Revision notes · Mathematics 0580
Mathematics 0580 formula bank
Every formula and rule from the notes, chapter by chapter. "Learn it" means it is not given in the exam.
Arithematics
Types of number
Prime numberLearn itexactly two factors: 1 and itself1 is NOT prime; 2 is the only even prime.Use it when "write down a prime number".
Prime testLearn itdivide by primes up to \(\sqrt n\)none divides exactly → prime.Use it when is 91 prime?
Squares and cubesLearn it\(1^2\ldots15^2=225\); cubes 1, 8, 27, 64, 125, 1000learn them.Use it when picking from a list.
RationalLearn it\(\dfrac ab\) with integers \(a,b\) (\(b\ne0\))its decimal stops or recurs.Use it when rational or irrational?
IrrationalLearn ite.g. \(\pi\), \(\sqrt2\), \(\sqrt{12}\)\(\sqrt n\) is irrational unless n is a perfect square.Use it when rational or irrational?
ReciprocalLearn it\(\dfrac1n\); for a fraction, turn it upside downnumber × reciprocal = 1; 0 has none.Use it when "reciprocal of".
Order of operations, negative numbers and ordering
Order of operationsLearn itBrackets → Indices → ÷ × → + −÷ and × have equal rank (left to right); so do + and −.Use it when any calculation with more than one operation.
Fraction lineLearn ita fraction line is a bracket round the top and the bottom\(\dfrac{12+8}{7-3}=(12+8)\div(7-3)\).Use it when typing fractions into a calculator.
Negatives: + and −Learn it+ (−a) = − a; − (−a) = + aUse it when adding or subtracting negatives.
Negatives: × and ÷Learn itsame signs → +; different signs → −\((-3)(-4)=12\); \(-20\div4=-5\).Use it when multiplying or dividing negatives.
Powers of negativesLearn it\((-3)^{2}=9\) but \(-3^{2}=-9\)the bracket decides what is squared.Use it when a negative with a power.
Fractions
Mixed → improperLearn it\(a\tfrac bc=\dfrac{a\times c+b}{c}\)\(4\tfrac35=\tfrac{23}5\).Use it when before ANY operation.
Add / subtractLearn it\(\dfrac ab\pm\dfrac cd=\dfrac{ad\pm cb}{bd}\) (or use the LCM)same-size parts first: a common denominator.Use it when + or −.
MultiplyLearn it\(\dfrac ab\times\dfrac cd=\dfrac{ac}{bd}\)cancel any top with any bottom first.Use it when ×, or "of".
DivideLearn it\(\dfrac ab\div\dfrac cd=\dfrac ab\times\dfrac dc\)keep, change, flip (the second one only).Use it when ÷.
Fraction of an amountLearn it\(\dfrac ab\) of N = N ÷ b × aUse it when "find 3/8 of 40".
Whole from a partLearn itwhole = part ÷ \(\dfrac ab\)Use it when "12 is 3/13 of the number".
Decimals
Add / subtractLearn itline up the decimal pointsfill gaps with zeros: 12.50 − 3.78.Use it when + or − with decimals.
MultiplyLearn itmultiply as whole numbers; answer has as many decimal places as both numbers together\(0.3\times0.02\): 3 × 2 = 6, 3 d.p. → 0.006.Use it when × with decimals.
DivideLearn itmake the divisor whole: × both numbers by 10, 100, …\(7.2\div0.06=720\div6\).Use it when ÷ by a decimal.
Decimal → fractionLearn itover 10, 100, 1000, then simplify\(0.45=\tfrac{45}{100}=\tfrac9{20}\).Use it when "write as a fraction".
Calculator fractionsLearn itbrackets round the whole top and the whole bottom(6.5 + 2.3) ÷ (4.1 − 1.7).Use it when typing a fraction.
Recurring decimals
Dot notationLearn it\(0.\dot4=0.444\ldots\); \(0.\dot4\dot7=0.4747\ldots\); \(0.1\dot7=0.1777\ldots\)a dot over each repeating digit, or over the first and last of a longer block.Use it when reading a recurring decimal.
Terminates or recurs?Learn itsimplest-form denominator with only 2s and 5s → terminates\(\tfrac38\) terminates; \(\tfrac56\) recurs (the 3).Use it when "does it terminate?".
Recurring → fractionLearn it\(x=\ldots\); multiply by \(10^{k}\); subtract; solvek = digits in the repeating block.Use it when "show that 0.47 recurring = …".
Check (not working)Learn itpure block over 9s: \(0.\dot4\dot7=\dfrac{47}{99}\)use only to check your answer.Use it when checking.
Differences, halfway points and number lines
DifferenceLearn itbigger − smalleralways positive: it is a distance.Use it when "find the difference", "how much warmer".
HalfwayLearn it\(\dfrac{a+b}{2}\)the mean of the two numbers.Use it when "halfway", "midway".
Missing endLearn it\(b=2\times\text{middle}-a\)Use it when "5 is halfway between 2 and x".
Equal stepsLearn itstep \(=\dfrac{b-a}{n}\); the kth mark \(=a+k\times\text{step}\)Use it when a number line split into n equal parts.
Using a calculator, and mixed arithmetic
Fractions on the calculatorLearn it( top ) ÷ ( bottom )or use the fraction template.Use it when any fraction with sums in it.
NegativesLearn ituse the (−) key; bracket before squaring: ( (−) 3 ) x²(−) 3 x² gives −9.Use it when a negative number with a power.
Decimal hoursLearn itminutes = decimal part × 607.2 h = 7 h 12 min.Use it when a time answer.
EstimateLearn itround every number to 1 significant figure60 × 0.3 ÷ 2 = 9.Use it when checking, or "estimate".
Indices
The rules of indices
Multiply: addLearn it\(a^m\times a^n=a^{m+n}\)same base only.Use it when powers multiplied.
Divide: subtractLearn it\(a^m\div a^n=a^{m-n}\)same base only.Use it when powers divided.
Power of a power: multiplyLearn it\((a^m)^n=a^{mn}\)Use it when a power outside a bracket.
Zero indexLearn it\(a^0=1\) \((a\neq0)\)Use it when a power of 0.
Power of a product / fractionLearn it\((ab)^n=a^nb^n\), \(\left(\dfrac ab\right)^n=\dfrac{a^n}{b^n}\)EVERYTHING inside gets the power.Use it when a bracket with a number and letters.
Negative, zero and fractional indices
Negative indexLearn it\(a^{-n}=\dfrac1{a^n}\), \(\left(\dfrac ab\right)^{-n}=\left(\dfrac ba\right)^{n}\)means one over (flip); never a negative answer.Use it when a minus sign in the index.
Unit fraction indexLearn it\(a^{\frac1n}=\sqrt[n]a\)denominator = root.Use it when \(a^{\frac12}\), \(a^{\frac13}\).
Fraction indexLearn it\(a^{\frac mn}=\left(\sqrt[n]a\right)^{m}\)root first, then power.Use it when \(16^{\frac34}\).
Moving across the lineLearn it\(\dfrac{k}{x^{n}}=kx^{-n}\)only the x part moves; its index changes sign.Use it when "write in the form \(ax^n\)".
Solving equations: the unknown in the power
Equal powers of one baseLearn it\(a^{p}=a^{q}\ \Rightarrow\ p=q\)only when the bases are the SAME.Use it when the unknown is in the index.
Useful rewritesLearn it\(\tfrac1{a^n}=a^{-n}\), \(\sqrt a=a^{\frac12}\), \(1=a^{0}\)Use it when fractions, roots or 1 on one side.
Changing the baseLearn it\(4^{x+1}=(2^{2})^{x+1}=2^{2(x+1)}\)multiply the WHOLE index - keep the bracket.Use it when 4, 8, 9, 27, 25 … in an equation.
Solving equations: the unknown in the base
Undo a powerLearn it\(x^{n}=k\Rightarrow x=k^{\frac1n}\); \(x^{\frac mn}=k\Rightarrow x=k^{\frac nm}\)raise both sides to the reciprocal power.Use it when the unknown is the base.
Two answersLearn iteven power: \(x=\pm\)odd power: one answer, same sign as k.Use it when \(x^2\), \(x^4\), \(x^{-2}\), \(x^{\frac23}\).
Roots as powersLearn it\(\sqrt[n]{x^{m}}=x^{\frac mn}\), \(\dfrac1{\sqrt[n]{x^m}}=x^{-\frac mn}\)Use it when "find n" with roots.
Indices: word problems, tough and challenge questions
Comparing powersLearn itmake the bases OR the indices the same\(2^{30}=8^{10}\), \(3^{20}=9^{10}\).Use it when "which is bigger" without a calculator.
Copies of one powerLearn it\(2^n+2^n=2^{n+1}\), \(3^n+3^n+3^n=3^{n+1}\)Use it when the same power added several times.
Repeated multiplierLearn it\(\text{start}\times(\text{multiplier})^{n}\)doubling × 2, tripling × 3, halving × ½.Use it when "doubles every hour".
SubstitutionLearn it\(8^a=(2^a)^3\), \(2^{a+2}=2^a\times4\)Use it when "given that 2ᵃ = 3".
Standard Form
Writing numbers in standard form
Standard formLearn it\(A\times10^{n}\), \(1\le A<10\), n an integerone non-zero digit before the point.Use it when very big or very small numbers.
Sign of the powerLearn itnumber \(\ge10\): \(n>0\); number \(<1\): \(n<0\); 1 to 9.99…: \(n=0\)Use it when every conversion.
