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
- 25 parts to scroll through
- 4 quick checks
- Marked quiz at the end
- Gentle pace: short sittings with pauses
Words to know
Scroll down — the lesson carries on below
Watch & discover
Part 1 of 25
Visual introductionPicture this
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.
In a nutshell
Interpret and compare numbers in standard form A x 10^n
What you already know
Part 2 of 25
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.
Key words
Part 3 of 25
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.
Reset break
Part 4 of 25
Pause
That's sitting 1 of 6 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.
Learning cycle
Part 5 of 25
Learning cycle 1 of 2
Writing numbers in standard form
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 6 of 25
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.
Explore the idea
Part 7 of 25
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.
Reset break
Part 8 of 25
Pause
That's sitting 2 of 6 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.
Quick check
Part 9 of 25
Quick check
Part 10 of 25
Learning cycle
Part 11 of 25
Learning cycle 2 of 2
Comparing and calculating
A short piece of teaching, then a check to make sure it has landed.
Reset break
Part 12 of 25
Pause
That's sitting 3 of 6 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.
Explore the idea
Part 13 of 25
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.
Explore the idea
Part 14 of 25
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.
Quick check
Part 15 of 25
Reset break
Part 16 of 25
Pause
That's sitting 4 of 6 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.
Quick check
Part 17 of 25
Common mix-ups
Part 18 of 25
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.
Challenge round
Part 19 of 25
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
Reset break
Part 20 of 25
Pause
That's sitting 5 of 6 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.
Challenge round
Part 21 of 25
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.
Challenge round
Part 22 of 25
Game · Recall cards
Write 52 000 in standard form.
Card 1 of 4
Exit quiz
Part 23 of 25
Exit quiz
Show what you've learned
6 questions, marked together at the end. Nothing is timed.
1. Standard form requires A to be...
2. 0.00081 in standard form is...
3. 3.7 x 10^6 in ordinary form is...
4. (9 x 10^8) ÷ (3 x 10^2) =
5. How many orders of magnitude bigger is 10^9 than 10^6?
6. 0.5 x 10^7 written correctly in standard form is...
Mastery quiz
Part 24 of 25
Marked quiz
End of lesson quiz: Standard form: very big and very small
4 questions, marked with the reasoning shown. No timer.
1. Write 52 000 in standard form.
2. Write 4.6 x 10^-3 in ordinary form.
3. Which is larger?
4. (4 x 10^3) x (2 x 10^6) =
Lesson round-up
Part 25 of 25
Lesson round-up
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Luna: 0 out of 4 on the practice checks. Only if you feel up to it — one more?
Ask LunaPart 1 of 25 · Watch & discover