KS3 · Maths

Ordering positive and negative fractions and decimals

Which is bigger: −8 or −6? And 0.15 or 0.2? If your gut said −8 and 0.15, you're in good company — and wrong twice. Both look bigger. Neither is.

Start here

Where does each number live?

First read the scale: this line runs from −1 to 1, and every small tick is 0.1. Now drag each number to where you think it lives, then check your line.

Place −3/4Place −0.7Place −0.2Place 0.2Place 3/5

Negative numbers

−8 or −6: which is greater?

One freezer is at −8 °C. Another is at −6 °C.

Which is greater, −8 or −6? Pick the idea closest to what you think right now.

−8 is greater, because 8 is bigger than 6.Neither. They're both below zero, so they're both just 'less than nothing'.−6 is greater, because it is closer to zero.−6 is greater — and by the same rule, 7 is smaller than −2, because 7 is further from zero.

How sure are you?

I'm sureI think soI'm guessing

−8 is greater, because 8 is bigger than 6.

This is a common misconception

8 is bigger than 6 — but −8 isn't 8. The minus sign puts it 8 steps to the LEFT of zero, and −6 is only 6 steps left. The distance from zero is called the absolute value: −8 has absolute value 8, −6 has absolute value 6. For negative numbers, the bigger the absolute value, the further left the number sits, so the smaller it is. −8 < −6. The freezer at −8 °C is the colder one.

Neither. They're both below zero, so they're both just 'less than nothing'.

This is a common misconception

Being below zero doesn't make two numbers the same, any more than 5 and 9 are the same because they're both above zero. Each negative has its own place: −6 is 6 steps left of zero, −8 is 8 steps left. Different places, different values — and the one further left, −8, is smaller.

−6 is greater, because it is closer to zero.

This matches the evidence

Yes. −6 is 6 away from zero and −8 is 8 away, so −6 sits further right. Further right is greater: −6 > −8. Temperatures agree — −6 °C is warmer than −8 °C.

−6 is greater — and by the same rule, 7 is smaller than −2, because 7 is further from zero.

This is a common misconception

The first half is spot on, but the rule has a boundary. 'Bigger distance from zero means smaller' only works when you compare negatives with each other. 7 is on the right of zero and −2 is on the left, and anything right of zero is greater than anything left of it. So −2 < 7.

Absolute value is distance from zero. Comparing two negatives? The bigger distance wins the title of smaller number: −8 < −6. Comparing a negative with a positive? The positive is always greater.

Decimals

Spot the slip

Put these decimals in ascending order: 0.2, 0.15, 0.09, 0.3

Sam's answer — which line goes wrong?

Ascending means smallest to largest.
0.09 has 0 tenths and the others all have at least 1 tenth, so 0.09 is the smallest.
0.15 has more digits than 0.2, so 0.15 is bigger than 0.2.
0.3 has 3 tenths, the most tenths of all, so 0.3 is the largest.
Order: 0.09, 0.2, 0.15, 0.3

Two shortcuts that point opposite ways

Same denominatorvsSame numerator

Read down each column: what is the same, what decides it, and which way the rule goes.

Focus

Example

Same denominator

11/17 and 13/17

Same numerator

11/17 and 11/19

The insight

Always check which part matches before you pick a rule.

What's the same

Same denominator

The size of each part: both are cut into seventeenths

Same numerator

The number of parts: both have 11

What decides it

Same denominator

The numerator: how many parts you have

Same numerator

The denominator: how big each part is

Which is greater?

Same denominator

13/17 > 11/17 — more parts of the same size

Same numerator

11/17 > 11/19 — the same number of bigger parts

When they don't match yet

Same denominator

Make a common denominator using the lowest common multiple: 2/3 = 10/15 and 3/5 = 9/15, so 3/5 < 2/3

Same numerator

Make a common numerator when that's quicker: 2/5 = 6/15 and 3/7 = 6/14, so 2/5 < 3/7

Choose your method

What's the quickest way to compare each pair?

Put each pair under the method that compares it most efficiently.

Still to sort

Make a common denominator (0)

One denominator is a multiple of the other, or their lowest common multiple is small.

Where the line is: If the NUMERATORS are the easy ones to match, a common numerator is quicker.

Make a common numerator (0)

One numerator is a multiple of the other, so the tops are easy to match.

Where the line is: Once the numerators match, the fraction with the bigger denominator is the smaller one.

Compare the gap to 1 (0)

Both fractions are the same number of parts short of a whole.

Where the line is: This only works when the number of parts missing is the same for both fractions.

Convert to decimals (0)

One number is already a decimal, or the fraction is a familiar half, quarter or fifth.

Where the line is: Familiar fractions such as 1/2 = 0.5, 3/4 = 0.75 and 2/5 = 0.4 need no working at all.

7 of 7 still to sort.

Look at each pair before you calculate anything. Which method does the pair practically hand you?

Your turn

Order a mixed list — you fill the gaps

Write these numbers in ascending order, as one inequality chain: 7/10, −0.35, −2/5, 0.65, 1/2

Split the list. Negatives: −0.35 and −2/5. Positives: 7/10, 0.65 and 1/2. Every negative will come before every positive.Sorting the signs first turns one hard problem into two easy ones.
missing step

Which line is step 2?

