ZETA MATHS National 5 Mathematics Revision Checklist Contents: Expressions & Formulae Page Rounding Surds. Indices.... Algebra... Algebraic Fractions. Volumes. Gradient. 3 Circles.. 3 Relationships The Straight Line.. 4 Solving Equations/Inequations.. 4 Simultaneous Equations. 4 Change the Subject... 5 Quadratic Functions.. 5 Properties of Shapes. 7 Similar Shapes... 8 Trigonometr (Graphs and Equations). 8 V. Applications Triangle Trigonometr... Vectors.. Percentages Fractions.. Statistics..
Expressions and Formulae Topic Skills Extra Stud / Notes Rounding Round to decimal places e.g. 5.4 5. to d.p. 34.676 34.68 to d.p. Round to Significant Figures e.g. 76 300 to sig. figs. 0.06356 0.064 to sig. figs. 37,684 37,700 to 3 sig. figs. 0.00583 0.00583 to 3 sig. figs. Surds Simplifing Learn Square Numbers: 4, 9, 6, 5, 36, 49, 64, 8, 00,, 44, 69. Use square numbers as factors: Add/Subtract Multipl/Divide Rationalise Denominator e.g. 50 = 5 = 5 e.g. 50 + 8 = 5 + 4 = 5 + = 7 e.g. 5 5 = 5 5 = 75 = 5 3 = 5 3 48 = 48 3 3 = 6 = 4 Remove surd from denominator. e.g. = 3 = 3 = 3 3 3 9 3 3 Indices Use Laws of Indices. a x a = a x+ e.g. a a 3 = a +3 = a 5 Scientific Notation / Standard Form Evaluate using indices Algebra. a x a = a x a 7 a 4 = a 7 4 = a 3 3. (a x ) = a x (a 4 ) 5 = a 4 5 = a 0 4. a = x a x 3 = a 3 a 5. a 0 = a 0 = The first number is alwas between and 0. e.g. 54,600 = 5.46 0 4 0.000978 = 9.78 0-4 (.3 0 5 ) (8 0 3 ) = 0.4 0 8 =.04 0 9 3 3 e.g. 7 = 7 = 3 = 9 Expand Single Bracket 3 ( x + 4) = 3x + Expand Two Brackets Simplif Expression Factorise Common Factor Use FOIL (Firsts Outsides Insides Lasts) or another suitable method (x + 3)(x ) = x + 3x x 6 = x + x - 6 Know that ever term in the first bracket must multipl ever term in the second. e.g. (x + )(x 3x 4) = x 3 3x 4x + x 6x 8 = x 3 x 0x 8 Put together the terms that are the same: e.g. x + 4x + 3 x + 8 = x + x + a a a = a 3 Take the factors each term has in common outside the bracket: e.g. 4x + 8x = 4x(x + ) NB: Alwas look for a common factor first. www.zetamaths.com Zeta Maths Limited
Factorise Difference of Two Squares Factorise Trinomial (simple) Factorise Trinomial (hard) Alwas takes the same form, one square number take awa another. Eas to factorise: e.g. x 9 = (x + 3)(x 3) 5x 5 = 5(x 5) (Common factor first) = 5(x + 5)(x 5) Use an appropriate method to factorise: e.g. Opposite of FOIL: Factors of first term are Firsts in brackets. Lasts multipl to give last term and add to give middle term. x x 6 = (x 3)(x + ) This is more difficult. Use suitable method. Using opposite of FOIL above with trial and error. NB: The Outsides add Insides give a check of the correct answer: e.g. 3x 3x 0 = (3x 5)(x + ) Check: 3x + (-5) x = 6x - 5x = -x = (3x + )(x 5) Check: 3x (-5) + x = -5x + x = -3x If the answer is wrong, score out and tr alterative factors or positions. Keep a note of the factors ou have tried. Complete the Square e.g. x + 8x -3 = (x + 4) 3 6 = (x + 4) - 9 Algebraic Fractions Simplifing Algebraic Fractions Step : Factorise expression Step : Look for common factors. Step 3: Cancel and simplif 6x x x + x 6 = 6x(x ) (x +3)(x ) = 6x x +3 Add and Subract Fractions Multipl Fractions Divide Fractions Volumes Volume of a prism Volume of a clinder Volume of a cone Find a common denominator. This can be done either b working out the lowest common denominator, or b using Smile and Kiss 5 a 3d 0ac 3bd 0ac + 3bd + = + = b c bc bc bc Multipl top with top, bottom with bottom: 3a 7c 4b 5d = ab 35cd Invert second fraction and multipl: 6x 4x 6x 3z 8x z 9xz = = = 7 3z 7 4x 8x 4 V = Area of base x height V = πr h V = 3 πr h Volume of a sphere V = 4 3 πr 3 www.zetamaths.com Zeta Maths Limited
