Number Operations & Calculator · Notes

1.Lesson overview

Syllabus focus

Cambridge IGCSE syllabus reference

1.5 Ordering
1.6 The four operations
1.14 Using a calculator

Edexcel IGCSE syllabus reference

1.1 Integers
1.11 Electronic calculators

AQA IGCSE syllabus reference

1.1 Structure and calculation

A correct numerical answer depends on operation order, signed-number notation and faithful calculator entry. Keep exact values through intermediate steps, then round once and use an estimate or inverse operation to check.

By the end of this lesson

Order quantities by magnitude and demonstrate familiarity with the symbols =, $\neq =$ , >, <, $⩾$ and $⩽$ .
Use the four operations for calculations with integers, fractions and decimals, including correct ordering of operations and use of brackets.
1 Use a calculator efficiently.
2 Enter values appropriately on a calculator.
3 Interpret the calculator display appropriately.

2.Learning map

Read this lesson in the order compare → calculate exactly → enter safely → estimate-check. That sequence prevents both order-of-operations errors and plausible-looking calculator slips.

Exact calculation workflow

Ordering and operation structure

Use the number classifications from the previous lesson when reasoning about negative values and fractions. Big idea. A calculator evaluates what you enter, not what you intended: brackets and unrounded values protect the…

Comparison and Ordering Symbols

There are six key symbols used to compare quantities. You must be completely familiar with each one.

Comparing Integers

Integers are whole numbers including negatives, zero, and positives. The number line is the best way to visualise their order: numbers increase from left to right.

Comparing Fractions

Comparing fractions requires a common basis. There are two main methods:

3.Core idea

Use the number classifications from the previous lesson when reasoning about negative values and fractions.

Big idea. A calculator evaluates what you enter, not what you intended: brackets and unrounded values protect the mathematics.

Key relationship

brackets \to powers \to multiply/divide \to add/subtract

Worked example

Example. Evaluate $18 - 3 (4 - 7)^{2} \div 9$ .

First, $4 - 7 = - 3$ , so $18 - 3 \times 9 \div 9 = 18 - 3 = 15$ . Do not subtract 3 before completing the multiplication and division.

4.Comparison and Ordering Symbols

There are six key symbols used to compare quantities. You must be completely familiar with each one.

Comparison & Ordering Symbols

Symbol	Name	Meaning	Example
=	Equal to	Both sides have the same value	$3 + 4 = 7$
≠	Not equal to	The sides have different values	$5 \neq = 8$
>	Greater than	Left side is larger than right side	$9 > 4$
<	Less than	Left side is smaller than right side	$2 < 7$
⩾	Greater than or equal to	Left side is larger than or equal to right side	$x ⩾ 5$ means $x$ is 5 or more
⩽	Less than or equal to	Left side is smaller than or equal to right side	$y ⩽ 10$ means $y$ is 10 or less

Reading Tip

The symbol always "points" to the smaller value. Think of the symbol as the open mouth of a crocodile — it always opens towards the bigger number!

Strict vs Non-Strict Inequalities

Strict inequalities use < and > — the value cannot be equal.
Non-strict inequalities use ⩽ and ⩾ — the value can be equal.

For example, $x > 3$ means $x$ can be 3.01, 4, 100, but not 3 itself.
$x ⩾ 3$ means $x$ can be 3, 3.01, 4, 100 — it includes 3.

5.Comparing Integers

Integers are whole numbers including negatives, zero, and positives. The number line is the best way to visualise their order: numbers increase from left to right.

A number line from minus 6 to 6. A pink arrow runs six steps right from −6 to 0 to show adding 6, and an orange arrow runs four steps left from 6 to 2 to show subtracting 4. — Figure 1: Numbers get larger as you move right along the line and smaller as you move left, and the order carries on unbroken through zero.

