Using a Calculator

What you'll learn

How to enter roots, powers, brackets and fractions correctly.
How to copy all digits from the calculator display when asked.
How to round to decimal places and significant figures.
How to check angle mode for sin, cos and tan questions.

1. Reading the question and the display

An expression is a calculation written using numbers and operation symbols, such as add, subtract, square or square root.

The display is the calculator screen. If a question says “write down all the figures on your calculator display”, it means do not round yet.

Calculator display showing why every digit, including a final zero, must be copied exactly.

Example

Copying all the display figures

Type this expression into your calculator:
$4.7 + 2.35$
Press equals. The display shows 7.05.
If asked for all the figures, write 7.05 exactly, including the zero.

Tip

Decimal display

If your calculator gives a fraction but the question asks for display figures, press the decimal-change key, often labelled S-D or S⇔D.

2. Square roots and powers

A square root asks: “What number was squared to get this?” The symbol $1.44\sqrt{1.44}$ means the square root of 1.44.

Square root reverses squaring, so 1.2 squared gives 1.44 and the square root of 1.44 gives 1.2.

A power tells you repeated multiplication. For example, $1.35^2$ means $1.35 \times 1.35$ . The little 2 is called an index.

Definition

Square root and square

$1.44=1.2\sqrt{1.44}=1.2$ because $1.2 \times 1.2=1.44$ .
$1.45^2$ means 1.45 squared.

Example

Using square root and square buttons

To find $1.21\sqrt{1.21}$ , press the square root key, enter 1.21, then press equals.
The display shows 1.1, so $1.21=1.1\sqrt{1.21}=1.1$ .
To find $1.45^2$ , enter 1.45 and use the square button, or enter power 2.
The display shows 2.1025, so $1.45^2=2.1025$ .

3. Brackets and the order of operations

Brackets show which part of a calculation must be done first.

The order of operations is the rule calculators follow: brackets first, then powers and roots, then multiplication and division, then addition and subtraction.

Key Idea

Type it as it is written

Brackets and fraction templates tell your calculator which parts belong together. If in doubt, add brackets around the top, bottom, or anything being squared.

Example

A bracket being squared

Work out $3.20-0.45)^2-5.18$ .

The brackets show that 3.20 − 0.45 is calculated first, then the whole result is squared.

Type the full expression, including the brackets:
$3.20 - 0.45)^2 - 5.18$
The bracket is worked out first:
$3.20 - 0.45 = 2.75$
Then that bracket answer is squared:
$2.75^2 = 7.5625$
Finally subtract 5.18:
$7.5625 - 5.18 = 2.3825$

Common Mistake

Squaring only one number

For $3.20-0.45)^2$ , the whole bracket is squared. If you type $3.20-0.45^2$ , only 0.45 is squared, which gives a different answer.

4. Fractions on a calculator

The numerator is the top of a fraction. The denominator is the bottom of a fraction. The fraction bar means “divide the top by the bottom”.

$A fraction has a numerator on top, a denominator on the bottom, and the fraction bar means divide.$

For longer fractions, use the fraction template if your calculator has one.

Example

A fraction with a square in the numerator

Work out $3.18+2.622.07\frac{3.18+2.6^2}{2.07}$ .

The whole numerator 3.18 + 2.6² must be kept together above the denominator 2.07.

The numerator is the top part:
$3.18 + 2.6^2$
Work out the numerator:
$3.18 + 2.6^2 = 9.94$
Now divide by the denominator:
$9.942.07=4.801932367…\frac{9.94}{2.07}=4.801932367\ldots$
If your calculator display shows 4.801932367, copy exactly 4.801932367.

Tip

Use brackets without a fraction key

If you do not use a fraction template, put the whole numerator in brackets before dividing. This stops the calculator dividing only the last number.

5. Square root of a whole calculation

Sometimes the square root sign covers a whole fraction or a whole bracket. Everything under the root sign must be calculated before taking the square root.

Example

Square root of a fraction

Work out $18.6−2.44.1+2.2\sqrt{\frac{18.6-2.4}{4.1+2.2}}$ .