Fixing ALearn itA ÷ 10 → \(n+1\); A × 10 → \(n-1\)\(45\times10^5=4.5\times10^6\).Use it when A not between 1 and 10.
Calculating in standard form
Standard formLearn it\(A\times10^{n}\), \(1\le A<10\)\(A\) = a number from 1 up to (not including) 10; \(n\) = a whole number.Use it when the question says "give your answer in standard form".
MultiplyLearn it\((A\times10^{m})(B\times10^{n})=AB\times10^{m+n}\)multiply the numbers, ADD the powers.Use it when two standard form numbers are multiplied.
DivideLearn it\(\dfrac{A\times10^{m}}{B\times10^{n}}=\dfrac AB\times10^{m-n}\)divide the numbers, SUBTRACT the powers.Use it when one is divided by the other.
Add / subtractLearn it\(A\times10^{n}\pm B\times10^{n}=(A\pm B)\times10^{n}\)only once the powers are the SAME.Use it when a sum or difference: match the powers first.
Power of a numberLearn it\((A\times10^{n})^{k}=A^{k}\times10^{nk}\)the power goes on BOTH parts.Use it when squaring or cubing.
Square rootLearn it\(\sqrt{A\times10^{n}}=\sqrt A\times10^{n/2}\)make \(n\) even first.Use it when a square root without a calculator.
Tidy upLearn it\(A\div10\Rightarrow n+1\qquad A\times10\Rightarrow n-1\)the number and the power move in opposite directions.Use it when your \(A\) is 10 or more, or less than 1.
Standard form: word problems and challenge questions
Per / density / rateLearn it\(\text{amount}\div\text{the 'per' quantity}\)people per km² = population ÷ area.Use it when the word "per".
How many in / how many timesLearn it\(\text{total}\div\text{one}\); \(\text{bigger}\div\text{smaller}\)Use it when "how many", "how many times bigger".
PrefixesLearn itk \(10^{3}\), M \(10^{6}\), G \(10^{9}\), m \(10^{-3}\), μ \(10^{-6}\), n \(10^{-9}\)Use it when units with prefixes.
Square unitsLearn it1 km² = \(10^{6}\) m²; 1 m² = \(10^{4}\) cm²square the length factor.Use it when area in different units.
Significant Figures
Rounding: decimal places and significant figures
The rounding ruleLearn itlook at the NEXT digit only: 5 or more → round up; 4 or less → keepUse it when every rounding question.
Decimal placesLearn itcount from the decimal point7.3851 to 2 d.p. = 7.39.Use it when "to 2 d.p.".
Significant figuresLearn itcount from the FIRST NON-ZERO digit0.004 708 to 3 s.f. = 0.00471.Use it when "to 3 s.f.".
Keep the sizeLearn itdropped whole-number places become zeros83 651 to 1 s.f. = 80 000.Use it when big numbers.
Keep accuracy zerosLearn ita final zero that is part of the accuracy stays2.6963 to 2 d.p. = 2.70.Use it when a carry through 9.
Estimation
Exam estimation ruleLearn itevery number to 1 s.f., WRITE the rounded calculation, then work it outUse it when "by writing each number correct to 1 significant figure, estimate…".
Dividing by a decimalLearn it\(40\div0.5=80\)÷ by a number less than 1 makes it bigger.Use it when a decimal on the bottom.
Over or under?Learn itrounding UP a top number → too big; rounding UP the bottom → too smallUse it when "is your estimate more or less?".
Rounding to a sensible accuracy
Default accuracyLearn itexact, or 3 s.f.; angles 1 d.p.; money 2 d.p.Use it when no accuracy is stated.
Full value firstLearn itwrite the calculator value, then the rounded answer126.676 8… = 127.Use it when every calculator answer.
Round onceLearn itkeep full values in the working; round only the final answeruse Ans or memory.Use it when multi-step questions.
CountsLearn itround up or down to suit the story2.86 tins → 3 tins; 6.8 full boxes → 6.Use it when people, tins, buses, boxes.
Functions
Function notation: evaluating and solving
EvaluateLearn it\(f(a)\): replace every \(x\) by \((a)\)a = a number or an expression; keep the brackets.Use it when 'find f(3)', 'find f(x − 2)'.
SolveLearn it\(f(x)=k\): solve 'the rule = k'the OUTPUT is given; find the input.Use it when 'solve f(x) = …'.
Inverse functions
InverseLearn it\(y=f(x)\) → swap → \(x=f(y)\) → make \(y\) the subjectthe result is \(f^{-1}(x)\).Use it when 'find f⁻¹(x)'.
Inverse of a numberLearn it\(f^{-1}(a)\): solve \(f(x)=a\)Use it when 'find f⁻¹(10)'.
UndoingLearn it\(f^{-1}(f(x))=x\)the inverse takes you back to the start.Use it when checking your inverse.
Composite functions
CompositeLearn it\(fg(x)=f\big(g(x)\big)\)the function NEXT TO \(x\) acts first.Use it when two function letters side by side.
With a numberLearn it\(fg(3)\): work out \(g(3)\), then put the answer into \(f\)Use it when a number in the brackets.
Domain and range
DomainLearn itthe inputs x allowedexclude anything that divides by zero or square-roots a negative.Use it when 'which value must be excluded?'.
RangeLearn itthe outputs f(x) producedlinear: work out the two ends; quadratic: use the turning point.Use it when 'find the range'.
Completed squareLearn it\((x+p)^{2}+q\ge q\)the smallest output is q.Use it when range of a quadratic.
Reciprocal graphLearn it\(y=\dfrac a{x-b}+c\): domain \(x\ne b\), range \(y\ne c\)read them off the asymptotes.Use it when a curve with asymptotes.
Reciprocal and exponential graphs: asymptotes
Reciprocal graphLearn it\(y=\dfrac a{x-b}+c\)vertical asymptote \(x=b\); horizontal asymptote \(y=c\); the asymptotes cross at \((b,\ c)\).Use it when you are given a fraction with \(x\) on the bottom.
Which quadrantsLearn it\(a>0\): top right + bottom left. \(a<0\): top left + bottom rightmeasured from the point \((b,\ c)\) where the asymptotes cross.Use it when you sketch a reciprocal graph.
Exponential graphLearn it\(y=ar^{x}+b\)horizontal asymptote \(y=b\); y-intercept \(a+b\); no vertical asymptote.Use it when the \(x\) is in the power.
Domain and range (reciprocal)Learn itdomain \(x\ne b\); range \(y\ne c\)the asymptotes are the only values never used / never reached.Use it when asked for the domain or range.
Domain and range (exponential)Learn itdomain: all \(x\); range \(y>b\) (if \(a>0\)) or \(y<b\) (if \(a<0\))the curve stays on one side of its asymptote.Use it when asked for the range of an exponential.
Factorization
Factorising: common factors and the difference of two squares
Common factorLearn it\(ab+ac=a(b+c)\)a = the highest common factor of every term.Use it when every term shares a number or a letter.
Difference of two squaresLearn it\(a^{2}-b^{2}=(a+b)(a-b)\)a² and b² are both perfect squares; there is a MINUS between them.Use it when two terms, a minus, both squares.
Factorising quadratics
Simple quadraticLearn it\(x^{2}+bx+c=(x+p)(x+q)\)\(p\times q=c\) and \(p+q=b\).Use it when the \(x^{2}\) has no number in front.
Splitting the middle termLearn it\(ax^{2}+bx+c\): find \(m, n\) with \(mn=ac\), \(m+n=b\)then \(ax^{2}+mx+nx+c\) and group in pairs.Use it when a number in front of \(x^{2}\).
Perfect squaresLearn it\(a^{2}+2ab+b^{2}=(a+b)^{2}\), \(a^{2}-2ab+b^{2}=(a-b)^{2}\)first and last terms are squares; the middle is 2 × their product.
Factorising four terms by grouping
GroupingLearn it\(ax+ay+bx+by=a(x+y)+b(x+y)=(a+b)(x+y)\)the matching bracket is the common factor.Use it when four terms with no factor common to all four.
Solving quadratic equations by factorising
Zero product ruleLearn itif \(A\times B=0\) then \(A=0\) or \(B=0\)only works for ZERO on the other side.Use it when a factorised quadratic equals 0.
Statistics
Averages and range
MeanLearn it\(\text{mean}=\dfrac{\text{total of the values}}{\text{number of values}}\)Use it when 'mean', 'average' with a calculation.
MedianLearn itthe middle value in order: position \(\tfrac{n+1}{2}\)two middle values? halfway between them.Use it when 'median'.
ModeLearn itthe most common valuecan be more than one, or none.Use it when 'mode', 'most common'.
RangeLearn itlargest − smallesta measure of SPREAD.Use it when 'range', 'how spread out'.
Frequency table meanLearn it\(\dfrac{\sum fx}{\sum f}\)x = value, f = how many times.Use it when a frequency table.
QuartilesLearn itlower quartile at \(\tfrac{n+1}{4}\), upper at \(\tfrac{3(n+1)}{4}\); IQR = UQ − LQthe middle half of the data.Use it when 'interquartile range'.
Grouped data and comparing
MidpointLearn it\(x=\dfrac{\text{lower}+\text{upper}}{2}\)the value that stands for the whole class.Use it when grouped data.
Estimated meanLearn it\(\dfrac{\sum fx}{\sum f}\)an ESTIMATE: we don't know the exact values.Use it when 'calculate an estimate of the mean'.
Modal classLearn itthe class with the highest frequencyUse it when 'modal class'.
