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    Structure and calculation — AQA GCSE Mathematics

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    Structure and calculation explained

    Ordering numbers means putting them in size order, and you have to read the direction the question asks for.

    Read the full explanation

    Negatives run the opposite way to positives, so −7 is smaller than −2 because it sits further left on the number line. To compare decimals, line up the decimal points and read digit by digit from the left, which is why 0.4 is larger than 0.35: four tenths outweigh three tenths. To compare fractions, rewrite them over a common denominator, or turn each into a decimal, so 3/5 becomes 0.6 and beats 4/7 at roughly 0.571. The strict signs < and > say one side is definitely bigger; ≤ and ≥ also allow the two sides to match; ≠ says only that the values differ. A typical question hands you a mixed list of fractions, decimals and negatives to arrange.

    apply the four operations, including formal written methods, to integers, decimals and simple fractions (proper and improper), and mixed numbers – all both positive and negative understand and use place value (eg when working with very large or very small numbers, and when calculating with decimals)

    Formal written methods rely on place value; keep columns and decimal points aligned. For decimal multiplication, multiply as integers then reinsert the total number of decimal places, eg 0.4 × 0.03 becomes 4 × 3 = 12, then 0.012. To divide by a decimal, multiply both numbers by a power of ten to make the divisor an integer. For negative numbers, multiplying or dividing two numbers with the same sign gives a positive result; different signs give a negative. Subtracting a negative is equivalent to adding a positive, eg 5 – (–2) = 5 + 2 = 7. For fractions, always convert mixed numbers to improper fractions first. To add or subtract, find a common denominator, convert the fractions, then add or subtract the numerators, eg 2/3 + 1/4 = 8/12 + 3/12 = 11/12. To multiply, multiply the numerators and multiply the denominators. To divide, invert the second fraction and multiply.

    recognise and use relationships between operations, including inverse operations (eg cancellation to simplify calculations and expressions) use conventional notation for priority of operations, including brackets, powers, roots and reciprocals

    The operations come in pairs that undo one another: addition and subtraction, multiplication and division, squaring and taking a square root, and for any number that is not zero, taking the reciprocal twice returns you to the start. That is what makes cancelling valid, so in (14 × 9)/(7 × 9) the factor of 9 on the top and the bottom divides out before you work anything else out. Priority of operations comes next. Brackets first, then powers and roots, then multiplication and division from left to right, then addition and subtraction from left to right. A fraction bar and a root sign both group what they contain, so finish everything underneath before dividing or rooting. That gives 3 + 4 × 2² as 3 + 4 × 4, then 3 + 16, then 19.

    use the concepts and vocabulary of prime numbers, factors (divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple, prime factorisation, including using product notation and the unique factorisation theorem

    A factor (or divisor) of a number divides into it exactly, so the factors of 12 are 1, 2, 3, 4, 6 and 12. A multiple is what you get by multiplying the number by a whole number, so the multiples of 5 are 5, 10, 15, 20 and so on. A common factor belongs to both lists, for example 4 is a common factor of 12 and 20, and a common multiple appears in both, for example 60 is a common multiple of 12 and 20. A prime has exactly two factors, itself and 1, which is why 1 is not prime and 2 is the only even prime. Every whole number above 1 splits into a product of primes in one way only apart from order, the unique factorisation theorem. Build the product with a factor tree or repeated division, then write it with indices, for example 360 = 2³ × 3² × 5. The highest common factor is the product of the shared primes at the lower index; the lowest common multiple uses every prime at the higher index.

    apply systematic listing strategies

    A systematic list is one written in an order you decide before you start, so nothing is missed and nothing is written twice. The usual method is to hold one item fixed and run the other item through every possibility, then move the fixed item on. To list the two-digit numbers made from the digits 1, 4 and 7 without reusing a digit, fix the first digit as 1 and write 14 and 17, then fix 4 and write 41 and 47, then fix 7 and write 71 and 74, which gives six numbers. Other questions ask for every pair chosen from a group of people, every outcome when two dice are rolled, or every combination of a main course with a dessert. A grid or a two-way table is often the tidiest way to record them.

    including use of the product rule for counting (Higher tier only)

    This is Higher tier only. The product rule for counting says that if one choice can be made in m ways and a second, separate choice can be made in n ways, then the two together can be made in m × n ways, and the idea extends to any number of stages. A menu with 5 starters, 4 main courses and 3 desserts gives 5 × 4 × 3 = 60 meals. When the choices interfere with each other, the count drops at each stage: the number of four-digit codes using digits 0 to 9 with no digit repeated is 10 × 9 × 8 × 7 = 5040, because one digit is used up every time. Multiply, do not add. Adding counts the ways of making one choice or another, while multiplying counts the ways of making one choice and then the next.