Examples: Comparing Integers

1
$- 3 < 2$ (-3 is to the left of 2 on the number line)
2
$5 > - 1$ (5 is to the right of -1)
3
$- 7 < - 2$ (-7 is further left than -2, so it is smaller)
4
$0 > - 4$ (zero is always greater than any negative number)

Key Rules for Negative Numbers

Every positive number is greater than every negative number
Zero is greater than all negative numbers
For negative numbers, the one closer to zero is greater: $- 2 > - 5$
Think of it as: the "less negative" a number is, the greater it is

6.Comparing Fractions

Comparing fractions requires a common basis. There are two main methods:

Method 1: Common Denominator

Convert all fractions to equivalent fractions with the same denominator (the LCM). Then compare numerators — the larger numerator means the larger fraction.

Example: Which is larger,

\frac{3}{4}

\frac{5}{6}

1
LCM of 4 and 6 is 12.
2
$\frac{3}{4} = \frac{9}{12}$ and $\frac{5}{6} = \frac{10}{12}$
3
Since 10 > 9, we have $\frac{5}{6} > \frac{3}{4}$

Method 2: Convert to Decimals

Divide numerator by denominator for each fraction, then compare the decimal values.

Example: Order

\frac{2}{3}, \frac{5}{8}, \frac{7}{12}

from smallest to largest.

1
$\frac{2}{3} = 0$ . $\overline{6} \approx 0.667$
2
$\frac{5}{8} = 0.625$
3
$\frac{7}{12} = 0.58 \overline{3} \approx 0.583$
4
Order: $\frac{7}{12} < \frac{5}{8} < \frac{2}{3}$

Which Method to Choose?

Common denominator is great for exact comparisons, especially with simple fractions. Converting to decimals is often quicker when comparing many values or mixed types (fractions with decimals and percentages).

7.Comparing Mixed Numbers

When a question gives you a mix of fractions, decimals, and percentages, the key strategy is to convert everything to the same form — usually decimals.

Convert all values to decimals → Compare → Write the answer in the original forms

Example: Put these in ascending order:

\frac{3}{5}

, 0.55, 62%,

\frac{7}{12}

Convert to decimals:

1
$\frac{3}{5} = 0.6$
2
0.55 (already a decimal)
3
62% = 0.62
4
$\frac{7}{12} \approx 0.583$

Order the decimals: 0.55, 0.583, 0.6, 0.62

Write in original forms: $0.55 < \frac{7}{12} < \frac{3}{5} < 62%$

Exam Tip

Always write your final answer using the original forms given in the question — don't leave them as decimals unless the question asks for decimals.

Example: Insert the correct symbol (< or >) between

\frac{4}{7}

and 58%.

1
$\frac{4}{7} \approx 0.571 < 0.58 = 58%$
2
Since 0.571 < 0.580, we write: $\frac{4}{7} < 58$ %

Order mixed fraction, decimal and percentage formsOpen full screen

8.Challenge Questions

Three stepping-stone questions on ordering mixed forms, ordering negatives, and telling values apart that are close enough to need real precision.

Challenge 1 — a close comparison

Insert the correct symbol ( $<$ , $>$ or $=$ ) between $\frac{5}{9}$ and $0.56$ .

1
$\frac{5}{9} = 0. \overline{5} = 0.5555 \dots$
2
Comparing digit by digit: $0.555 \dots < 0.56$ , so $\frac{5}{9} < 0.56$ .

Challenge 2 — a double-ended inequality

A number $x$ satisfies $- 4 ⩽ x < 3$ . List every integer value $x$ could take.

1
$- 4 ⩽ x$ means $- 4$ is included (non-strict).
2
$x < 3$ means $3$ is not included (strict).
3
Integers: $- 4, - 3, - 2, - 1, 0, 1, 2$ .

Challenge 3 — four different forms

Order these values from smallest to largest: $0.42$ , $\frac{3}{8}$ , $41%$ , $\frac{5}{12}$ .

1
Convert every value to a decimal: $0.42 = 0.420$ ; $\frac{3}{8} = 0.375$ ; $41% = 0.410$ ; $\frac{5}{12} = 0.41 \overline{6} \approx 0.417$ .
2
Order the decimals: $0.375 < 0.410 < 0.417 < 0.420$ .
3
Answer, in original forms: $\frac{3}{8} < 41% < \frac{5}{12} < 0.42$ .

9.Order of Operations (BIDMAS)

When a calculation involves more than one operation, you must follow a strict order. This is called BIDMAS (or BODMAS).

