Teaching unit · UK Year 9 (Key Stage 3, ages 13 to 14)

Powers, roots and standard form

Evaluating integer powers, finding exact and approximate roots, and writing very large or very small numbers in standard form

About three lessons of 45 to 60 minutes

Student view

Start here · hook

Numbers too big, or too small, to write out

The Sun is about 150,000,000 km from Earth. A typical bacterium is about 0.000002 m wide. Both numbers are correct, but both are painful to read, write and multiply: count the zeros wrong by 1 and you are out by a factor of 10. Standard form fixes this by writing every number as a single digit (with decimals), times a power of 10 that does all the zero-counting for you: 1.5 x $10^{8}$ km, or 2 x $10^{-6}$ m.

Standard form leans on powers and roots, the other half of this unit. A power like $10^{8}$ is just 10 multiplied by itself 8 times, and a root undoes a power: the square root of a perfect square is always an exact whole number, but the square root of most other numbers is not exact at all, only a decimal approximation from a calculator. Knowing which is which matters just as much as calculating the number itself.

Distance to the Sun: 150,000,000 km= 1.5 x $10^{8}$ km in standard form
Width of a bacterium: 0.000002 m= 2 x $10^{-6}$ m in standard form
A square field with side 9 m has area $9^{2}$ = 81 $m^{2}$ and a square field of area 81 $m^{2}$ has an EXACT side length, 9 m
$\sqrt{50}$ = 7.0710678...50 is not a perfect square, so this is only ever a rounded decimal approximation

Learning objective

What students will be able to do

Students will evaluate integer powers, find exact square and cube roots and distinguish them from decimal approximations, convert between ordinary numbers and standard form (A x $10^{n}$ , 1 <= A < 10), and compare and order numbers written in standard form.

Success criteria

I can evaluate an integer power such as $2^{5}$ or $3^{4}$ .
I can find the exact square root or cube root of a perfect square or perfect cube.
I can find a decimal approximation for the square root of a number that is not a perfect square, and explain why it is not exact.
I can convert an ordinary number to standard form, and a standard-form number back to an ordinary number.
I can compare and order 2 or more numbers written in standard form.

Curriculum anchor

Standards this unit teaches

KS3 Maths: NumberUK National Curriculum (England)
Number
Statutory requirement (Department for Education, "National curriculum in England: mathematics programmes of study", updated 28 September 2021, Key stage 3, "Number" strand, https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study): pupils should be taught to "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 x $10^{n}$ , 1 <= A < 10, where n is a positive or negative integer or 0".

Before you start

Prior knowledge

This unit builds on skills students should already have met. Revisit any that are shaky first.

Key vocabulary

Words to teach and display

Power (exponent): a small raised number showing how many times a value is multiplied by itself, e.g. $2^{5}$ means 2 x 2 x 2 x 2 x 2
Square root: a number that, multiplied by itself, gives the original number; the inverse of squaring
Cube root: a number that, multiplied by itself 3 times, gives the original number; the inverse of cubing
Perfect square / perfect cube: a number that is exactly the square (or cube) of a whole number, so its root is exact, not rounded
Standard form: a way of writing a number as A x $10^{n}$ , where 1 <= A < 10 and n is a whole number (positive, negative or 0)
Mantissa: the leading number A in standard form, always between 1 and 10 (1 <= A < 10)

Teaching sequence

Teach it: concrete, pictorial, abstract

The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.

1. Powers and roots

Concrete

A power multiplies a number by itself repeatedly: $3^{4}$ means 3 x 3 x 3 x 3 = 81. A root undoes a power: the square root of 81 asks 'what number, squared, gives 81?', and the answer is 9, because 9 x 9 = 81.

When a number is a perfect square (like 81 = $9^{2}$ ) or a perfect cube (like 125 = $5^{3}$ ), its root is an EXACT whole number. Most numbers are not perfect squares or cubes, though: $\sqrt{50}$ cannot be written as an exact whole number at all, only estimated as a decimal using a calculator, $\sqrt{50}$ = 7.0710678..., usually rounded to a sensible number of decimal places.

The square numbers 1, 4, 9, 16, 25 plotted as (side length, area): every 1 is a perfect square, so its square root is exactly the side length shown.

Worked example

A square has a side length of 5 m. Find its area. Then a different square has an area of 81 $m^{2}$ . Find its side length, and state whether your answer is exact.

Area of the first square: $5^{2}$ = 5 x 5 = 25 $m^{2}$ .
Side length of the second square: $\sqrt{81}$ . Since 9 x 9 = 81, 81 is a perfect square.
$\sqrt{81}$ = 9 exactly, not a rounded decimal.