    use positive integer powers and associated real roots (square, cube and higher), recognise powers of 2, 3, 4, 5

    A power records repeated multiplication, so 2⁵ means 2 × 2 × 2 × 2 × 2, which is 32, and the matching root undoes it, making the fifth root of 32 equal 2. Several families are worth knowing on sight. The powers of 2 are 2, 4, 8, 16, 32, 64, 128 and 256; the powers of 3 are 3, 9, 27, 81 and 243; the powers of 4 are 4, 16, 64 and 256; the powers of 5 are 5, 25, 125 and 625. Learn the squares of the whole numbers up to 15 and the cubes of 2, 3, 4, 5 and 10 as well. Two details catch people out: a positive number has two square roots, one positive and one negative, while a cube root has a single real value that keeps the sign of the number, so the cube root of −8 is −2.

    estimate powers and roots of any given positive number (Higher tier only)

    This is Higher tier only. To estimate a root, trap the number between the two nearest values you already know. √50 lies between √49 and √64, so it is between 7 and 8, and much nearer 7 because 50 is only a little above 49. Cube roots work the same way: the cube root of 30 lies between 3 and 4 because 3³ = 27 and 4³ = 64, and it sits nearer 3. To estimate a power, round the base to something convenient before you raise it, so 5.2³ is roughly 5³ = 125, and 1.98⁶ is a little under 2⁶ = 64. Questions of this kind usually ask which two whole numbers a root lies between, or ask for an estimate you then compare with a calculator answer to show it is sensible.

    calculate with roots, and with integer indices

    The index laws let you handle powers without writing them out. Multiplying powers of the same base adds the indices, aᵐ × aⁿ = aᵐ⁺ⁿ. Dividing them subtracts, aᵐ ÷ aⁿ = aᵐ⁻ⁿ. A power of a power multiplies, (aᵐ)ⁿ = aᵐⁿ. Any base that is not zero raised to the power zero gives 1, and a negative index means a reciprocal, so 5⁻² = 1/5² = 1/25. Roots obey matching rules: √a × √b = √(ab), and √a ÷ √b = √(a/b) provided a is not negative and b is positive, which is why √3 × √12 = √36 = 6. A common question hands you something like (2³ × 2⁵)/2⁴ and asks for it as a single power, which here is 2⁴.

    calculate with fractional indices (Higher tier only)

    This is Higher tier only. A fractional index is a root written another way. An index of ½ means the square root, so 49^½ = 7, and an index of 1/n means the nth root. Where the numerator is not 1, the denominator gives the root and the numerator gives the power, so 8^(2/3) tells you to take the cube root of 8 and then square it, giving 2² = 4. Take the root first every time, because it keeps the numbers small enough to handle. A negative fractional index adds a reciprocal on top of that: 16^(−3/4) is 1 divided by 16^(3/4), and 16^(3/4) is the fourth root of 16, then cubed, which is 2³ = 8, so the answer is 1/8. Every index law you already use still holds for fractional indices.

    calculate exactly with fractions

    Working exactly means the answer stays a fraction rather than becoming a rounded decimal. To add or subtract, rewrite both fractions over a common denominator and then work on the numerators only, so 2/3 + 1/4 becomes 8/12 + 3/12, which is 11/12. To multiply, multiply the numerators and multiply the denominators, cancelling any common factor first to keep the numbers small. To divide, turn the second fraction over and multiply. Convert a mixed number into an improper fraction before multiplying or dividing, so 1½ becomes 3/2. Simplify at the end, and turn the result back into a mixed number if the question began with one. On a non-calculator paper an exact answer such as 7/12 earns the mark where a rounded 0.58 does not.

    calculate exactly with multiples of π

    An answer given in terms of π keeps the symbol in place instead of a rounded decimal, which makes it exact. Treat π like a letter in algebra: 3π + 5π = 8π, and 12π ÷ 4 = 3π. Circle work is where this appears most. The circumference = 2πr = πd, and the area of a circle = πr², so a circle of radius 5 cm has circumference 10π cm and area 25π cm². A semicircle of radius 6 cm has area 18π cm², because half of 36π is 18π. Arc lengths and sector areas behave the same way once you have taken the correct fraction of the whole circle. Only type π into a calculator at the very end, and only when the question asks for a rounded answer.

    calculate exactly with surds simplify surd expressions involving squares (eg √12 = √(4 × 3) = √4 × √3 = 2√3) and rationalise denominators (Higher tier only)