BIDMAS — Order of Operations

Letter	Stands For	Meaning	Example
B	Brackets	Do anything inside brackets first	$(3 + 2) \times 4 = 20$
I/O	Indices/Orders	Powers and roots	$2^{3} = 8$ .
M/D	Division/Multiplication	Left to right (equal priority)	$12 \div 4 = 3$ $3 \times 5 = 15$
A/S	Addition/Subtraction	Left to right (equal priority)	$7 + 3 = 10$

Key Rules

Multiplication and Division have the same priority — work left to right
Addition and Subtraction have the same priority — work left to right
Nested brackets: work from the innermost brackets outward

Worked Example

Evaluate $3 + 4 \times 2^{2} - (6 \div 3)$

1
Brackets: $6 \div 3 = 2$ → $3 + 4 \times 2^{2} - 2$
2
Indices: $2^{2} = 4$ → $3 + 4 \times 4 - 2$
3
Multiplication: $4 \times 4 = 16$ → $3 + 16 - 2$
4
Addition & Subtraction (left to right): $3 + 16 = 19$ , then $19 - 2 = 17$
5
Answer: 17

Common Mistakes

Adding before multiplying: $3 + 4 \times 2 = 11$ , not 14
Forgetting brackets change priority: $(3 + 4) \times 2 = 14$ because the brackets force addition first
Not working left to right for equal-priority operations: $12 \div 3 \times 2 = 4 \times 2 = 8$ , not 2

Worked Example: Nested Brackets

Evaluate $5 \times [3 + (8 - 2)]$ .

1
Inner brackets: $8 - 2 = 6$ → $5 \times [3 + 6]$
2
Outer brackets: $3 + 6 = 9$ → $5 \times 9$
3
Multiply: $5 \times 9 = 45$

Planned interactiverecommended

Number Operations, Estimation and Calculator Entry Lab

Parse an expression, predict its size and enter it safely.

Practise BIDMAS, exact fraction work, estimation and structured calculator entry.

Inputs & controls

expression builder
brackets and fraction templates
rounding level
exact or decimal

Learner actions

order operations
enter expression
estimate
check

Visible outputs

parse tree
step sequence
exact result
estimate
entry warning
percentage error

Implementation note

Use a visual calculator display rather than imitating any proprietary calculator model.

10.Operations with Integers

You must be confident with adding, subtracting, multiplying and dividing positive and negative integers.

Adding & Subtracting Signed Numbers

$(+) + (+) = +$ adding two positives gives a positive
$(-) + (-) = -$ adding two negatives gives a negative (add magnitudes, keep -)
Different signs: subtract the smaller magnitude from the larger, keep the sign of the larger
$a - (- b) = a + b$ — subtracting a negative is the same as adding
$a - (+ b) = a - b$ — subtracting a positive is normal subtraction

Examples: Addition & Subtraction

1
$- 5 + 3 = - 2$ because $5 - 3 = 2$ ; the larger magnitude is negative.
2
$- 4 + (- 7) = - 11$ ; so the answer is $- 11$ .
3
$6 - (- 3) = 6 + 3 = 9$ ; subtracting a negative is the same as adding.
4
$- 2 - 5 = - 7$ ; move further negative.

Rules for Multiplying

(+) \times (+) = +, (-) \times (-) = +, (+) \times (-) = -

Rules for Dividing

(+) \div (+) = +, (-) \div (-) = +, (+) \div (-) = -

Quick Rule

Same signs → Positive result
Different signs → Negative result

Examples: Multiplication & Division

1
$(- 3) \times (- 4) = 12$ (same signs → positive).
2
$(- 5) \times 6 = - 30$ (different signs → negative).
3
$(- 24) \div (- 6) = 4$ (same signs → positive).
4
$18 \div (- 3) = - 6$ (different signs → negative).

Practical Example: Temperature Changes

Directed numbers are used constantly for temperature. (a) The temperature at 6 a.m. is $- 5° C$ . It rises by $8° C$ by midday. What is the midday temperature? (b) The temperature at 6 a.m. was $- 5° C$ and by 2 p.m. it had risen to $9° C$ . By how many degrees did it rise?

