Using a calculator

Using a calculator efficiently

Papers 3 and 4 need a scientific calculator, and Papers 1 and 2 test everything except this subtopic without one. On the calculator papers, marks are often lost not because the method is wrong but because the calculator was used carelessly. This lesson is about the habits that stop that happening.

Type the whole calculation in one go where you can. A scientific calculator follows the order of operations (brackets, then powers and roots, then multiplication and division, then addition and subtraction), so it will give the right answer to a long calculation as long as you tell it exactly what you mean.

Use the keys that save time and accuracy:

• the fraction template, for calculations with fractions, which also gives answers as fractions;
• the Ans key, which recalls the last answer with every digit, so you never need to retype a rounded value;
• the memory keys, for a value you will use several times;
• the $\times {10}^x$ key for standard form, and the degrees, minutes and seconds key for times and angles.

Write down what the display shows before you round. If you then round wrongly, an examiner can still give the method mark, because they can see the unrounded value. And a quick estimate (round each number to 1 significant figure) tells you whether the display is sensible: if you expect about 2.5 and the calculator says 0.0251, something was typed wrongly.

Entering calculations correctly

Most calculator errors are entry errors. The calculator does exactly what you type, which is not always what you meant.

• Fraction lines are brackets. $\frac{12.6+{3.85}^2}{\sqrt{19.4}-2.7}$ must be typed with brackets around the whole top and the whole bottom, (12.6 + 3.85²) ÷ (√19.4 - 2.7), or with the fraction template. Without the brackets the calculator divides only ${3.85}^2$ by $\sqrt{19.4}$.
• Roots cover everything under them. For $\sqrt{{5.6}^2-{2.3}^2}$, make sure the subtraction is inside the root. Most calculators open a bracket after the root key; close it in the right place.
• Negative numbers and powers. $(-2.4{)}^2=5.76$, but typing $-{2.4}^2$ gives $-5.76$, because the calculator squares 2.4 first and then applies the minus. Put a negative number in brackets before raising it to a power.
• Standard form. Enter $3.2\times {10}^{-5}$ using the $\times {10}^x$ key, or as 3.2 × 10^(-5) with brackets round the power.

If a calculation is long, it is fine to work out the top and the bottom separately, as long as you keep every digit (store them in memory, or use Ans) and do not round either one.

Do not round until the end

The syllabus says you should know not to round values within a calculation, and to round only the final answer. The reason is that a small rounding error early on is carried through, and often multiplied, by every later step.

The exam papers say the same thing: to earn accuracy marks, avoid rounding until the final answer, and where a later part of a question uses an earlier answer, use the value before it was rounded. In practice:

• keep the full value on the display and carry straight on, or use Ans;
• if you must write an intermediate value down, write at least 4 or 5 significant figures, and keep the full value in the calculator;
• round once, at the end, to the accuracy the question asks for: 3 significant figures unless told otherwise, 1 decimal place for angles in degrees, and the nearest penny or cent for money.

Entering times

Times are not decimals. There are 60 minutes in an hour, so 2 hours 30 minutes is 2.5 hours, but 2 hours 20 minutes is not 2.2 hours: it is $2+\frac{20}{60}=2.333\dots$ hours.

There are two good ways to enter a time:

• Divide the minutes by 60 and add: type 2 + 20 ÷ 60. For seconds, divide by 3600.
• Use the degrees, minutes and seconds key (marked ° ' '' on most calculators). Enter 2 hours 30 minutes as 2° 30' 0'', and the calculator treats it as 2.5 in any calculation.

Never type 2.30 for 2 hours 30 minutes. That is 2 and 30 hundredths of an hour, which is only 18 minutes past 2 hours.

Interpreting the display

The calculator gives a number; you have to decide what it means in the question.

• Money. A display of 4.8 in a money calculation means £4.80 (or 4.80 dollars): always write money with two decimal places. A display of 12.433333 pounds is £12.43 to the nearest penny.
• Time. A display of 3.25 for a time in hours means 3 hours and $0.25\times 60=15$ minutes, not 3 hours 25 minutes. Subtract the whole hours and multiply the decimal part by 60. (The degrees, minutes and seconds key will also convert 3.25 to 3° 15' 0''.)
• Standard form. A display such as ${4.2}^{-05}$ or 4.2E-05 means $4.2\times {10}^{-5}=0.000042$, not 4.2.
• Rounding in context. Decide whether the situation needs rounding up or down. 5.18 coaches means you need 6 coaches; 26.7 biscuits' worth of butter means you can make 26 biscuits; if a bill must be covered, each person's share is rounded up to the next penny, not to the nearest.
• Fractions and decimals. The S⇔D key switches between a fraction and its decimal. If a question asks for an exact answer or a fraction, do not copy down a rounded decimal.