Answer: The first square has area 25 $m^{2}$ . The second square has side length 9 m, exact.

Check for understanding, ask

Evaluate $2^{6}$ .
Find $\sqrt{60}$ , correct to 2 decimal places. Is your answer exact?

2. Writing numbers in standard form

Pictorial

Standard form writes any number as A x $10^{n}$ , where 1 <= A < 10 (a single non-zero digit, then a decimal part if needed). The power n tells you how many places, and which direction, the decimal point has moved.

For a large number, count how many places the decimal point moves LEFT until only 1 digit remains before it: 340,000 becomes 3.4 (moved 5 places), so 340,000 = 3.4 x $10^{5}$ . For a small number less than 1, the decimal point moves RIGHT instead, and n is negative: 0.0034 = 3.4 x $10^{-3}$ .

340,000 has its first non-zero digit (3) 5 places left of the ones column, so 340,000 = 3.4 x

10^{5}

Worked example

Write 6,200,000 in standard form. Then write 4.5 x $10^{-3}$ as an ordinary number.

6,200,000: move the decimal point left until 1 digit remains: 6.2, moved 6 places.
6,200,000 = 6.2 x $10^{6}$ .
4.5 x $10^{-3}$ : the negative exponent means move the decimal point 3 places LEFT (a small number): 0.0045.

Answer: 6,200,000 = 6.2 x $10^{6}$ . 4.5 x $10^{-3}$ = 0.0045.

Check for understanding, ask

Write 0.00081 in standard form.
Write 7 x $10^{4}$ as an ordinary number.

3. Comparing and ordering numbers in standard form

Abstract

To compare 2 numbers in standard form, compare the exponent n first, since both mantissas already satisfy 1 <= A < 10. A bigger n always means a bigger number; only if the exponents are EQUAL do you compare the mantissas.

It is tempting to compare mantissas first, but a mantissa can only ever be between 1 and 10, so it can never make up for a smaller exponent: 9 x $10^{2}$ (= 900) is still far smaller than 1 x $10^{5}$ (= 100,000), even though 9 looks like the 'bigger' digit.

6 x

10^{3}

= 6,000 and 2.5 x

10^{4}

= 25,000: plotted on the same line, 2.5 x

10^{4}

is clearly the larger number.

Worked example

Which is larger, 6 x $10^{3}$ or 2.5 x $10^{4}$ ? Then order 3 x $10^{-2}$ , 3 x $10^{2}$ and 3 x $10^{-5}$ from smallest to largest.

6 x $10^{3}$ has exponent 3; 2.5 x $10^{4}$ has exponent 4. The larger exponent wins, so 2.5 x $10^{4}$ is larger (25,000 > 6,000).
Compare the 3 exponents: -5 < -2 < 2.
Smallest to largest: 3 x $10^{-5}$ , 3 x $10^{-2}$ , 3 x $10^{2}$ .

Answer: 2.5 x $10^{4}$ is larger. Order: 3 x $10^{-5}$ , 3 x $10^{-2}$ , 3 x $10^{2}$ .

Check for understanding, ask

Which is larger, 9 x $10^{-2}$ or 1 x $10^{-1}$ ?
Order 5 x $10^{3}$ , 5 x $10^{-3}$ and 5 x $10^{0}$ from smallest to largest.

Watch for

Common misconceptions and how to address them

Misconception9 x $10^{2}$ must be bigger than 1 x $10^{5}$ , because 9 is a bigger digit than 1.

Why it happens: Students compare the mantissas first, treating standard form like ordinary place value where the leftmost digit always dominates.

How to address it: Compare the EXPONENT first, always. A mantissa can only ever be between 1 and 10, so it can never outweigh a bigger power of 10: 9 x $10^{2}$ = 900, but 1 x $10^{5}$ = 100,000, which is far larger.

Misconception $\sqrt{50}$ should round to a whole number, because most roots asked at school 'come out nicely'.

Why it happens: Students over-generalise from a diet of perfect-square examples and expect every root to be exact.

How to address it: Check first whether the number is a perfect square (or cube): only then is the root exact. Most whole numbers are NOT perfect squares, so most square roots are genuine decimal approximations, not values to be rounded to a 'nicer' whole number.

MisconceptionTo convert to standard form, just count the total number of digits in the number.

Why it happens: Students confuse 'how many digits' with 'how many places the decimal point moves to leave exactly 1 digit before it', which are not the same count once trailing zeros are involved.