    This is Higher tier only. A surd is a root that cannot be written exactly as a fraction, such as √2. To simplify one, split the number under the root into a square factor and the rest, so √50 = √25 × √2 = 5√2. Like terms collect, giving 3√2 + 4√2 = 7√2, while √2 + √3 cannot be combined at all. Multiplication uses √a × √b = √(ab), and √2 × √2 = 2, which is what makes brackets expand tidily. Rationalising a denominator means clearing the root from the bottom: multiply top and bottom of 6/√3 by √3 to reach 6√3/3, which simplifies to 2√3. Where the denominator has two terms, multiply top and bottom by the same expression with the middle sign reversed, so 1/(3 + √5) becomes (3 − √5)/4.

    calculate with and interpret standard form A × 10ⁿ, where 1 ≤ A < 10 and n is an integer

    This notation keeps very large and very small numbers short by writing them as a value that is at least 1 and below 10 multiplied by a power of ten. The index counts how far the decimal point has moved, so 4500000 is 4.5 × 10⁶, and 0.00072 is 7.2 × 10⁻⁴, where the negative index shows a number between 0 and 1. To multiply, multiply the front values and add the indices; to divide, divide the front values and subtract the indices. Then check that the front value is still at least 1 and below 10, adjusting if it is not, so 40 × 10⁵ is rewritten as 4 × 10⁶. To add or subtract, either turn both numbers into ordinary form first, or rewrite them with the same index and then combine the front values.

    Your focus

    1. Write down which of =, ≠, <, >, ≤ and ≥ fits a pair of values, saying which two signs also allow the sides to match.
    2. Work out the order of a mixed list of negatives, decimals and fractions by first converting every value to one common form.
    3. Explain why −7 is smaller than −2 and why 0.4 is larger than 0.35, using number line position and place value.
    Show all 48 objectives
    1. Calculate with integers and decimals in column layouts, keeping place value columns and decimal points lined up.
    2. Work out a decimal multiplication by multiplying as whole numbers then restoring the decimal places, so 0.4 × 0.03 is 0.012.
    3. Calculate with mixed numbers by rewriting them as improper fractions first, then giving the answer back as a mixed number.
    4. Explain why dividing by a decimal begins by multiplying both numbers by a power of ten, and what subtracting a negative does.
    5. Write down the order operations are carried out in, and say what a fraction bar and a root sign each group together.
    6. Calculate the value of an expression such as 3 + 4 × 2², settling the power before the multiplication to reach 19.
    7. Explain how an inverse operation checks an answer, and justify cancelling the shared factor of 9 in (14 × 9)/(7 × 9).
    8. Explain why 1 is not prime and 2 is the only even prime, using the rule that a prime has exactly two factors.
    9. Use the vocabulary of factors (divisors), multiples, common factors and common multiples, for example listing the factors of 12 and the multiples of 5.
    10. Work out a number as a product of primes in index form, so 360 becomes 2³ × 3² × 5, using a factor tree or repeated division.
    11. Find the highest common factor by taking shared primes at the lower index and the lowest common multiple at the higher index.
    12. Justify a lowest common multiple by checking that both starting numbers divide into it exactly.
    13. Describe a systematic order for listing, such as holding one item fixed while the other runs through every possibility.
    14. Work out every two-digit number formed from given digits without reuse, listing all six for the digits 1, 4 and 7.
    15. Complete a sample space diagram for two events and explain how it shows no outcome was missed or written twice.
    16. Explain the difference between multiplying counts for one choice and then another and adding them for one choice or another.
    17. Calculate the total number of outcomes by multiplying the choices at each stage, so 5 starters, 4 mains and 3 desserts give 60 meals.
    18. Work out how many four-digit codes have no repeated digit, reducing the choice count by one at each stage to reach 5040.
    19. Write down the powers of 2, 3, 4 and 5 on sight, with the squares up to 15 and the cubes of 2, 3, 4, 5 and 10.
    20. Work out a power as a repeated multiplication and find the root that undoes it, so the fifth root of 32 is 2.
    21. Explain why a positive number has two square roots while a cube root has one that keeps the sign, so the cube root of −8 is −2.
    22. Write down the two whole numbers a root lies between, such as √50 between 7 and 8, and say which end it sits nearer.
    23. Estimate a power by rounding the base to a convenient value first, so 5.2³ is roughly 5³ = 125.
    24. Explain whether a calculated value is believable by comparing it with your estimate and saying why the estimate falls where it does.
    25. Write down the index laws for multiplying, dividing and raising a power to a power, and what a zero or negative index means.
    26. Simplify an expression such as (2³ × 2⁵)/2⁴ to a single power by combining the indices before any value is worked out.
    27. Work out √3 × √12 using the rule that √a × √b = √(ab), giving 6.
    28. Explain why 5⁻² is 1/25 rather than a negative number, treating the minus sign as a reciprocal.
    29. Explain what each part of a fractional index does, with the denominator giving the root and the numerator giving the power.
    30. Calculate a value such as 8^(2/3) by taking the cube root first and then squaring it, reaching 4.
    31. Justify taking the root before the power, and show that 16^(−3/4) is 1/8 rather than a negative value.
    32. Describe how to add, subtract, multiply and divide fractions, naming the common denominator and the inverted second fraction.
    33. Work out 2/3 + 1/4 exactly by rewriting both over a common denominator and combining the numerators only.
    34. Explain why an exact answer such as 7/12 earns the mark on a non-calculator paper where a rounded 0.58 does not.
    35. Calculate the circumference and the area of a circle in terms of π, so a radius of 5 cm gives 10π cm and 25π cm².
    36. Work out a semicircle or sector by taking the right fraction of the whole circle and collecting the multiples of π.
    37. Explain why an answer left in terms of π is exact, and say when a decimal value should be typed in instead.
    38. Simplify a surd by splitting the number under the root into a square factor and the rest, so √50 becomes 5√2.
    39. Work out 6/√3 and 1/(3 + √5) with the root cleared from the denominator, reaching 2√3 and (3 − √5)/4.
    40. Show that (2 + √3)(2 − √3) is 1 by expanding, and say why √3 × √3 removes the root.
    41. Explain why 3√2 + 4√2 collects to 7√2 while √2 + √3 cannot be combined at all.
    42. Write down a number in the form A × 10ⁿ with A at least 1 and below 10, so 0.00072 becomes 7.2 × 10⁻⁴.
    43. Calculate a product or quotient in standard form by combining the front values and adding or subtracting the indices.
    44. Work out an addition in standard form by rewriting both numbers with the same index, or as ordinary numbers first.
    45. Explain what a negative index says about the size of a number, and why 0.35 × 10⁷ must be adjusted to 3.5 × 10⁶.

