KS3 & KS4 Curriculum Model

8.13.2.1Gradient measures steepness: the vertical change divided by the horizontal change. The gradient of a straight line is the same wherever along it it is measured. A positive gradient rises from left to right and a negative gradient falls. A larger size of gradient means a steeper line, whatever its sign. A horizontal line has gradient 0.5

8.13.3.1The gradient of a distance-time graph is the speed. Its units come from the axes: distance units divided by time units. A steeper section is a faster speed, and a horizontal section a speed of zero. Average speed over a whole journey is total distance divided by total time, including any time spent stopped. Average speed is not the average of the separate speeds, unless each stage took the same time.5

9.2.1.1A coordinate pair gives horizontal position then vertical position, (x, y), across all four quadrants. A vertical line has equation x = a and a horizontal line y = b. Gradient measures steepness: the vertical change divided by the horizontal change. A positive gradient rises from left to right and a negative gradient falls. The gradient of a straight line is the same wherever along it it is measured.5

9.2.6.1In a real context both axes carry units, so the gradient carries units too: y units per x unit. On a distance-time graph the gradient is the speed. The intercept is the value when the x variable is zero, such as a fixed charge before any usage. A straight line means a constant rate of change, which is what makes the model linear.5

10.3.4.1A gradient can be negative (the line falls left to right) or a fraction (the line rises or falls less steeply than a whole-number gradient) - the same vertical-change-over-horizontal-change definition applies in both cases.4

10H.13.7.1In a real context both axes carry units, so the gradient carries units too: y units per x unit. On a distance-time graph the gradient is the speed. The intercept is the value when the x variable is zero, such as a fixed charge before any usage. A straight line means a constant rate of change, which is what makes the model linear.5

FM.3.4-3.5.1To 'draw' or 'plot' a graph means calculating and plotting points accurately, whereas to 'sketch' means showing correct shape and position with axis crossings only. The gradient between two points (x1,y1) and (x2,y2) is (y2-y1)/(x2-x1); a horizontal line y = b has gradient zero and a vertical line x = a has an undefined (infinite) gradient. The gradient m and intercept c fix a line y = mx + c completely, and any linear equation can be rearranged into that form or the general form px + qy + r = 0.4

9.2.4.1In y = mx + c, m is the gradient and c is the y-intercept. The y-intercept is where the line crosses the y-axis, which is the value of y when x = 0. A line can be sketched from its equation alone: mark c on the y-axis, then step out using the gradient. Parallel lines have the same gradient, so their equations share the same m and differ only in c. A negative m gives a line falling from left to right.5

9.2.5.1The gradient and intercept can only be read off directly when the equation is in the form y = mx + c. Any linear equation can be rearranged into that form, using the same inverse operations as solving. Dividing through by the coefficient of y has to be applied to every term. A negative coefficient of x makes m negative, which is easily lost during the rearrangement.5

10.15.3.1Michaelmas topic 3 already gives a line's gradient and y-intercept from its graph and from its equation, so this lesson starts from that and adds the four things it did not cover. An equation written ax + by = c is rearranged into y = mx + c before it is drawn, using the same balancing moves as solving an equation. A horizontal line has gradient 0 and an equation of the form y = k; a vertical line x = k has no gradient at all, because its horizontal change is 0 and nothing can be divided by 0. When m is known but the line's crossing point is not on the grid, c is found by substituting the coordinates of any point on the line into y = mx + c and solving for c. A point lies on a line exactly when its coordinates satisfy the equation, and that is the check on every answer here.4

10.3.2.1In y = mx + c, m is the gradient (steepness: vertical change divided by horizontal change) and c is the y-intercept, the value of y where the line crosses the y-axis (when x = 0).4

10.3.3.1Just as an arithmetic sequence's common difference and first term give its nth-term rule directly (dn + c) without listing every term, a straight line's gradient and y-intercept give its equation directly (y = mx + c) without plotting a table of points.4

10H.13.2.1The gradient and intercept can only be read off directly when the equation is in the form y = mx + c. Any linear equation can be rearranged into that form, using the same inverse operations as solving. Dividing through by the coefficient of y has to be applied to every term. A negative coefficient of x makes m negative, which is easily lost during the rearrangement.5

10.17.5.1A quadratic function contains an x^2 term and its graph is a smooth curve called a parabola, never a set of straight segments. A positive x^2 coefficient gives a curve shaped like a valley with a minimum; a negative one gives a hill with a maximum. A table of values gives the y for each x, and squaring a negative x gives a positive result, which is what makes the curve symmetrical.4

10.17.6.1The x-coordinates of the points where a curve crosses the x-axis are the solutions of y = 0, and they are called the roots. A quadratic graph may cross the x-axis twice, touch it once, or not meet it at all, which is why an equation can have two, one or no solutions. A root read off a plotted graph is an estimate limited by the scale, while a root found by factorising is exact.4

10.17.7.1A parabola is symmetrical about a vertical line through its turning point, so points at equal horizontal distances either side have equal y values. The turning point is the minimum of a valley-shaped curve or the maximum of a hill-shaped one. When the curve crosses the x-axis at two roots, the line of symmetry lies exactly halfway between them.5

10.17.8.1The solutions of f(x) = k are the x-coordinates of the points where the curve y = f(x) meets the horizontal line y = k. Drawing that horizontal line across the graph is therefore the method, and each intersection gives one solution. A solution read from a graph is an estimate, and it is quoted to the accuracy the grid supports.4