Median classLearn itthe class holding the \(\tfrac{n+1}{2}\)th valueUse it when 'which class contains the median'.
Charts and scatter diagrams
Pie angleLearn it\(\text{angle}=\dfrac{\text{frequency}}{\text{total}}\times360^\circ\)Use it when drawing a pie chart.
Back from an angleLearn it\(\text{frequency}=\dfrac{\text{angle}}{360}\times\text{total}\)Use it when reading a pie chart.
PictogramLearn itcount = number of symbols × what one symbol stands forUse it when a key like '☺ = 4 people'.
CorrelationLearn itpositive: up together; negative: one up, other down; none: no patternUse it when 'describe the correlation'.
Cumulative frequency
Cumulative frequencyLearn itrunning total of the frequenciesplotted at the UPPER end of each class.Use it when 'cumulative frequency'.
MedianLearn itread across at \(\tfrac{n}{2}\)on a curve use n/2, not (n+1)/2.Use it when 'median' from a curve.
QuartilesLearn itLQ at \(\tfrac{n}{4}\), UQ at \(\tfrac{3n}{4}\); IQR = UQ − LQUse it when 'interquartile range'.
PercentileLearn itthe pth percentile is at \(\tfrac{p}{100}\times n\)Use it when '90th percentile'.
Histograms
Frequency densityLearn it\(\text{frequency density}=\dfrac{\text{frequency}}{\text{class width}}\)the HEIGHT of each bar.Use it when drawing a histogram.
FrequencyLearn it\(\text{frequency}=\text{frequency density}\times\text{class width}\)the AREA of each bar.Use it when reading a histogram.
Part of a classLearn it\(\dfrac{\text{part width}}{\text{class width}}\times\text{frequency}\)assumes values spread evenly.Use it when 'estimate how many between…'.
2D Trigonometry
Pythagoras and right-angled trigonometry
PythagorasLearn it\(a^{2}+b^{2}=c^{2}\)c = hypotenuse (longest side, opposite the right angle).Use it when two sides known, third side wanted.
SineLearn it\(\sin x=\dfrac{\text{opp}}{\text{hyp}}\)Use it when opposite and hypotenuse involved.
CosineLearn it\(\cos x=\dfrac{\text{adj}}{\text{hyp}}\)Use it when adjacent and hypotenuse involved.
TangentLearn it\(\tan x=\dfrac{\text{opp}}{\text{adj}}\)Use it when opposite and adjacent involved.
AngleLearn it\(x=\sin^{-1}(\ldots),\ \cos^{-1}(\ldots),\ \tan^{-1}(\ldots)\)the SHIFT key on the calculator.Use it when an angle wanted.
Elevation, depression and bearings
ElevationLearn it\(\tan(\text{elevation})=\dfrac{\text{height}}{\text{distance}}\)measured UP from the horizontal.Use it when looking up at something.
DepressionLearn itangle of depression = angle of elevation (alternate angles)measured DOWN from the horizontal.Use it when looking down from a height.
Bearing partsLearn iteast \(=d\sin\theta\), north \(=d\cos\theta\)\(\theta\) = bearing (under 90°), d = distance travelled.Use it when 'how far north/east'.
Back bearingLearn itbearing ± 180°Use it when 'bearing of A from B'.
The sine and cosine rules
Sine rule (side)On the formula sheet\(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\)a side over the sine of its OPPOSITE angle.Use it when a side and its opposite angle are known.
Sine rule (angle)Learn it\(\dfrac{\sin A}{a}=\dfrac{\sin B}{b}\)flip it when an angle is wanted.Use it when an angle wanted, opposite pair known.
Cosine rule (side)On the formula sheet\(a^{2}=b^{2}+c^{2}-2bc\cos A\)A is the angle BETWEEN sides b and c.Use it when two sides and the angle between them.
Cosine rule (angle)Learn it\(\cos A=\dfrac{b^{2}+c^{2}-a^{2}}{2bc}\)a is the side opposite the angle you want.Use it when all three sides known.
Obtuse sineLearn it\(\sin(180^\circ-x)=\sin x\)so a sine gives two angles: x and 180° − x.Use it when the ambiguous case.
Area of a triangle and multi-step problems
Area of a triangleOn the formula sheet\(\text{Area}=\tfrac12ab\sin C\)C is the angle BETWEEN sides a and b.Use it when two sides and the included angle.
Angle from an areaLearn it\(\sin C=\dfrac{2\times\text{Area}}{ab}\)can give two angles: C and 180° − C.Use it when area given, angle wanted.
Side from an areaLearn it\(a=\dfrac{2\times\text{Area}}{b\sin C}\)Use it when area given, side wanted.
Mensuration
Units, perimeter and area
RectangleLearn it\(A=lw\)Use it when rectangles and squares.
TriangleOn the formula sheet\(A=\tfrac12bh\)h is the PERPENDICULAR height.Use it when triangles.
ParallelogramLearn it\(A=bh\)perpendicular height, not the slanting side.Use it when parallelograms.
TrapeziumLearn it\(A=\tfrac12(a+b)h\)a and b are the parallel sides.Use it when trapezia.
Unit conversionsLearn it1 m² = 100² cm² = 10 000 cm²; 1 m³ = 100³ cm³ = 1 000 000 cm³; 1 cm³ = 1 ml; 1000 cm³ = 1 litre; 1 m³ = 1000 litresUse it when changing area or volume units.
Circles, arcs and sectors
CircumferenceOn the formula sheet\(C=2\pi r=\pi d\)Use it when distance round a circle.
Area of a circleOn the formula sheet\(A=\pi r^{2}\)
Arc lengthOn the formula sheet\(\dfrac{\theta}{360}\times2\pi r\)θ = the angle at the centre.Use it when part of the circumference.
Sector areaOn the formula sheet\(\dfrac{\theta}{360}\times\pi r^{2}\)Use it when a 'slice' of the circle.
Prisms and cylinders
PrismOn the formula sheet\(V=\) area of cross-section \(\times\) lengthany solid with the same cross-section all the way through.Use it when prisms, cuboids, cylinders.
CuboidLearn it\(V=lwh\); surface area \(=2(lw+lh+wh)\)Use it when boxes.
CylinderOn the formula sheet\(V=\pi r^{2}h\); curved surface area \(=2\pi rh\)total surface area \(=2\pi rh+2\pi r^{2}\) (closed).Use it when cylinders.
CapacityLearn it1 cm³ = 1 ml; 1000 cm³ = 1 litreUse it when 'how many litres'.
Pyramids, cones and spheres
PyramidOn the formula sheet\(V=\tfrac13\times\text{base area}\times h\)h = perpendicular height.Use it when pyramids.
ConeOn the formula sheet\(V=\tfrac13\pi r^{2}h\); curved surface \(=\pi rl\)l = slant height; \(l^{2}=r^{2}+h^{2}\).Use it when cones.
SphereOn the formula sheet\(V=\tfrac43\pi r^{3}\); surface \(=4\pi r^{2}\)Use it when spheres.
HemisphereLearn it\(V=\tfrac23\pi r^{3}\); curved \(=2\pi r^{2}\); solid total \(=3\pi r^{2}\)half a sphere (+ the flat circle if solid).Use it when hemispheres.
Compound solids and frustums
Compound volumeLearn it\(V=V_1+V_2\) (or \(V_1-V_2\) for a hole)split into shapes you know.Use it when joined or hollowed solids.
FrustumLearn it\(V=V_{\text{big cone}}-V_{\text{small cone}}\)the small cone was cut off the top.Use it when a bucket, lampshade or cut cone.
Same volumeLearn it\(V_{\text{before}}=V_{\text{after}}\)melting, pouring or reshaping keeps the volume.Use it when 'melted', 'recast', 'poured'.
Water riseLearn it\(\text{rise}=\dfrac{\text{volume added}}{\text{base area of tank}}\)the water forms a thin prism.Use it when 'the water level rises by'.
Time
Time calculations
UnitsLearn it60 s = 1 min, 60 min = 1 h, 24 h = 1 dayUse it when any time question.
12-hour → 24-hourLearn itpm: add 12 to the hour; 12.xx am → 00 xxUse it when am / pm times.
Decimal hours → minutesLearn itdecimal part \(\times60\)2.35 h = 2 h 21 min.Use it when a decimal number of hours.
Minutes → decimal hoursLearn itminutes \(\div60\)3 h 24 min = 3.4 h.Use it when before a speed calculation.
Timetables, time zones and schedules
TimetableLearn itone COLUMN = one bus; one ROW = one stopUse it when a table of times.
Ahead / behindLearn itahead → ADD the gap; behind → SUBTRACT itUse it when converting to another city's clock.
Flight arrival (local)Learn itdeparture + flight time (departure clock), then convertUse it when time zones.
Average speedLearn it\(\text{distance}\div\text{time in decimal hours}\)Use it when a journey with clock times.
Kinematics
Speed, distance and time
SpeedLearn it\(\text{speed}=\dfrac{\text{distance}}{\text{time}}\)also distance = speed × time, time = distance ÷ speed.Use it when any journey.
Average speedLearn it\(\dfrac{\text{total distance}}{\text{total time}}\)NOT the mean of the speeds.Use it when a journey with stages or a stop.
UnitsLearn itkm/h \(\div3.6\) = m/s; m/s \(\times3.6\) = km/h; minutes \(\div60\) = hoursUse it when mixed units.