1
(a) Apply the rise directly: $- 5 + 8 = 3° C$ .
2
(b) A rise is the later value minus the earlier one: $9 - (- 5) = 9 + 5 = 14° C$ — subtracting a negative starting temperature is the same as adding it.

11.Adding & Subtracting Fractions

To add or subtract fractions, the denominators must be the same. If they are different, find the lowest common denominator (LCD).

Same Denominator

\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}

Example: Same Denominator

1
$\frac{3}{7} + \frac{2}{7} = \frac{3 + 2}{7} = \frac{5}{7}$

Different Denominators

\frac{a}{b} + \frac{c}{d} = \frac{a d + b c}{b d} (or u se L C M o f ban dd)

Worked Example: Different Denominators

Calculate

\frac{2}{3} + \frac{3}{5}

1
Find LCD: 3 and $5 = 15$
2
Convert: $\frac{2}{3} = \frac{10}{15}$ and $\frac{3}{5} = \frac{9}{15}$
3
Add numerators: $\frac{10}{15} + \frac{9}{15} = \frac{19}{15} = 1 \frac{4}{15}$

Worked Example: Mixed Numbers

Calculate

2 \frac{1}{4} - 1 \frac{2}{3}

1
Convert to improper fractions: $2 \frac{1}{4} = \frac{9}{4}$ and $1 \frac{2}{3} = \frac{5}{3}$
2
Find LCD: 4 and $3 = 12$
3
Convert: $\frac{9}{4} = \frac{27}{12}$ and $\frac{5}{3} = \frac{20}{12}$
4
Subtract: $\frac{27}{12} - \frac{20}{12} = \frac{7}{12}$

Remember

Always simplify your answer if possible
Convert mixed numbers to improper fractions before operating
Convert the answer back to a mixed number if the question uses mixed numbers

12.Multiplying & Dividing Fractions

Multiplying Fractions

\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}

Multiply numerators together and denominators together. No common denominator needed. Simplify before or after multiplying.

Example

1
$\frac{3}{5} \times \frac{4}{7} = \frac{3 \times 4}{5 \times 7} = \frac{12}{35}$

Dividing Fractions

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} (K ee p, C han g e, F l i p)

To divide by a fraction: keep the first fraction, change

\div

\times

, flip (reciprocal of) the second fraction.

Example

1
$\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$

Worked Example: Mixed Numbers

Calculate $1 \frac{3}{4} \times 2 \frac{2}{5}$ .

1
Convert to improper: $1 \frac{3}{4} = \frac{7}{4}$ and $2 \frac{2}{5} = \frac{12}{5}$
2
Multiply: $\frac{7}{4} \times \frac{12}{5} = \frac{84}{20} = \frac{21}{5} = 4 \frac{1}{5}$

Worked Example: Dividing Mixed Numbers

Calculate $3 \frac{1}{2} \div 1 \frac{1}{4}$ .

1
Convert: $\frac{7}{2} \div \frac{5}{4}$
2
Keep, Change, Flip: $\frac{7}{2} \times \frac{4}{5} = \frac{28}{10} = \frac{14}{5} = 2 \frac{4}{5}$

Exam Tip

Always convert mixed numbers to improper fractions before multiplying or dividing. Cross-cancel common factors before multiplying to keep numbers small.

13.Operations with Decimals

Being able to add, subtract, multiply and divide decimals without a calculator is an essential skill.

Adding & Subtracting Decimals

Line up the decimal points vertically, then add or subtract as normal. Fill empty decimal places with zeros.

Example: 12.6 + 3.47

1
12.60
2
+ 3.47
3
———
4
16.07

Multiplying Decimals

1
Ignore the decimal points and multiply the whole numbers
2
Count the total number of decimal places in both original numbers
3
Place the decimal point in the answer so it has that many decimal places

Example: 0.3

\times

0.04

1
Multiply: $3 \times 4 = 12$
2
Decimal places: $1 + 2 = 3$ (one from 0.3 and two from 0.04).
3
Place decimal: 0.012

Example: 2.5

\times

1.3

1
Multiply: $25 \times 13 = 325$
2
Decimal places: $1 + 1 = 2$
3
Answer: 3.25

Dividing Decimals

Make the divisor a whole number by multiplying both numbers by the same power of 10, then divide as normal.

