Number - Structure and calculation — Edexcel GCSE Mathematics

Number - Structure and calculation covers the fundamental rules of arithmetic, including the order of operations (BIDMAS), factors, multiples, primes, powers, roots, and standard form. Mastery of these core skills is essential for solving problems across all areas of Pearson Edexcel GCSE Mathematics, from algebra to geometry and statistics.

This topic covers the essential building blocks of number work: the order of operations (BIDMAS), factors, multiples, primes, powers, roots, and standard form. You will learn to perform calculations accurately and efficiently, both with and without a calculator, and to apply these skills to solve real-world problems. Understanding these concepts is crucial because they underpin almost every other area of mathematics, including algebra, ratio, and geometry.

In the Pearson Edexcel GCSE Mathematics exam, questions on structure and calculation often appear as short, straightforward calculations or as part of multi-step problems. You may be asked to find the highest common factor (HCF) or lowest common multiple (LCM), simplify expressions involving powers, or convert between ordinary numbers and standard form. Mastering these skills will not only secure marks in the Number strand but also boost your confidence and accuracy across the entire exam.

• →BIDMAS: The order of operations - Brackets, Indices, Division and Multiplication (left to right), Addition and Subtraction (left to right).
• →Factors, Multiples, Primes: A factor divides exactly into a number; a multiple is in the times table of a number; a prime has exactly two factors (1 and itself).
• →Powers and Roots: Powers (indices) represent repeated multiplication; roots are the inverse operations. Know square numbers, cube numbers, and their roots up to at least 15 and 5 respectively.
• →Standard Form: A way of writing very large or very small numbers as A x 10^n, where 1 ≤ A < 10 and n is an integer.
• →HCF and LCM: The highest common factor is the largest number that divides into two or more numbers; the lowest common multiple is the smallest number that is a multiple of two or more numbers.

• converting every value into a single form, all decimals or all fractions over a common denominator, even if the final order is wrong
• a correct comparison shown as a chain, such as −4 < −0.5 < 1/3 < 0.75
• the finished list written in the direction the question asked for, smallest first or largest first as stated
• choosing the correct symbol when you are asked to complete a statement between two given values
• Correct use of a formal written method, such as column addition or long division, with clear layout.
• Correctly handling decimal points in calculations, either by scaling for division or counting places for multiplication.
• Converting mixed numbers to improper fractions before multiplying or dividing.
• Finding a common denominator before adding or subtracting fractions.
• Applying the correct sign rules for calculations involving negative numbers across all number types.
• Applying the priority of operations in the correct order (BIDMAS/BODMAS).
• Correctly identifying and cancelling a common numerical or algebraic factor.
• Using inverse operations to reverse a chain of steps, such as in solving an equation.
• Finding the correct reciprocal of an integer, fraction or decimal.
• Correctly listing factors or multiples of a number.
• A complete factor tree or repeated division method showing only prime numbers at the end.
• The prime factorisation written correctly as a product, using index notation for repeated primes.
• Selecting the correct prime powers for the HCF (lowest powers of common primes) or LCM (highest powers of all primes).
• Evaluating the final product for the HCF or LCM correctly.
• starting a systematic list, with the first element held fixed while the rest vary
• a list containing every possibility, with nothing repeated
• the count or the particular outcome the question asks for, read off your own list
• a count that follows through from your own list, provided the order you worked in is clear
• Demonstrating a clear, systematic method to list all outcomes (e.g., using a table, tree diagram, or ordered list).
• Identifying the correct number of independent choices available at each stage of a process.
• Correctly multiplying the number of choices for each stage to find the total number of outcomes.
• Reducing the number of choices at subsequent stages when items are selected without replacement (no repeats).
• writing a power out as a repeated multiplication where that is the step being tested
• the correct value of a power or a root, even where a later step then goes wrong
• recognising a given number as a power of a small base, such as writing 243 as 3⁵
• handling a power and a root in the same expression in the right order, indices before multiplying and dividing
• naming the two known powers the value sits between, such as 49 and 64 for a square root near 50
• rounding the base to a value whose power you know, such as using 3⁵ in place of 2.9⁵
• an estimate that lies in the correct interval, even where it is not the closest available value
• saying whether the true value is above or below the estimate, where the question asks for that
• the correct index law used, adding the indices for a product and subtracting them for a quotient
• a correct intermediate power such as 2⁷, even where it is not then evaluated
• evaluating that power to a number where the question wants a value
• rewriting a negative index as a fraction, or a fraction as a negative index, whichever direction the question asks for
• a correct root value, including cube roots such as ∛27 = 3, not only square roots
• Correctly interpreting a fractional index as a root and a power.
• Evaluating the root part of the calculation correctly.
• Correctly applying the reciprocal rule for a negative index.
• Simplifying an expression into a single power with the correct base and index.
• Calculating the final exact numerical value where required.
• Finding a correct common denominator when adding or subtracting fractions, e.g. using 12 for 1/4 + 2/3.
• Multiplying fractions by multiplying numerators together and denominators together, then simplifying.
• Dividing fractions by inverting the second fraction and multiplying, e.g. 3/5 ÷ 2/7 = 3/5 × 7/2 = 21/10.
• Leaving an answer in terms of π when an exact value is required, e.g. writing 16π rather than 50.3.
• Simplifying a fraction to its simplest form by dividing numerator and denominator by a common factor.
• calculating exactly with fractions, for example 2/3 + 1/4 = 11/12, without converting to a rounded decimal
• leaving an answer as a multiple of π, such as 6π for the circumference of a circle of radius 3 cm, rather than a decimal
• splitting a surd at its largest square factor, such as writing √50 as √25 × √2 = 5√2
• multiplying numerator and denominator by the same root, or by the conjugate of a two-term denominator, to rationalise
• a fully simplified exact answer, with no root left in the denominator
• a number converted correctly, with the front number at least 1 and below 10
• adding the indices when multiplying, or subtracting them when dividing
• a correct but untidied answer such as 0.4 × 10⁵ before it is adjusted
• the final answer in standard form, which needs both the front number and the index right

