KS3 · Maths
Powers of 10 and exponential column headings
Writing one million means a 1 and six zeros, which is easy to miscount. So we shrink it to 10⁶. But what does that tiny 6 actually count?
Place-value headings on one ×10 axis
Tap any heading. Each step along the axis multiplies by 10; each step back divides by 10. (This axis runs small to large, the opposite way round to a place-value table.)
1/1000
thousandths heading
The thousandths heading, 1/1000: 1/100 divided by 10 once more. The pattern doesn't stop at the ones column.
Why 10⁰ = 1
Reason it through
Why is the ones column 10⁰, and why does that equal 1?
First link · your turn
Start at 1000. How many times has 1 been multiplied by 10 to make it?
Maths · Fractional headings
Which column is it in?
Sort each value into the column heading it belongs to.
Still to sort
Tenths (1/10) (0)
Values counted in tenths.
Hundredths (1/100) (0)
Values counted in hundredths.
Thousandths (1/1000) (0)
Values counted in thousandths.
WHAT YOU'VE LEARNED
A quick recap of today's lesson.
the column headings, from a million down to a thousandth
What you need to know
- A power counts how many 10s are multiplied together: 10³ = 10 × 10 × 10 = 1000.
- The whole-number column headings are 10⁰ = 1, 10¹ = 10, 10² = 100, 10³ = 1000, 10⁴ = 10 000, 10⁵ = 100 000 and 10⁶ = one million.
- 10⁰ = 1 because 1 is 1 not multiplied by 10 at all.
- Dividing by 10 past the ones column gives the fractional headings: 1/10 (tenths), 1/100 (hundredths) and 1/1000 (thousandths).
- A number's digits can be written as digit × heading, with a 0 holding every empty column.
The big picture
Every place-value column heading is ten times the one before it, so the whole-number headings are powers of ten: 1 = 10⁰, 10 = 10¹, 100 = 10², up to one million = 10⁶. Keep dividing by ten past the ones column and you reach the fractional headings 1/10, 1/100 and 1/1000, which also tell you which of two decimals is bigger.
Key points
Worked example
Problem
Write 100 000 as a power of 10, and show why it is not 10 × 5.
⚠ Watch out
Reading 10³ as 10 × 3 = 30. The power says how many 10s to multiply together: 10³ = 10 × 10 × 10 = 1000. The same goes for 2³ = 2 × 2 × 2 = 8, not 2 × 3 = 6.
Memory hook
Count the tens, not the times: the little number says how many 10s are multiplied together. The ones column has no tens multiplied in at all: 10⁰ = 1.
Check yourself
Write 10⁴ as a full number, then say why it isn't 40. (10⁴ = 10 × 10 × 10 × 10 = 10 000: the 4 counts the 10s.)
Flashcards
(13)What does the power in 10ⁿ count?
What is 10⁰, and why?
Which column is 10¹?
Write one million as a power of 10.
Why write 10⁶ instead of 1 000 000?
Moving one heading further from the ones column towards the tenths, what happens to the value?
Name the three fractional column headings after the ones column.
When is a number in fractional form?
In 0.352, what are the values of the 3, the 5 and the 2 as fractions?
Which fraction has the same value as 0.01?
Write 8352 using powers of ten.
What job does the 0 do in 827 006?
Which is greater, 3/10 or 3/100, and what does that tell you about 0.3 and 0.03?
Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.
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