Example: 4.56

\div

0.3

1
Multiply both by 10: $45.6 \div 3$
2
$45.6 \div 3 = 15.2$

Example: 0.126

\div

0.07

1
Multiply both by 100: $12.6 \div 7$
2
$12.6 \div 7 = 1.8$

14.The Four Operations — Challenge Questions

Three questions that combine BIDMAS with negative indices, fractions in context, and a decimal calculation you check by reversing it.

Challenge 1 — the notation trap meets BIDMAS

Evaluate $- 3^{2} + (- 2)^{3} \div 4$ .

1
Indices: $- 3^{2} = - (3^{2}) = - 9$ (no brackets around $- 3$ , so only the 3 is squared).
2
$(- 2)^{3} = - 8$ (brackets mean the whole $- 2$ is cubed).
3
Division: $- 8 \div 4 = - 2$ .
4
Addition: $- 9 + (- 2) = - 11$ .

Challenge 2 — fractions in a real recipe

A recipe needs $2 \frac{1}{4}$ cups of flour per batch. How many cups are needed for $1 \frac{1}{3}$ batches?

1
Convert to improper fractions: $2 \frac{1}{4} = \frac{9}{4}$ and $1 \frac{1}{3} = \frac{4}{3}$ .
2
Multiply: $\frac{9}{4} \times \frac{4}{3} = \frac{36}{12} = 3$ .
3
$3$ cups of flour are needed — the 4s cancel neatly, which is worth spotting before multiplying out in full.

Challenge 3 — decimals with a check

Calculate $3.6 \div 0.15$ , then check your answer using the inverse operation.

1
Multiply both numbers by 100 to make the divisor whole: $360 \div 15$ .
2
$360 \div 15 = 24$ .
3
Check: $24 \times 0.15 = 3.6$ ✓

15.Key Calculator Buttons

Scientific calculators have many buttons beyond the basic operations. Here are the most important ones for the IGCSE course.

Essential Scientific Calculator Functions

Button	Function	Example
√ or $x$	Square root	$144 = 12$
$x^{2}$	Square a number	$7^{2} = 49$
$x^{y}$ or ^	$P o w er s / in d i ces$	$2^{5} = 32$
$n x$ or $x^{1/ y}$	nth root	$364 = 4$
( )	Brackets — controls order of operations	$(3 + 4) \times 5 = 35$
a $\frac{b}{c}$ or ▸	Fraction key — enters fractions and mixed numbers	$\frac{3}{4} + \frac{1}{2}$
$\times 1 0^{x}$ or EXP	Enter numbers in standard form	$3.2 \times 1 0^{5}$
Ans	Recalls the last answer	Use in chain calculations
M+, M-, MR	Memory store, subtract, recall	Store intermediate results
S⇔D	Toggle between $s u r d / f r a c t i o n$ and decimal form	$\frac{1}{3} \leftrightarrow 0.333$ ...
(-) or +/-	Negative sign (different from subtraction)	$(- 3)^{2} = 9$

Standard Form Shortcut

To enter $4.6 \times 1 0^{- 3}$ , type 4.6 EXP (-)3. Do NOT type $4.6 \times 10$ ^ (-)3 — this takes longer and creates room for error.

Negative Numbers vs Subtraction

Use the (-) or +/- key for negative numbers, not the subtract key. For example, to calculate $(- 5)^{2}$ , press ( (-) 5 ) $x^{2}$ .

16.Entering Complex Expressions

The most common source of calculator errors is entering expressions incorrectly. The key is understanding how your calculator interprets what you type.

Golden Rule

If the numerator or denominator contains more than one term, you must use brackets.

Translate written structure into calculator structure, then estimate

1
Read grouping
identify fractions, powers and outer operation
2
Enter faithfully
use brackets/templates · inspect display
3
Calculate
keep full precision · store intermediate values
4
Check independently
round first · compare sign and size

A calculator follows the entered syntax, not the intended syntax; inspect grouping before pressing equals.