• 💡Underline whether the list is wanted smallest first or largest first before you write anything down.
• 💡Turn a mixed list into decimals to compare it, then give the final answer using the numbers as they were printed unless the question says otherwise.
• 💡Write each converted value beside the original so the examiner can follow the comparison you made.
• 💡Estimate the answer first by rounding each number to one significant figure. This helps you check if your final answer is a reasonable magnitude.
• 💡Show your full working for formal methods, as marks are awarded for the correct process even if you make a calculation error.
• 💡On a calculator paper, always use brackets for negative numbers in operations like squaring or multiplication, e.g., enter `(-3.5)²` not `-3.5²`.
• 💡Write each stage of a priority-of-operations calculation on a new line to reduce the chance of errors spreading.
• 💡Check a numerical answer by applying the inverse operations.
• 💡Cancel common factors before you multiply fractions; the numbers stay smaller and the arithmetic is easier.
• 💡When finding both HCF and LCM from prime factorisations, a Venn diagram can help. The intersection gives the HCF, and the product of all numbers in the diagram gives the LCM.
• 💡Multiply your prime factorisation back out as a check before you use it for anything else.
• 💡Read the question carefully to see if the answer is wanted as a product of primes or as a single number.
• 💡Set the list out in columns headed by the first choice; a short column is a sign that cases are missing.
• 💡Count the finished list twice before you write the total down.
• 💡When only a few items are involved, listing them all is usually quicker than trying to reason the count out.
• 💡If a question asks you to 'list all possible combinations' and the number is small, use a systematic approach. A two-way table or tree diagram can help organise your thinking.
• 💡If a question asks 'how many' and the number of outcomes is likely to be large, the product rule is needed. Draw a box for each choice, write the number of options in it, and multiply them together.
• 💡Learn the squares as far as 15² and the cubes as far as 5³ so you recognise them on sight.
• 💡On a calculator paper, find the power key and the root key before you need them and test one value you already know.
• 💡Write a repeated multiplication out in full if you are unsure; it costs one line and stops you multiplying the base by the index.
• 💡Write down the two exact powers you trapped the value between, because that is the step being credited.
• 💡Round to one significant figure first, then adjust up or down and say which way you moved.
• 💡Square or cube your estimate back and check it lands near the number you started with.
• 💡Check the bases match before you add or subtract any indices; if they do not, work each power out separately.
• 💡Where the answer must be a single power, leave it as a power and do not evaluate it.
• 💡Turn a negative index into a fraction as your first move, because the arithmetic is easier from there.
• 💡For a cube root, ask what number multiplied by itself three times gives the value, so ∛64 = 4 because 4 × 4 × 4 = 64.
• 💡When dealing with a negative fractional index, handle the negative sign first by writing the reciprocal, then evaluate the root and power.
• 💡To keep calculations manageable, always perform the root operation before applying the power.
• 💡The words 'exact' or 'in terms of π' are instructions not to convert your answer to a rounded decimal.
• 💡Check whether your final fraction can be simplified; for example, 6/20 should be written as 3/10.
• 💡Learn the square numbers as far as 144 so you can spot the largest square factor quickly.
• 💡Keep fractions, surds and π exact the whole way through; rounding partway loses the accuracy mark.
• 💡After rationalising, check that no root is left in the denominator before writing the final line.
• 💡Check the index against the size of the number: a positive number smaller than 1 always takes a negative index.
• 💡When adding or subtracting and the indices are close, converting to ordinary numbers is often faster than matching the powers.
• 💡Give the answer in standard form whenever the question asks for it, even when the numbers you were given were written out in full.
• 💡Always show your working, especially for multi-step calculations. Method marks are often available even if the final answer is wrong.
• 💡Check whether a question asks for an exact answer or a rounded one. For standard form, ensure your value of A is between 1 and 10.
• 💡Use your calculator efficiently: learn how to input powers, roots, and standard form using the EXP or x10^x button.

