Ordering numbers and using inequality symbols

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Every number has a place on the number line, and that place tells you its size.

Numbers get bigger as you move to the right and smaller as you move to the left.

Ordering is just reading the line.

This section is examined on both tiers, and a quick ordering question often opens a Foundation paper.

You need to be able to "order positive and negative integers, decimals and fractions; use the symbols =, ≠, <, >, ≤, ≥".

Negative numbers on the number line

Zero sits in the middle.

Positive numbers go to the right of it and negative numbers go to the left, like a mirror.

On a thermometer, −13 °C is colder than −6 °C, because −13 is further below zero.

So −13 is the smaller number, even though 13 is bigger than 6.

A number line from −15 to 6 with dots at −13, −6, −1, 0, 2 and 5. The further left a number is, the smaller it is, so −13 is the smallest and 5 the largest.

A quick rule: the further left, the smaller.

Every negative number is smaller than zero, and zero is smaller than every positive number.

Between two negative numbers, the one with the bigger digits is the smaller number: −20 < −2.

Ordering decimals

Decimals are ordered by place value, one column at a time, starting from the left.

Line the numbers up by their decimal points and compare the units, then the tenths, then the hundredths, and so on.

The first column where the digits differ decides the order.

Number

Units

Tenths

Hundredths

Thousandths

0.62

0.6

0.602

0.26

0.612

Writing zeros in the empty columns does not change a number (0.6 = 0.600) but it makes the comparison fair.

Now 0.26 is clearly smallest (2 tenths).

The others all have 6 tenths, so look at the hundredths: 0, 0, 1, 2.

Then 0.6 and 0.602 differ in the thousandths.

So the order is 0.26, 0.6, 0.602, 0.612, 0.62.

A number line from 0.58 to 0.64 in steps of 0.01. Dots mark 0.6, then 0.602 just to its right, then 0.612 and 0.62, so 0.602 is smaller than 0.62.

Why a longer decimal is not always bigger. 0.602 has more digits than 0.62, but it is smaller.

The digits after the point are tenths, hundredths and thousandths, not a whole number.

0.62 is 62 hundredths, which is 620 thousandths; 0.602 is only 602 thousandths.

Ordering fractions

To compare fractions, make them comparable.

There are two good ways.

A common denominator. Write each fraction with the same denominator, then compare the numerators. Same-sized pieces can be counted.

Change to decimals. Divide each numerator by its denominator. This is quick on a calculator paper, and some fractions you should know already: $\frac{1}{4} = 0.25$ , $\frac{1}{2} = 0.5$ , $\frac{3}{4} = 0.75$ , $\frac{1}{5} = 0.2$ , $\frac{1}{8} = 0.125$ .

For unit fractions (numerator 1) there is a shortcut you can reason out: $\frac{1}{9}$ is smaller than $\frac{1}{7}$ , because cutting a whole into 9 pieces gives smaller pieces than cutting it into 7.

Recent papers have also mixed fractions, decimals and a percentage in one list: change them all to decimals first.

The six symbols

Symbol

Meaning

Example

is equal to

$0.5 = \frac{1}{2}$

≠

is not equal to

$0.3 \ne \frac{1}{3}$

is less than

$-8 < -3$

is greater than

$0.7 > 0.67$

≤

is less than or equal to

$n \le 4$ means 4 or anything below it

≥

is greater than or equal to

$t \ge -2$ means −2 or anything above it

The open end of < and > always faces the bigger number.

Read $-2 \le n < 3$ in two halves: $n$ is at least −2, and $n$ is less than 3.

If $n$ is an integer, its possible values are −2, −1, 0, 1 and 2.

The 3 is not included, because the sign at that end is <, not ≤.

The number halfway between two numbers

The number halfway between two numbers is their total divided by 2, even when one of them is negative.

Halfway between −9 and 15 is $(-9 + 15) \div 2 = 6 \div 2 = 3$ .

