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Key Stage 3 maths · Number

KS3 powers, roots and standard form: a guide for parents

In Years 7, 8 and 9 pupils use all four operations with whole numbers, decimals, fractions and mixed numbers, both positive and negative. They follow the correct order of operations, work with powers and roots, tell exact values from decimal approximations, and learn to read and compare very large and very small numbers written in standard form.

Powers, roots and standard form, explained

Pupils apply formal written methods, such as column addition, long multiplication and long division, to integers and decimals, and extend them to proper fractions, improper fractions and mixed numbers, which can be negative as well as positive. The rules for negatives are practised until they are secure: −3 × −4 = 12, −12 ÷ 4 = −3 and 5 − (−2) = 7. With fractions, pupils add and subtract using a common denominator and multiply across the top and across the bottom. To divide, they multiply by the reciprocal of the second fraction, so 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6. Mixed numbers are first turned into improper fractions: 1½ × 2⅔ = 3/2 × 8/3 = 24/6 = 4.

Pupils use the conventional order of operations: brackets first, then powers and roots, then multiplication and division, then addition and subtraction, working left to right within each level. Many schools teach this with a mnemonic such as BIDMAS or BODMAS. So 3 + 4 × 2² = 3 + 4 × 4 = 3 + 16 = 19. The reciprocal of a number is 1 divided by that number, so the reciprocal of 4 is 1/4 and of 2/3 is 3/2, and any number multiplied by its reciprocal gives 1. Pupils also use inverse operations to check and to rearrange, so if 18 ÷ 3 = 6 then 6 × 3 = 18, and squaring and square rooting undo each other.

A power, or index, tells you how many copies of a number to multiply together: 2³ = 2 × 2 × 2 = 8. A root is the inverse of a power. The square root of 49 is 7, the cube root of 27 is 3, and the fourth root of 81 is 3, because 3⁴ = 81. Pupils learn to recognise the early powers of 2, 3, 4 and 5, such as 2, 4, 8, 16, 32, 64; 3, 9, 27, 81; 4, 16, 64, 256; and 5, 25, 125, 625. They also learn the difference between an exact value and an approximation. The exact value of the square root of 2 is written √2. A calculator shows 1.41421356…, which goes on for ever without repeating, so 1.414 is only a close approximation. Writing √2 is the exact way to say it.

Standard form is a way to write very large or very small numbers compactly as A × 10ⁿ, where A is at least 1 and less than 10, and n is a whole number, which may be negative or zero. So 3,400,000 is 3.4 × 10⁶ and 0.00052 is 5.2 × 10⁻⁴. A positive power means a large number, and a negative power a small one. In Key Stage 3, pupils interpret and compare numbers in standard form, as scientists do for distances in space or the size of cells. The quickest comparison is to look at the powers of 10 first, and then at A if the powers match.

Words your child will hear

Power (index)
The small raised number that says how many copies of a number are multiplied together. In 2³ the power is 3, so 2³ = 2 × 2 × 2.
Root
The inverse of a power. The square root of 25 is 5 because 5² = 25.
Reciprocal
1 divided by a number. The reciprocal of 5 is 1/5, and the reciprocal of 2/3 is 3/2.
Inverse operation
An operation that undoes another: addition and subtraction, multiplication and division, squaring and square rooting.
Order of operations
The agreed order for working out a calculation, often remembered as BIDMAS or BODMAS.
Standard form
A number written as A × 10ⁿ, with A at least 1 and less than 10, such as 6.5 × 10⁶.
Exact value
A value written without rounding, such as √2 or 1/3, as opposed to a decimal approximation.

What the National Curriculum says

In Key Stage 3, pupils should be taught to:

  • Use the 4 operations, including formal written methods, applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative
  • Use conventional notation for the priority of operations, including brackets, powers, roots and reciprocals
  • Recognise and use relationships between operations including inverse operations
  • Use integer powers and associated real roots (square, cube and higher), recognise powers of 2, 3, 4, 5 and distinguish between exact representations of roots and their decimal approximations
  • Interpret and compare numbers in standard form A × 10ⁿ, 1 ≤ A < 10, where n is a positive or negative integer or 0

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

Write 83,000,000 and 0.0052 in standard form. Then decide which is larger: 4.2 × 10⁻³ or 9.1 × 10⁻⁴.

