ALGEBRA! I!can!factorise!simple! expressions.!

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1 EHughes2013 A* Icanmanipulate algebraicfractions. Icanusethe equationofacircle. Icansolvesimultaneous equationsalgebraically, whereoneisquadraticand oneislinear. Icantransform graphs,includingtrig graphs. Icandrawandrecogniseanexponentialgraph. A Icansimplify algebrainvolving powers. Icanrearrange formulaewiththe subjectinmorethan once. Icansolve quadraticsby usingthe formula, completingthe square,and factorising. Icansolve trigonometry: cosx=0.5 andrecognise triggraphs. Icanprove thingsusing algebra. Icanfindtheequationofalinethatgoes throughapoint,andisperpendiculartoanother line. B Icanfactoriseandexpandcomplex expressions. Icansolve simultaneous equations algebraicallyand graphically. Icansolve inequalities algebraically andgraphically. Icanusemy knowledgeofy= mx+ctoworkout theequationofa line. Icansolvecubicand quadraticgraphs graphically. IcanfactoriseQuadratics Icanusey=mx+c tofindthegradient ofaline. Icanrecognisecubicand reciprocalgraphs,and matchequationsto graphs. IcanrecognisetheDifferenceofTwo Squares(D.O.T.S) C Icansubstituteinto complexformulae. Icansolve equationswith unknownson bothsides: 2x+3=3xO2 Icansolve inequalities. Icaninterpret realolife graphs. Icanfindthen th termofa sequence. Icandrawquadratic graphsusingtheruleto findthecooordinates. Icanrearrange formulae D Icanexpand bracketsandsimplify myanswer. Icansubstitutein negativenumbersto formulae. Icansolveand rearrange equations ALGEBRA Icanfactorisesimple expressions.

2 You can collect terms together if they are the same letter, with the same power. 7x + 3x = 10x 4x + 2y = 4x + 2y 5x x = 8x + 2 4y + 2y² + 3y = 7y + 2y² (different letters) (letters and numbers are separate) (y and y² are different powers, so can t be put together) To solve equations, you must always do the same to both sides. To get rid of something, you do the opposite - eg - to get rid of a +3, you -3. to get rid of a x2, you 2 Keep going until you have what you want on its own. Sequences 3, 7, 11, 15 Goes up by 4 each time, so we write 4n as the first part of your rule. To find the second part, follow the pattern back from the first term. You get -1, so you write that on the end of your rule. 4n -1 This means it is one less than the 4 times table each time. 6, 11, 16, 21 = 5n + 1 (goes up in 5s, back 5 would be +1) -2, 0, 2, 4 = 2n - 4 (goes up in 2s, back 2 would be -4) 10, 7, 4, 1 = - 3n + 13 (goes up in -3 s, back 3 would be 13) Factorising quadratics To factorise - underline the expression. List underneath all the things that multiply to give each part. eg - 10y = 2 x 5 x y Circle anything in both lists. These go outside the bracket. Anything left goes inside the bracket, on the correct side. Factorising 10x²y + 2xy² x x y x y y 2xy (5x + y) 5 miles = 8km You can leave answers as fractions, like above, if it doesn t give a whole number answer. Remember, one step at a time, trying to get the x on its own. This also works when rearranging formulae - use the same steps - it s just that you ll end up with a different letter on its own than you started with. Linear Quadratic Cubic Straight line (linear) graphs y=2x+1 The y value is double the x value, plus 1. eg - (0,1), (1,3), (2,5) With the general y=mx+c, the line cuts the y-axis at c, and for every 1 you go across (right), you go m up.

3 A* Icanmanipulatecomplexindicesandsurds. Icanfindupperandlowerboundsinareaandvolume. A Icanrationalise surds. Icancalculate withfractional indices. NUMBER Icanfindupperandlowerboundsofnumbers. B Icancalculateusingstandardform. Icancalculatewithnegative indices. Icandofractioncalculationsstarting withmixednumbers. Icancalculatecompound interest. Icanchangebetweenrecurring decimalsandfractions. Icandoreverse percentages. C Icanx& by10, 100,1000and0.1, 0.01etc. Icanbreakdowna numberintoprime factors. Icansolve equationswith trialand improvement. Icanmultiply anddivideby numbersless than1. Icanmultiply anddivideby decimals. Icancalculatewith fractionsand ratios. Icanworkoutsimple compoundinterest. Icanuseindexlaws withnumbers. Icanusemycalculatortoefficiently workoutcomplexcalculations. D Icanestimatethe answerstoa calculation. Icanworkoutratiosin recipes. Icancalculateprofit andloss. Icanworkoutsimpleproportion. Icanincreaseordecreasebyapercentage. Icandosimplefractioncalculations.