Rearrange each of the formulae to find an unknown e.g. Clinder has volume 400cm 3 and radius 6cm, find the height V = πr h h = 400 π 6 V πr = h Volume of composite shapes These are two of the above combined: Label them V and V e.g. V V V = 4 3 πr 3 V = V = πr h V = Gradient Find the gradient of a line joining two points Circles Length of Arc Area of Sector Know that gradient is represented b the letter m Step : Select two coordinates Step : Label them (x, ) (x, ) Step 3: Substitute them into gradient formula x x e.g. (-4, 4), (, -8) m = x x = ( 8) 4 ( 4) = 3 6 = This finds the length of the arc of a sector of a circle: LOA = angle 360 πd LOA or πd = angle 360 For harder questions rearrange formula to find angle AOS = angle AOS 360 πr or πr = angle 360 For harder questions rearrange formula to find angle www.zetamaths.com 3 Zeta Maths Limited
Relationships Topic Skills Notes Straight Line Gradient Represented b m Measure of steepness of slope Positive gradient the line is increasing Negative gradient the line is decreasing Y-intercept Represented b c Shows where the line cuts the -axis Find b making x = 0 Find the gradient of a line joining two points Find equation of a line (from gradient and - intercept) Find equation of a line (from two points) Rearrange equation to find gradient and - intercept Sketch lines from their equations Solving Equations / Inequations Solving Equations Solving inequations Simultaneous Equations Solve b sketching lines Solve b substitution Know that gradient is represented b the letter m Step : Select two coordinates Step : Label them (x, ) (x, ) Step 3: Substitute them into gradient formula x x e.g. (-4, 4), (, -8) ( 8) 4 3 m = = = = x x ( 4) 6 Step : Find gradient m Step : Find -intercept c Step 3: Substitute into = mx + c (see above for definitions) Use this when there are onl two points (i.e. no -intercept) Step : Find gradient Step : Substitue into b = m(x a) where (a, b) are taken from either one of the points e.g. 3 + 6x = 3 = -6x + = -x + 4 m = -, c = 4 Step : Rearrange equation to the form = mx + c (see note above) Step : Draw a table of points Step 3: Plot points on coordinate axes Use suitable method: e.g. 5(x + 4) = (x 5) 5x + 0 = x 0 5x = x 30 3x = -30 x = -0 Solve the same wa as equations. NB: When dividing b a negative change the sign: e.g. -3x 5 x -5 Step : Rearrange lines to form = mx + c Step : Sketch lines using table of points (as above) Step 3: Find coordinate of point of intersection This works when one or both equations are of the form = ax + b e.g. Solve 3x + = 7 = x + Sub equation into : 3x + (x + ) = 7 5x + = 7 x = 3 so = 3 + = 4 www.zetamaths.com 4 Zeta Maths Limited
Simultaneous Equations Contd. Solve b Elimination Step : Scale equations to make one unknown equal with opposite sign. Step : Add Equations to eliminate equal term and solve. Step 3: Substitute number to find second term. Form Equations Change the Subject Linear Equations Equations with powers or roots Quadratic Functions Quadratics and their equations e.g. 4a + 3b = 7 a b = -4 x 8a + 6b = 4 3 x 3 6a 6b = -4 4 3 + 4 4a = -8 a = - substitute a = - into 4(-) + 3b = 7 3b = 5 b = 5 Ans. a = -, b = 5 Form equations from a variet of contexts to solve for unknowns Rearrange equations change the subject: e.g. e.g. D = 4C 3 [C] D + 3 = 4C D + 3 = C 4 D + 3 C = 4 V = πr h [r] V = r πh r = V πh = x = -x 5 = 5(z + 6) [z] = z + 6 5 6 = z z = 6 5 = x = x + 5 = (x 3) = (x + ) - 3 www.zetamaths.com 5 Zeta Maths Limited