Distance-time graphs
Speed from the graphLearn itspeed = gradient = \(\dfrac{\text{change in distance}}{\text{change in time}}\)change minutes to hours for km/h.Use it when a straight sloping section.
Average speedLearn it\(\dfrac{\text{total distance}}{\text{total time}}\)include stops.Use it when the whole journey.
Speed-time graphs
AccelerationLearn it\(a=\dfrac{\text{change in speed}}{\text{time}}\) = gradientunits m/s²; a negative gradient is a deceleration.Use it when a sloping section.
DistanceLearn itdistance = area under the speed-time graphsplit into triangles, rectangles, trapezia.Use it when 'how far'.
TrapeziumLearn it\(\tfrac12(a+b)h\)a and b: the parallel sides; h: the distance between them.Use it when a whole speed-time journey.
Circles
Parts of a circle and the right-angle theorems
Angle in a semicircleLearn itthe angle in a semicircle is 90°the triangle's longest side is a diameter.Use it when a diameter and a point on the circle.
Tangent and radiusLearn itthe angle between a tangent and a radius is 90°Use it when a tangent touching the circle.
Two radiiLearn ittwo radii make an isosceles trianglethe base angles are equal.Use it when the centre O joined to two points.
Circle theorems
CentreLearn itthe angle at the centre is twice the angle at the circumferenceboth standing on the same arc.Use it when an angle at O and one on the circle from the same two points.
Same segmentLearn itangles in the same segment are equalsame chord, same side.Use it when a 'bow-tie' shape.
Cyclic quadrilateralLearn itopposite angles of a cyclic quadrilateral add up to 180°all four corners on the circle.Use it when a quadrilateral inside a circle.
Alternate segmentLearn itthe angle between a tangent and a chord equals the angle in the alternate segmentUse it when a tangent and a chord from the touching point.
Chords and tangents
ChordLearn it\(r^{2}=d^{2}+\left(\tfrac{\text{chord}}2\right)^{2}\)d = distance of the chord from the centre.Use it when a chord length or its distance.
Perpendicular bisector of a chordLearn itthe perpendicular from the centre bisects the chord
Two tangentsLearn ittangents from an external point are equalOAPB is a kite with right angles at A and B.Use it when two tangents from one point.
Simultaneous
Simultaneous equations by elimination
Elimination ruleLearn itsame signs → subtract; different signs → addapplies to the letter you want to remove, once its numbers match.Use it when two linear equations, letters lined up.
Substitution, and a line with a curve
SubstitutionLearn itif \(y=2x-1\), replace every \(y\) by \((2x-1)\)keep the brackets.Use it when one equation already gives x = … or y = ….
Co-Ordinate Geometry
Gradient, midpoint and length
GradientLearn it\(m=\dfrac{y_2-y_1}{x_2-x_1}\)change in y ÷ change in x; subtract in the same order top and bottom.Use it when two points given.
MidpointLearn it\(\left(\dfrac{x_1+x_2}2,\ \dfrac{y_1+y_2}2\right)\)the average of the x's and of the y's.Use it when 'midpoint'.
LengthLearn it\(\sqrt{(x_2-x_1)^{2}+(y_2-y_1)^{2}}\)Pythagoras on the change in x and change in y.Use it when 'length' or 'distance'.
The equation of a straight line
Straight lineLearn it\(y=mx+c\)m = gradient, c = where it crosses the y-axis.Use it when any non-vertical line.
Special linesLearn it\(x=k\) (vertical), \(y=k\) (horizontal, gradient 0)Use it when lines parallel to an axis.
Point on a line?Learn itsubstitute its x; does the equation give its y?Use it when 'does (6, −5) lie on the line?'.
Parallel and perpendicular lines
ParallelLearn it\(m_2=m_1\)same gradient, different y-intercept.Use it when 'parallel to'.
PerpendicularLearn it\(m_1\times m_2=-1\), so \(m_2=-\dfrac1{m_1}\)flip the fraction and change the sign: 2 ↔ −½, −⅔ ↔ 3/2.Use it when 'perpendicular to' / 'at right angles'.
Perpendicular bisector of ABLearn itthrough the midpoint of AB, gradient −1 ÷ (gradient of AB)Use it when 'perpendicular bisector'.
Number Sequences
Linear sequences and the nth term
nth term of a linear sequenceLearn it\(dn+c\)\(d\) = common difference; \(c\) = first term − \(d\) (the 'zero term').Use it when the terms go up or down by the same amount.
Is it a term?Learn itsolve \(dn+c=\text{number}\)it is a term only if n is a positive whole number.Use it when "is 99 a term?".
First term above a valueLearn itsolve \(dn+c>\text{value}\), round n UPUse it when "first term greater than …".
Square, cube, triangular and geometric sequences
Square numbersLearn it\(n^{2}\): 1, 4, 9, 16, 25differences 3, 5, 7, … (odd numbers).Use it when squares or shifted squares.
Cube numbersLearn it\(n^{3}\): 1, 8, 27, 64, 125Use it when cubes.
Triangular numbersLearn it\(\dfrac{n(n+1)}{2}\): 1, 3, 6, 10, 15differences 2, 3, 4, 5.Use it when triangles of dots.
GeometricLearn it\(ar^{n-1}\)\(a\) = first term, \(r\) = common ratio (multiply by it each time).Use it when each term × the same number.
Fraction sequenceLearn it\(\dfrac{\text{nth term of the tops}}{\text{nth term of the bottoms}}\)Use it when a list of fractions.
Quadratic, cubic and combined sequences
Quadratic sequenceLearn it\(an^{2}+bn+c\), with second difference \(=2a\)constant SECOND differences.Use it when first differences go up by the same amount.
The three-equation methodLearn it\(2a=\) 2nd difference; \(3a+b=\) 1st first-difference; \(a+b+c=\) 1st termUse it when an alternative to subtracting \(an^{2}\).
Cubic / combinationLearn itcompare with \(n^{3}\) or \(2^{n}\) term by termUse it when neither linear nor quadratic.
Sequences from patterns and tables
Pattern → formulaLearn itcount Patterns 1, 2, 3 → table → nth termUse it when a sequence of diagrams.
Which pattern uses X?Learn itsolve nth term \(=X\)n must be a whole number, otherwise no pattern uses X.Use it when "which pattern has 60 tiles".
Largest pattern you can makeLearn itsolve nth term \(\le X\), round n DOWNUse it when "she has 150 dots".
Square with a ringLearn itinside \(n^{2}\), whole \((n+2)^{2}\), ring \(4n+4\)Use it when a square of tiles with a border.
Graphical Solutions
Solving equations with graphs
IntersectionsLearn itwhere \(y=f(x)\) meets \(y=g(x)\): \(f(x)=g(x)\)the x-coordinates of the crossing points are the solutions.Use it when any 'use your graph to solve'.
The line to drawLearn itline \(=\) curve \(-\) (new equation's left side, when it equals 0)e.g. curve \(x^{2}-2x-3\), new \(x^{2}-3x-1=0\): line \(y=x-2\).Use it when 'by drawing a suitable straight line'.
Gradients of curves: tangents
GradientLearn it\(\dfrac{\text{change in }y}{\text{change in }x}=\dfrac{y_2-y_1}{x_2-x_1}\)use two points ON THE TANGENT, far apart.Use it when estimating a curve's gradient.
Rate of changeLearn itgradient = y-units per x-unitdistance-time: speed; speed-time: acceleration.Use it when a graph in context.
Linear Inequalities And Shading
Solving linear inequalities
The one new ruleLearn itmultiply or divide by a NEGATIVE → reverse the sign\(-2x>8\Rightarrow x<-4\)Use it when the number in front of x is negative.
Number lineLearn it\(<,\ >\): open circle; \(\le,\ \ge\): filled circleopen = not included, filled = included.Use it when drawing or reading a number line.
Inequalities on graphs: regions and shading
Line styleLearn it\(\le,\ \ge\): solid line; \(<,\ >\): broken linesolid = points on the line are included.Use it when drawing any boundary.
Test pointLearn itput a point not on the line into the inequalitytrue → that side is wanted; false → the other side.Use it when deciding which side to shade.
Inequalities in context: best profit and least cost
Words to symbolsLearn itat least → ≥; at most / no more than → ≤; more than / fewer than → > or <Use it when turning sentences into inequalities.
Profit or costLearn ite.g. \(P=20x+15y\)work it out at each candidate integer point.Use it when 'greatest profit' / 'least cost'.
Symmetry
Line and rotational symmetry
Line symmetryLearn ita mirror line: folding along it, the halves matchUse it when 'lines of symmetry'.
Rotational symmetryLearn itorder = how many times it looks the same in one full turnno symmetry = order 1 (never 0).Use it when 'order of rotational symmetry'.
Regular polygonLearn itn sides → n lines of symmetry and order nUse it when a regular shape.
Symmetry of solids
Prism (regular n-gon end)Learn itn + 1 planes; order n about its long axisn through the length + 1 across the middle.Use it when prisms.
Pyramid (regular n-gon base)Learn itn planes; order n about its axisno plane across the middle.Use it when pyramids.
Special solidsLearn itcuboid 3; cube 9; cylinder and cone: infinitely many
Polygons
Angle facts and reasons
Straight lineLearn itangles on a straight line add up to 180°Use it when angles meeting on a line.
PointLearn itangles at a point add up to 360°Use it when angles all the way round.
Vertically oppositeLearn itvertically opposite angles are equalUse it when two straight lines crossing.
Triangle / quadrilateralLearn itangles in a triangle add up to 180°; in a quadrilateral, 360°Use it when angles inside a shape.