    Structure and calculation exam tips

    Quick Revision Summary (Key Takeaway)

    Structure and calculation covers the fundamental rules of number: place value, the four operations, order of operations (BIDMAS), rounding and estimation, factors, multiples, primes, powers, roots, and standard form. Mastery of these skills is essential for all other GCSE Mathematics topics, including algebra, ratio, and problem solving.

    Topic Overview

    Structure and calculation is the foundation of GCSE Mathematics. It covers the rules and conventions for working with numbers, including the order of operations, rounding, estimation, factors, multiples, primes, powers, roots, and standard form. These skills are essential for solving problems in algebra, geometry, ratio, proportion, and statistics.

    This topic also develops numerical fluency and problem-solving skills. You will learn to perform calculations accurately, make sensible estimates, and express numbers in different forms. A strong grasp of structure and calculation is crucial for success in both calculator and non-calculator exams, and it underpins many real-world applications.

    Key Concepts
    • →Order of operations (BIDMAS): Brackets, Indices, Division and Multiplication (left to right), Addition and Subtraction (left to right).
    • →Place value and rounding: Understanding the value of each digit, rounding to decimal places and significant figures, and using estimation to check answers.
    • →Factors, multiples, primes, and prime factorisation: Finding factors and multiples, identifying prime numbers, and expressing a number as a product of its prime factors.
    • →Powers and roots: Calculating squares, cubes, and higher powers, and finding square roots, cube roots, and other roots, including using index laws.
    • →Standard form: Writing very large or very small numbers as A x 10^n where 1 ≤ A < 10 and n is an integer, and performing calculations with numbers in standard form.
    Marking Points
    • converting every value to the same form, all decimals or all fractions over a common denominator, even if the final order is wrong
    • a correct comparison written as a chain, such as −5 < −1 < 0.3 < 4/5
    • the finished list in the direction the question asked, smallest first or largest first as stated
    • choosing the correct symbol when a question asks you to fill a gap between two values
    • a written method set out in columns with the place value columns lined up, even if there is an arithmetic slip
    • finding a correct common denominator before adding or subtracting fractions
    • converting a mixed number into an improper fraction before performing a calculation
    • a correct final answer in the form and accuracy requested
    • Use place value at different scales: in 3,400,000 the 3 represents three million and the 4 four hundred thousand; in 0.004 the 4 represents four thousandths.
    • For long multiplication, align the partial products by place value: 23 × 14 = 92 + 230 = 322. For 156 ÷ 12, subtract 120 (ten lots), then 36 (three lots), giving 13.
    • evaluating the bracket first and writing the reduced expression before going on
    • dealing with the power or the root before the multiplication or division
    • showing a cancelled form, such as dividing top and bottom by a common factor, even if the arithmetic after it is wrong
    • the correct final value
    • a complete factor tree or repeated division that ends with primes only
    • the product of primes written in index form, for example 360 = 2³ × 3² × 5
    • selecting the shared primes with the lower index for the highest common factor
    • a lowest common multiple that both original numbers divide into exactly
    • correct use of the vocabulary of factors (divisors), multiples, common factors and common multiples in a definition or listing question
    • a list that shows a clear order, such as all the options starting with the smallest value first
    • a complete list with no repeated outcome
    • a correctly labelled sample space diagram or two-way table where two events are involved
    • the count or the probability that the finished list leads to
    • identifying how many choices are available at each stage
    • multiplying those counts together rather than adding them
    • reducing the count at each stage when an item cannot be reused
    • the final total
    • writing a power out as a repeated multiplication when its value is not known by sight
    • the correct value of the power or of the root
    • giving both the positive and the negative square root when a question asks for every value
    • using a recognised power in a related step, such as seeing 64 as 2⁶ and as 4³
    • naming the two known powers or roots that the value lies between
    • the rounded value used in the estimate, such as replacing 5.2 by 5
    • the estimate itself, together with which end it is closer to where the question asks for that
    • a sensible comment comparing the estimate with the calculated value
    • using the correct index law and showing the combined index before any value is worked out
    • rewriting a negative index as a reciprocal
    • a correct root simplification, even if the arithmetic that follows goes wrong
    • the final single power, or for the numerical value where that is what was asked for
    • rewriting the fractional index as a root and a power
    • evaluating the root correctly, even if the power applied afterwards is wrong
    • treating the negative sign as a reciprocal rather than as a negative answer
    • the exact final value, left as a fraction where it is not a whole number
    • a correct common denominator with both numerators changed to match it
    • converting a mixed number to an improper fraction before multiplying or dividing
    • inverting the second fraction in a division and then multiplying
    • the final fraction in its simplest form, or as a mixed number where the question used those
    • substituting the radius into the correct circle formula
    • an answer left in terms of π rather than converted to a decimal
    • collecting the multiples of π correctly when several parts are added together
    • the correct units, squared units for an area and ordinary length units for a perimeter
    • writing the number under the root as a square factor multiplied by the remainder