Equations of quadratics = kx Step : Identif coordinate from graph Step : Substitute into = kx Step 3: Solve to find k e.g. Coordinate: (, ) Substitution: = k() = 4k k = 0.5 Quadratic: = 0.5x Sketching Quadratics = k(x + a) + b Sketching Quadratics (Harder) = (x + a)(x b) Solving Quadratics (finding roots) Algebraicall Solving Quadratics (finding roots) Graphicall Step : Identif shape, if k = then graph is +ve or if k = - then the graph is -ve Step : Identif turning point (-a, b) Step 3: Sketch axis of smmetr x = -a Step 5: Find -intercept (make x = 0) Step 4: Sketch information Step : Identif shape (+ve or -ve) Step : Identif roots (x-intercepts) x = -a, x = b Step 3: Find -intercept (make x = 0) Step 4: Identif turning point e.g. = (x + 4)(x ) +ve graph Minimum turning point Roots: x =, x = -4 -intercept: = (0 + 4)(0 ) = -8 Turning Point (-, -9) (see below) NB: Turning point is halfwa between roots. x-coord = ( + (-4)) = - -coord = (- + 4)(- ) = -9 Step : Equate to zero Step : Factorise quadratic Step 3: Set each factor equal to zero Step 4: Solve each factor to find roots e.g. = x + 4x = x 5x 6 x(x + 4) = 0 (x 6)(x + ) = 0 x = 0 or x + 4 = 0 x 6 = 0 or x + = 0 x = 0 or x = -4 x = 6, x = - Read roots from graph x =, x = - Solving Quadratics Quadratic Formula When asked to solve a quadratic to a number of decimal places use the quadratic formula: b ± b 4ac x = a where = ax + bx + c www.zetamaths.com 6 Zeta Maths Limited
Discriminant Using the Discriminant Properties of Shapes Circles e.g. Solve = x 6x + to d.p. a = b = -6 c = ( 6) ± x = 6 ± x = 8 ( 6) 4 6 + 8 x = x = x = 5.6 x = 0.4 6 8 b 4ac where = ax + bx + c The discriminant describes the nature of the roots b 4ac > 0 two real roots b 4ac = 0 equal roots (tangent to axis) b 4ac < 0 Example : Determine the nature of the roots of the quadratic = x + 5x + 4 Solution: a =, b = 5, c = 4 b 4ac = 5 4 4 = 5 6 = 9 Since b 4ac > 0 the quadratic has two real roots. Example : Determine p, where x + 8x + p has equal roots Solution: b 4ac = 0 8 4 p = 0 64 4p = 0 64 = 4p P = 6 Pthagoras Use Pthagoras Theorem to solve problems involving circles and 3D shapes. e.g. Find the depth of water in a pipe of radius 0cm. r is the radius 0cm x 8cm x = 0-9 x = x = 4.4cm Depth = 0 4.4 = 5.6cm www.zetamaths.com 7 Zeta Maths Limited
Similar Shapes Linear Scale Factor Linear.Scale.Factor = New.Length Original.Length Area Scale Factor New.Length Area.Scale.Factor = Original.Length Volume Scale Factor 3 New.Length Volume.Scale.Factor = Original.Length Trigonometr Trig Graphs Sine = asin bx + c Curve a = maxima and minima of graph b = no. of waves between 0 and 360⁰ c = movement of graph verticall = sin x maxima and minima and -, period = 360⁰ - = sin x 80⁰ 360⁰ x - = sin 3x 80⁰ 360⁰ x - = sin x + 4 60⁰ 0⁰ x - = -sin x 80⁰ 360⁰ x - 80⁰ 360⁰ x www.zetamaths.com 8 Zeta Maths Limited
= sin (x - 30⁰) Trig Graphs Cosine Curve - 30⁰ 0⁰ 360⁰ x = acos bx + c a = maxima and minima of graph b = no. of waves between 0 and 360⁰ c = movement of graph verticall = cos x maxima and minima and -, period = 360⁰ - 80⁰ 360⁰ x Trig Graphs Tan Curve The same transformations appl for Cosine as Sine (above) = tan x no maxima or minima, period = 80⁰ -90⁰ 90⁰ x Solving Trig Equations Know the CAST diagram Memor Aid: All Students Take Care Use the diagram above to solve trig equations: Example : Solve sinx = 0 sinx = sin x = ½ x = sin - (½) x = 30 ⁰, 80 ⁰ - 30 ⁰ x = 30 ⁰, 50 ⁰ Example : Solve 4tan x + 5 = 0 4tan x = -5 tan x = -5/4 NB: tan x is negative so there will be solutions in the second and fourth quadrant x = = tan - (5/4) x acute = 5.3 ⁰ x = 80 ⁰ - 5.3 ⁰, 360 5.3 ⁰ x = 8.7 ⁰, 308.7 ⁰ www.zetamaths.com 9 Zeta Maths Limited
Trig Identities Know: sin x + cos x = sin x = - cos x and cos x = - sin x and tan x = sinx cosx Use the above facts to show one trig function can be another. Start with the left hand side of the identit and work through until it is equal to the right hand side. www.zetamaths.com 0 Zeta Maths Limited