Convert −2/5 to a decimal: 2 ÷ 5 = 0.4, so −2/5 = −0.4.Convert −2/5 to a decimal: 5 ÷ 2 = 2.5, so −2/5 = −2.5.Convert −2/5 to a decimal: write the digits after the point, so −2/5 = −0.25.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Every number has exactly one place on the number line. Put it there, and the order reads itself: left to right, smallest to largest.

What you need to know

Every number has one place on the number line. Further right means greater; further left means smaller.
The absolute value of a number is its distance from zero, so 5 and −5 both have absolute value 5.
Decimals are compared digit by digit, left to right, using place value — never by how many digits they have.
Fractions can be compared with a common denominator, a common numerator, their gap from 1, or by converting them to decimals.
Ascending means smallest to largest; descending means largest to smallest. Write the result with =, ≠, <, >, ≤ or ≥.

The big picture

Every number — positive or negative, fraction or decimal — has one place on the number line, and further right always means greater. Negatives with a bigger distance from zero sit further left, decimals are compared digit by digit using place value, and fractions are compared with a common denominator, a common numerator, their gap from 1, or by turning them into decimals. Then the order is written with <, >, =, ≠, ≤ or ≥.

Key points

1Further right on the number line is always greater: −0.75 < −0.7 < 0.2 < 0.6.

2Always read a number line's scale before placing or reading anything on it.

3Comparing negatives: the one further from zero is smaller, so −10 < −5. Any negative is less than any positive: −2 < 7.

4Decimals: compare place-value columns from the left. 0.2 > 0.15, because 2 tenths beat 1 tenth.

5A fraction is a division: 2/5 = 2 ÷ 5 = 0.4. If the denominator is a factor of 10, 100 or 1000, write an equivalent fraction instead: 3/25 = 12/100 = 0.12.

6Same denominator: compare numerators (11/17 < 13/17). Same numerator: the bigger denominator is the smaller fraction (11/19 < 11/17).

7Mixed lists: split negatives from positives, convert to a common format where you need to, order each group, then write the chain with the right symbols.

.we-_R_18slubsnpfiv5ubrb_ [data-we-step]{display:flex !important}.we-_R_18slubsnpfiv5ubrb_ [data-we-solution]{display:block !important}.we-_R_18slubsnpfiv5ubrb_ [data-we-reveal]{display:none !important}

Worked example

Problem

Write these numbers in descending order: 0.3, −1/2, 2/5, −0.45, 1/4

⚠ Watch out

Judging a number by how it looks instead of where it sits. A longer decimal is not automatically bigger (0.15 < 0.2), and when two fractions share a numerator, the bigger denominator gives the SMALLER fraction (1/8 < 1/5).

🧠

Memory hook

Right is greater, colder is smaller. A thermometer turned on its side is a number line: −10 °C is colder than −5 °C, so −10 < −5.

✓

Check yourself

Without a calculator, explain why −0.9 < −0.4, why 0.4 < 0.45 and why 3/8 < 3/7. Your three reasons: further from zero, compare the hundredths, and eighths are smaller parts.

Flashcards

(15)

On a number line, what does 'further right' tell you about a number?

It is greater. Further right is always greater; further left is always smaller.

What is the absolute value of a number?

Its distance from zero. 5 and −5 both have an absolute value of 5.

Which is smaller: −8 or −6? Why?

−8. Among negatives, the larger the absolute value, the smaller the number, because it is further left of zero.

Is a negative number ever greater than a positive number?

No. Every negative is less than every positive, e.g. −2 < 7.

What do ascending and descending order mean?

Ascending: smallest to largest. Descending: largest to smallest.

What should you do before placing numbers on a number line?

Work out its scale: what each tick is worth, for example by dividing one unit into equal parts.

How do you compare two decimals?

Compare digits in the same place-value column, working from the left. The first column where they differ decides it.

How do you turn a fraction into a decimal by dividing?

Divide the numerator by the denominator: 3/8 = 3 ÷ 8 = 0.375.

When can you convert a fraction to a decimal without dividing?

When the denominator is a factor of 10, 100 or 1000: write an equivalent fraction over that number, e.g. 7/20 = 35/100 = 0.35.

Two fractions have the same denominator. How do you compare them?

Compare the numerators: more parts of the same size is bigger, e.g. 4/9 < 7/9.

Two fractions have the same numerator. Which is bigger?

The one with the SMALLER denominator, because its parts are bigger, e.g. 5/8 < 5/6.

How do you choose a common denominator for two fractions?

Use the lowest common multiple of the two denominators, e.g. 12 for quarters and sixths.

Why doesn't multiplying the top and bottom of a fraction by 3 change its value?

You are multiplying by 3/3, which equals 1. The fraction looks different but is the same size.

How can the gap from 1 compare 5/6 and 7/8?

Both are 1 part short of a whole. Eighths are smaller parts than sixths, so 7/8 is closer to 1: 5/6 < 7/8.

What do ≠, ≤ and ≥ mean?

≠: not equal to. ≤: less than or equal to. ≥: greater than or equal to.

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.

Learning with Lightbulb is opening soon

You can use this lesson now. Join the waitlist and we'll let you know when the full Lightbulb experience is ready.

Keep me posted

Where does each number live?

−8 or −6: which is greater?

Spot the slip

What's the quickest way to compare each pair?

Order a mixed list — you fill the gaps

More KS3 Maths topics