IsoscelesLearn itbase angles of an isosceles triangle are equalthe angles opposite the equal sides.Use it when two equal sides (ticks).
Exterior angleLearn itexterior angle of a triangle = sum of the two interior opposite anglesUse it when a triangle side extended.
Angles in parallel lines
CorrespondingLearn itcorresponding angles are equalF-shape: same position at each crossing.Use it when parallel lines (arrows).
AlternateLearn italternate angles are equalZ-shape: opposite sides of the transversal, between the lines.Use it when parallel lines.
Co-interiorLearn itco-interior angles add up to 180°C-shape: same side, between the lines.Use it when parallel lines.
Angles in polygons
Interior angle sumLearn it\((n-2)\times180^\circ\)n = number of sides; any polygon.Use it when the total of the inside angles.
Exterior angle sumLearn it\(360^\circ\)any polygon.Use it when exterior angles.
Regular polygonLearn itexterior \(=\dfrac{360^\circ}n\); interior \(=180^\circ-\) exteriorbackwards: \(n=\dfrac{360^\circ}{\text{exterior}}\).Use it when all sides and angles equal.
Daily Maths
Percentages, fractions and decimals
Per centLearn it\(x\%=\dfrac{x}{100}\)per cent means "out of 100". The whole is always 100%.Use it when turning any percentage into a fraction or decimal.
Percentage → decimalLearn it\(\div100\)move the point 2 places left: 7% = 0.07.Use it when you need a multiplier for the calculator.
Decimal or fraction → percentageLearn it\(\times100\)\(0.3=30\%\); \(\tfrac38\times100=37.5\%\).Use it when "write as a percentage".
Percentage of an amountLearn it\(\dfrac{x}{100}\times\text{amount}\)"of" means multiply.Use it when "find 35% of 240".
One amount as a percentage of anotherLearn it\(\dfrac{\text{part}}{\text{whole}}\times100\)the whole is the amount after the word "of"; same units first.Use it when "write 42 as a percentage of 60".
Percentage increase and decrease
Increase by r%Learn it\(\text{new}=\text{original}\times\left(1+\dfrac r{100}\right)\)+15% → × 1.15; +3% → × 1.03.Use it when tax, pay rise, growth, "increase by".
Decrease by r%Learn it\(\text{new}=\text{original}\times\left(1-\dfrac r{100}\right)\)−12% → × 0.88.Use it when discount, sale, loss, "decrease by".
Percentage changeLearn it\(\dfrac{\text{change}}{\text{original}}\times100\)change = new − old; original = the value BEFORE.Use it when "find the percentage increase/decrease".
Repeated changeLearn it\(\text{original}\times m_1\times m_2\); same change n times: \(\times m^{n}\)multiply the multipliers, never add the percentages.Use it when "then", "a further", "each year".
Reverse percentages: finding the original
Reverse percentageLearn it\(\text{original}=\dfrac{\text{new amount}}{\text{multiplier}}\)the multiplier is the one that went FROM the original TO the new amount.Use it when you know the amount after the change.
After an increase of r%Learn itnew = \((100+r)\%\) of the originale.g. a price including 15% tax is 115% of the price before tax.Use it when tax, pay rise, profit.
After a decrease of r%Learn itnew = \((100-r)\%\) of the originale.g. a sale price after 15% off is 85% of the original.Use it when sale, loss, depreciation.
Given the part or the changeLearn it\(\text{whole}=\dfrac{\text{part}}{r/100}\)"$18 is 12% of the price" → 18 ÷ 0.12.Use it when the number given IS the r%.
Profit, loss, discount and other money problems
Profit / lossLearn itprofit \(=SP-CP\); loss \(=CP-SP\)CP = cost price (what the seller paid), SP = selling price.Use it when buying and selling.
Percentage profit or lossLearn it\(\dfrac{\text{profit or loss}}{CP}\times100\)always out of the COST price.Use it when "find the percentage profit".
Selling priceLearn itprofit \(p\%\): \(SP=CP\times\left(1+\dfrac p{100}\right)\); loss: \(\times\left(1-\dfrac p{100}\right)\)Use it when a target profit or a given loss.
Sale priceLearn it\(\text{marked}\times\left(1-\dfrac d{100}\right)\)d% discount off the marked (label) price.Use it when a sale.
Percentage discountLearn it\(\dfrac{\text{marked}-\text{sale}}{\text{marked}}\times100\)out of the MARKED price.Use it when "find the percentage discount".
InstalmentsLearn ittotal = deposit + number × paymentextra = total − cash price.Use it when "pay monthly" plans.
Percentage word problems: strategy and full-depth practice
Percentage OFLearn itwhole \(\times\dfrac r{100}\)Use it when you know the whole and the %.
AS a percentageLearn it\(\dfrac{\text{part}}{\text{whole}}\times100\)Use it when you know the part and the whole.
Increase / decreaseLearn it\(\times\left(1\pm\dfrac r{100}\right)\)Use it when you know the start and the % change.
Percentage changeLearn it\(\dfrac{\text{change}}{\text{old}}\times100\)Use it when you know old and new.
ReverseLearn it\(\div\left(1\pm\dfrac r{100}\right)\)Use it when you know the amount AFTER the change.
Percentage of a percentageLearn it\(\dfrac{r_1}{100}\times\dfrac{r_2}{100}\)e.g. 15% of the 60% who are adults = 9% of everyone.Use it when the second % is of a smaller group.
Ratio and proportion
One partLearn it\(\text{one part}=\dfrac{\text{amount}}{\text{number of parts it is}}\)total ÷ sum of parts; a share ÷ its parts; a difference ÷ difference in parts.Use it when every sharing question.
Ratio → fractionLearn itin \(a:b\), the first share is \(\dfrac a{a+b}\) of the wholeboys : girls = 3 : 5 → boys are \(\tfrac38\).Use it when "what fraction is ...".
Increase / decrease in a ratioLearn itold : new \(=a:b\Rightarrow\text{new}=\text{old}\times\dfrac ba\)Use it when "increase 40 in the ratio 5 : 8".
Unitary methodLearn itamount for 1 = amount ÷ number; then × the number you wantUse it when recipes and proportion.
Best valueLearn itprice per unit (per 100 g, per item)lowest price per unit = best value.Use it when comparing packs.
Rates and unit conversion
Big ↔ small unitLearn itbig → small: ×; small → big: ÷1 km = 1000 m, 1 m = 100 cm, 1 kg = 1000 g, 1 litre = 1000 ml.Use it when any unit change.
Area and volume unitsLearn itarea: × (factor)\(^{2}\); volume: × (factor)\(^{3}\)\(1\text{ m}^2=10\,000\text{ cm}^2\); \(1\text{ m}^3=1\,000\,000\text{ cm}^3\).Use it when squared or cubed units.
CapacityLearn it\(1\text{ cm}^3=1\text{ ml}\); \(1000\text{ cm}^3=1\) litreUse it when volume of a container in litres.
RateLearn itrate = amount ÷ time; amount = rate × time; time = amount ÷ rate"per" means divide.Use it when flow, pay, fuel.
DensityLearn itdensity = mass ÷ volumeg/cm³ or kg/m³.Use it when mass and volume.
PressureLearn itpressure = force ÷ areaN/m².Use it when force on an area.
Population densityLearn itpeople ÷ areapeople per km².Use it when towns and countries.
km/h ↔ m/sLearn itkm/h → m/s: × 1000 ÷ 3600 (÷ 3.6)m/s → km/h: × 3.6.Use it when speeds in different units.
Currency conversion and exchange rates
Exchange rateLearn it1 A = r Br = how many B for ONE A.Use it when every currency question.
A → B (away from the 1)Learn it\(\times r\)$1 = €0.92: $250 → 250 × 0.92.Use it when you have the currency with the 1.
B → A (towards the 1)Learn it\(\div r\)€161 → 161 ÷ 0.92.Use it when you have the other currency.
Rate per 100Learn itrate for 1 = rate for 100 ÷ 100Use it when a table "for 100 SGD".
Find the rateLearn it\(r=\dfrac{\text{amount of B}}{\text{amount of A}}\)divide by the side that should be 1.Use it when "find the exchange rate".
Maps and scales
Map → realLearn itreal = map \(\times n\)do it in cm, then change units.Use it when you measured on the map.
Real → mapLearn itmap = real (in cm) \(\div n\)Use it when you know the real distance.
Key factLearn it1 km = 100 000 cm1000 m, each 100 cm.Use it when every map question.
Area scaleLearn it1 cm² stands for (what 1 cm stands for)\(^{2}\)1 : 50 000: 1 cm = 0.5 km, so 1 cm² = 0.25 km².Use it when areas on maps.
Simple interest
Simple interestLearn it\(I=\dfrac{P\times R\times T}{100}\)\(P\) = amount invested or borrowed, \(R\) = % rate per year, \(T\) = years.Use it when "simple interest".
Total valueLearn it\(A=P+I\)the money at the end.Use it when "value of the investment", "total owed".
Find the rateLearn it\(R=\dfrac{100I}{PT}\)find I first (total − P).Use it when Type 2.
Find the timeLearn it\(T=\dfrac{100I}{PR}\)Use it when Type 3.
Find the principalLearn it\(P=\dfrac{100I}{RT}\)Use it when Type 3.
Compound interest and repeated percentage change
Compound interestLearn it\(A=P\left(1+\dfrac r{100}\right)^{n}\)\(A\) = value after \(n\) years, \(P\) = amount invested, \(r\) = % rate.Use it when "compound interest", "each year".