Check it on the number line: from −9 up to 3 is 12 steps, and from 3 up to 15 is also 12 steps.

Key terms

Integer: a whole number. It can be positive, negative or zero, such as −7, 0 or 12.

Place value: the value a digit has because of its column; in 0.62 the 6 is worth 6 tenths.

Inequality: a statement that two values are not equal, written with <, >, ≤ or ≥.

Ascending order: smallest first. Descending order: largest first.

How the marks work

B is a mark for an answer, a fact or a drawing on its own; a B2 or B3 can give part credit.

M is a method mark: you earn it for a correct method carried out, even if you then slip with the arithmetic. P is a process mark: one for each correct step of a problem you have to plan yourself, written with its value.

A is the accuracy mark for the correct answer from that method or process. C is a communication mark, for a complete, correct statement with nothing contradicting it: a reason, a decision with its figures, or the last line of a Show that.

"dep" means a mark needs the mark before it; "ft" means a wrong earlier value carried forward correctly still earns marks; "oe" means any equivalent form is accepted.

Where working is not demanded, a correct answer alone usually scores full marks, but a wrong answer alone scores nothing, so always write the method.

Where the question says "You must show all your working." or "You must show how you get your answer.", an answer with no working scores 0.

Examiners look for method in your working when the answer line is wrong or empty, and a correct answer from clearly wrong working scores 0. With two methods and no answer chosen you get the lower mark, so cross out the one you do not want.

Wrong further work after a correct answer can lose the mark, and rounding too early costs the accuracy mark.

Worked example 1: ordering integers (non-calculator)

Write these temperatures in order.

Start with the lowest.

2 °C, −13 °C, 0 °C, 5 °C, −6 °C, −1 °C

Negatives first, the one furthest from zero first: −13, −6, −1

Then zero, then the positives: 0, 2, 5

Answer: −13 °C, −6 °C, −1 °C, 0 °C, 2 °C, 5 °C B1 (all six in the right order)

Count the numbers in your answer: six in the question, six in your list.

Leaving one out loses the mark.

Worked example 2: ordering decimals (non-calculator)

Put these numbers in order of size, smallest first.

0.74, 0.7, 0.047, 0.704, 0.4

Fill the gaps with zeros: 0.740, 0.700, 0.047, 0.704, 0.400

Compare as thousandths: 740, 700, 47, 704, 400

Answer: 0.047, 0.4, 0.7, 0.704, 0.74 B1

Once every number has three decimal places, the order is the same as ordering 47, 400, 700, 704 and 740.

Worked example 3: ordering fractions (non-calculator)

Write these fractions in order of size.

Start with the smallest.

$\frac{3}{5}$ , $\frac{2}{3}$ , $\frac{7}{10}$ , $\frac{5}{8}$

A common denominator of 5, 3, 10 and 8 is 120: $\frac{72}{120}$ , $\frac{80}{120}$ , $\frac{84}{120}$ , $\frac{75}{120}$ M1 (all four written with one common denominator, or all four as decimals 0.6, 0.666..., 0.7, 0.625)

Order the numerators: 72, 75, 80, 84

Answer: $\frac{3}{5}$ , $\frac{5}{8}$ , $\frac{2}{3}$ , $\frac{7}{10}$ A1

Give the answer in the fractions you were given, not in hundred-and-twentieths.

Worked example 4: the symbols, and integers in a range (non-calculator)

(a) Write <, > or = between each pair: −4 and −9; 0.3 and $\frac{3}{10}$ .

(b) $n$ is an integer and $-4 < n \le 1$ . Write down all the possible values of $n$ .

(a) −4 is to the right of −9, so −4 > −9; 0.3 is 3 tenths, so $0.3 = \frac{3}{10}$ B1 (both)

(b) −4 is not included (<); 1 is included (≤)

Answer: (b) −3, −2, −1, 0, 1 B2 (B1 for four correct values with no wrong ones, or all five with one wrong value)

A common slip is to include −4.