8.14.6.1Expanding double brackets multiplies every term in the first bracket by every term in the second, giving four products before simplifying. (x + a)^2 means (x + a)(x + a), so it expands to x^2 + 2ax + a^2, not x^2 + a^2. (x + a)(x - a) gives x^2 - a^2, the difference of two squares, because the two middle terms cancel. Triple brackets are expanded two at a time, and that result is then multiplied by the third. A bracket in a factorised expression is itself a factor of the whole expression.6

9.7.1.1Expanding double brackets multiplies every term in the first bracket by every term in the second, giving four products before simplifying. (x + a)^2 means (x + a)(x + a), so it expands to x^2 + 2ax + a^2, not x^2 + a^2. Several single brackets are expanded one at a time, then simplified by collecting like terms. A negative outside a bracket changes the sign of every term inside it.5

9.7.2.1(x + a)(x - a) gives x^2 - a^2, the difference of two squares, because the two middle terms cancel. (x + a)^2 gives x^2 + 2ax + a^2, and (x - a)^2 gives x^2 - 2ax + a^2. Recognising these forms saves expanding them term by term every time. A perimeter adds all the side lengths and an area multiplies two of them, so both give expressions to expand and simplify.5

10.17.2.1To expand (x + a)(x + b) every term in the first bracket multiplies every term in the second, giving four products. Collecting the two x terms gives x^2 + (a + b)x + ab, so the x coefficient is the SUM of a and b and the constant is their PRODUCT. (x + a)^2 means (x + a)(x + a) and expands to x^2 + 2ax + a^2; it is not x^2 + a^2.4

7.7.3.1An expression with several brackets is expanded one bracket at a time, then like terms are collected. Subtracting a whole bracket changes the sign of every term in it: 5 - (x - 2) = 5 - x + 2. Forming an expression means naming the unknown with a letter and writing the relationship in algebra. Perimeter adds all the side lengths, and '3 more than double n' is 2n + 3.4

8.14.5.1An expression with several brackets is expanded one bracket at a time, then simplified by collecting like terms. A negative outside a bracket changes the sign of every term inside. Subtracting a whole bracket changes the sign of every term in it: 5 - (x - 2) = 5 - x + 2. A perimeter adds all the side lengths and an area multiplies two of them, so both give expressions to expand and simplify.5

8.2.5.1A formula is a rule relating two or more variables, with one of them as the subject. Substituting known values into a formula leaves an equation in the remaining unknown. Squaring and taking a square root are inverses, as are cubing and the cube root. An unknown inside a power is freed by rooting both sides; an unknown under a root is freed by raising both sides to that power. A square root has a positive and a negative value, so x^2 = 25 gives x = 5 or x = -5.5

8.6.3.1Changing the subject rearranges a formula so that a different variable stands alone. It uses the same inverse operations as solving an equation, but the result is a formula rather than a number. The operations are undone in the reverse of the order they were applied to the variable wanted as subject. Squaring and rooting are inverses, so a squared subject is freed by rooting both sides. A rearranged formula answers for any values, whereas substituting first answers for only one set.5

8.10.1.1Changing the subject rearranges a formula so that a different variable stands alone on one side. It applies the same inverse operations to both sides as solving an equation does. The operations are undone in the reverse of the order they were applied to the wanted subject. The result is a formula rather than a number, so it answers for any set of values.5

8.10.2.1Squaring and taking a square root are inverse operations, as are cubing and the cube root. A squared subject is freed by rooting both sides; a subject under a root is freed by raising both sides to that power. The root or power applies to the whole of the other side, so that side is bracketed first. Rearranging before substituting keeps the working shorter and avoids rounding partway through.5

10.8.5.1A formula is a rule connecting quantities, with one of them written on its own as the subject. Substituting means replacing each letter by its given value and then calculating, keeping to the order of operations. Changing the subject uses the same balancing moves as solving an equation, but the answer is an expression rather than a number. A square is undone by a square root and a square root by squaring, so r can be made the subject of A = pi x r^2.5

9.8.1.1A quadratic has a squared term, and a monic quadratic is one where the coefficient of x^2 is 1. x^2 + bx + c factorises into (x + p)(x + q), where p + q = b and pq = c. This is the reverse of expanding: the middle term comes from p + q and the constant from pq. The factor pairs of c are listed first, then the pair that adds to b is chosen.5

9.8.2.1The signs of b and c together fix the signs of p and q. A positive c means p and q share a sign, and that sign is b's. A negative c means p and q have different signs, and the larger of the two takes b's sign. Expanding the factorised form back out checks both the numbers and the signs.5

9.8.3.1x^2 - a^2 factorises as (x + a)(x - a), because expanding it cancels the two middle terms. It is recognised by two square terms separated by a minus sign, with no x term. A common factor is always taken out before factorising any further. Removing a common factor can reveal a difference of two squares that was hidden: 2x^2 - 18 = 2(x^2 - 9) = 2(x + 3)(x - 3).5

10.17.3.1Factorising reverses expanding: x^2 + bx + c factorises as (x + p)(x + q) where p + q = b and pq = c. So the two numbers are found by listing the factor pairs of c and choosing the pair that adds to b. If c is positive both numbers share the sign of b; if c is negative they have opposite signs. A common factor in every term is taken out first, because 2x^2 + 10x + 12 = 2(x^2 + 5x + 6).4

10.2.2.1The nth term rule of an arithmetic sequence has the form dn + c, where d is the common difference and c is found by working back one step from the first term (c = first term - d). Substituting a position number into the rule generates that term directly, without working out every term before it.4

10.2.3.1Substituting a term's value into the nth term rule and solving for n tests whether that value is actually a term of the sequence - it is a term only if n comes out as a positive whole number. The same equation, solved for n, gives the position of a known term.4