• treating −8 as bigger than −3 because 8 is bigger than 3
• judging a decimal by how many digits it has, so calling 0.125 larger than 0.4
• assuming the fraction with the bigger denominator is the bigger fraction, so ranking 1/8 above 1/3
• reading ≥ as strictly greater, and so dropping a value that is allowed to match
• Adding fraction numerators and denominators without finding a common denominator (e.g., 1/2 + 1/3 = 2/5). Correction: Find a common denominator first: 3/6 + 2/6 = 5/6.
• Scaling only the divisor when dividing by a decimal (e.g., 7.5 ÷ 0.25 becomes 7.5 ÷ 25). Correction: Scale both numbers by the same power of 10 to maintain equivalence: 750 ÷ 25.
• Incorrectly applying sign rules, especially with multiple negatives (e.g., 5 − (−2) = 3 or −4 × −3 = −12). Correction: Two negatives make a positive in multiplication, division, and when subtracting a negative: 5 + 2 = 7 and −4 × −3 = 12.
• Working strictly from left to right, so calculating 2 + 3 × 4 as 5 × 4 = 20 rather than 2 + 12 = 14.
• Calculating −3² as (−3) × (−3) = 9, when the index applies only to the 3, so the value is −(3 × 3) = −9.
• Incorrectly cancelling terms instead of factors in an expression, e.g. simplifying (x+2)/2 to x+1 instead of x/2 + 1.
• Confusing the reciprocal with the negative of a number.
• Confusing factors and multiples, e.g. listing multiples of 12 when asked for factors.
• Calling 1 a prime number, or not recognising that 2 is the only even prime.
• Stopping a factor tree at a composite branch such as 6 or 10 and treating it as a prime.
• Swapping the methods for HCF and LCM, for example by using the highest powers for the HCF.
• listing in a random order, losing track, and writing the same pair twice
• missing the cases that begin with the last item because the list was abandoned part way through
• counting an arrangement twice when the question treats the two orders as the same choice
• including an option the question ruled out, such as reusing a digit when repeats are not allowed
• Creating a disorganised list which misses outcomes or includes duplicates. Correction: Fix the first item and cycle through all options for the second, then change the first item and repeat the process.
• Adding the number of options at each stage instead of multiplying them (e.g., 4 + 5 + 3 instead of 4 × 5 × 3 for the menu problem). Correction: The product rule applies to sequential choices, so you must multiply.
• Forgetting to reduce the number of available options when repeats are not allowed. Correction: If an item is used, it cannot be used again, so the pool of options for the next choice is smaller by one.
• reading 3⁴ as 3 × 4 and answering 12 rather than 81
• giving only the positive square root where a question about x² = 25 needs both 5 and −5
• confusing a cube root with dividing by three, so giving the cube root of 27 as 9
• assuming a square root is always smaller than the number, which fails between 0 and 1 where √0.25 = 0.5
• reaching for the calculator on a question that says estimate, which throws away the marks for the bounds
• rounding the base up and then claiming the estimate is too small when it is too large
• assuming a root sits halfway between the two whole numbers around it, so calling √50 about 7.5
• estimating the square root of a number between 0 and 1 as smaller than the number, when in that range the root is the larger of the two
• multiplying the indices when the powers are multiplied, so writing 2³ × 2⁴ as 2¹²; the indices are added, giving 2⁷
• reading a negative index as a negative answer, so giving 2⁻³ as −8; the correct value is 1/8
• combining indices when the bases are different, so turning 2³ × 3² into a single power; work each power out separately instead