$The square root sign covers the whole fraction, so both the top and bottom are inside the root.$

Work out the top and bottom separately:
$18.6−2.4=16.24.1+2.2=6.3\begin{aligned} 18.6 - 2.4 &= 16.2\\ 4.1 + 2.2 &= 6.3 \end{aligned}$
Put these into the fraction:
$16.26.3=2.571428571…\frac{16.2}{6.3}=2.571428571\ldots$
Take the square root of the whole fraction:
$2.571428571…=1.603567451…\sqrt{2.571428571\ldots}=1.603567451\ldots$
If asked for all the figures, write every digit shown on your calculator display.

Common Mistake

Rooting only the top

In $18.6−2.44.1+2.2\sqrt{\frac{18.6-2.4}{4.1+2.2}}$ , the square root is around the whole fraction, not just the top line.

6. Rounding calculator answers

A decimal place is a digit after the decimal point. For example, 2 decimal places means two digits after the decimal point.

A significant figure is a counting digit starting from the first non-zero digit. For example, in 0.587, the first significant figure is 5.

Example

Rounding the display

Suppose your calculator display is 0.587104236.

For 2 decimal places, keep 0.58 and use the next digit, 7, to round up.

For 2 decimal places, keep the first two digits after the decimal point: 0.58.
The next digit is 7, so round up to 0.59.
For 2 significant figures, start counting at the first non-zero digit, which is 5.
Keep 5 and 8, then look at the next digit, 7, so the answer is 0.59.

Common Mistake

Dropping a final zero

If an answer to 2 decimal places is 1.60, write 1.60, not 1.6. The zero shows you have rounded to 2 decimal places.

7. Angle buttons: sin, cos and tan

Some calculator questions use $sin⁡\sin$ , $cos⁡\cos$ and $tan⁡\tan$ . These are trigonometric functions, which are calculator rules connected to angles.

A degree is a unit for measuring angles. In GCSE questions, angles are usually in degrees.

Common Mistake

Radians mode

If you are using sin, cos or tan and your calculator says RAD, your answer will be wrong for degree questions. Change it to DEG mode.

GCSE trigonometry questions in degrees need the calculator set to DEG mode, not RAD mode.

Example

Using sin and cos

Work out $sin⁡30∘+cos⁡60∘cos⁡30∘−sin⁡20∘\frac{\sin 30^\circ+\cos 60^\circ}{\cos 30^\circ-\sin 20^\circ}$ , then round to 2 decimal places.

Check your calculator is in DEG mode.
Type the fraction carefully:
$sin⁡30∘+cos⁡60∘cos⁡30∘−sin⁡20∘\frac{\sin 30^\circ+\cos 60^\circ}{\cos 30^\circ-\sin 20^\circ}$
Press equals. A common display is 1.908377034.
To round to 2 decimal places, look at the third decimal digit, which is 8.
The answer to 2 decimal places is 1.91.

Exam technique

In the exam

Type the calculation exactly as written, using brackets or fraction templates for long tops and bottoms.
If the question says “all the figures”, copy every digit on your display before rounding.
Only round at the end, and keep final zeros for decimal places, such as 1.60.

Self review

Check yourself

Did you put the whole numerator over the denominator, not just the last number?
Can you explain the difference between 2 decimal places and 2 significant figures?
Is your calculator in DEG mode when using sin, cos or tan with angles?

Using a Calculator

What you'll learn

How to enter roots, powers, brackets and fractions correctly.
How to copy all digits from the calculator display when asked.
How to round to decimal places and significant figures.
How to check angle mode for sin, cos and tan questions.

1. Reading the question and the display

An expression is a calculation written using numbers and operation symbols, such as add, subtract, square or square root.

The display is the calculator screen. If a question says “write down all the figures on your calculator display”, it means do not round yet.

Calculator display showing why every digit, including a final zero, must be copied exactly.

Example

Copying all the display figures

Type this expression into your calculator:
$4.7 + 2.35$
Press equals. The display shows 7.05.
If asked for all the figures, write 7.05 exactly, including the zero.

