smaller unit needs a bigger number.
  • Count the steps of 1 000 between the prefixes: giga to kilo is 2 steps, so multiply by 1 000 × 1 000 = 10⁶.
  • Watch the symbols: M (mega) is a million, but m (milli) is a thousandth! Capital and small letters matter.
  • Choose a unit that gives a sensible number: 384 Mm is easier to read than 384 000 000 m.
Hands-on activity

Prefix hunt and zoom model

You need: some packages, labels and devices at home (food packages, a vitamin bottle, a phone or tablet settings screen, a light bulb box, a router or charger label), a metre stick or ruler, paper and a pencil.

  1. Hunt for units with prefixes. Look for MB, GB, TB (storage), MHz or GHz (radio and computer speeds), µg or mg (vitamin labels), W or kW (power). Record each one in a table: measurement, prefix, meaning as a power of 10.
  2. Write each one in the base unit using exponents. For example, 2.4 GHz = 2.4 × 10⁹ Hz.
  3. Zoom model: look at the millimetre marks on your ruler. Imagine one millimetre stretched to the length of the metre stick. On that stretched millimetre, each millimetre mark would be 1 µm. Use this to explain to someone how small a micrometre is.
  4. Find the biggest and the smallest measurements in your hunt.

What to look for: correct prefixes and powers of 10 (giga = 10⁹, mega = 10⁶, micro = 10⁻⁶). FM radio stations in Canada are between 88 and 108 MHz, so a station like 99.1 MHz is 99 100 000 Hz.

Just for fun

A century is 100 years, so a microcentury is 10⁻⁶ of a century: about 52.6 minutes. The physicist Enrico Fermi is said to have joked that a lecture should last one microcentury. If class feels long, it might only be a microcentury.

Your turn

1

How many metres are in 1 megametre? Write it as a power of 10.

2

How many nanometres are in 1 micrometre?

3

Convert 3 GB to megabytes.

4

Convert 2 500 µg to milligrams.

5

Convert 7 TB to gigabytes.

6

Write 1 picometre in metres, as a decimal and as a power of 10.

7

Put in order from smallest to largest: 1 Gm, 1 Tm, 1 Mm.

More practice: warm-up

1

What is the symbol for micro?

2

What does kilo mean?

3

What does mega mean?

4

How many micrometres are in 1 mm?

5

What is 10³?

More practice: on your own

1

An FM radio station broadcasts at 99.9 MHz. Write this in hertz, as a number and in scientific notation.

2

Red light has a wavelength of about 650 nm. Write this in metres.

3

A 128 GB memory card holds photos of about 4 MB each. About how many photos fit?

4

Earth is about 150 Gm from the Sun. Write this in kilometres.

5

Convert 0.002 s to milliseconds and to microseconds.

Challenge

1

How many times larger is 1 terabyte than 1 kilobyte? Use exponents.

2

Light travels about 300 000 km in one second. How far does it travel in 1 nanosecond?

3

Make a model: if you enlarge things so that 1 µm looks like 1 mm, how wide would a hair 100 µm wide look? How long would a light wave of 500 nm look?

Check your understanding

Quiz

1.

Which length is the largest?

2.

1 µm is equal to:

3.

5 GB is equal to:

4.

What does the prefix nano mean?

5.

4 000 nm is equal to:

6.

What is 3.84 × 10⁸ written as a whole number?

For parents and teachers

Big idea: The Ontario curriculum (E2.1) asks Grade 8 students to represent very large (mega, giga, tera) and very small (micro, nano, pico) metric units using models, base ten relationships and exponential notation. Each of these prefixes is 1 000 times the next.

Common difficulties:

  • Multiplying when they should divide (and the reverse). Ask: is the new unit bigger or smaller? Smaller units need bigger numbers.
  • Negative exponents. 10⁻⁶ is not a negative number; it is a very small positive number, 1/1 000 000.
  • Mixing up M (mega) and m (milli).

How to help: The prefix hunt is easy to do at home: phone storage, internet speeds, radio stations and vitamin labels all use these prefixes. Ask your child to say what each one means as a power of 10. This lesson also builds A1 skills: representing numbers in different ways and making connections to everyday life.