Common Mistakes with Standard Form in GCSE Maths

You can understand powers of ten and still lose marks through one misplaced decimal point. The most common mistakes with standard form in GCSE maths are choosing the wrong power, leaving the first number outside its allowed range, and using multiplication rules when adding.

The fix is to check two things separately: does your answer have the right value, and is it written in standard form? Those are different questions.

Imagine a student reviewing a marked worksheet. Instead of writing “careless” beside every correction, they identify the exact rule that went wrong. Follow that approach below: each mistake becomes something you can recognise, repair and check next time.

Your standard form checklist

For positive numbers, standard form means writing

$$a\times 10^n,1\le a<10,$$

where $n$ is an integer. The number $a$ is called the coefficient.

Before moving on from an answer, check:

• Coefficient: is it at least $1$ but less than $10$?
• Power: does its sign match the size of the original number?
• Operation: have you used the correct rule for the calculation?
• Size: is your answer approximately what you expected?

For a positive number greater than or equal to $10$, the power is positive. For a positive number smaller than $1$, it is negative. Numbers from $1$ up to, but not including, $10$ use the power $0$.

Common mistakes with standard form in GCSE maths

Leaving the coefficient too large

Our imagined student writes:

$$72000=72\times 10^3.$$

Why it happens: the value is correct, so the answer feels finished. But $72$ is too large to be the coefficient in standard form.

Correct method: first choose the coefficient, then find the power needed to recover the original number.

$$72000=7.2\times 10^4.$$

The coefficient became ten times smaller, so the power-of-ten factor must become ten times larger.

This also matters after calculations. An intermediate answer can be numerically correct without being in standard form. Check the final line, not just the working.

Choosing the wrong sign for a small number

The next answer reads:

$$0.00063=6.3\times 10^4.$$

Why it happens: the student counts decimal places correctly but forgets what the power means. Multiplying by $10^4$ makes $6.3$ much larger, not smaller.

Correct method: establish the coefficient, then think about the multiplier needed.

$$0.00063=6.3\times 0.0001=6.3\times 10^{-4}.$$

A negative power does not make the answer negative. It gives a small positive power-of-ten factor:

$$10^{-4}=\frac{1}{10^4}.$$

If that distinction feels uncertain, MathsGenie's indices revision lesson explains the underlying rules.

Counting zeros instead of decimal-place movements

Another correction involves converting back:

$$4.08\times 10^{-3}=\mathrm{0.000408.}$$

Why it happens: counting zeros is unreliable because the digits already present affect how many zeros appear.

Correct method: multiplying by $10^{-3}$ means dividing by $1000$. Move the decimal point three places left:

$$4.08\to 0.408\to 0.0408\to \mathrm{0.00408.}$$

Therefore,

$$4.08\times 10^{-3}=\mathrm{0.00408.}$$

Keep the zero between $4$ and $8$. It belongs to the number and cannot be discarded. Counting movements rather than visible zeros gives you a method that works consistently.

Once you can check the coefficient and power separately, use the Lessons and Question Bank on MathsGenie's standard form topic page to practise applying those checks without prompts.

Stick-figure comic about replacing a dramatic revision strategy with a calmer checking habit

Multiplying correctly but stopping too soon

The student calculates:

$$(6\times 10^4)(3\times 10^{-2})=18\times 10^2.$$

Why it happens: both calculation steps are correct, so the student stops before checking the required format.

Correct method: multiply the coefficients and add the powers:

$$6\times 3=18,4+(-2)=2.$$

Then rewrite the result in standard form:

$$18\times 10^2=1.8\times 10^3.$$

For division, divide the coefficients and subtract the powers. Take particular care when subtracting a negative:

$$\frac{8\times 10^3}{2\times 10^{-2}}=4\times 10^{3-(-2)}=4\times 10^5.$$

MathsGenie's explanation of calculations with numbers in standard form reinforces this distinction between the coefficients and powers.

Adding powers when adding numbers

Here is a different kind of error:

$$(3\times 10^5)+(2\times 10^4)=5\times 10^9.$$

Why it happens: a familiar multiplication rule gets applied to an addition question. Index laws depend on the operation; they are not interchangeable shortcuts.

Correct method: make the powers equal before combining the coefficients.

$$2\times 10^4=0.2\times 10^5.$$

Now add:

$$(3\times 10^5)+(0.2\times 10^5)=3.2\times 10^5.$$

You could also convert both numbers into ordinary notation first. For subtraction, use the same matching-powers approach. An intermediate coefficient such as $0.2$ is acceptable during working; the final answer must satisfy the standard form rule.

Misreading a calculator display

A calculator may use an exponent indicator such as E when displaying a number in scientific notation. The exact appearance depends on the model.

Why it happens: the display looks different from the notation used on paper, so the student copies it without interpreting it.

If the display represents $7.2\times 10^{-5}$, write that expression when standard form is requested. Do not drop the negative sign or treat the exponent indicator as a variable.

Before entering a calculation, predict its approximate size. After entering it, compare the display with that prediction. This catches many input mistakes that pressing the same buttons again would miss.

Exam-style practice: test the repairs

Try these without looking at the answers. Show working for the calculations and give every answer in standard form.

• Write $580000$ in standard form.
• Write $0.000092$ in standard form.
• Calculate $(4\times 10^3)(6\times 10^{-5})$.
• Calculate $(7\times 10^6)+(5\times 10^5)$.

Answers and checks

The first answer is $5.8\times 10^5$. Check that multiplying $5.8$ by $100000$ recovers the original number.

The second is $9.2\times 10^{-5}$. The original number is positive and smaller than $1$, so a negative power makes sense.

For multiplication:

$$(4\times 10^3)(6\times 10^{-5})=24\times 10^{-2}=2.4\times 10^{-1}.$$

For addition:

$$(7\times 10^6)+(0.5\times 10^6)=7.5\times 10^6.$$

For further exam-style material, the standard form exam booklet provides questions with answers. When checking, label the cause of each error rather than simply copying the correction.

Turn corrections into reliable habits

Our imagined student's useful discovery is not that they need to “be more careful”. It is that each mistake has a specific check: coefficient, power, operation or size. Knowing those checks is the start; remembering them independently takes practice.

Your turn: start with MathsGenie's standard form Lessons and Question Bank, and apply all four checks to each answer.