    • collecting like surds once each term has been simplified
    • multiplying top and bottom by the correct expression when rationalising
    • the exact simplified answer with the fraction reduced
    • a number correctly converted, with the front value at least 1 and below 10
    • adding or subtracting the indices when powers of ten are multiplied or divided
    • adjusting a result such as 0.35 × 10⁷ into proper standard form
    • the final answer written in standard form rather than as an ordinary number when that is what the question asked for
    Examiner Tips
    • 💡Check the direction before you write anything; if it says smallest first, confirm your first value really is the smallest.
    • 💡Turn a mixed list into decimals to compare it, then write the answer using the original numbers unless the question says otherwise.
    • 💡Mark each converted value next to the original in your working so the examiner can follow the comparison.
    • 💡Set out columns clearly so the calculation and place values can be followed. Correct layout alone does not establish a correct method.
    • 💡Estimate first by rounding each number to one significant figure, so an answer ten times too big shows up straight away.
    • 💡On a non-calculator paper, check a division by multiplying your answer back up to the number you started with.
    • 💡Rewrite the calculation one line at a time, doing a single operation per line, so the order you used is visible.
    • 💡Put your own brackets round the whole numerator and the whole denominator when typing a fraction into a calculator.
    • 💡When a question asks you to insert brackets to make a statement true, test each position rather than guessing.
    • 💡Write both numbers in index form before you start; the two answers then read straight off the primes.
    • 💡Check that your highest common factor divides both numbers exactly, and that both numbers divide into your lowest common multiple exactly.
    • 💡In a word problem about repeating events meeting again, it is the lowest common multiple you want, not the highest common factor.
    • 💡Decide the order before you write your first outcome, then follow it without jumping about.
    • 💡Count your list at the end and compare it with a quick check of how many there should be.
    • 💡Cross out a duplicate rather than rubbing it out, so the examiner can still see the method.
    • 💡Draw an empty box for each stage, write the number of choices in each box, then multiply across the row.
    • 💡Look for the words 'no repeats' or 'cannot be the same', because they change every count after the first.
    • 💡If the total looks impossibly large, check whether the question restricted any of the stages.
    • 💡Learn the powers of 2 up to 256; they appear again in index questions, standard form and sequences.
    • 💡On a calculator paper use the power key rather than repeated multiplication, so you cannot lose count of the factors.
    • 💡Check a root by raising your answer back to that power before you move on.
    • 💡Write down the two square numbers or cube numbers on either side first; the estimate follows from them.
    • 💡Round to one significant figure unless told otherwise, and state what you rounded to.
    • 💡Use the estimate as a check: if the calculator answer is nowhere near it, retype the calculation.
    • 💡Rewrite every term as a power of the same base before you reach for any index law.
    • 💡If the question says 'give your answer as a power of 2', stop at the power rather than working the number out.
    • 💡Deal with a bracket raised to a power by applying the index to every factor inside it.
    • 💡Write the index as a root sign before doing any arithmetic; it turns an abstract question into one you can do in stages.
    • 💡Check an answer by raising it to the reciprocal index and seeing whether you get back to the original number.
    • 💡Where a question mixes indices, convert everything to the same base first and then combine.
    • 💡Use the lowest common denominator rather than multiplying the denominators together; the simplifying at the end is then much shorter.
    • 💡When the question says 'exact', leave the fraction in place and do not reach for a decimal.
    • 💡Write the common denominator line out even when you could add the fractions mentally, because it carries the method mark.
    • 💡The phrase 'give your answer in terms of π' means no decimal is wanted; the π stays in the answer.
    • 💡Work with the number in front of π and leave the symbol to one side until the last line.
    • 💡Add the straight edge separately when a shape has one, because the perimeter of a semicircle is not only the curved part.
    • 💡Learn the square numbers so you can spot the largest square factor at once; √72 then goes straight to 6√2 rather than through 2√18.
    • 💡Keep everything in surd form to the last line, because a decimal partway through loses the exactness being tested.
    • 💡Expand brackets containing surds term by term in the usual way, then collect the like surds.
    • 💡Say out loud which way the decimal point moved; a number between 0 and 1 always gives a negative index.
    • 💡Use the standard form key on your calculator, and check the display is not silently showing an ordinary number.
    • 💡To order numbers written this way, compare the indices first and only compare the front values when the indices match.
    • 💡Always show your working, especially in multi-step calculations. Method marks are often available even if the final answer is wrong.
    • 💡When using a calculator, write down intermediate values to avoid rounding errors and to show your method.
    • 💡Check whether the question asks for an exact answer, an estimate, or a rounded answer. Pay attention to the required form (e.g., standard form, significant figures).
    Common Mistakes
    • treating −8 as larger than −3 because 8 is larger than 3
    • comparing decimals by how many digits they have, so calling 0.125 bigger than 0.4
    • assuming the fraction with the larger denominator is the larger fraction
    • reading ≤ as strictly less than, and so leaving out a value that is allowed to match
    • lining up the last digits instead of the decimal points for addition, eg in 12.3 + 4.56
    • multiplying decimals and putting the decimal point in the wrong place, eg 0.6 × 0.7 = 4.2 instead of 0.42
    • adding the denominators as well as the numerators when adding two fractions, eg 1/3 + 1/4 = 2/7
