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SavannahYear 8
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MathsYear 9 Maths: standard form, modelling and Pythagoras30 min★★★ difficulty

Standard form: very big and very small

The mass of an electron and the distance to the Sun cannot sit comfortably on the same page as ordinary numbers. Standard form writes both in the same compact way, so you can compare them at a glance.

Lesson overview

What you'll learn in this lesson

Interpret and compare numbers in standard form A x 10^n

Key learning points

  • • The rule behind the form
  • • Converting both ways
  • • Comparing sizes
  • • Multiplying and dividing

This lesson at a glance

  • 30 minutes
  • 19 parts to scroll through
  • 4 quick checks
  • Marked quiz at the end
  • Gentle pace: short sittings with pauses

Words to know

interpretnumbersstandardbehindconverting

Scroll down — the lesson carries on below

Part 1 of 19 · Discover5%
1

Watch & discover

Part 1 of 19

Watch first

Standard form: very big and very small

An English teaching video on this topic from Cognito. Watch it, then scroll down for the lesson.

In a nutshell

Interpret and compare numbers in standard form A x 10^n

2

What you already know

Part 2 of 19

Before we start

What you already know

You should already be confident with powers of ten, place value and multiplying or dividing by 10, 100 and 1000.

Recap: Expressions: collecting, expanding and factorising
3

Key words

Part 3 of 19

Key words

Words you'll need today

Standard form
A number written as A x 10^n, where A is at least 1 and less than 10.
Index
The power that ten is raised to.
Ordinary form
The number written out in full, without powers.
Order of magnitude
How many times bigger, in powers of ten.
4

Learning cycle

Part 4 of 19

Learning cycle 1 of 2

Writing numbers in standard form

A short piece of teaching, then a check to make sure it has landed.

5

Explore the idea

Part 5 of 19

Learn

The rule behind the form

A number in standard form looks like A x 10^n. A must be at least 1 and less than 10, so 43 x 10^3 is not yet in standard form even though it is a correct value; 4.3 x 10^4 is. The index n counts how far the decimal point has moved: positive for large numbers, negative for small ones. 0.00072 becomes 7.2 x 10^-4 because the point moves four places right to make 7.2.

6

Explore the idea

Part 6 of 19

Learn

Converting both ways

To go from standard form to ordinary form, move the decimal point n places: right for a positive index, left for a negative one, filling with zeros. 6.05 x 10^5 is 605 000. 3.1 x 10^-3 is 0.0031. Going the other way, place the point after the first non-zero digit and count how far it travelled. Counting digits carefully matters more than speed here.

7

Quick check

Part 7 of 19

Quick check

Write 52 000 in standard form.

8

Quick check

Part 8 of 19

Quick check

Write 4.6 x 10^-3 in ordinary form.

9

Reset break

Part 9 of 19

Pause

That's sitting 1 of 2 done

Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.

Stop here for now
10

Learning cycle

Part 10 of 19

Learning cycle 2 of 2

Comparing and calculating

A short piece of teaching, then a check to make sure it has landed.

11

Explore the idea

Part 11 of 19

Learn

Comparing sizes

Compare the index first, because it decides the size band. 9.9 x 10^6 is smaller than 1.2 x 10^7 even though 9.9 looks bigger, because the second number is in a higher band. Only when two indices match do you compare the A values. This is what people mean by an order of magnitude: 10^7 is one order of magnitude bigger than 10^6, so it is ten times larger.

12

Explore the idea

Part 12 of 19

Learn

Multiplying and dividing

To multiply, multiply the A values and add the indices: (3 x 10^4) x (2 x 10^5) = 6 x 10^9. To divide, divide the A values and subtract the indices. If the A value ends up outside the 1-to-10 range, adjust: 0.6 x 10^9 becomes 6 x 10^8. Adding and subtracting is different — the indices must be made equal first, which is why it is usually easier in ordinary form.

13

Quick check

Part 13 of 19

Quick check

Which is larger?

14

Quick check

Part 14 of 19

Quick check

(4 x 10^3) x (2 x 10^6) =

15

Common mix-ups

Part 15 of 19

Common mix-ups

Things people often get wrong

  • People often write 43 x 10^3 and call it standard form. The first part must be at least 1 and below 10.
  • People often think a negative index means a negative number. It means a small positive number, less than one.
16

Challenge round

Part 16 of 19

Game · Sort it

Which of these are true?

Drag each card into the right column. Tap a card first if dragging is fiddly.

True

Not true

17

Challenge round

Part 17 of 19

Game · Fill the gaps

Finish the sentences

Choose the word that belongs in each gap.

A number in ____ form looks like A x 10^n.

The index n counts how far the decimal point has moved: positive for large ____, negative for small ones.

18

Mastery quiz

Part 18 of 19

Marked quiz

End of lesson quiz: Standard form: very big and very small

6 questions, marked with the reasoning shown. No timer.

  1. 1. Standard form requires A to be...

  2. 2. 0.00081 in standard form is...

  3. 3. 3.7 x 10^6 in ordinary form is...

  4. 4. (9 x 10^8) ÷ (3 x 10^2) =

  5. 5. How many orders of magnitude bigger is 10^9 than 10^6?

  6. 6. 0.5 x 10^7 written correctly in standard form is...

19

Lesson round-up

Part 19 of 19

Lesson round-up

Ready when you are

Quiz score

Not sat

Games

Not played

Points this lesson

0

Best run

0 in a row

Luna: 0 out of 4 on the practice checks. Only if you feel up to it — one more?

Ask Luna

Do the work

Your written work for this lesson

This is the part you'd do in your book at school. Pick the pieces below, write here or on paper, and photograph anything you draw or work out.

Article

The cost of a school trip

A school is planning a trip to a wildlife park. The coach costs £480 however many people go. Entry costs £9.50 for each pupil and each adult goes free. The school says every pupil must pay the same amount, and the trip must not lose money.

If 60 pupils go, the coach works out at £8 each, so the total for each pupil is £17.50. If only 40 pupils go, the coach works out at £12 each, so the price rises to £21.50. The teacher explains that the coach is a fixed cost and the entry is a cost per person — which is why the price per pupil changes when the number changes.

The school decides to charge £20 and to use anything left over for a second trip later in the year.

Written for Good Learning Co.

Reading questions

  1. Work out the total cost of the trip if 50 pupils go.
  2. Show why the price per pupil falls when more pupils go, using numbers.
  3. At £20 each, how many pupils must go before the trip makes money? Show your working.

  • Working out

    Working out

    Do the practice questions on paper, showing every step — not just the answer. Mark them yourself and circle anything you got wrong.

    10–15 questions

  • Worked example, explained

    Writing

    Copy one question out and write underneath, in words, why each step works.

    1 question, 4–5 sentences

  • Problem in words

    Working out

    Write your own real-life problem that uses today's method, then solve it.

    1 problem + solution

Saved as you type.

Part 1 of 19 · Watch & discover

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