How to address it: Ignore digit count. Instead, find where the decimal point needs to land so exactly 1 non-zero digit sits before it, then count how many places it moved from its original position (at the end of a whole number). That count is n.

MisconceptionA negative exponent means the number itself is negative.

Why it happens: Students see a minus sign in ' $10^{-3}$ ' and read it as a sign on the whole number, rather than a direction for the decimal point.

How to address it: A negative exponent means a SMALL number (less than 1), not a negative one: 3 x $10^{-3}$ = 0.003, which is small and positive. The exponent's sign says which way the decimal point moves, not whether the number is negative.

Do it together

Guided practice (with answers)

1. Evaluate $5^{3}$ .
Answer: 125, because 5 x 5 x 5 = 125.
2. Find $\sqrt{64}$ . Is your answer exact?
Answer: 8, exact, because 8 x 8 = 64.
3. Find $\sqrt{20}$ , correct to 2 decimal places. Is your answer exact?
Answer: 4.47, a decimal approximation, because 20 is not a perfect square.
4. Write 45,000 in standard form.
Answer: 4.5 x $10^{4}$ , because the decimal point moves 4 places left.
5. Write 2 x $10^{-4}$ as an ordinary number.
Answer: 0.0002, moving the decimal point 4 places left (a small number).
6. Which is larger, 3 x $10^{5}$ or 8 x $10^{4}$ ?
Answer: 3 x $10^{5}$ , because its exponent (5) is larger than 8 x $10^{4}$ 's exponent (4).

On their own

Independent practice worksheets

Practise powers, roots and standard form with computed, never-wrong answer keys.

Year 9 powers & roots worksheetevaluating powers and finding exact/approximate roots, the core skill from section 1 Year 9 standard form conversion worksheetconverting between ordinary numbers and standard form, the core skill from section 2 Year 9 comparing standard form worksheetcomparing, ordering and validity-checking standard-form numbers, the core skill from section 3

Reach every student

Differentiation

Support

Keep a printed list of the first 12 square numbers and first 6 cube numbers visible so 'is this a perfect square?' is a quick lookup, not a guess.
For standard form, physically count decimal-point moves on paper with an arrow for every place, rather than trying to do it mentally.
Practise converting 'friendly' powers of 10 (10, 100, 1000) before mixing in mantissas with decimals.
When comparing standard-form numbers, always underline the exponent first and compare those before looking at anything else.

Extension

Investigate higher roots (4th root, 5th root) and connect them to the corresponding powers of 2, 3, 4, 5 named in the curriculum.
Multiply 2 numbers in standard form (multiply the mantissas, add the exponents), adjusting the mantissa back into the 1 to 10 range if the product is 10 or more.
Research a real scientific measurement (a distance in astronomy, or a size in biology) and justify why standard form is the practical choice over the ordinary number.
Explore why $\sqrt{2}$ is called irrational, and connect it to the never-terminating, never-repeating decimal a calculator shows.

Check it stuck

Assessment: exit ticket

A three-question exit ticket sampling powers, standard-form conversion, and standard-form comparison.

1. Evaluate $4^{3}$ .
Answer: 64, because 4 x 4 x 4 = 64.
2. Write 520,000 in standard form.
Answer: 5.2 x $10^{5}$ , because the decimal point moves 5 places left.
3. Which is larger, 7 x $10^{-3}$ or 2 x $10^{-2}$ ?
Answer: 2 x $10^{-2}$ , because its exponent (-2) is larger than 7 x $10^{-3}$ 's exponent (-3).

For the teacher

Teacher notes and timings

Rough timing across 3 lessons: Lesson 1 powers and roots (section 1), Lesson 2 standard-form conversion (section 2), Lesson 3 comparing/ordering standard form plus the exit ticket (section 3).
Every worksheet answer for standard form is generated by shifting DIGITS through a string, never by multiplying floats, so a value like 1 x $10^{-7}$ is always exact, matching this site's 'never wrong' answer-key policy.
Language to keep repeating: check 'is it a perfect square/cube?' before claiming an exact root; compare the EXPONENT first in standard form; a negative exponent means small, not negative.
The functionGraph figure in section 1 deliberately plots the square numbers as (side length, area) so students connect 'squaring' to the everyday idea of a square's area, not just an abstract exponent rule.
Use Student view to project this lesson. Print saves the full teacher unit, including answers and teacher notes; use the linked independent-practice worksheets for student handouts.

All teaching units Make a worksheet