    • confusing the rules for negative numbers, eg calculating 5 - (-2) as 3 instead of 7
    • working strictly left to right, so reading 2 + 3 × 5 as 25 rather than 17
    • squaring the product instead of the single term, so reading 3 × 4² as (3 × 4)²
    • missing the grouping in a fraction, so treating the line in (8 + 4)/2 as if only the 4 were divided
    • using the reciprocal key twice on a calculator and undoing the step that was wanted
    • counting 1 as prime, or calling 9 prime because it is odd
    • stopping a factor tree at a composite number such as 4 or 6 and writing that into the answer
    • multiplying the two numbers together for the lowest common multiple, which is only right when they share no factor above 1
    • swapping the two answers over, so giving a highest common factor that is larger than both numbers
    • confusing factors with multiples, for example listing 2, 4, 6 as the factors of 6
    • writing outcomes in a random order and missing one or two near the end
    • counting the same pair twice because it has been written in both orders when order does not matter
    • including repeats such as 11 when the question says the digits must be different
    • stopping after a handful of outcomes instead of working through every case
    • adding the number of options at each stage instead of multiplying them
    • keeping the count the same at every stage when repeats are banned, using 10 × 10 × 10 where 10 × 9 × 8 is needed
    • treating two arrangements as different when the question counts them as the same, such as picking a team where order does not matter
    • ignoring a restriction, for example a code that is not allowed to begin with 0
    • reading 3⁴ as 3 × 4 and writing 12 instead of 81
    • giving only the positive root when x² = 49 asks for all values of x
    • confusing squaring with doubling, or cubing with trebling
    • assuming that a negative number has no cube root
    • halving instead of rooting, so giving √50 as 25
    • placing a root exactly halfway between the two whole numbers, ignoring how close the number is to each square
    • rounding after working the power out rather than before, which removes the point of estimating
    • presenting an estimate as though it were the exact answer
    • multiplying the indices when the powers are multiplied, so writing 2³ × 2⁵ as 2¹⁵
    • assuming a negative index makes the answer negative, so writing 3⁻² as −9 rather than 1/9
    • adding the numbers under two separate root signs, so treating √9 + √16 as √25
    • applying an index law to two powers that do not share the same base
    • reading 8^(2/3) as two thirds of 8
    • raising to the power first and then trying to root a very large number
    • letting a negative index turn the answer negative instead of producing a reciprocal
    • taking the numerator as the root and the denominator as the power, which is the wrong way round
    • adding the denominators as well as the numerators, so writing 1/2 + 1/3 as 2/5
    • multiplying whole number parts and fraction parts separately when multiplying two mixed numbers
    • cancelling across an addition, when cancelling is only valid across a multiplication
    • giving a rounded decimal where the question asked for an exact value
    • putting the diameter where the formula needs the radius, which makes an area four times too big
    • rounding to 3.14 partway through a question that asked for an exact answer
    • dropping the π from the final line after using it correctly in the working
    • forgetting to halve a semicircle, or halving the radius rather than the area
    • writing √a + √b as √(a + b), so turning √9 + √16 into 5 instead of 7
    • choosing a factor that is not a square, so writing √50 as √10 × √5 and getting no further
    • leaving a root on the bottom of a fraction when the question asked for a rationalised denominator
    • reversing the wrong sign when multiplying by the expression that clears a two-term denominator
    • leaving the front value outside the allowed range, so writing 35 × 10³ and stopping there
    • counting zeros instead of decimal point moves, so writing 0.0004 as 4 × 10⁻³
    • adding the front values as well as the indices when two numbers are being multiplied
    • trying to add two numbers straight away when their indices are different
    • Students often think that multiplication always comes before division in BIDMAS, but they have equal priority and should be done left to right. Similarly for addition and subtraction.
    • When rounding to significant figures, students may start counting from the first zero after the decimal point, but significant figures start from the first non-zero digit.
    • Students sometimes believe that 1 is a prime number, but a prime number must have exactly two distinct factors: 1 and itself. 1 has only one factor, so it is not prime.
    Revision Plan
    1. 1Day 1-2: Revise order of operations (BIDMAS) and practice with negative numbers and indices. Complete a set of mixed problems.
    2. 2Day 3-4: Focus on rounding, estimation, and significant figures. Practice rounding to decimal places and significant figures, and use estimation to check calculations.
    3. 3Day 5-6: Study factors, multiples, primes, and prime factorisation. Practice finding HCF and LCM using prime factors.
    4. 4Day 7-8: Learn powers, roots, and index laws. Practice calculations involving squares, cubes, and roots, including fractional and negative indices.
    5. 5Day 9-10: Master standard form: converting between ordinary and standard form, and multiplying/dividing numbers in standard form. Complete past paper questions on all topics.
    Exam Question Types
    • 📋Straightforward calculation questions: e.g., 'Calculate the value of ...' or 'Work out ...'. Show all steps clearly.
    • 📋Problem-solving with number properties: e.g., 'Find the highest common factor of ...' or 'Express ... as a product of prime factors'. Use systematic methods.
    • 📋Estimation and rounding: e.g., 'Estimate the value of ...' or 'Round ... to ... significant figures'. Pay attention to the required degree of accuracy.
    • 📋Standard form: e.g., 'Write ... in standard form' or 'Calculate ... giving your answer in standard form'. Ensure the coefficient is between 1 and 10.
    Command Word Expectations (AQA)
    Calculate