8.6.1.1However complicated an equation looks, the method is unchanged: expand any brackets, collect the unknown onto one side, then apply inverse operations. Brackets on both sides are expanded before anything is collected. Collecting the smaller unknown term onto the other side avoids a negative coefficient. A solution may be negative or fractional, and is checked the same way, by substituting it back.5

8.6.2.1An equation containing a fraction is cleared by multiplying every term by the denominator. Where there is more than one denominator, multiplying by their LCM clears them all at once. Every term must be multiplied, including any term that is not itself a fraction. An unknown in the denominator is cleared the same way, by multiplying both sides by that denominator. A worded or geometric context becomes an equation once the unknown is named and the relationship written down.5

10.8.3.1When the letter appears on both sides, the terms in the letter are collected onto one side and the numbers onto the other. Adding or subtracting the same term from both sides is the same balancing move as adding or subtracting a number. Moving the smaller term in the letter keeps its coefficient positive, which avoids a negative coefficient to handle at the end.4

10H.7.4.1An equation with the unknown on both sides is solved by expanding any brackets, collecting the unknown terms on one side by doing the same to both sides, then using inverse operations. Collecting on the side with the larger coefficient keeps the coefficient positive. A worded or geometric problem becomes an equation when a letter names one unknown quantity and two expressions for the same quantity are set equal, such as two equal perimeters or two ages after the same number of years. The value of the letter is not always the answer, so the question is read again once the equation is solved.5

10.17.4.1If a product of two factors is zero then at least one factor is zero, so (x + p)(x + q) = 0 gives x = -p or x = -q. A quadratic equation must therefore be rearranged so that one side is zero before it is factorised. A quadratic equation usually has two solutions, and both are stated.5

10H.9.1.1If two numbers multiply to make zero, at least one of them is zero: if ab = 0 then a = 0 or b = 0. So a product of brackets equal to zero is solved by setting each bracket equal to zero in turn: (x - 3)(x + 5) = 0 gives x = 3 or x = -5. A factor of x on its own gives the solution x = 0, so x(x - 4) = 0 gives x = 0 or x = 4. A bracket with a coefficient of x gives a fraction: 2x - 3 = 0, so x = 3/2. The principle works only for zero: (x - 1)(x - 2) = 6 does not give x - 1 = 6 or x - 2 = 6.5

10H.9.2.1The zero-product principle needs zero on one side, so every term of a quadratic equation is collected on one side first, keeping the x^2 term positive. x^2 + bx + c = 0 is factorised as (x + p)(x + q) = 0, where p + q = b and pq = c, and each bracket is then set to zero. The solutions are the numbers in the brackets with their signs changed: (x + 3)(x - 5) = 0 gives x = -3 or x = 5. Both sides are never divided by x, because that loses the solution x = 0. Each solution can be checked by substituting it into the original equation.6

9.14.3.1Adding or subtracting two true equations produces a third equation that is also true. If a variable has the same coefficient in both, subtracting removes it; if the coefficients are equal but opposite, adding removes it. Removing one variable leaves a single equation in the other, which solves as usual. The value found is substituted back into either original equation to give the second unknown. Same signs subtract, different signs add.5

9.14.4.1Multiplying an entire equation by any number keeps it true, since both sides change equally. So one or both equations can be scaled until a variable's coefficients match. Every term must be multiplied, including the constant on the right-hand side. The LCM of the two coefficients gives the smallest numbers to work with. Substituting the answer into both original equations checks it, since it has to satisfy both.6

9.14.5.1Either variable may be eliminated, and choosing the one with the easier coefficients reduces the arithmetic. A solution can be negative or fractional, and is checked in exactly the same way. A worded problem gives two conditions, and each condition becomes one equation. The two unknowns are named explicitly before either equation is written.6

9.14.6.1Where one equation already gives a variable on its own, that expression can replace the variable in the other equation. The result is a single equation in one unknown. The substituted expression is bracketed, so that any multiplication reaches all of it. Substitution suits a pair where one equation is already rearranged; elimination suits a pair where neither is.6

9.14.7.1Two unknowns need two independent equations, which a worded problem supplies as two separate conditions. Two equations are independent only if neither is a rearrangement or a multiple of the other. Costs and totals give the commonest pair: one equation for the quantities, one for the money. The answer is interpreted back in context, with units, and checked for sense.6

11H.3.1.1Where one equation already gives a variable on its own, that expression can replace the variable in the other equation. The result is a single equation in one unknown. The substituted expression is bracketed, so that any multiplication reaches all of it. Substitution suits a pair where one equation is already rearranged; elimination suits a pair where neither is.6

9.14.1.1Simultaneous equations are two equations in two unknowns that must both hold at once. Every point on a line satisfies that line's own equation, so the point where two lines cross satisfies both. That crossing point is therefore the solution, read off as a coordinate pair. A line can only be drawn once its equation gives y from x, so an equation with x and y on the same side is rearranged first.5

8.1.2.1A square number has only even powers in its prime factorisation, because squaring doubles every power. A cube number has only powers that are multiples of three, for the same reason. The smallest multiplier making a number square is the product of the primes whose powers are odd. For three numbers, the HCF takes the lowest power of each prime shared by all three and the LCM the highest power of every prime appearing in any of them. A calculator's FACT function gives a prime factorisation directly, which matters for numbers too large to tree by hand.5

9.1.2.1A square number has only even powers in its prime factorisation, because squaring doubles every power. A cube number has only powers that are multiples of three, for the same reason. The smallest multiplier making a number square is the product of the primes whose powers are odd. The smallest multiplier making it a cube brings every power up to the next multiple of three.5