  1. For 83,000,000, A must be at least 1 and less than 10, so A = 8.3. To get from 8.3 back to 83,000,000 you multiply by 10 seven times, so the digits move 7 places and the power of 10 is 7.
    83,000,000 = 8.3 × 10⁷
  2. For 0.0052, A = 5.2. To get from 5.2 to 0.0052 you divide by 10 three times, so the digits move 3 places the other way. The number is less than 1, so the power is negative.
    0.0052 = 5.2 × 10⁻³
  3. Both numbers to compare are in standard form already, because 4.2 and 9.1 are each between 1 and 10. Compare the powers of 10 first: −3 is greater than −4, so 4.2 × 10⁻³ is larger. A being bigger in the second number doesn’t change this.
  4. Check by writing them out in full: 4.2 × 10⁻³ is 0.0042 and 9.1 × 10⁻⁴ is 0.00091. 0.0042 is more than 4 thousandths, while 0.00091 is less than 1 thousandth, so the first is bigger.
    0.0042  >  0.00091

83,000,000 = 8.3 × 10⁷ and 0.0052 = 5.2 × 10⁻³. The larger number is 4.2 × 10⁻³.

Common mistakes and how to help

Working out 2³ as 6

The raised 3 looks like a multiplication instruction, so 2 is multiplied by 3 instead of three 2s being multiplied together.

What helps: Write it out in full: 2³ = 2 × 2 × 2. Build the pattern 2, 4, 8, 16 from doubling, and keep a list of squares and cubes within reach.

Working left to right, so 3 + 4 × 2 becomes 14

Reading from left to right is natural, and a basic calculator may do the same.

What helps: Use the order of operations: multiplication before addition, so 3 + 8 = 11. Underline the part to do first, and use brackets if you want a different order.

Thinking a negative power gives a negative number, so 10⁻² = −100

The minus sign suggests a negative answer, and the connection with dividing is not obvious.

What helps: Notice that dividing by 10 moves down the powers: 100, 10, 1, 1/10, 1/100. Then 10⁻² = 1/100 = 0.01, a small positive number.

Writing 45 × 10³ or 0.5 × 10⁴ and calling it standard form

The 10ⁿ part is right but A isn’t between 1 and 10, so the number is not in standard form.

What helps: Always check A first. 45 × 10³ is 4.5 × 10⁴, and 0.5 × 10⁴ is 5 × 10³.

Treating a calculator’s rounded root as the exact answer

The screen shows a decimal, so 1.414 for √2 looks final. Squaring 1.414 gives 1.999396, not 2.

What helps: Write √2 when an exact answer is wanted, and use a decimal only when the question asks for rounding, such as to 2 decimal places.

How to help at home

  • Order-of-operations puzzles. Give four numbers, such as 2, 3, 4 and 5, and ask your child to use them once each, with any operations and brackets, to make a target such as 20. Different bracket positions give different answers, which makes the point.
  • Doubling story. Imagine saving 1p on day 1, 2p on day 2, 4p on day 3, and doubling each day. Ask how much goes in on day 10. That is 2⁹ = 512p, or £5.12.
  • Estimating roots. Ask for the square root of 50 without a calculator. 7² = 49 and 8² = 64, so it is just over 7. A calculator then gives 7.07, which is a quick check on the idea.
  • Very big and very small. There are about 31,500,000 seconds in a year, or 3.15 × 10⁷. Find other large or small numbers, such as the population of the UK or the width of a hair, and write them in standard form.
  • Fractions in the kitchen. How many ¾-cup servings are in 6 cups of rice? Dividing by 3/4 means multiplying by 4/3, so there are 8. Ask your child to check by multiplying: 8 × ¾ = 6.

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