    Work out the value of something, showing all necessary steps. A correct answer with no working may still gain full marks, but working is expected.

    Estimate

    Give an approximate value by rounding numbers to 1 significant figure (unless otherwise specified). Show the rounded values and the calculation.

    Write

    Express a number or expression in a specific form, e.g., standard form or prime factor form. Ensure the final answer meets the required format.

    How Students Lose Marks (Examiner Pitfalls)
    Pitfall: Misapplying the order of operations (BIDMAS) when brackets and indices are combined, especially with negative numbers and powers.
    ❌ Weak Answer (Loses Marks):For 3 + 2^2 x 4, a student might calculate 3 + 2 = 5, then 5^2 = 25, then 25 x 4 = 100.
    Example improved answer:First evaluate the index: 2^2 = 4. Then multiply: 4 x 4 = 16. Finally add: 3 + 16 = 19.
    Examiner Tip: Always underline or circle the part to evaluate first. Remember that indices apply only to the number or bracket immediately before them, not to the whole expression.
    Pitfall: Incorrectly rounding numbers when estimating, particularly with decimals and significant figures, leading to inaccurate answers.
    ❌ Weak Answer (Loses Marks):Estimating 4.7 x 2.1 by rounding 4.7 to 5 and 2.1 to 2, then calculating 5 x 2 = 10, but then giving the exact answer as 10 instead of an estimate.
    Example improved answer:Round each number to 1 significant figure: 4.7 ≈ 5 and 2.1 ≈ 2. Then multiply: 5 x 2 = 10. State that the estimate is 10, and the actual answer is approximately 10.
    Examiner Tip: When estimating, always round to 1 significant figure unless told otherwise. Clearly show the rounded values and state that your answer is an estimate, not the exact value.
    Step-by-Step Worked Solutions

    Question: Calculate the exact value of (3.2 x 10^4) x (2.5 x 10^3). Give your answer in standard form.