10.6.1.1An index says how many times the base is used as a factor (e.g. 2 to the power 5 = 2x2x2x2x2); multiplying powers of the same base adds the indices (a^m x a^n = a^(m+n)) and dividing subtracts them (a^m / a^n = a^(m-n)). Any non-zero number to the power 0 is 1.3

10.6.2.1A negative index means a reciprocal, for any base, not only base 10: a^-n = 1/a^n. This extends the same pattern already met for 10^-n = 1/10^n to any number.4

10H.5.1.1An index says how many times the base is used as a factor: 2^5 = 2 x 2 x 2 x 2 x 2. Multiplying powers of the same base adds the indices, a^m x a^n = a^(m+n); dividing subtracts them, a^m / a^n = a^(m-n); a power of a power multiplies them, (a^m)^n = a^(mn). Any non-zero number to the power 0 is 1, since a^n / a^n is both 1 and a^0. A power outside a bracket applies to every factor inside, coefficients included: (ab)^n = a^n x b^n, so (3x^2)^3 = 27x^6. The laws need the same base, so 2^3 x 3^2 cannot be combined into one power.4

10H.6.1.1A base that is itself a power can be rewritten: 8 = 2^3, so 8^4 = (2^3)^4 = 2^12 by the power law. A reciprocal or a root of the base is a power of it too: 1/8 = 2^-3 and the square root of 2 = 2^(1/2). Once every part is written over one base, a product or a quotient becomes a single power: 4^3 x 8 = 2^6 x 2^3 = 2^9. Two powers of the same base are equal only when their indices are equal, so 2^x = 2^9 means x = 9.6

9.5.4.1In a reverse percentage problem the amount given is the result of the change, not the original. The original is found by dividing by the multiplier, never by applying the percentage to the amount given. A price including 20% VAT is 120% of the pre-VAT price, so dividing by 1.2 recovers it. A sale price after a 30% reduction is 70% of the original, so dividing by 0.7 recovers it. The wording tells which is which: 'before' or 'original' signals a reverse problem, 'after' an ordinary one.5

9.5.2.1Successive percentage changes multiply their multipliers together. Two changes therefore combine into a single multiplier, applied once. The combined effect is not the sum of the percentages: a 10% rise then a 10% fall is x1.1 x 0.9 = x0.99, not a return to the original. The order the changes are applied in does not matter, since multiplication can be done in any order.5

9.5.5.1Simple interest adds the same amount each period, always calculated on the original. Over n periods simple interest totals n times the interest for one period. Compound interest applies the multiplier to the running total, so each period earns interest on the interest already added. After n periods the compound amount is the original x multiplier^n. Compound growth therefore accelerates, and beyond the first period always exceeds simple interest at the same rate.5

10.13.4.1Compound interest is calculated on the amount at the start of each period, including interest already added, so the multiplier is applied once per period. After n periods at r% per period, amount = starting amount x ((100 + r) / 100)^n. Depreciation is the same calculation with a decrease, so the multiplier is (100 - r) / 100. Simple interest, by contrast, adds the same amount each period and is not compounded.4

9.5.3.1Percentage change is the change divided by the original amount, then multiplied by 100. The original is always the denominator, which is why a rise and the matching fall are different percentages. Profit and loss are percentage changes on the cost price, not on the selling price. A multiplier turns the original into the new amount, so dividing the new amount by that multiplier recovers the original.5

9.15.4.1Knowing the value of one share gives the value of one part, by dividing by that share's number of parts. Every other share, and the total, then follow from that one part value. This is the multiplicative reading of a ratio, and it is what solves problems where the total is unknown. A given difference between two shares works the same way, over (a - b) parts instead of one share's parts.5

8.5.4.1Two quantities are in inverse proportion when multiplying one divides the other by the same factor. Their product therefore stays constant, so one known pair of values gives that constant for every other pair. Doubling one halves the other, which is the opposite of direct proportion. More of one meaning less of the other, such as workers and time taken, signals inverse proportion; more meaning more signals direct.5

9.15.6.1A unit cost makes different pack sizes comparable, so the best buy is the lowest cost per unit. An exchange rate is a direct proportion, so it works the same way in both directions. In direct proportion, multiplying one quantity multiplies the other by the same factor, so their ratio stays constant. In inverse proportion, multiplying one divides the other by the same factor, so their product stays constant. More of one meaning less of the other signals inverse proportion; more meaning more signals direct.5

10.22.2.1Two quantities are in inverse proportion when doubling one halves the other, so their PRODUCT stays the same rather than their ratio. Six workers taking 10 days and twelve workers taking 5 days is inverse proportion, because 6 x 10 = 12 x 5. The constant product is found from one complete pair and then used to find any missing value.5

10H.18.1.1A unit cost makes different pack sizes comparable, so the best buy is the lowest cost per unit. An exchange rate is a direct proportion, so it works the same way in both directions. In direct proportion, multiplying one quantity multiplies the other by the same factor, so their ratio stays constant. In inverse proportion, multiplying one divides the other by the same factor, so their product stays constant. More of one meaning less of the other signals inverse proportion; more meaning more signals direct.5

8.9.3.1A negative index means a reciprocal: 10^-n = 1/10^n. So multiplying by 10^-n divides, shifting the digits n places to the right. A negative index in standard form therefore represents a number smaller than 1. The index counts place shifts, not zeros, so 4.2 x 10^-3 = 0.0042. A stays between 1 and 10 whatever the sign of the index.5

8.9.4.1Numbers in standard form are ordered first by index, and only by A when the indices are equal. A larger index always means a larger number, whatever the A values are. A negative index is always smaller than a positive one. A number is not in standard form unless 1 <= A < 10. Adjusting A back into range changes the index to compensate: 34 x 10^5 = 3.4 x 10^6, and 0.7 x 10^-3 = 7 x 10^-4.5

