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Miss Bβs Maths DIRT Bank Created by teachers for teachers, to help improve the work life balance and also the consistent quality of feedback our students receive. This aims to be a continuously evolving document. Please contribute to the DIRT Bank simply by emailing your DIRT question(s) youβve created and your name/twitter handle to missbsresources@gmail.com and I will update weekly. www.missbsresources.com Template Sub Topic www.missbsresources.com Template Subtopic www.missbsresources.com Contents Number Topic Contributor Twitter Handle Multiplying 3 digit by 1 digit (grid) Danielle Bartram @Missbsresources Multiplying 3 digit by 1 digit (Long) Danielle Bartram @Missbsresources Multiplying 3 digit by 2 digit (grid) Danielle Bartram @Missbsresources Multiplying 3 digit by 2 digit (Long) Danielle Bartram @Missbsresources Multiplying by scalars of 10 Danielle Bartram @Missbsresources Multiplying decimals Danielle Bartram @Missbsresources Dividing decimals Danielle Bartram @Missbsresources Functional division 1 Danielle Bartram @Missbsresources Functional division 2 Danielle Bartram @Missbsresources Addition and subtraction 1 Danielle Bartram @Missbsresources Addition and subtraction 2 Danielle Bartram @Missbsresources Directed numbers four rules Danielle Bartram @Missbsresources Functional Temperature 1 Danielle Bartram @Missbsresources Functional Temperature 2 Danielle Bartram @Missbsresources Ordering Decimals Danielle Bartram @Missbsresources Rounding nearest 10 Danielle Bartram @Missbsresources Significant Figures Danielle Bartram @Missbsresources Rounding using a calculator 1 Danielle Bartram @Missbsresources Rounding using a calculator 2 Danielle Bartram @Missbsresources Laws of Indices Danielle Bartram @Missbsresources Inequality Notation 1 Danielle Bartram @Missbsresources Inequality Notation 2 Danielle Bartram @Missbsresources Solving Inequalities Danielle Bartram @Missbsresources Bounds nearest unit Danielle Bartram @Missbsresources Bounds nearest (1dp) Danielle Bartram @Missbsresources Bounds nearest (1dp) Danielle Bartram @Missbsresources Bounds Area Danielle Bartram @Missbsresources www.missbsresources.com Contents Number Topic Contributor Twitter Handle Simplifying Fractions Peter Hall @MathsAST Simplifying Fractions Peter Hall @MathsAST Calculate missing numerator Peter Hall @MathsAST Calculate missing numerator Peter Hall @MathsAST www.missbsresources.com Contents Geometry Topic Contributor Twitter Handle Area and Perimeter (ππ2 ππππ) Danielle Bartram @Missbsresources Area and Perimeter half squares (ππ2 ππππ) Danielle Bartram @Missbsresources Area of a Triangle Danielle Bartram @Missbsresources Area of a Parallelogram Danielle Bartram @Missbsresources Area of a Trapezium Danielle Bartram @Missbsresources Parts of a Circle Danielle Bartram @Missbsresources Circumference of a Circle Danielle Bartram @Missbsresources Area of a Circle Danielle Bartram @Missbsresources Arc Length Danielle Bartram @Missbsresources Area of a Sector 1 Danielle Bartram @Missbsresources Area of a Sector 2 Danielle Bartram @Missbsresources Perimeter of Rectilinear Shapes Danielle Bartram @Missbsresources Area of Rectilinear Shapes Danielle Bartram @Missbsresources Area of Compound Shapes Danielle Bartram @Missbsresources Volume β Counting Cubes Danielle Bartram @Missbsresources Volume β Cuboid Danielle Bartram @Missbsresources Volume β Triangular Prism Danielle Bartram @Missbsresources Volume β Cylinder Danielle Bartram @Missbsresources Volume β Hemisphere Danielle Bartram @Missbsresources Volume β Sphere/Cone (Ice Cream) Danielle Bartram @Missbsresources Dimensions Danielle Bartram @Missbsresources Surface Area β Cuboid Danielle Bartram @Missbsresources Surface Area β Cylinder Danielle Bartram @Missbsresources Pythagoras β Identify Hypotenuse Danielle Bartram @Missbsresources Pythagoras β Missing Hypotenuse 1 Danielle Bartram @Missbsresources Pythagoras β Missing Hypotenuse 2 Danielle Bartram @Missbsresources Pythagoras β Missing Short Side Danielle Bartram @Missbsresources Pythagoras β 3D Cuboid Danielle Bartram @Missbsresources Pythagoras β Isosceles Triangle Danielle Bartram @Missbsresources Danielle Bartram @Missbsresources Trigonometry β 3D Square-based pyramid www.missbsresources.com Contents Geometry Topic Contributor Twitter Handle Missing Angle inside a Triangle Peter Hall @MathsAST Missing Angle inside a Quadrilateral Peter Hall @MathsAST Missing Angle on a Straight Line Peter Hall @MathsAST Missing Angle around a point Peter Hall @MathsAST www.missbsresources.com Contents Algebra Topic Contributor Twitter Handle Sequences β Worded Rule Caroline Beale @cbeale83 Sequences β Missing Number Caroline Beale @cbeale83 Sequences β Finding term numbers from given rule. Caroline Beale @cbeale83 Sequences β Geometric and Arithmetic Caroline Beale @cbeale83 Sequences β nth term rule Caroline Beale @cbeale83 Forming an expression (add/subtract) Danielle Bartram @Missbsresources Forming an expression (multiply/divide) Danielle Bartram @Missbsresources Simplifying Expressions Danielle Bartram @Missbsresources Expanding single brackets Danielle Bartram @Missbsresources Expand and Simplify single brackets Danielle Bartram @Missbsresources Expanding Single brackets Danielle Bartram @Missbsresources Expanding double brackets Danielle Bartram @Missbsresources Expanding double brackets Danielle Bartram @Missbsresources Factorising Single Brackets (HCF) Danielle Bartram @Missbsresources Factorising Single Brackets (Division) Danielle Bartram @Missbsresources Factorising Single Bracket (Variable) Danielle Bartram @Missbsresources Factorising double brackets Danielle Bartram @Missbsresources Algebraic Fractions - Multiply Danielle Bartram @Missbsresources Algebraic Fractions - Add Danielle Bartram @Missbsresources Algebraic Fractions - Factorise Danielle Bartram @Missbsresources Substitute into an Expression Danielle Bartram @Missbsresources Substitute into worded formula Danielle Bartram @Missbsresources Substitute and reverse worded formula Danielle Bartram @Missbsresources with a coefficient. www.missbsresources.com Contents Algebra Topic Contributor Twitter Handle Substitute into science formula Danielle Bartram @Missbsresources Deduce possible variables. Danielle Bartram @Missbsresources Write a formula in words Danielle Bartram @Missbsresources Substitute into scientific formula Danielle Bartram @Missbsresources Form and solve Equations Perimeter Danielle Bartram @Missbsresources Form and solve Equations Simple Functional Danielle Bartram @Missbsresources Form and solve Equations I think of a numberβ¦ Danielle Bartram @Missbsresources Form and solve Equations Rectangles have same Area. Danielle Bartram @Missbsresources Solving one-step equations Add/Subtract Danielle Bartram @Missbsresources Solving one-step equations Multiply/Divide Danielle Bartram @Missbsresources Solving two-step equations Danielle Bartram @Missbsresources Solving two-step equations With Brackets Danielle Bartram @Missbsresources Solve Equations Unknowns on both sides Danielle Bartram @Missbsresources Solve Equations Unknowns on both sides Danielle Bartram @Missbsresources Solve Equations with bracket Unknowns on both sides Danielle Bartram @Missbsresources Solve