DepreciationLearn it\(A=P\left(1-\dfrac r{100}\right)^{n}\)a value falling by r% each year.Use it when cars, machines, population falling.
Interest earnedLearn it\(A-P\)the formula gives the VALUE.Use it when "how much interest".
Amount investedLearn it\(P=A\div(\text{multiplier})^{n}\)Use it when you know the final value.
RateLearn it\(1+\dfrac r{100}=\sqrt[n]{\dfrac AP}\)Use it when "find r".
Pie charts
Degrees per itemLearn it\(\dfrac{360}{\text{total}}\)the one number that unlocks every question.Use it when drawing or reading a pie chart.
Sector angleLearn it\(\dfrac{\text{frequency}}{\text{total}}\times360°\)Use it when you know the frequency.
FrequencyLearn it\(\dfrac{\text{angle}}{360}\times\text{total}\)Use it when you know the angle.
Angles round the centreLearn itall sector angles add up to 360°Use it when missing angles; checking your angles.
Proportionality
Direct and inverse proportion
Direct proportionLearn it\(y\propto x\ \Rightarrow\ y=kx\)\(\dfrac yx=k\) is constant; graph a straight line through (0, 0).Use it when "directly proportional", more → more.
Inverse proportionLearn it\(y\propto\dfrac1x\ \Rightarrow\ y=\dfrac kx\)\(xy=k\) is constant; graph a curve that never meets the axes.Use it when "inversely proportional", more → less.
Workers and daysLearn itworkers × days = total work (constant)Use it when same rate, different team size.
Proportion with powers and roots
Proportional to a power or rootLearn it\(y=kx^{2},\ y=k\sqrt x,\ y=kx^{3},\ y=k\sqrt[3]x\)Use it when "proportional to the square / square root / cube / cube root".
Inversely proportionalLearn it\(y=\dfrac k{x^{2}},\ \dfrac k{\sqrt x},\ \dfrac k{x^{3}}\)Use it when "inversely proportional to the …".
Of a bracketLearn it\(y=k(x+3)^{2}\), \(y=\dfrac k{\sqrt{x+1}}\)the power is on the WHOLE bracket.Use it when "the square of (x + 3)".
Effect of multiplying x by mLearn itdirect: \(y\times m^{n}\); inverse: \(y\div m^{n}\)n is the power in the relationship.Use it when "what happens to y when x is doubled".
Limits Of Accuracy
Upper and lower bounds
BoundsLearn itvalue \(\pm\tfrac12\) of the rounding unitnearest 1 → ± 0.5; nearest 10 → ± 5; nearest 5 → ± 2.5; nearest 0.1 → ± 0.05.Use it when "upper / lower bound".
Error intervalLearn it\(\text{LB}\le x<\text{UB}\)≤ at the bottom, < at the top.Use it when "complete … ≤ m < …".
Unit from s.f. / d.p.Learn itthe place value of the LAST figure given3.40 to 3 s.f.: unit 0.01, so ± 0.005.Use it when "to 3 s.f.", "to 2 d.p.".
Totals of n equal itemsLearn itLB total \(=n\times\)LB; UB total \(=n\times\)UBUse it when perimeter of a regular shape, several bags.
Bounds in calculations
Add / multiplyLearn itUB = UB + UB (or × ); LB = LB + LB (or ×)same direction for both.Use it when a sum or product (area, perimeter).
SubtractLearn itUB = UB\(_a\) − LB\(_b\); LB = LB\(_a\) − UB\(_b\)cross over: take away the least for the biggest answer.Use it when a difference.
DivideLearn itUB = UB\(_a\) ÷ LB\(_b\); LB = LB\(_a\) ÷ UB\(_b\)cross over: divide by the smallest for the biggest answer.Use it when speed, density, a rearranged formula.
Certain to fitLearn itworst case: LB capacity ÷ UB item, round DOWNUse it when "certain to carry".
Appropriate accuracyLearn itround LB and UB until they agreeUse it when "to an appropriate degree of accuracy".
Hcf Lcm
Prime factors, HCF and LCM
Prime factorisationLearn ite.g. \(360=2^{3}\times3^{2}\times5\)every whole number above 1 has exactly one.Use it when "product of prime factors".
HCFLearn itprimes in BOTH numbers, each to the LOWER powerthe biggest number dividing both.Use it when "highest common factor", sharing into equal groups.
LCMLearn itevery prime in EITHER number, each to the HIGHER powerthe smallest number both divide into.Use it when "lowest common multiple", things happening together.
CheckLearn it\(\text{HCF}\times\text{LCM}=a\times b\)for two numbers.Use it when checking your answers.
Making squares and cubes: smallest n problems
Perfect squareLearn itevery index in the prime factorisation is EVEN\(2^{4}\times3^{2}\times7^{2}=84^{2}\).Use it when "is a square number".
Perfect cubeLearn itevery index is a MULTIPLE OF 3\(3^{3}\times5^{3}=15^{3}\).Use it when "is a cube number".
Square / cube rootLearn ithalve every index / divide every index by 3Use it when a root without a calculator.
Smallest multiplierLearn itthe fewest extra primes to fix every indexUse it when "smallest n such that 180n is a square".
HCF and LCM problems
Use the LCMLearn ithappen together again; same number of each; smallest number of packsthe answer is BIGGER than the numbers.Use it when cycles, packs.
Use the HCFLearn itequal pieces as long as possible; largest tiles; most identical groupsthe answer is SMALLER than the numbers.Use it when cutting, sharing, tiling.
Numbers from HCF and LCMLearn itnumbers \(ha\), \(hb\) with \(ab=\dfrac{\text{LCM}}{\text{HCF}}\), \(a,b\) sharing no factorUse it when "find the two numbers".
Algebraic termsLearn itletters behave like primes: HCF lower powers, LCM higher powersHCF of \(12a^{3}b\) and \(20a^{2}b^{2}\) is \(4a^{2}b\).Use it when terms with letters.
Graphs Of Functions
Graph shapes and sketching
y-interceptLearn itput \(x=0\)Use it when every sketch.
x-intercepts (roots)Learn itput \(y=0\) and solvefactorised: each bracket = 0.Use it when every sketch.
Shape from the highest powerLearn it\(x\): line; \(x^{2}\): parabola; \(x^{3}\): S-shape; \(\tfrac1x\): two branches; \(a^{x}\): exponentiala negative coefficient flips the shape.Use it when recognising a graph.
Quadratic and cubic graphs
Line of symmetryLearn it\(x=\) midway between the roots, or \(x=-\dfrac b{2a}\)Use it when a parabola's mirror line or turning point.
Turning pointLearn it\(y=(x+p)^{2}+q\) → \((-p,\ q)\)minimum if the \(x^{2}\) is positive, maximum if negative.Use it when after completing the square.
Equation from rootsLearn it\(y=a(x-p)(x-q)\)use one more point to find a.Use it when a graph showing two roots.
Reciprocal, exponential and mixed graphs
ReciprocalLearn it\(y=\dfrac ax\) (so \(xy=a\))asymptotes: the x- and y-axes. \(a>0\): quadrants 1 and 3; \(a<0\): quadrants 2 and 4.Use it when a number divided by x.
ExponentialLearn it\(y=a^{x}\) (\(a>0\))passes through (0, 1); always above the x-axis.Use it when x is a power.
Exponential with a multiplierLearn it\(y=k\times a^{x}\)passes through (0, k).
Domain and rangeLearn it\(y=\dfrac ax\): \(x\ne0\), \(y\ne0\); \(y=a^{x}\): \(y>0\)read from the asymptotes.Use it when 'state the range'.
Drawing and reading graphs accurately
One small squareLearn itvalue of one small square = (value of one big square) ÷ (number of small squares in it)work this out on each axis before plotting.Use it when every graph.
Solving f(x) = kLearn itdraw the line y = k; read the x-values where it meets the curvethere may be more than one.Use it when 'use your graph to solve …'.
Algebra
Expressions, substitution and expanding brackets
Single bracketLearn it\(a(b+c)=ab+ac\)the outside term multiplies EVERY inside term.Use it when one bracket with something in front.
Two bracketsLearn it\((a+b)(c+d)=ac+ad+bc+bd\)every term × every term (4 products), then collect.Use it when two brackets multiplied.
Squared bracketLearn it\((a+b)^{2}=a^{2}+2ab+b^{2}\), \((a-b)^{2}=a^{2}-2ab+b^{2}\)always a middle term.Use it when a bracket squared.
Index lawsLearn it\(x^{m}\times x^{n}=x^{m+n}\), \(x^{m}\div x^{n}=x^{m-n}\)Use it when multiplying or dividing terms.
Linear equations and changing the subject
Inverse operationsLearn it+ ↔ −, × ↔ ÷, square ↔ square rootundo what was done to x, in reverse order.Use it when solving or rearranging.
Clearing fractionsLearn itmultiply EVERY term by the LCM of the denominatorsput brackets round numerators with two terms.Use it when an equation with fractions.
Algebraic fractions
CancellingLearn it\(\dfrac{a\times c}{b\times c}=\dfrac ab\)cancel a FACTOR (something multiplied), never a term (something added).Use it when simplify.
Multiply / divideLearn it\(\dfrac ab\times\dfrac cd=\dfrac{ac}{bd}\), \(\dfrac ab\div\dfrac cd=\dfrac ab\times\dfrac dc\)Use it when × or ÷ of fractions.