8.9.6.1Multiplying in standard form multiplies the A values and adds the indices. Dividing divides the A values and subtracts the indices. The result is then adjusted if its A falls outside 1 <= A < 10. Adding and subtracting need a common index first, because only matching powers of ten combine. Alternatively both numbers can be written in ordinary form, added or subtracted, and converted back.5

10.10.1.1A number is in standard form when it is written as a x 10^n, where a is at least 1 and less than 10, and n is a whole number. A positive power of 10 makes the number larger and a negative power makes it smaller, so 1.5 x 10^8 = 150,000,000 and 1.5 x 10^-3 = 0.0015. The power counts the places the decimal point moves, not the number of zeros written. A power can also be rewritten with a different base, because (a^m)^n = a^(m x n).4

10.10.2.1Two numbers in standard form are compared by their powers of 10 first, and only by the value of a when the powers are equal. A number such as 34 x 10^5 is not in standard form, because 34 is not less than 10; it is rewritten as 3.4 x 10^6. The power goes up by one for each place the decimal point moves left, and down by one for each place it moves right, so 0.62 x 10^4 = 6.2 x 10^3.4

10.10.3.1Multiplying two numbers in standard form multiplies the values of a and adds the powers of 10; dividing divides the values of a and subtracts the powers. The answer is rewritten afterwards if a is no longer between 1 and 10. Adding or subtracting needs both numbers written with the SAME power of 10 first, or written out in full. Both papers allow a calculator, which has a standard-form key, and its display may show an answer in standard form without writing the x 10 out.4

10H.6.2.1A number is in standard form when it is written as A x 10^n, where 1 <= A < 10 and n is an integer. Moving the decimal point of A one place left raises n by one, and one place right lowers it by one, so the value is unchanged: 34 x 10^5 = 3.4 x 10^6 and 0.62 x 10^-3 = 6.2 x 10^-4. Only numbers with the same power of 10 can be added or subtracted directly, because only matching place values combine: 4.2 x 10^5 + 3 x 10^4 = 4.2 x 10^5 + 0.3 x 10^5 = 4.5 x 10^5. The powers of 10 are never added when the numbers are added; the answer is rewritten if A has left 1 <= A < 10.5

10H.6.3.1Multiplying in standard form multiplies the A values and adds the powers of 10, because the factors can be taken in any order: (3 x 10^4) x (5 x 10^6) = 15 x 10^10 = 1.5 x 10^11. Dividing divides the A values and subtracts the powers: (2 x 10^3) / (8 x 10^-2) = 0.25 x 10^5 = 2.5 x 10^4. A power of a number in standard form raises both parts: (3 x 10^4)^2 = 9 x 10^8; to take a square root, first make the power of 10 even: 1.6 x 10^9 = 16 x 10^8, whose square root is 4 x 10^4. The answer is rewritten at the end if A has left 1 <= A < 10.5

9.10.2.1A compound unit converts by converting each of its parts in turn. m/s to km/h multiplies by 3600 to turn seconds into hours and divides by 1000 to turn metres into kilometres, so overall it multiplies by 3.6. The unit on the bottom converts the opposite way round to the one on top. Every quantity must be in the units the answer is wanted in before any calculation starts.5

9.10.3.1Pressure is force divided by area, measured in N/m2 or N/cm2. It follows the same triangular relationship as speed and density, so force = pressure x area and area = force / pressure. Any 'per' quantity works this way: a rate is one measure divided by another. A multi-step problem finds an intermediate quantity first, keeping it exact until the end.5

9.10.4.1Two compound measures combine when the output of one is the input of the other, as volume gives mass through density. The units show the chain: the unit that cancels between the two is the shared quantity. On a curved graph the rate of change is not constant, so it can only be read at a single point. The gradient of a tangent drawn at that point gives the rate of change at that instant.6

9.3.4.1Where a polygon's sides meet parallel lines, the alternate and corresponding angle rules apply as usual. A polygon's own angle facts and the parallel line facts combine within a single diagram. Similar shapes have equal corresponding angles and corresponding sides in the same ratio. The scale factor is the ratio of a length on one shape to the matching length on the other, and multiplying by it gives a missing length.5

9.4.4.1Congruent shapes are identical in both size and shape, whereas similar shapes share angles but not size. Two triangles are congruent if they satisfy SSS, SAS (the angle between the two sides), ASA or RHS. AAA proves similarity only, because equal angles fix a shape's proportions but not its size. SSA is not a condition, since two different triangles can share two sides and a non-included angle. Corresponding parts of congruent triangles are equal, which is what makes congruence a tool for proving lengths and angles equal.5

7.19.2.1Volume is the space a solid occupies, measured in cubic units such as cm3. The volume of a cuboid is length x width x height. All three dimensions must be in the same unit before multiplying. A missing dimension is found by dividing the volume by the product of the other two.3

7.19.3.1Converting a length unit changes a volume unit by the cube of that factor: 1 cm3 = 1000 mm3 and 1 m3 = 1000000 cm3. This is because all three dimensions are converted, not just one. 1 cm3 holds 1 ml, so 1000 cm3 is 1 litre. A compound solid is split into cuboids whose volumes are added.5

8.19.2.1A prism has the same cross-section along its whole length. The volume of any prism is cross-sectional area x length. A cylinder is a prism with a circular cross-section, so its volume is pi x r^2 x h. All dimensions must be in the same unit before multiplying, and a volume unit converts by the cube of the length factor: 1 m^3 = 1000000 cm^3. A missing dimension is found by dividing the volume by the rest of the formula.5