Tip

Decimal display

If your calculator gives a fraction but the question asks for display figures, press the decimal-change key, often labelled S-D or S⇔D.

2. Square roots and powers

A square root asks: “What number was squared to get this?” The symbol $1.44\sqrt{1.44}$ means the square root of 1.44.

Square root reverses squaring, so 1.2 squared gives 1.44 and the square root of 1.44 gives 1.2.

A power tells you repeated multiplication. For example, $1.35^2$ means $1.35 \times 1.35$ . The little 2 is called an index.

Definition

Square root and square

$1.44=1.2\sqrt{1.44}=1.2$ because $1.2 \times 1.2=1.44$ .
$1.45^2$ means 1.45 squared.

Example

Using square root and square buttons

To find $1.21\sqrt{1.21}$ , press the square root key, enter 1.21, then press equals.
The display shows 1.1, so $1.21=1.1\sqrt{1.21}=1.1$ .
To find $1.45^2$ , enter 1.45 and use the square button, or enter power 2.
The display shows 2.1025, so $1.45^2=2.1025$ .

3. Brackets and the order of operations

Brackets show which part of a calculation must be done first.

The order of operations is the rule calculators follow: brackets first, then powers and roots, then multiplication and division, then addition and subtraction.

Key Idea

Type it as it is written

Brackets and fraction templates tell your calculator which parts belong together. If in doubt, add brackets around the top, bottom, or anything being squared.

Example

A bracket being squared

Work out $3.20-0.45)^2-5.18$ .

The brackets show that 3.20 − 0.45 is calculated first, then the whole result is squared.

Type the full expression, including the brackets:
$3.20 - 0.45)^2 - 5.18$
The bracket is worked out first:
$3.20 - 0.45 = 2.75$
Then that bracket answer is squared:
$2.75^2 = 7.5625$
Finally subtract 5.18:
$7.5625 - 5.18 = 2.3825$

Common Mistake

Squaring only one number

For $3.20-0.45)^2$ , the whole bracket is squared. If you type $3.20-0.45^2$ , only 0.45 is squared, which gives a different answer.

4. Fractions on a calculator

The numerator is the top of a fraction. The denominator is the bottom of a fraction. The fraction bar means “divide the top by the bottom”.

$A fraction has a numerator on top, a denominator on the bottom, and the fraction bar means divide.$

For longer fractions, use the fraction template if your calculator has one.

Example

A fraction with a square in the numerator

Work out $3.18+2.622.07\frac{3.18+2.6^2}{2.07}$ .

The whole numerator 3.18 + 2.6² must be kept together above the denominator 2.07.

The numerator is the top part:
$3.18 + 2.6^2$
Work out the numerator:
$3.18 + 2.6^2 = 9.94$
Now divide by the denominator:
$9.942.07=4.801932367…\frac{9.94}{2.07}=4.801932367\ldots$
If your calculator display shows 4.801932367, copy exactly 4.801932367.

Tip

Use brackets without a fraction key

If you do not use a fraction template, put the whole numerator in brackets before dividing. This stops the calculator dividing only the last number.

5. Square root of a whole calculation

Sometimes the square root sign covers a whole fraction or a whole bracket. Everything under the root sign must be calculated before taking the square root.

Example

Square root of a fraction

Work out $18.6−2.44.1+2.2\sqrt{\frac{18.6-2.4}{4.1+2.2}}$ .

$The square root sign covers the whole fraction, so both the top and bottom are inside the root.$

Work out the top and bottom separately:
$18.6−2.4=16.24.1+2.2=6.3\begin{aligned} 18.6 - 2.4 &= 16.2\\ 4.1 + 2.2 &= 6.3 \end{aligned}$
Put these into the fraction:
$16.26.3=2.571428571…\frac{16.2}{6.3}=2.571428571\ldots$
Take the square root of the whole fraction:
$2.571428571…=1.603567451…\sqrt{2.571428571\ldots}=1.603567451\ldots$
If asked for all the figures, write every digit shown on your calculator display.