Example 1: Calculate

\frac{3.7 + 2.8}{4.1 - 1.6}

1
Key sequence: ( 3.7 + 2.8 ) ÷ ( 4.1 - 1.6 )
2
Or using the fraction key: ▸ 3.7 + 2.8 ▾ 4.1 - 1.6
3
$\frac{6.5}{2.5} = 2.6$

Common Mistake

Typing 3.7 + 2.8 ÷ 4.1 - 1.6 without brackets gives a completely wrong answer because the calculator follows BIDMAS — it divides 2.8 by 4.1 first!

Example 2: Calculate

(3.2)^{2} + (4.7)^{2}

1
Key sequence: √( 3.2 $x^{2}$ + 4.7 $x^{2}$ )
2
$10.24 + 22.09 = 32.33 \approx 5.686$

Example 3: Calculate

\frac{2}{3} + \frac{5}{7}

1
Key sequence: ▸ 2 ▾ 3 + ▸ 5 ▾ 7
2
Or: ( 2 ÷ 3 ) + ( 5 ÷ 7 )
3
$\frac{14}{21} + \frac{15}{21} = \frac{29}{21} = 1 \frac{8}{21}$

Example 4: Calculate

1.0 5^{12}

1
Key sequence: 1.05 x^y 12 =
2
$1.0 5^{12} \approx 1.795856$

Example 5: Calculate

327000

1
Key sequence: 3 ⁿ√x 27000 or 27000 x^y ( 1 ÷ 3 )
2
$327000 = 30$

Exam Tip

Always count your brackets — every opening bracket ( needs a matching closing bracket ). Missing a bracket is the single most common calculator error.

17.Time Calculations

Calculators can work with time in hours, minutes, and seconds using the degrees/minutes/seconds button (often labelled ° ' " or DMS).

The ° ' " Button

This button lets you enter and convert between hours:minutes:seconds and decimal hours. On most calculators:

Enter hours, press ° ' ", enter minutes, press ° ' ", enter seconds, press ° ' "
Press ° ' " to toggle between sexagesimal and decimal display

Example 1: Convert 3.75 hours to hours and minutes.

1
$0.75 hours = 0.75 \times 60 = 45 minutes$
2
So $3.75 hours$ = $3 hours 45 minutes$
3
On calculator: Type 3.75 then press ° ' " $\Rightarrow$ displays 3°45'0"

Example 2: Convert 2.4 hours to hours, minutes, and seconds.

1
$0.4 hours = 0.4 \times 60 = 24 minutes$
2
So $2.4 hours$ = $2 hours 24 minutes$

Example 3: Convert 2 hours 15 minutes to decimal hours.

1
$15 min = \frac{15}{60} hours = 0.25 hours$
2
So $2 h 15 min$ = $2.25 hours$

Example 4: Convert 1 hour 40 minutes 30 seconds to decimal hours.

1
$40 min = \frac{40}{60} = 0.6 \overline{6} = hours$
2
$30 sec = \frac{30}{3600} = 0.008 \overline{3} = hours$
3
$Total = 1 + 0.6 \overline{6} + 0.008 \overline{3} = 1.675 hours$ .
4
On calculator: 1 ° ' " 40 ° ' " 30 ° ' " = then press ° ' " again to see decimal.

Quick Conversions: Minutes to Decimal Hours

Minutes	Decimal	Minutes	Decimal
5	0.083	35	0.583
10	0.167	40	0.667
15	0.25	45	0.75
20	0.333	50	0.833
25	0.417	55	0.917
30	0.5	60	1.0

Common Mistake

2 hours 30 minutes is NOT 2.30 hours — it is 2.5 hours. Always divide minutes by 60 to convert.

Reading the Display: Money vs Time

A calculator does not know what your answer represents — it strips trailing zeros and shows a plain decimal, so you must interpret the display correctly for the context.

Money: a display of 4.8 in a money calculation means £4.80, not "£4 and 8 pence" — always write money to two decimal places.

Time: a display of 3.25 in a time calculation means 3 hours 15 minutes, not "3 hours 25 minutes" — the decimal part is a fraction of an hour ( $0.25 \times 60 = 15$ minutes), not minutes read directly off the display.