National 5 Learning Checklist - Applications Topic Skills Extra Stud / Notes Triangle Trigonometr Triangle Label Triangle C Area of a Triangle Sine Rule A c B A = absinc a b c = = sina sinb sinc Use Sine Rule to find a side Use Sine Rule to find an angle. NB: sina = A = sin - ( ) Coosine Rule Use a = b + c bccos A to find a side b + c a Use cos A = to find an angle bc NB: cosa = A = cos - ( ) Bearings Use knowledge of bearings to solve trig problems. Including knowledge of Corresponding, Alternate and Supplementar angles. NB: Extend right north arrow and use Z-angles b a Vectors D Line Segments Add or subtract D line Segments Vectors end-to-end Arrows in same direction a a+b b Position Vectors Finding a Vector from Two Coordinates 3D Vectors The position vector of a coordinate is the vector from the origin to the coordinate. E.g. A (4, -3) has 4 the position vector a = 3 Know that to find a vector between two points A and B then AB = b a NB: Vector notation for a vector between two points A and B is AB x Determine coordinates of a point from a diagram representing a 3D object -a A b b - a B Look at difference in x, and z axes individuall e.g. Find the coordinates of C z C (5, 9, 0) A (4, 5, 0) x B (5, 9, 6) www.zetamaths.com Zeta Maths Limited C
Vector Components Percentages Compound Interest Percentage increase/decrease Reverse the Change Add and Subtract D and 3D vector components. 3 + 3 a = and b = a + b = + 4 5 4 + 5 Multipl vector components b a scalar a = = 4 8 Find the magnitude of a D or 3D vector: x For vector u =, u = x + x For vector v =, v = x + + z z Calculate multiplier from percentage: e.g. 5% increase 00% + 5% = 05% =.05 Use multiplier to calculate compound interest / depreciation. e.g. 500 with 5% interest for 3 ears.05 3 x 500 % Increase/decrease = difference 00 original Find initial amount. e.g. Watch reduced b 30% to 4. 70% = 4, % = 0.60, 00% = 60 or 4 0.7 = 60 Fractions Add and Subract Fractions Find a common denominator 3 + 4 5 = 0 Add and Subract Mixed Numbers Multipl Fractions Multipl Mixed Numbers Divide Fractions Statistics Comparing Data 5 + 5 Add or subtract whole numbers, or make an improper fraction: 3 +3 4 5 = 50 5 + 5 or 3 +3 4 5 = 8 3 + 9 5 Multipl top with top, bottom with bottom: 3 7 4 5 = 35 Make top heav fraction then as above: 3 3 7 4 5 = 3 7 4 5 = 9 35 Invert second fraction and multipl: 6 7 3 = 6 7 3 = 8 0 = 9 5 sum of data Calculate the mean: x = number of terms Find five figure summar: L = lowest term, Q = lower quartile, Q = Median, Q3 = upper quartile, h = highest term Interquartile range: IQR = Q3 Q middle 50% of data www.zetamaths.com Zeta Maths Limited
Comparing Data (Contd.) Semi-Interquartile range: SIQR = Q3 Q Calculate Standard Deviation: s = Σx (Σx) n n or s = Σ(x x) n Know that IQR, SIQR and standard deviation are a measure of the spread of data. Lower value means more consistent data. When comparing data, alwas compare the measure of average and the measure of spread i.e. compare the medians or the means and then compare the SIQR or the standard deviation. Sa what each comparison means in the context. Line of Best Fit e.g. On average John exercises more because his mean exercise time is greater, but Zahid is more consistent as his standard deviation is smaller. Use knowledge of straight line to find the equation of a line of best fit: = mx + c or b = m(x a) Use equation of line of best fit to find estimate for new value. Usuall do so b substituting value for x into equation. Draw best fitting line: In line with direction of points Roughl the same number of points above and below line. www.zetamaths.com 3 Zeta Maths Limited