KS3 · Maths

Rounding to significant figures

7,003,729 and 0.00300858 are full of zeros that look empty. Some are. Others are doing real work — and that's the whole trick.

Start here

Which thousand is it closest to?

Each of these numbers has its first significant figure in the thousands column. Drag each one to where it belongs on the line. The small unlabelled ticks are the halfway marks: 2,500, 4,500 and so on. When you check, look at which whole thousand each number is closer to.

Place 2,795Place 6,210Place 4,567Place 8,500

Which digits count?

Significant or not?

For each highlighted digit, decide: is it a significant figure or not? Commit before you read the reason.

Still to sort

Significant (0)

The first non-zero digit, and every digit after it.

Where the line is: The line falls at the first non-zero digit. Zeros before it aren't significant. Every digit after it is, zeros included.

Not significant (0)

Leading zeros: zeros that come before the first non-zero digit.

Where the line is: A leading zero can still be needed to hold the other digits in their columns. That doesn't make it significant.

7 of 7 still to sort.

Everything depends on finding the first significant figure. Here's how to spot it, and which zeros count after it.

Keep the size

What is 87,392 to 1 significant figure?

The first significant figure of 87,392 is the 8, in the ten thousands column. So rounding to 1 significant figure means rounding to the nearest ten thousand.

Which answer is closest to what you'd write?

90,000980,000

How sure are you?

I'm sureI think soI'm guessing

90,000

This matches the evidence

Yes. 87,392 sits between 80,000 and 90,000 and it's past the halfway mark of 85,000, so it's closer to 90,000. Here's a quicker way to decide without drawing a line: look at the digit just to the right of the one you're rounding. Here that's the 7. If it's less than 5, the digit stays the same. If it's 5 or more, the digit goes up by one. 7 is 5 or more, so the 8 becomes 9, and zeros fill the rest.

This is a common misconception

You rounded the 8 up to 9 correctly, but then the other digits disappeared. The 9 needs to stay in the ten thousands column, and the only way to keep it there is to write zeros in the columns after it. Those zeros are placeholders. Without them, 87,392 has shrunk to 9. This is a really common mistake.

80,000

This is a common misconception

You kept the 8 and swapped the rest for zeros, which cuts the number off instead of rounding it. 87,392 is past the halfway mark of 85,000, so it's closer to 90,000. The digit after the 8 is a 7, which is 5 or more, so the 8 goes up by one.

Round the digit, then keep the size: placeholder zeros hold every digit in its place value. 87,392 to 1 significant figure is 90,000, never 9.

Predict, then check

The digit after the 9 is a 6, so the 9 has to go up by one. But what is one more than 9 in that column?

What is 968 rounded to 1 significant figure?

More than one significant figure

Your turn to supply the steps

Round each number to 3 significant figures: 23,924, then 10.392, then 0.06076. The method is the same every time: find the first significant figure, count on to the 3rd, look at the next digit, round, then keep the place value.

23,924: the first significant figure is the 2, in the ten thousands column.
missing step

Which line is step 2?

Count on three significant figures: 2, 3, 9. The 3rd is the 9, in the hundreds column, so we round to the nearest hundred.Count three digits from the right: 9, 2, 4. So we round to the nearest thousand.Count on three significant figures: 2, 3, 9. The 3rd is the 9, so we round to the nearest ten.

Spot the slip

Where does this answer go wrong?

Round 7,003,729 to 4 significant figures.

A student's answer — which line goes wrong?

The first significant figure is the 7, in the millions column.
Count four significant figures: 7, 3, 7, 2. The 4th is the 2, in the tens column.
The next digit is 9, which is 5 or more, so the 2 goes up to 3.
Answer: 7,003,730

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

What you need to know

What significant figures are, and how to find the first significant figure of any number.
Why leading zeros aren't significant, but zeros after the first significant figure are.
How to round to 1 significant figure by rounding to the place value of the first significant figure.
How to round integers and decimals to any number of significant figures, keeping the place value with placeholders.

The big picture

Significant figures are the digits that contribute to a number's accuracy. The first significant figure is the first non-zero digit, and its column tells you roughly how big the number is, so rounding 4,567 to 1 significant figure is rounding to the nearest thousand: 5,000. For more significant figures, count on from the first one, zeros included, and round in the column of the last one you need. The next digit decides: under 5 stays, 5 or more goes up. Placeholder zeros keep every digit in its place.

Key points

1Significant figures are the digits in a number that contribute to its accuracy.

2The first significant figure is the first non-zero digit. It tells you most about the size of the number: in 123,456,789 it's the 1, in the hundred millions column.

3Leading zeros are not significant. In 0573 the first significant figure is the 5, and in 0.00483 it's the 4.

4Every digit after the first significant figure is significant, including zeros. In 203,568 the third significant figure is the 3.

5To round to a number of significant figures, round to the place value of the last significant figure you need. To 1 significant figure, 4,567 is 5,000, the same as rounding to the nearest thousand.

6Look at the next digit to the right: less than 5 and the digit stays the same, 5 or more and it goes up by one. Exactly halfway rounds up: 650 to 1 significant figure is 700.

7Use placeholder zeros to keep the place value: 87,392 to 1 significant figure is 90,000, not 9. For numbers smaller than 1, the leading zeros must still be written: 0.0068 to 1 significant figure is 0.007.

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Worked example

Problem

Round 145,508 to 2 significant figures.

⚠ Watch out

Starting the count too early on a number smaller than 1. In 0.0068 the zeros before the 6 are leading zeros, so the first significant figure is the 6, not the 0 at the front. To 1 significant figure, 0.0068 is 0.007, with the leading zeros still written in.

🧠

Memory hook

First non-zero, then count on, zeros and all. Round in that column, then fill the gaps with zeros so the number keeps its size.

✓

Check yourself

Without looking back: round 30,559 to 3 significant figures. Then explain, in one sentence, why the 0 counts as a significant figure.

Flashcards

(13)

What are significant figures?

The digits in a number that contribute to its accuracy.

How do you find the first significant figure?

It's the first non-zero digit. In 123,456,789 it's the 1, in the hundred millions column.

Why does the first significant figure matter most?

Its column tells you roughly how big the number is, even without the other digits.

Are leading zeros significant?

No. In 0.00483 the zeros before the 4 aren't significant, so the first significant figure is the 4.

Do zeros after the first significant figure count?

Yes, every digit after it counts, zeros included. In 203,568 the 0 is the second significant figure.

Rounding 4,567 to 1 significant figure is the same as rounding it to the nearest…?

Thousand, because its first significant figure is in the thousands column. The answer is 5,000.

How does a number line help you round?

Mark the multiples below and above the number and the midpoint between them. See which multiple the number is closer to.

What happens to a number that's exactly halfway?

It rounds up. 650 to 1 significant figure is 700.

The digit-to-the-right rule

Look at the digit just after the last significant figure you need. Less than 5: it stays the same. 5 or more: it goes up by one.

Why do you need placeholder zeros?

To keep every digit in its place value. 87,392 to 1 significant figure is 90,000; writing 9 is a really common mistake.

What happens when you round a 9 up?

It exchanges into the next column. 968 to 1 significant figure is 1,000.

Rounding a number smaller than 1: which zeros must you still write?

The leading zeros. They aren't significant, but they hold the place value: 0.123 to 1 significant figure is 0.1.

How do you round to 2 or more significant figures?

Round to the place value of the last significant figure you need. 547 to 2 significant figures is 550.

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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Which thousand is it closest to?

Significant or not?

What is 87,392 to 1 significant figure?

Your turn to supply the steps

Where does this answer go wrong?

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