Add / subtractLearn it\(\dfrac ab+\dfrac cd=\dfrac{ad+bc}{bd}\)use the lowest common denominator if there is one.Use it when 'as a single fraction'.
Equations that lead to quadratics
Quadratic formulaOn the formula sheet\(x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}\)for \(ax^{2}+bx+c=0\).Use it when it won't factorise, or 'give answers to 2 d.p.'.
Clearing two fractionsLearn it\(\dfrac A{P}+\dfrac B{Q}=C\ \Rightarrow\ AQ+BP=CPQ\)multiply EVERY term by \(PQ\).Use it when x in the denominators.
Completing the square
Completing the squareLearn it\(x^{2}+bx+c=\left(x+\tfrac b2\right)^{2}-\left(\tfrac b2\right)^{2}+c\)halve b for the bracket, then subtract that number squared.Use it when 'write in the form \((x+p)^{2}+q\)'.
Turning pointLearn it\(y=(x+p)^{2}+q\) has turning point \((-p,\ q)\)minimum value \(q\) at \(x=-p\) (maximum if the \(x^{2}\) is negative).Use it when 'minimum value', 'turning point'.
The quadratic formula
Quadratic formulaOn the formula sheet\(x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}\)for \(ax^{2}+bx+c=0\); the WHOLE top is divided by \(2a\).Use it when it won't factorise, or decimals are asked for.
Forming and solving equations from words
TimeLearn it\(\text{time}=\dfrac{\text{distance}}{\text{speed}}\)Use it when a journey with an unknown speed.
Consecutive numbersLearn it\(n,\ n+1,\ n+2\); odd or even: \(n,\ n+2,\ n+4\)Use it when 'consecutive'.
RectangleLearn itarea \(=lw\), perimeter \(=2(l+w)\)Use it when a shape with sides in terms of x.
Sets
Set notation: listing, union, intersection, complement
UnionLearn it\(A\cup B\)in A OR B (or both).Use it when "∪".
IntersectionLearn it\(A\cap B\)in A AND B.Use it when "∩".
ComplementLearn it\(A'\)everything in ξ that is NOT in A.Use it when a dash after the letter.
Number of elementsLearn it\(n(A)\)a COUNT, not a list.Use it when "find n(…)".
Counting a unionLearn it\(n(A\cup B)=n(A)+n(B)-n(A\cap B)\)shared elements are counted once.Use it when counting either set.
SubsetLearn it\(C\subseteq A\)every element of C is in A.Use it when "is a subset of".
Shading and describing Venn diagram regions
RegionsLearn it2 sets → 4 regions; 3 sets → 8 regionsevery expression is a choice of regions.Use it when any shading question.
A onlyLearn it\(A\cap B'\)Use it when in A, not in B.
Outside bothLearn it\((A\cup B)'=A'\cap B'\)Use it when neither.
All but the overlapLearn it\((A\cap B)'=A'\cup B'\)Use it when not in both.
Venn diagram problems and probability
OverlapLearn it\(n(A\cap B)=n(A)+n(B)-n(A\cup B)\)the counting rule; n(A ∪ B) = total − neither.Use it when two circle totals and a total.
'Only' regionLearn itcircle total − everything else inside that circleUse it when after the overlap.
Smallest overlapLearn it\(n(A)+n(B)-n(\xi)\) (or 0)Use it when "smallest possible number in both".
Largest overlapLearn itthe smaller of n(A) and n(B)Use it when "largest possible".
ProbabilityLearn it\(\dfrac{\text{number in the region}}{\text{number you choose FROM}}\)"chosen from those who …" changes the bottom.Use it when probability from a Venn diagram.
Probability
Probability of an event
Single eventLearn it\(P(A)=\dfrac{\text{number of ways A can happen}}{\text{total number of outcomes}}\)outcomes must be equally likely.Use it when picking at random.
Not happeningLearn it\(P(A')=1-P(A)\)Use it when 'not', 'does not'.
All outcomesLearn itprobabilities of all outcomes add to 1Use it when a table with a missing probability.
Relative frequencyLearn it\(\dfrac{\text{number of times it happened}}{\text{number of trials}}\)an ESTIMATE of the probability.Use it when results of an experiment.
Expected frequencyLearn it\(P(A)\times\text{number of trials}\)Use it when 'how many times would you expect'.
Sample spaces and Venn diagrams
Sample spaceLearn it\(P=\dfrac{\text{cells you want}}{\text{all cells}}\)two dice: 6 × 6 = 36 cells.Use it when two things happen together.
And (Venn)Learn it\(P(A\cap B)\) = the overlapUse it when 'both'.
Or (Venn)Learn it\(P(A\cup B)\) = everything inside A or B (or both)Use it when 'A or B', 'at least one'.
NotLearn it\(P(A')\) = everything outside AUse it when 'not A'.
Tree diagrams and combined events
AND (along the branches)Learn it\(P(A\text{ and }B)=P(A)\times P(B)\)for independent events (or using the new probability on the second branch).Use it when 'both', 'then'.
OR (different routes)Learn it\(P(\text{route 1})+P(\text{route 2})\)only for outcomes that can't both happen.Use it when 'either', 'one of each'.
At least oneLearn it\(1-P(\text{none})\)Use it when 'at least one'.
Branches from a pointLearn itadd to 1Use it when checking a tree.
Conditional probability
Given thatLearn it\(P(\text{A given B})=\dfrac{\text{number in A and B}}{\text{number in B}}\)the 'given' group becomes the new total.Use it when 'given that', 'of those who…', 'if … what is the probability'.
With probabilitiesLearn it\(P(\text{A given B})=\dfrac{P(A\cap B)}{P(B)}\)same idea with probabilities instead of counts. You don't need the notation P(A|B).Use it when Venn or tree with decimals.
Vectors
Column vectors and magnitude
Add / subtractLearn it\(\begin{pmatrix}a\\b\end{pmatrix}\pm\begin{pmatrix}c\\d\end{pmatrix}=\begin{pmatrix}a\pm c\\b\pm d\end{pmatrix}\)top with top, bottom with bottom.Use it when sums of vectors.
Scalar multipleLearn it\(k\begin{pmatrix}a\\b\end{pmatrix}=\begin{pmatrix}ka\\kb\end{pmatrix}\)Use it when 2a, −3b.
MagnitudeLearn it\(\left|\begin{pmatrix}x\\y\end{pmatrix}\right|=\sqrt{x^{2}+y^{2}}\)Pythagoras.Use it when 'length' or '|a|'.
Between two pointsLearn it\(\overrightarrow{AB}=\) B − Aend minus start.Use it when vector from A to B.
ParallelLearn it\(\mathbf{u}=k\mathbf{v}\)one is a multiple of the other.Use it when 'show that … parallel'.
Vector geometry
RouteLearn it\(\overrightarrow{AB}=\overrightarrow{AO}+\overrightarrow{OB}=-\mathbf{a}+\mathbf{b}\)going backwards along a vector = minus it.Use it when any vector you don't know.
MidpointLearn it\(\overrightarrow{OM}=\tfrac12(\mathbf{a}+\mathbf{b})\)M is the midpoint of AB.Use it when 'M is the midpoint'.
RatioLearn itAP : PB = m : n \(\Rightarrow\overrightarrow{AP}=\dfrac{m}{m+n}\overrightarrow{AB}\)Use it when 'P divides AB in the ratio…'.
ParallelLearn it\(\overrightarrow{PQ}=k\,\overrightarrow{RS}\)Use it when 'show that PQ is parallel to RS'.
CollinearLearn it\(\overrightarrow{AC}=k\,\overrightarrow{AB}\) AND a shared point AUse it when 'show that A, B, C lie on a straight line'.
Transformations
Reflection and translation
Reflect in x = aLearn it\((x,y)\to(2a-x,\ y)\)the y-axis is x = 0.Use it when a vertical mirror.
Reflect in y = bLearn it\((x,y)\to(x,\ 2b-y)\)the x-axis is y = 0.Use it when a horizontal mirror.
Reflect in y = xLearn it\((x,y)\to(y,\ x)\)swap.Use it when a diagonal mirror.
Reflect in y = −xLearn it\((x,y)\to(-y,\ -x)\)swap and change both signs.Use it when the other diagonal.
TranslationLearn it\(\begin{pmatrix}a\\b\end{pmatrix}\): a right (left if negative), b up (down if negative)Use it when a slide.
Rotation
90° anticlockwise about OLearn it\((x,y)\to(-y,\ x)\)Use it when a quarter turn left.
90° clockwise about OLearn it\((x,y)\to(y,\ -x)\)same as 270° anticlockwise.Use it when a quarter turn right.
180° about OLearn it\((x,y)\to(-x,\ -y)\)direction does not matter.Use it when a half turn.
Any other centre CLearn itwork from C: (point − C) → turn → + CUse it when a centre that is not the origin.
Enlargement
Scale factorLearn it\(k=\dfrac{\text{image length}}{\text{object length}}\)Use it when finding k.
Image of a pointLearn itimage = centre + k × (point − centre)distance from the centre is multiplied by k.Use it when drawing or calculating.
Fraction kLearn it\(0<k<1\): the image is SMALLERstill called an enlargement.Use it when 'scale factor ½'.
Negative kLearn itimage on the OTHER side of the centre, upside downUse it when 'scale factor −2'.
AreaLearn itarea of image \(=k^{2}\times\) area of objectUse it when an area after enlarging.
Combined transformations
Two reflections, axesLearn itin the y-axis then the x-axis = rotation 180° about OUse it when mirrors at right angles.