9.13.3.1Converting a length unit changes a volume unit by the cube of that factor: 1 m^3 = 1000000 cm^3 and 1 cm^3 = 1000 mm^3. This is because all three dimensions are converted, not just one. An area unit converts by the square of the factor, for the same reason. 1 cm^3 holds 1 ml, so 1000 cm^3 is 1 litre. A missing dimension is found by dividing the volume, or by rearranging the surface area formula, before solving.5

10H.10.4.11 cm = 10 mm, 1 m = 100 cm and 1 km = 1000 m. Area has two length dimensions, so the conversion factor is squared: 1 cm2 = 10^2 mm2 = 100 mm2 and 1 m2 = 100^2 cm2 = 10000 cm2. Volume has three, so it is cubed: 1 cm3 = 10^3 mm3 = 1000 mm3 and 1 m3 = 100^3 cm3 = 1000000 cm3. 1 litre = 1000 cm3.5

9.13.4.1A prism has the same cross-section along its whole length. The volume of any prism is cross-sectional area x length. A cylinder is a prism with a circular cross-section, so its volume is pi x r^2 x h. The cross-section is the face that repeats, which is not always the face the solid appears to stand on. A contextual problem may need the cross-sectional area found first, from a compound 2D shape.5

10.11.3.1A compound solid is split into prisms and cylinders whose volumes are added, or treated as a larger solid with a piece removed. When the volume is known and a length is not, the volume formula is treated as an equation and the unknown length is found by rearranging it. Undoing a squared radius needs a square root, so a cylinder's radius is the square root of the whole of V divided by pi x h. The answer is checked by substituting the length back into the volume formula.5

8.8.2.1pi is the ratio of any circle's circumference to its diameter, and is the same for every circle. So C = pi x d, which is the same as C = 2 x pi x r. The diameter is found from the circumference by dividing by pi, and the radius by dividing by 2 x pi. Leaving an answer in terms of pi keeps it exact, whereas a decimal answer has been rounded. One full revolution of a wheel covers one circumference.4

8.8.3.1The area of a circle is A = pi x r^2. The radius is squared first and then multiplied by pi, since the index applies only to r. Given the diameter, the radius is halved before the formula is used. The radius is found from the area by dividing by pi and then taking the square root. Area is measured in square units and circumference in units of length, which is why the two formulas cannot be swapped.4

8.8.4.1A semicircle is half a circle and a quarter circle a quarter, so their areas are the whole circle's area halved or quartered. Their perimeters are not a simple fraction of the circumference, because the straight edges must be added. A semicircle's perimeter is half the circumference plus the diameter; a quarter circle's is a quarter of the circumference plus two radii. A compound shape is split into circles, rectangles and triangles whose areas are added, or subtracted where a piece has been removed. Only the outside edges count towards a compound shape's perimeter.5

10.5.2.1The circumference of a circle is C = pi x d (or C = 2 x pi x r) and its area is A = pi x r squared; the radius is squared before multiplying by pi, since the index applies only to r. A semicircle's area is half the full circle's area, but its perimeter is half the circumference plus the diameter - the straight edge must be added separately.4

10H.4.1.1The circumference of a circle is C = pi x d (or C = 2 x pi x r) and its area is A = pi x r squared; the radius is squared before multiplying by pi, since the index applies only to r. A semicircle's area is half the full circle's area, but its perimeter is half the circumference plus the diameter - the straight edge must be added separately.4

8.12.2.1A compound shape is split into rectangles, triangles, parallelograms and trapeziums whose areas are added. It can instead be treated as a larger shape with a piece subtracted. Only the outside edges count towards the perimeter, and missing lengths are deduced from the ones given. All lengths must be in the same unit before any area is calculated. Converting an area between units squares the conversion factor, so 1 m^2 = 10000 cm^2.5

10.5.4.1An area or perimeter formula can be applied backwards, using inverse operations, to find a missing length: for a triangle, base = 2 x area / height; for a rectangle or parallelogram, a missing side is the area divided by the known side; for a circle, the radius is found from the area by dividing by pi and taking the square root, or from the circumference by dividing by 2 x pi. Where a length appears more than once, or the shape is compound, the situation is written as an equation and solved.4

7.18.6.1Transformations can be applied one after another, each acting on the image the previous one produced. The result of two transformations can sometimes be described as a single transformation. Two reflections in parallel mirror lines give a translation; two reflections in perpendicular mirror lines give a rotation of 180 degrees. The order in which two transformations are applied can change the result.5

10H.12.6.1Transformations can be applied one after another, each acting on the image the previous one produced. The result of two transformations can sometimes be described as a single transformation. Two reflections in parallel mirror lines give a translation; two reflections in perpendicular mirror lines give a rotation of 180 degrees. The order in which two transformations are applied can change the result.5

8.4.1.1Similar shapes have equal corresponding angles and corresponding sides in the same ratio. The scale factor is the ratio of a length on one shape to the matching length on the other. A scale factor above 1 enlarges and one below 1 reduces. A missing length is found by multiplying the corresponding length by the scale factor. Corresponding sides are matched by their position in the shape, not by how they look on the page.4

8.4.2.1Triangles formed between a pair of parallel lines are similar, because alternate and corresponding angles make their angles equal. Two triangles with all three angles equal are similar whatever their size. In the hourglass and arrowhead arrangements it is the equal angles that identify which sides correspond.6

8.16.5.1Proving two triangles congruent establishes that every pair of corresponding sides and angles is equal. That is what makes congruence a tool for proving two lengths or angles equal, rather than an end in itself. Corresponding parts are matched by the order the vertices are named, so ABC congruent to DEF means AB = DE. A structured proof states each fact with its reason, then the condition satisfied, then the conclusion drawn.6