4 Division 79 5 = 15.8 First, how many 5 s go into 7? 1, remainder 2. The 1 goes on top, the 2 carries over in front of the 9 to make it 29. For 10 cookies: 120ml milk 90g sugar 60g flour 24g butter For 15 cookies: ( ) ml milk ( ) g sugar ( ) g flour ( ) g butter Significant figures: Works the same as rounding to a given decimal place etc, just a different way to describe where to round. The first number that isn t a zero is the first significant figure. Everything after that counts. Eg - rounded to 1 s.f: Now how many 5 s go into 29? 5, remainder 4. The 5 goes on top, the 4 carries over. We can always add a.0 (and then as many 0 s as we want) after a number, to deal with remainders. We finally do: how many 5 s go into 40? 8 with no remainder. The 8 goes on top, making the answer We can stop now, as there is no remainder left. Don t forget to put the decimal in the answer too Multiplication (grid method) 73 x 356 To get from 10 to 15, you need 5 more. 5 is half of 10, so just halve each ingredient and add it on... If you re not sure, divide by the total to see how much of each ingredient you need for 1 cookie, then multiply by how many you actually need. Easy % For 17.5% (used for VAT) Divide total by 10 = 10% Halve it = 5% Halve it = 2.5% Add them up = 17.5% Prime factorisation (prime factor trees) 4753 rounds to rounds to rounds to 0.04 To estimate, round each number to 1 s.f, and do the sum. This will give you a rough answer (an estimate) Remember, you only needed to do 7 x 3 for the first bit, then add on the three 0 s from the 300 and the 70 to make

5 A* A Icanprovecircle theorems Iknowconstructionproofs. Icansolve3D trigonometryproblems. Icanusethesineandcosinerulestofind trianglemeasurements. Icanusecircle theorems Icanusesimilarityinlength, areaandvolumes. Icansolve3D Pythagorasproblems. Icanfindarc lengths,andareas ofsectorsand segmentsof circles. Icanfindthesurface areaandvolumeof solids. Icanusefractionalscalefactors inenlargements. B Icanprovecongruency. Icanuse½absinC. Icanusesomeofthe circletheorems. Iunderstandwhentwoshapes aremathematicallysimilar. Icansolvemulti@stage trigonometryproblems. Icanworkoutthe dimensionsof formulae. Icanuseinteriorand exterioranglesto solveproblems. Icandescribetransformations Icansolveinterior angleproblems. Icando enlargementswith negativescale factors. Icandrawloci. Icansolve problemswith bearings. IcanuseTrigonometry tofindmissingsidesor anglesinright@angled triangles. Icansaywhetherameasurementinofa length,areaorvolumefromtheunits. C Icanconstructaperpendicular bisector,andaccuratetriangles. IcanusePythagorasto findthemissingsideof aright@angledtriangle. Icanfindthearea and circumferenceof acircle,giventhe diameter. Icanworkoutthe volumeofa3d shape. Icananswer questionsabout polygons. Icando/recognise rotations, reflections, translationsand enlargements. Icando isometric drawings. Icandraw andmeasure bearings. Icanfindthearea and circumferenceof acircle,giventhe radius. Icanchangem 2 to cm 2 etc D Icanfindtheareaof atriangle,regular polygons,andother shapes. Icandrawplans andelevations. Icanfind anglesusing parallellines. GEOMETRY Icanuse measurementsof similartrianglesto findmissingedges.