Common Mistake

Rooting only the top

In $18.6−2.44.1+2.2\sqrt{\frac{18.6-2.4}{4.1+2.2}}$ , the square root is around the whole fraction, not just the top line.

6. Rounding calculator answers

A decimal place is a digit after the decimal point. For example, 2 decimal places means two digits after the decimal point.

A significant figure is a counting digit starting from the first non-zero digit. For example, in 0.587, the first significant figure is 5.

Example

Rounding the display

Suppose your calculator display is 0.587104236.

For 2 decimal places, keep 0.58 and use the next digit, 7, to round up.

For 2 decimal places, keep the first two digits after the decimal point: 0.58.
The next digit is 7, so round up to 0.59.
For 2 significant figures, start counting at the first non-zero digit, which is 5.
Keep 5 and 8, then look at the next digit, 7, so the answer is 0.59.

Common Mistake

Dropping a final zero

If an answer to 2 decimal places is 1.60, write 1.60, not 1.6. The zero shows you have rounded to 2 decimal places.

7. Angle buttons: sin, cos and tan

Some calculator questions use $sin⁡\sin$ , $cos⁡\cos$ and $tan⁡\tan$ . These are trigonometric functions, which are calculator rules connected to angles.

A degree is a unit for measuring angles. In GCSE questions, angles are usually in degrees.

Common Mistake

Radians mode

If you are using sin, cos or tan and your calculator says RAD, your answer will be wrong for degree questions. Change it to DEG mode.

GCSE trigonometry questions in degrees need the calculator set to DEG mode, not RAD mode.

Example

Using sin and cos

Work out $sin⁡30∘+cos⁡60∘cos⁡30∘−sin⁡20∘\frac{\sin 30^\circ+\cos 60^\circ}{\cos 30^\circ-\sin 20^\circ}$ , then round to 2 decimal places.

Check your calculator is in DEG mode.
Type the fraction carefully:
$sin⁡30∘+cos⁡60∘cos⁡30∘−sin⁡20∘\frac{\sin 30^\circ+\cos 60^\circ}{\cos 30^\circ-\sin 20^\circ}$
Press equals. A common display is 1.908377034.
To round to 2 decimal places, look at the third decimal digit, which is 8.
The answer to 2 decimal places is 1.91.

Exam technique

In the exam

Type the calculation exactly as written, using brackets or fraction templates for long tops and bottoms.
If the question says “all the figures”, copy every digit on your display before rounding.
Only round at the end, and keep final zeros for decimal places, such as 1.60.

Self review

Check yourself

Did you put the whole numerator over the denominator, not just the last number?
Can you explain the difference between 2 decimal places and 2 significant figures?
Is your calculator in DEG mode when using sin, cos or tan with angles?

Using a Calculator

What you'll learn

1. Reading the question and the display

2. Square roots and powers

3. Brackets and the order of operations

4. Fractions on a calculator

5. Square root of a whole calculation

6. Rounding calculator answers

7. Angle buttons: sin, cos and tan

Recap questions

You've reached the end

Using a Calculator

What you'll learn

1. Reading the question and the display

2. Square roots and powers

3. Brackets and the order of operations

4. Fractions on a calculator

5. Square root of a whole calculation

6. Rounding calculator answers

7. Angle buttons: sin, cos and tan

Recap questions

You've reached the end

Using a Calculator

What you'll learn

1. Reading the question and the display

2. Square roots and powers

3. Brackets and the order of operations

4. Fractions on a calculator

5. Square root of a whole calculation

6. Rounding calculator answers

7. Angle buttons: sin, cos and tan

Recap questions

You've reached the end

Using a Calculator Revision Guide

Using a Calculator

What you'll learn

1. Reading the question and the display

2. Square roots and powers

3. Brackets and the order of operations

4. Fractions on a calculator

5. Square root of a whole calculation

6. Rounding calculator answers

7. Angle buttons: sin, cos and tan

Recap questions

You've reached the end

Using a Calculator Revision Guide