• halving instead of rooting, so giving the square root of 36 as 18; the correct value is 6
• confusing a cube root with a square root, so giving ∛27 as 9; the correct value is 3 because 3³ = 27, whereas √27 is not 9 (9 is (∛27)² or 27 ÷ 3)
• Multiplying the base by the index, for example calculating 25^½ as 12.5 instead of √25 = 5.
• Confusing the numerator and denominator of the index, for example treating 8^(2/3) as (√8)³ instead of (³√8)².
• Incorrectly handling a negative index by making the answer negative, for example calculating 9^(−1/2) as −3 instead of 1/3.
• Adding or subtracting denominators as well as numerators, e.g. 1/3 + 1/4 = 2/7. Correction: find a common denominator, so 1/3 + 1/4 = 4/12 + 3/12 = 7/12.
• Using a calculator to give a rounded decimal when an exact answer is requested. Correction: leave answers as simplified fractions or as multiples of π.
• Forgetting to invert the second fraction when dividing, e.g. 3/5 ÷ 2/7 = 3/5 × 2/7. Correction: division requires multiplying by the reciprocal, so 3/5 × 7/2 = 21/10.
• Treating π as 3.14 and giving a decimal answer when asked for an exact value. Correction: keep π as a symbol, e.g. 16π.
• writing √a + √b as √(a + b), which already fails for √9 + √16, since that is 7 and not 5
• multiplying only the denominator by the root when rationalising, which changes the value of the fraction
• rounding a multiple of π or a surd partway through, so the final answer is no longer exact
• changing the sign of only one term when multiplying by the conjugate, so the roots fail to cancel
• leaving the front number outside the range, so offering 0.45 × 10⁴ as a final answer
• counting digits rather than place value shifts, so writing 0.0035 as 3.5 × 10⁻² rather than 3.5 × 10⁻³
• adding the front numbers as they stand when the two powers of ten are different
• copying a calculator display in its own shorthand instead of rewriting it in standard form
• Misconception: In BIDMAS, division always comes before multiplication. Correction: Division and multiplication have equal priority and are performed left to right.
• Misconception: 1 is a prime number. Correction: 1 is not prime because it has only one factor (itself), not two distinct factors.
• Misconception: When writing a number like 0.00056 in standard form, the power of 10 is positive. Correction: For numbers less than 1, the power is negative because you move the decimal point to the right.

1. 1Day 1-2: Revise BIDMAS and practice evaluating expressions with multiple operations, including brackets and indices.
2. 2Day 3-4: Learn and practice finding factors, multiples, HCF, and LCM using prime factor decomposition.
3. 3Day 5-6: Master powers and roots, including negative and fractional indices (for Higher tier).
4. 4Day 7-8: Study standard form: converting between ordinary numbers and standard form, and performing calculations with numbers in standard form.
5. 5Day 9-10: Complete past paper questions on Number - Structure and calculation, timing yourself to build exam technique.

• 📋Short calculation questions: e.g., 'Evaluate 5 + 3 x 2^2'. Advice: Apply BIDMAS carefully and show each step.
• 📋HCF/LCM problems: e.g., 'Find the highest common factor of 24 and 36'. Advice: Use prime factor trees or listing method, and double-check your answer.
• 📋Standard form conversion: e.g., 'Write 0.00034 in standard form'. Advice: Ensure the number is in the form A x 10^n with 1 ≤ A < 10.
• 📋Problem-solving with standard form: e.g., 'Calculate (3 x 10^4) x (2 x 10^5), giving your answer in standard form'. Advice: Multiply the numbers and add the indices, then adjust to standard form if needed.