6 N Bearings always start from North and go clockwise. They always have 3 digits. Eg: Centre Scale factor Enlargement Mirror line Centre Angle Direction Reflection Rotation Translation Vector - eg 10 mm = 1 cm 100 cm = 1 m 1000 m = 1 km 1000g = 1kg 60 seconds = 1min 60 min = 1 hr 365 days = 1yr 52 weeks = 1 yr Polygon (many sided shape) 3 = triangle 4 = quadrilateral 5 = pentagon 6 = hexagon 7 = heptagon 8 = octagon 9 = nonagon 10 = decagon Exterior Interior Exterior angles add to 360 Do 360 number of sides. Exterior + interior = 180 Angle bisector Pythagoras Perpendicular bisector Equidistant from a point (Loci) Equidistant from a line (Loci) Trigonometry a² + b² = c² Areas of shapes base x height ½ base x height ½(a+b)h r² r Circumference d d Parallel lines Alternate Corresponding Opposite

7 A Icanconstructand interprethistograms. Iunderstandstratified sampling. Icanfindtheprobabilityofcombinedevents,using multiplicationandadditionofprobabilities. B Icanfindthemedianand interquartilerangefromcumulative frequency. Icananalyseboxplots. Icananalysedatavs theoreticalprobability. Icanusetreediagrams. C Icanfindthemean andmedianfrom groupeddata. Icanexplain myuseof averages. Icandrawboxplots. Icandesign questionnaires. HANDLINGDATA D Icanidentifythe modalclass. Icandrawastem<and<leafdiagram, includingthekey. Icanexplainwhatis wrongwitha questionnaire. Icanfindthe relative frequencyofan event. Icanfind missing probabilities fromatable. Icanlistthe possible outcomesof events. Icanfindthemean ofasetofdata. Icandrawascatterdiagram,describea relationshiporcorrelationfromit,anduse alineofbestfittoestimate. Iknowwhatmakesa goodsample.

8 Scatter Diagrams Hey Diddle diddle, the median s the middle, You add then divide for the mean. The mode is the value that comes up the most, and the range is the difference between *modal means the same as mode. We use it when there is grouped data. Stem and leaf diagrams To find the median, keep crossing off the smallest and largest numbers until you find the middle. (If there are 2 numbers left in the middle, find the middle of those two numbers.) To find the mean of grouped data, find the midpoint of each group, and multiply by the frequency. Questionnaires The three key things to design a good question are: Give a time frame (where appropriate) Make sure your options don t overlap Allow all possible choices (eg, none, other, more than) For example: How much money do you spend on sweets each week? Less than 1 1 to to or more? Probabilities always add up to 1. That means if the probability you pick a red ball is 0.6 P(red) = 0.6 then the probability you don t pick a red ball is (1-0.6) so P(not red) = 0.4 If the probability of something happening is 0.4, and you do the experiment 200 times, you d expect it to happen 0.4 x 200 times = 80 times. This is called relative frequency. Number of lengths Class midpoint (m) Frequency (f) F x m 1 to 5 6 to to to to x 12 = 36 8 x 33 = x 27 = x 6 = x 2 = 46 Totals The mean is now Lowest value Box plots Lower quartile Median Upper quartile The width of the box shows the interquartile range Highest value Each set of branches adds to 1. Read the question very carefully in case the probabilities change for the second set of branches.

9 Histograms Upper and Lower bounds To find the upper and lower bounds, it is the rounded value ± half the unit of rounding. Learn the conditions for congruency: Frequency density = frequency class width 100cm to the nearest cm is 100 ± 0.5cm 500g to the nearest 10g is 500 ± 5g SSS ASA SAS RHS The frequency is the area of the bar. For arc length, you need to work out what fraction of your circumference it is by doing θ 360. Then multiply the circumference by this fraction to get the arc length. You do the same with the area of a sector - find what fraction of the whole are you need. Circle Theorems Stratified Sampling Work out what fraction of the total population your sample is. For each subgroup, you want that fraction of it. Eg - sample size 50, population 1000 You want 50/1000 of each subgroup If there were 700 boys and 300 girls, you would do 700 x 50/1000 = 35 boys, and 300 x 50/1000 = 15 girls. Surds To rationalise the denominator, multiply the whole fraction by the denominator again You can use the rules to simplify surds by splitting them into their factors (and looking for square factors).

10 Graph transformations Trig Graphs Equation of a circle Exponential graphs On formula page Completing the square If a question is asking for a diagonal length in a cuboid, it is a 3D Pythagoras question. In a cuboid measuring a x b x c, with a 3D diagonal d, a² + b² + c² = d² 3D Pythagoras d x² + bx b ( 2) 2