    1. 1.Step 1: Multiply the coefficients: 3.2 x 2.5 = 8.
    2. 2.Step 2: Multiply the powers of 10: 10^4 x 10^3 = 10^(4+3) = 10^7.
    3. 3.Step 3: Combine: 8 x 10^7. This is already in standard form because 8 is between 1 and 10.
    Final Answer: 8 x 10^7

    Question: Write 360 as a product of its prime factors. Give your answer in index form.

    1. 1.Step 1: Divide by the smallest prime: 360 ÷ 2 = 180.
    2. 2.Step 2: Continue dividing by 2: 180 ÷ 2 = 90, 90 ÷ 2 = 45.
    3. 3.Step 3: 45 is not divisible by 2, so divide by next prime 3: 45 ÷ 3 = 15, 15 ÷ 3 = 5.
    4. 4.Step 4: 5 is prime, so stop. The prime factors are 2, 2, 2, 3, 3, 5.
    5. 5.Step 5: Write in index form: 2^3 x 3^2 x 5.
    Final Answer: 2^3 x 3^2 x 5
    Active Recall Memory Test
    What does BIDMAS stand for and why is it important?
    Key Fact: BIDMAS stands for Brackets, Indices, Division and Multiplication (left to right), Addition and Subtraction (left to right). It gives the correct order to evaluate expressions so that everyone gets the same answer.
    How do you round a number to 3 significant figures?
    Key Fact: Start from the first non-zero digit and count three digits. Look at the next digit: if it is 5 or more, round up the third digit; if less than 5, keep it the same. Replace any digits after with zeros if needed, but keep place value.
    What is the difference between a factor and a multiple?
    Key Fact: A factor of a number divides exactly into it without remainder. A multiple is the result of multiplying the number by an integer. For example, factors of 12 are 1,2,3,4,6,12; multiples of 12 are 12,24,36,...
    How do you write a number in standard form?
    Key Fact: Write it as A x 10^n where A is between 1 and 10 (not including 10) and n is an integer. For large numbers, n is positive; for small numbers (less than 1), n is negative.
    Frequently Asked Questions
    What is BIDMAS and how do I use it?
    BIDMAS is an acronym for the order of operations: Brackets, Indices, Division and Multiplication, Addition and Subtraction. You evaluate expressions in that order, doing division and multiplication from left to right, and addition and subtraction from left to right. For example, in 2 + 3 x 4, you do 3 x 4 = 12 first, then add 2 to get 14.
    How do I round numbers to significant figures?
    To round to a given number of significant figures, start counting from the first non-zero digit. For example, to round 0.004567 to 2 significant figures, the first significant figure is 4, the second is 5. Look at the next digit (6), which is 5 or more, so round up the 5 to 6, giving 0.0046. Remember to keep the place value correct.
    What is standard form and when is it used?
    Standard form is a way of writing very large or very small numbers as A x 10^n, where 1 ≤ A < 10 and n is an integer. It is used in science and engineering to handle numbers like the speed of light (3 x 10^8 m/s) or the size of a virus (2 x 10^-7 m). It makes calculations and comparisons easier.
    How do I find the prime factorisation of a number?
    To find the prime factorisation, divide the number by the smallest prime (2, then 3, then 5, etc.) repeatedly until you are left with a prime. For example, 60 ÷ 2 = 30, 30 ÷ 2 = 15, 15 ÷ 3 = 5, so 60 = 2 x 2 x 3 x 5 = 2^2 x 3 x 5. Write the factors in index form.
    What is the difference between HCF and LCM?
    HCF (Highest Common Factor) is the largest number that divides exactly into two or more numbers. LCM (Lowest Common Multiple) is the smallest number that is a multiple of two or more numbers. For example, for 12 and 18, HCF = 6 and LCM = 36. You can find them using prime factorisation or listing.
    How do I estimate a calculation?
    To estimate, round each number to 1 significant figure (or as instructed), then perform the calculation. For example, to estimate 19.7 x 4.2, round to 20 x 4 = 80. This gives a rough answer to check your exact calculation. Always state that it is an estimate.

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