The circle at −4 on a number line would be open.

Worked example 5: halfway between (non-calculator)

Find the number that is halfway between −11 and 5.

$-11 + 5 = -6$ , and $-6 \div 2 = -3$ B1

Answer: −3

Check: from −11 to −3 is 8 steps, and from −3 to 5 is 8 steps.

Exam tip

Read which end the order starts from ("Start with the smallest" or "largest first"), copy every number into your answer and write them in the same form as the question.

An answer in the wrong order, with a number missing or a number changed into a decimal, scores 0 on a 1-mark item.

Common mistakes

Thinking −13 is bigger than −6. Picture the thermometer: −13 °C is colder. The bigger the digits after a minus sign, the smaller the number.

Thinking a longer decimal is bigger. 0.602 is smaller than 0.62. Fill the gaps with zeros and compare column by column.

Comparing fractions by their denominators alone. $\frac{5}{8}$ is not smaller than $\frac{2}{3}$ just because 8 is bigger than 3; you need a common denominator or decimals.

Mixing up < and ≤. $n < 3$ does not include 3; $n \le 3$ does.

Leaving a number out, or ordering the wrong way round. Count, and reread the instruction.

Check your understanding

1. Put these numbers in order of size, smallest first.

4, −7, 0, −2, 9, −10

..................................................

Total [1]

2. (Non-calculator)

Put these numbers in order of size, smallest first.

0.48, 0.408, 0.5, 0.084, 0.45

..................................................

Total [1]

3. (a) Write the correct symbol, < or >, between these two numbers.

−5 ......

−8

(1)

(b) $n$ is an integer.

$-1 \le n < 4$

Write down all the possible values of $n$ .

..................................................

(2)

Total [3]

4. Find the number that is halfway between −14 and 6.

..................

Total [1]

5. Ali says,

"0.35 is bigger than 0.4 because 35 is bigger than 4."

Is Ali correct?

Give a reason for your answer.

Total [1]

6. (Non-calculator)

Write these fractions in order of size.

Start with the smallest.

$\frac{3}{4}$ , $\frac{5}{6}$ , $\frac{7}{12}$ , $\frac{2}{3}$

..................................................

Total [2]

7. Here are four cards.

$\frac{1}{8}$ , 0.2, $\frac{1}{6}$ , 0.15

Write the cards in order of size.

Start with the largest.

..................................................

Total [2]

Answers

1. −10, −7, −2, 0, 4, 9 B1

2. 0.084, 0.408, 0.45, 0.48, 0.5 B1 (as thousandths: 84, 408, 450, 480, 500)

3. (a) −5 > −8 B1

(b) −1, 0, 1, 2, 3 B2 (B1 for four correct values with no wrong ones, or all five with one wrong value; 4 is not included)

4. −14 + 6 = −8, and −8 ÷ 2 = −4 B1

Answer: −4

5. No, because 0.35 has 3 tenths and 0.4 has 4 tenths, so 0.4 is bigger (0.4 = 0.40, and 40 hundredths is more than 35 hundredths). C1 (a decision with a reason about place value; "No, 0.4 is bigger" with no reason is not enough)

6. $\frac{9}{12}$ , $\frac{10}{12}$ , $\frac{7}{12}$ , $\frac{8}{12}$ M1 (all four with a common denominator, or as decimals)

$\frac{7}{12}$ , $\frac{2}{3}$ , $\frac{3}{4}$ , $\frac{5}{6}$ A1

7. As decimals: $\frac{1}{8} = 0.125$ , 0.2, $\frac{1}{6} = 0.1666...$ , 0.15 M1 (at least two fractions changed to decimals correctly)

0.2, $\frac{1}{6}$ , 0.15, $\frac{1}{8}$ A1 (largest first, written as the cards)

Ordering numbers and using inequality symbols

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