18.Checking Answers

It is essential to check whether your calculator answer is reasonable. The exam often awards marks for showing checking.

Methods for Checking

1
Estimation — round numbers to 1 s.f. and do a rough mental calculation
2
Inverse operations — work backwards from the answer
3
Reasonableness — does the answer make sense in context?
4
Re-entering — redo the calculation or enter it differently

Example: Check

\frac{48.7 \times 3.12}{0.489}

1
Calculator gives: 310.6...
2
Estimate: $\frac{50 \times 3}{0.5} = \frac{150}{0.5} = 300$
3
The estimate (300) is close to 310.6, so the answer is reasonable.

Example: You calculated

576 = 24. C h ec k t hi s .

1
Inverse: $2 4^{2} = 576$ Correct:
2
The answer checks out.

Example: You calculated

378 \div 14 = 27

. Check this.

1
Inverse: $27 \times 14 = 378$ , so the answer is correct.

Example: A car travels for 2.5 hours at 60

km/h

. The calculator gives 15,000 km.

1
This is clearly unreasonable — a car cannot travel 15,000 km in 2.5 hours.
2
Correct: $2.5 \times 60 = 150$ km. You likely typed an extra zero or used the wrong operation.

Exam Tip

If an exam question says "show that" or "verify", you must show a check. Estimation is usually the fastest method.

Estimate by rounding before calculating exactlyOpen full screen

19.Using a Calculator — Challenge Questions

Three questions on reading calculator notation correctly, using the time-conversion button, and checking a displayed answer against an estimate.

Challenge 1 — reading the display

A calculation gives the display 4.803838...⁻⁰⁵. Write this in standard form, then as an ordinary decimal.

1
The small raised number is the power of 10, so in standard form: $4.803838 \dots \times 1 0^{- 5}$ .
2
As an ordinary decimal, a negative power of 10 this size shifts the digits four places right of a leading zero: $0.00004803838 \dots$
3
Interpreting the display correctly matters just as much as calculating — an exam answer copied straight off the screen without this conversion can be marked wrong.

Challenge 2 — using the DMS button

Convert 5 hours 12 minutes into decimal hours using the ° ' " function, and state the key sequence.

1
Key sequence: 5 ° ' " 12 ° ' " 0 ° ' " =, then press ° ' " again to toggle to decimal.
2
By hand: $12 min = \frac{12}{60} = 0.2$ hours.
3
So $5 h 12 min = 5.2$ hours.

Challenge 3 — is the display reasonable?

A calculation of $\frac{124.8}{0.024}$ gives $5200$ on the display. Verify this is reasonable using estimation, then confirm with the inverse operation.

1
Estimate: round to $\frac{125}{0.025} = 5000$ , which is close to $5200$ — the answer is plausible.
2
Inverse check: $5200 \times 0.024 = 124.8$ ✓, confirming the answer exactly.

20.Exam-Style Worked Examples

Worked example 1 — order of operations (3 marks)

Question. Evaluate $\frac{5 + 3 \times 4 ^{2}}{2 \times ( 9 - 5 )}$ .

1
Numerator: indices first, $4^{2} = 16$ ; then $3 \times 16 = 48$ ; then $5 + 48 = 53$ .
2
Denominator: brackets first, $9 - 5 = 4$ ; then $2 \times 4 = 8$ .
3
$\frac{53}{8} = 6.625$ .

Marking. 1 mark per point. A fraction bar acts as a bracket — evaluate top and bottom completely before dividing.

Worked example 2 — negative numbers (3 marks)

Question. Work out (a) $- 7 - (- 3)$ , (b) $- 6 \times - 4$ , (c) $\frac{- 20}{4} - (- 5)$ .

1
(a) Subtracting a negative is the same as adding: $- 7 + 3 = - 4$ .
2
(b) Two negatives multiplied give a positive: $- 6 \times - 4 = 24$ .
3
(c) $\frac{- 20}{4} = - 5$ , then $- 5 - (- 5) = - 5 + 5 = 0$ .