10.12.1.1Two shapes are similar when one is an enlargement of the other: corresponding angles are equal and corresponding lengths are all in the same ratio. That ratio is the scale factor k, found by dividing a length on the image by the length it corresponds to on the object. A length on the image = k x the matching length on the object, so k > 1 enlarges and k < 1 reduces.4

10H.10.1.1Similar shapes have equal corresponding angles and corresponding sides in the same ratio. The length scale factor k is a length on one shape divided by the matching length on the other, and every length on the second shape, its perimeter included, is k times its partner. Corresponding sides are matched by the equal angles they sit opposite, not by where they sit on the page. Going from the larger shape back to the smaller, each length is divided by k. When one triangle sits inside another, sharing an angle, with the third sides parallel, the two triangles are similar, and the scale factor uses the whole sides, never only the parts.5

8.4.3.1Congruent shapes are identical in both size and shape, whereas similar shapes share angles but not size. Two triangles are congruent if they satisfy SSS (three sides), SAS (two sides and the angle between them), ASA (two angles and a corresponding side) or RHS (right angle, hypotenuse and one other side). AAA proves similarity only, because equal angles fix a shape's proportions but not its size. SSA is not a condition, since two different triangles can share two sides and a non-included angle. Corresponding parts of congruent triangles are equal, which is what makes congruence useful for proving two lengths or angles equal.5

10.12.2.1A scale of 1 : n means that 1 unit on the drawing stands for n of the same units in real life. A map scale of 1 : 25000 means 1 cm on the map is 25000 cm, which is 250 m, in real life. Real length = drawing length x n, and drawing length = real length / n, so the two are inverse operations.3

8.11.2.1The same theorem finds a shorter side, rearranged to a^2 = c^2 - b^2. Finding a shorter side subtracts, whereas finding the hypotenuse adds. Identifying the hypotenuse first is what decides which of the two applies. The hypotenuse is opposite the right angle and is always the largest of the three values. A shorter side coming out larger than the hypotenuse signals the subtraction was done the wrong way round.5

9.6.2.1With the unknown on the numerator, the ratio is an equation solved in one step: multiply both sides by the denominator. sin(x) = opp/hyp with the hypotenuse known therefore gives opp = hyp x sin(x). The working is set out as a labelled equation before any numbers are substituted. The calculator must be in degrees mode or every ratio will be wrong. Rounding is done once, at the end, to the accuracy the question states.5

9.6.3.1With the unknown on the denominator, the ratio takes two steps: multiply both sides by the unknown, then divide. sin(x) = opp/hyp with the opposite known therefore gives hyp = opp / sin(x). The unknown sits on the denominator when finding the hypotenuse with sin or cos, or the adjacent with tan. Which case applies is decided from the labelled diagram before any calculation starts. The hypotenuse must come out larger than either other side, which checks the answer.5

10.15.2.1A line segment drawn on a coordinate grid is the hypotenuse of a right-angled triangle whose shorter sides are the horizontal and vertical differences between the end points. Those differences are x2 - x1 and y2 - y1, and a negative difference is used as its positive size, because a length cannot be negative. Pythagoras' theorem then gives length^2 = (x2 - x1)^2 + (y2 - y1)^2.5

10.9.1.1A right-angled triangle has one angle of exactly 90 degrees, and the side opposite that angle is the hypotenuse, which is always its longest side. Pythagoras' theorem says the square drawn on the hypotenuse has the same area as the squares on the two shorter sides added together: a^2 + b^2 = c^2, where c is the hypotenuse. The theorem is true only for right-angled triangles, so the right angle is found in the diagram before the theorem is used. A length is never negative, so only the positive square root is taken.5

10.9.2.1Rearranging a^2 + b^2 = c^2 for a shorter side gives a^2 = c^2 - b^2, so the two squares are SUBTRACTED, not added. Which of the two operations is needed is decided by whether the unknown side is the hypotenuse or one of the shorter sides. A shorter side must come out smaller than the hypotenuse, and that is the check on the answer.5

9.6.4.1Given two sides, the ratio is known but the angle is not. The inverse functions sin-1, cos-1 and tan-1 recover the angle from its ratio. The ratio is formed first from the two known sides, and the inverse function applied to it. Choosing the ratio and choosing the solving route are separate decisions: which two sides are known, then where the unknown sits. Any angle other than the right angle is less than 90 degrees, which checks the answer.5

10.19.2.1Once the ratio is chosen, the unknown side is found by rearranging the ratio: from sin x = opposite / hypotenuse, opposite = hypotenuse x sin x and hypotenuse = opposite / sin x. Whether the unknown is multiplied or divided depends on whether it is on the top or the bottom of the ratio. An answer is checked against the triangle: the hypotenuse must be the longest side.5

10.19.3.1When both sides of a ratio are known, the angle is recovered with the inverse function, written sin^-1, cos^-1 or tan^-1. The inverse undoes the ratio, so from sin x = 0.5 comes x = sin^-1(0.5) = 30 degrees. Because both papers allow a calculator, exact trigonometric values are not required at any point.5

10.19.4.1Pythagoras' theorem is used when three SIDES are involved and no angle; a trigonometric ratio is used when an angle is involved as well as sides. The choice is made by listing what the diagram gives and what it asks for. A problem may need both in turn: a side found by Pythagoras' theorem can then be used in a ratio, and an angle found from a ratio can then be used elsewhere. Because both papers allow a calculator, answers are given to a stated accuracy rather than as exact values.5