Two reflections, parallel linesLearn it= a translation, twice the gap between the linesUse it when parallel mirrors.
InverseLearn itreflection: the same; rotation: same angle, opposite way; translation: minus the vector; enlargement: 1/k, same centreUse it when 'undo', 'maps B back onto A'.
Similarity
Similar shapes: lengths
Scale factorLearn it\(k=\dfrac{\text{length on the new shape}}{\text{matching length on the old shape}}\)every length × k; angles stay the same.Use it when two similar shapes.
Similar trianglesLearn ittwo pairs of equal anglesthe third pair then matches too.Use it when showing triangles are similar.
Similar shapes: areas and volumes
Length, area, volumeLearn itlength × \(k\); area × \(k^{2}\); volume × \(k^{3}\)k = the LENGTH scale factor.Use it when similar shapes or solids.
Back to kLearn it\(k=\sqrt{\text{area ratio}}\) or \(k=\sqrt[3]{\text{volume ratio}}\)Use it when given two areas or two volumes.
Same heightLearn ittriangles with the same height: areas in the ratio of their basesUse it when triangles sharing a vertex and a base line.
Congruent
Congruent shapes
SSSLearn itthree pairs of equal sidesUse it when all three sides given.
SASLearn ittwo sides and the angle BETWEEN themthe angle must be the included one.Use it when two sides and an angle.
ASA (AAS)Learn ittwo angles and a matching sideUse it when two angles and a side.
RHSLearn itright angle, hypotenuse and one other sideUse it when right-angled triangles.
Surds
Surds: simplifying, adding and expanding
MultiplyLearn it\(\sqrt a\times\sqrt b=\sqrt{ab}\)\(\sqrt6\times\sqrt3=\sqrt{18}\).Use it when multiplying surds; simplifying.
DivideLearn it\(\dfrac{\sqrt a}{\sqrt b}=\sqrt{\dfrac ab}\)\(\dfrac{\sqrt{48}}{\sqrt3}=4\).Use it when dividing surds.
SquareLearn it\((\sqrt a)^{2}=a\)\((3\sqrt2)^{2}=18\).Use it when a surd times itself.
Not for + or −Learn it\(\sqrt a+\sqrt b\ne\sqrt{a+b}\)only LIKE surds add: \(3\sqrt5+2\sqrt5=5\sqrt5\).Use it when sums of surds.
Difference of two squaresLearn it\((a+\sqrt b)(a-\sqrt b)=a^{2}-b\)the surds cancel.Use it when rationalising (next page).
Rationalising the denominator
Surd on the bottomLearn it\(\dfrac{k}{\sqrt a}\times\dfrac{\sqrt a}{\sqrt a}=\dfrac{k\sqrt a}{a}\)multiplying by 1 changes nothing but the look.Use it when a bottom like \(\sqrt5\) or \(2\sqrt6\).
ConjugateLearn itconjugate of \(a+\sqrt b\) is \(a-\sqrt b\)change the sign in the middle.Use it when a bottom like \(2-\sqrt5\).
Why it worksLearn it\((a+\sqrt b)(a-\sqrt b)=a^{2}-b\)no surd is left.Use it when every conjugate question.
3D Trigonometry
Cuboids, diagonals and the angle with a plane
Space diagonalLearn it\(d=\sqrt{a^{2}+b^{2}+c^{2}}\)a, b, c = length, width, height.Use it when corner to opposite corner of a cuboid.
Base diagonalLearn it\(\sqrt{a^{2}+b^{2}}\)Use it when the line under the space diagonal.
Line and planeLearn it\(\tan\theta=\dfrac{\text{height}}{\text{base diagonal}}\)θ is between the line and its 'shadow' on the plane.Use it when 'angle between AG and the base'.
Pyramids and prisms in 3D
Centre of a square baseLearn it\(AM=\tfrac12\times\text{diagonal}=\tfrac12\sqrt{a^{2}+b^{2}}\)M = where the diagonals cross; the apex is above M.Use it when a right pyramid.
Slant edgeLearn it\(VA=\sqrt{AM^{2}+VM^{2}}\)VM = vertical height.Use it when edge length.
Edge to baseLearn it\(\tan\theta=\dfrac{VM}{AM}\)Use it when angle between VA and the base.
Exponential Growth And Decay
Exponential growth and decay
Exponential modelLearn it\(\text{value}=\text{start}\times\left(1\pm\dfrac r{100}\right)^{n}\)+ for growth, − for decay; n = number of periods (years, hours, days).Use it when "increases / decreases exponentially at r% per …".
MultiplierLearn itgrowth \(1+\tfrac r{100}\); decay \(1-\tfrac r{100}\)+12% → × 1.12; −2.4% → × 0.976.Use it when any percentage change.
Going back in timeLearn it\(\text{earlier value}=\text{value}\div\text{multiplier}^{n}\)Use it when "what was it worth … ago".
Finding the rateLearn it\(\text{multiplier}=\sqrt[n]{\dfrac{\text{final}}{\text{start}}}\)Use it when two values and the time between them.
Finding the number of years (or days)
The condition to testLearn it\(\text{start}\times m^{n}\ \ge\ \text{target}\) (or \(<\) for decay)try whole numbers n.Use it when "how many complete years …".
Doubling timeLearn itsmallest n with \(m^{n}\ge2\)the starting amount does not matter.Use it when "double".
Halving timeLearn itsmallest n with \(m^{n}\le0.5\)Use it when "half".
Constructions And Scale Drawings
Constructing triangles and nets
Triangle from three sidesLearn itbase, then an arc from each end, join to the crossingleave all arcs.Use it when 'ruler and compasses only'.
Will it make a triangle?Learn itthe two shorter sides must add to MORE than the longest5, 6, 8 works; 3, 4, 9 doesn't.Use it when checking lengths.
NetLearn itsurface area of the solid = total area of its netUse it when nets and surface area.
Scale drawings and bearings
ScaleLearn itreal length = drawing length × scaleconvert units at the end (100 000 cm = 1 km).Use it when scale drawings and maps.
Map areasLearn itreal area = map area × (scale)²Use it when an area on a map.
BearingLearn itangle clockwise from north, three figures (065°)Use it when directions.
Back bearingLearn itbearing ± 180°add 180 if under 180°, subtract if over.Use it when 'bearing of A from B'.
Trigonometric Graphs
Graphs of sin, cos and tan
Same sineLearn it\(\sin x=\sin(180^\circ-x)\)Use it when a second angle with the same sine.
Same cosineLearn it\(\cos x=\cos(360^\circ-x)\)Use it when a second angle with the same cosine.
Same tangentLearn it\(\tan x=\tan(x+180^\circ)\)tan repeats every 180°.Use it when a second angle with the same tangent.
Exact valuesLearn it\(\sin30^\circ=\tfrac12,\ \sin45^\circ=\tfrac{\sqrt2}2,\ \sin60^\circ=\tfrac{\sqrt3}2\); \(\tan45^\circ=1,\ \tan60^\circ=\sqrt3\)cos runs the sin list backwards.Use it when 'exact value' (no calculator).
Solving trigonometric equations
Second solutionLearn itsin: \(180^\circ-x\); cos: \(360^\circ-x\); tan: \(x+180^\circ\)x = the calculator's answer.Use it when every trig equation.
Into rangeLearn itadd 360° (sin, cos) or 180° (tan) to a negative angleUse it when the calculator gives a negative angle.
No solutionsLearn it\(\sin x\) or \(\cos x\) = a number above 1 or below −1the calculator says Math ERROR.
Differentiation
Differentiation: the gradient function
The ruleLearn it\(y=ax^{n}\ \Rightarrow\ \dfrac{dy}{dx}=nax^{n-1}\)in words: multiply by the power, then take 1 off the power.Use it when each term with \(x\) to a power.
A term in xLearn it\(y=kx\ \Rightarrow\ \dfrac{dy}{dx}=k\)the \(x\) disappears, the number stays.Use it when a term like \(7x\).
A number on its ownLearn it\(y=k\ \Rightarrow\ \dfrac{dy}{dx}=0\)a flat line has gradient 0.Use it when a constant like \(+9\).
Gradient at a pointLearn itgradient at \(x=a\) = value of \(\dfrac{dy}{dx}\) when \(x=a\)Use it when "find the gradient of the curve at ...".
Using differentiation: gradients and tangents
Gradient at a pointLearn itput \(x=a\) into \(\dfrac{dy}{dx}\)the answer is the gradient of the curve there.Use it when "find the gradient at x = ...".
Point with a given gradientLearn itsolve \(\dfrac{dy}{dx}=k\)then y from the CURVE's equation.Use it when "find the point where the gradient is ...".
Straight lineLearn it\(y=mx+c\)\(m\) = gradient, \(c\) = where it crosses the y-axis.Use it when you need the equation of a tangent.
Turning points: maxima and minima
Turning pointLearn it\(\dfrac{dy}{dx}=0\)the gradient is 0 at the top of a hill or the bottom of a valley.Use it when "find the turning point(s)".
Max or min (gradient either side)Learn it\(+\ \to\ 0\ \to\ -\) maximum; \(-\ \to\ 0\ \to\ +\) minimumrising then falling = top of a hill.Use it when "determine whether it is a maximum or minimum".
Max or min (quick check)Learn it\(\dfrac{d^{2}y}{dx^{2}}<0\) maximum; \(>0\) minimumdifferentiate \(\dfrac{dy}{dx}\) again and put the x-value in.Use it when the same question, if you prefer it.