Marking. 1 mark per point. The rule for signs differs between $+ / -$ and $\times / \div$ — deal with one operation at a time.

Worked example 3 — using a calculator efficiently (4 marks)

Question. Calculate $\frac{18.4 + 7.6}{3. 2 ^{2} - 1.4}$ , giving your answer correct to 3 significant figures.

1
Numerator: $18.4 + 7.6 = 26$ , so $26 = 5.0990 \dots$
2
Denominator: $3. 2^{2} = 10.24$ , so $10.24 - 1.4 = 8.84$ .
3
$\frac{5.0990 \dots}{8.84} = 0.57681 \dots$
4
To 3 significant figures: $0.577$ .

Marking. 1 mark per point. Keep the unrounded value in the calculator until the final step — rounding early loses accuracy marks.

21.Exam Tips & Common Misconceptions

Exam Tips

Use $B I D M A S / B O D M A S$ consistently — Brackets, Indices, $D i v i s i o n / M u l t i pl i c a t i o n$ (left to right), $A dd i t i o n / S u b t r a c t i o n$ (left to right).
When ordering negatives, the larger the magnitude, the smaller the value: $- 7 < - 3$ .
Write down intermediate calculator displays — you'll lose marks if you give only the final answer in a multi-step question.
On the calculator, use brackets liberally around numerators, denominators and any expression with a unary minus.
For "≥" or "≤" inequalities, remember equality is allowed — use a closed circle on a number line.

Common Misconceptions

“ $- 5^{2} = 25$ ” is false. Without brackets, $- 5^{2} = - (5^{2}) = - 25$ by BIDMAS; $(- 5)^{2} = 25$ .
Doing addition before division in $6 + 4 \div 2$ — answer is 8, not 5.
" $0.2 > 0.15$ because $2 < 15$ " — compare place value, not digit count.
Rounding too early in long calculations introduces accumulated error; round only at the end.

22.Worked method — Preserve fraction bars and brackets when entering a calculation

Structured entry, exact working and a quick estimate protect against lost brackets and implausible answers.

Transfer the method

Reasoning prompt. A calculator produces an unexpected answer. What should you inspect before assuming the calculator is wrong?

1
Rewrite the calculation with brackets and operation signs visible.
2
Compare that expression with the order of operations.
3
Re-enter the complete expression, then round only the final result.

Self-check

A rough estimate should have the same sign and a similar size as the final answer.

23.Exact calculation, safe entry and independent checking

A four-stage calculator workflow

1
Model
Write the mathematical expression from the words, including units and negative signs.
2
Structure
Add brackets around a multi-term numerator, denominator, power base or negative number.
3
Evaluate
Enter the complete expression and retain the unrounded display for later steps.
4
Verify
Estimate the sign and size, or use an inverse operation; round only the requested final answer.

Worked example: a unary minus is not a bracket

Evaluate $- (3 - 5)^{2} + 18 \div (- 3)$ .

Brackets: $3 - 5 = - 2$ .

Power: $(- 2)^{2} = 4$ , so the leading minus gives $- 4$ .

Division: $18 \div (- 3) = - 6$ .

Therefore $- 4 + (- 6) = - 10$ .

Notation trap

$- 2^{2} = - (2^{2}) = - 4$ , whereas $(- 2)^{2} = 4$ . The brackets decide whether the negative sign is part of the base.

24.Summary

Key Points

Central principle. A calculator evaluates what you enter, not what you intended: brackets and unrounded values protect the mathematics
There are six key symbols used to compare quantities. You must be completely familiar with each one.
Left side is larger than or equal to right side
Left side is smaller than or equal to right side
The symbol always "points" to the smaller value. Think of the symbol as the open mouth of a crocodile — it always opens towards the bigger number!
Key relationship: $brackets \to powers \to multiply/divide \to add/subtract$
Multiplying and dividing signs: same signs give a positive result; different signs give a negative result.
Same Denominator: $\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$
Different Denominators: $\frac{a}{b} + \frac{c}{d} = \frac{a d + b c}{b d} (or u se L C M o f ban dd)$
Multiplying Fractions: $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$
Dividing Fractions: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} (K ee p, C han g e, F l i p)$