Powers, powers of ten and orders of magnitude

What is it about?

A power is a short way to write a product of equal factors: $10^6 = 10 \cdot 10 \cdot 10 \cdot 10 \cdot 10 \cdot 10 = 1\,000\,000$ . The lower number is the base, the small raised number the exponent – it tells you how many times the base is used as a factor. Likewise $2^5 = 32$ and $3^4 = 81$ .

$10^n$ is a $1$ followed by $n$ zeros: $10^3 = 1\,000$ , $10^6$ is one million, $10^9$ one billion. By definition $10^0 = 1$ – each step down divides by $10$ .

Every step in the exponent makes a length 10 times as large: $1\,\text{mm} = 10^{-3}\,\text{m}$ (grain of sand), $1\,\text{m}$ (child), $100\,\text{m}$ (football pitch), $1\,\text{km} = 10^3\,\text{m}$ , $10^7\,\text{m}$ (the Earth is about $12\,742\,\text{km}$ across). So $1\,\text{km} = 10^3\,\text{m}$ and $1\,\text{m} = 10^3\,\text{mm}$ .

Scientific notation writes numbers as $a \cdot 10^n$ with $1 \le a < 10$ : $35\,000 = 3.5 \cdot 10^4$ , $1\,270\,000 = 1.27 \cdot 10^6$ . Outlook: for numbers smaller than $1$ the point moves to the right and the exponent becomes negative: $10^{-1} = 0.1$ , $10^{-3} = 0.001$ and $0.0042 = 4.2 \cdot 10^{-3}$ . In year 5 this is an outlook; in year 6 you will learn it in more detail.

Try it

Start the journey and watch the scale bar at the top left. What happens each time the exponent grows by $1$ ? Load example
Set $1$ square $= 1\,\text{m}$ . How many squares tall is the child? Then press View × 10: how big is one square now, and what happens to the child? Child
Drag left or right in the picture, or tap an object on the size line. Which objects are only one step apart, which are several steps apart?
Switch to Power as a product and raise the exponent step by step. How often are new cubes added? Powers
Set the base to $2$ . How does the cube model change at each step now? Powers of two
In Scientific notation, enter the Earth’s diameter in metres and let the decimal point move. Earth

Tasks

Task 1: Working out powers

Work out $10^4$ , $2^5$ and $3^4$ . Write down the product first each time. Load task

Show solution

$10^4 = 10 \cdot 10 \cdot 10 \cdot 10 = 10\,000$ ; $2^5 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32$ ; $3^4 = 3 \cdot 3 \cdot 3 \cdot 3 = 81$ . Show in the simulation

Task 2: Million and billion

Write one million and one billion as powers of ten. How many zeros do the numbers have? How many millions make a billion? Load task

Show solution

$1\,000\,000 = 10^6$ (6 zeros), $1\,000\,000\,000 = 10^9$ (9 zeros). $10^9 = 10^6 \cdot 10^3$ , so $1\,000$ millions make a billion. Show in the simulation

Task 3: Estimating sizes

Set $1$ square $= 10\,\text{m}$ . Use the squares to estimate how tall the house is. How many children standing on each other would be as tall? Load task

Show solution

The house is about $10\,\text{m} = 10^1\,\text{m}$ tall. The child is about $1.4\,\text{m}$ tall; $10 \div 1.4 \approx 7$ , so about seven children standing on each other would be as tall as the house. Show in the simulation

Task 4: Converting units

Write as a power of ten in metres: $1\,\text{km}$ , $1\,\text{cm}$ and $1\,\text{mm}$ . How many millimetres are $1\,\text{km}$ ?

Show solution

$1\,\text{km} = 1\,000\,\text{m} = 10^3\,\text{m}$ , $1\,\text{cm} = 0.01\,\text{m} = 10^{-2}\,\text{m}$ , $1\,\text{mm} = 0.001\,\text{m} = 10^{-3}\,\text{m}$ (negative exponents as an outlook). $1\,\text{km} = 10^3 \cdot 10^3\,\text{mm} = 10^6\,\text{mm} = 1\,000\,000\,\text{mm}$ . Show in the simulation

Task 5: Scientific notation

Write $35\,000$ and $1\,270\,000$ in the form $a \cdot 10^n$ with $1 \le a < 10$ . Load task

Show solution

$35\,000 = 3.5 \cdot 10^4$ (point moves 4 places to the left), $1\,270\,000 = 1.27 \cdot 10^6$ (6 places to the left). Check: $1.27 \cdot 1\,000\,000 = 1\,270\,000$ . Show in the simulation

Task 6: Outlook – very small numbers

Write $0.0042$ as $a \cdot 10^n$ . Which way does the decimal point move? Load task

Show solution

The point moves 3 places to the right, so the exponent is negative: $0.0042 = 4.2 \cdot 10^{-3}$ . Check: $4.2 \cdot 0.001 = 0.0042$ . Show in the simulation

Ideas for the classroom

Use: grade 5 (powers and powers of ten with large numbers), grade 6 for revision, scientific notation and negative exponents such as $10^{-3} = 0.001$ (depending on the curriculum also later).
The journey works well as a whole-class starter; the cube models show that “exponent + 1” means “times the base”. With results hidden, pupils can estimate sizes from the grid first.
Objects are drawn to scale; only the airliner and the space station are enlarged symbols.
Common stumbling blocks: $10^3$ calculated as $10 \cdot 3$ ; mixing up $2^5$ and $5^2$ ; thinking $10^0 = 0$ ; writing $35 \cdot 10^3$ instead of $3.5 \cdot 10^4$ .