10.19.1.1In a right-angled triangle the sides are named relative to a chosen angle: the hypotenuse is opposite the right angle, the opposite side faces the chosen angle and the adjacent side is between the chosen angle and the right angle. sin x = opposite / hypotenuse, cos x = adjacent / hypotenuse and tan x = opposite / adjacent. The ratio to use is the one whose two sides are the side known and the side wanted. The calculator must be in degrees.5

10.4.2.1A probability read from a Venn diagram is the number of elements in a region divided by n(xi); because every element appears exactly once, the region counts add up to n(xi).4

10.4.4.1The structure of a problem decides which representation to use: a Venn diagram suits overlapping categories, a two-way table suits two characteristics at once, and a tree diagram suits a sequence of events. Whichever is used, every representation must account for the whole sample space, so its probabilities sum to 1.4

7.17.2.1The probabilities of all possible outcomes sum to 1. The complement of A is everything else, so P(not A) = 1 - P(A). Mutually exclusive events cannot both happen at the same time. The probability that one of several mutually exclusive events occurs is found by adding their probabilities.4

9.16.1.1Two events are independent when one happening does not change the probability of the other. They are dependent when it does, as when an item is taken and not replaced. They are mutually exclusive when they cannot both happen at once. Mutually exclusive is not the same as independent: mutually exclusive events are strongly dependent, since one occurring makes the other impossible. The probabilities of all possible outcomes sum to 1, so P(not A) = 1 - P(A).4

9.16.3.1For mutually exclusive events, P(A or B) = P(A) + P(B). Where the events can both happen, the overlap has been counted twice, so P(A or B) = P(A) + P(B) - P(A and B). In set notation 'A or B' is the union, A union B, and the overlap is the intersection. 'And' multiplies and 'or' adds, which is the quickest way to tell which rule a question needs.5

7.17.4.1A probability distribution lists every outcome with its probability, and these must total 1. A missing probability is found by subtracting the others from 1. A sample space diagram lists every possible outcome of two events, so probabilities can be counted from it. Two events are independent when one does not affect the other. For independent events, the probability of both is found by multiplying their probabilities.5

9.16.2.1For two independent events, P(A and B) = P(A) x P(B). Multiplying makes the answer smaller, which fits: both happening is less likely than either alone. The rule extends to more than two independent events by multiplying all of their probabilities. In set notation 'A and B' is the intersection, A intersect B.5

7.2.4.1Each set is a closed region inside a rectangle representing the universal set. Every element appears once, in the region matching all the sets it belongs to; elements in no set sit outside the circles but inside the rectangle. Two sets make four regions; three sets make eight. An Euler diagram shows only the relationships that occur, so it is used when sets share nothing or one lies inside another.4

7.2.5.1A shaded region is the set of all elements the expression describes. Read a compound expression inside out: brackets, then union and intersect, with ' applying to whatever it follows. A' intersect B means 'in B but not A'; (A union B)' means 'in neither'. Different expressions can give the same region: (A union B)' = A' intersect B'.5

8.7.2.1In a Venn diagram each set is a closed region inside a rectangle representing the universal set. Every element appears exactly once, in the region matching all the sets it belongs to. Elements in none of the sets sit inside the rectangle but outside every circle. Two sets create four regions and three sets create eight, each region corresponding to one set expression. An Euler diagram is used instead when the sets are nested or share no elements.4

8.7.4.1A probability read from a Venn diagram is the number of elements in the region divided by n(xi). The region is identified from the set notation first, and its elements counted second. Because every element appears exactly once, the region counts add up to n(xi). A given probability works backwards: multiplying it by n(xi) gives the number of elements in that region.5

8.17.3.1Grouping data loses the individual values, so only the class each value fell into is known. The mean can therefore only be estimated, using the midpoint of each class to stand in for its values. The estimated mean is the total of midpoint x frequency, divided by the total frequency. The modal class is the class with the highest frequency, and the median is reported as the class containing the middle value. Both answers are classes rather than single values, which is why they are named rather than calculated.5

8.17.4.1Continuous data can take any value in a range, so its classes are written as inequalities rather than separate values. Inequality boundaries leave no gaps, so every possible value falls into exactly one class. The midpoint of a class is halfway between its two boundaries. Because the total is mean x count, the original total can always be recovered from a given mean. A value added, removed or corrected changes that total and possibly the count, and the new mean follows from both.6

9.16.7.1Grouping data loses the individual values, so only the class each value fell into is known. The mean can therefore only be estimated, using each class midpoint to stand in for its values. The estimated mean is the total of midpoint x frequency, divided by the total frequency. The modal class is the class with the highest frequency, and the median is the class containing the middle value. Both are classes rather than single values, which is why they are named rather than calculated.5

9.16.8.1Continuous data can take any value in a range, so its classes are written as inequalities. Inequality boundaries leave no gaps, so every value falls into exactly one class. The midpoint of a class is halfway between its two boundaries. A conditional probability restricts the sample space to the outcomes where a known event has already happened. On a tree diagram that means using only the branches following the known outcome, rescaled so they sum to 1.6

10.18.3.1When data are grouped the individual values are lost, so the mean can only be estimated. The midpoint of a class is taken as representative of every value in it, and midpoint = (lower bound + upper bound) / 2. Estimated mean = (sum of midpoint x frequency) / (total frequency). The estimate is not the true mean, and it would only be exact if the values in every class were symmetrically spread about its midpoint.4

10H.16.1.1Grouping data loses the individual values, so only the class each value fell into is known. The mean can therefore only be estimated, using each class midpoint to stand in for its values. The estimated mean is the total of midpoint x frequency, divided by the total frequency. The modal class is the class with the highest frequency, and the median is the class containing the middle value. Both are classes rather than single values, which is why they are named rather than calculated.5