Equations with brackets Unknowns on both sides Danielle Bartram @Missbsresources www.missbsresources.com Contents Algebra Topic Contributor Twitter Handle Solving Inequalities Unknowns on both sides Danielle Bartram @Missbsresources Represent the solution set Danielle Bartram @Missbsresources Solving Inequalities with brackets Unknowns on both sides Danielle Bartram @Missbsresources Solving quadratics - Factorising Danielle Bartram @Missbsresources Simultaneous Equations Multiply both Danielle Bartram @Missbsresources Simultaneous Equations Worded Danielle Bartram @Missbsresources Trial and improvement Danielle Bartram @Missbsresources Change the subject β simple Danielle Bartram @Missbsresources Change the subject Square and Square Root Danielle Bartram @Missbsresources Change the subject Factorise Danielle Bartram @Missbsresources Reading Coordinates (1 quadrant) Danielle Bartram @Missbsresources Plotting Coordinates (1 quadrant) Danielle Bartram @Missbsresources Read and Plot (4 Quadrants) Danielle Bartram @Missbsresources Mid Point of Coordinates Danielle Bartram @Missbsresources Identify the rule of a line Danielle Bartram @Missbsresources Plotting rules of a line Danielle Bartram @Missbsresources Plotting straight line graphs Danielle Bartram @Missbsresources Identifying graphs and features Danielle Bartram @Missbsresources Plotting Quadratic graphs Danielle Bartram @Missbsresources Equation of a line. Danielle Bartram @Missbsresources Identifying parallel lines from an equation Danielle Bartram @Missbsresources Gradient and equation of line Danielle Bartram @Missbsresources www.missbsresources.com Contents Algebra Topic Contributor Twitter Handle Equation of line parallel when given coordinate Danielle Bartram @Missbsresources Equations of perpendicular lines Danielle Bartram @Missbsresources Identifying quadratic, cubic, reciprocal and exponential graphs Danielle Bartram @Missbsresources Plotting exponential graphs Danielle Bartram @Missbsresources www.missbsresources.com Contents Data Handling Topic Contributor www.missbsresources.com Twitter Handle Contents Algebra Topic Contributor www.missbsresources.com Twitter Handle Contents Problem Solving and Functional Topic Contributor www.missbsresources.com Twitter Handle Number Multiplication Calculate 273 x 4 Calculate 247 x 3 × 4 800 70 3 Calculate 247 × 3 Calculate 273 × 4 2 7 3 1 4 × 2 Complete the multiplication and add the exchanged (carried) numbers. Calculate 354 × 27 × Calculate 634 × 52 20 300 2100 1000 4 Calculate 354 × 27 3 5 4 2 7 2 4 7 8 0 × + Calculate 634 × 52 Complete the multiplication and add the exchanged (carried) numbers. www.missbsresources.com Number Multiplication Calculate 3.9 × 100 Th Calculate 2.7 × 1000 H T U . 1 10 3 3 . 9 Calculate 0.97 × 100 . Fill in the gaps. Calculate 15 × 0.6 9 x 0.3 Show all working out. 9 x 3 = 27 1dp in the question, so 1dp in the answer Calculate 0.15 × 0.6 9 x 0.3 = 2.7 Show all working out. www.missbsresources.com Number Division The school minibus seats 14 children. 60 children need to go to the cricket tournament. How many times must the bus make the trip. 60 ÷ 14 = 14 14 14 14 ? The school minibus seats 14 children. 365 children need to go to the cricket tournament. How many times must the bus make the trip. 365 ÷ 14 = 50 people are going to a meeting in the school hall. We need chocolate brownies for everyone but they come in packs of 6. How many packs do we need to buy? 359 people are going to a meeting in the school hall. We need chocolate brownies for everyone but they come in packs of 6. How many packs do we need to buy? 14 Answer: Calculate 518 ÷ 0.7 Calculate 245 ÷ 0.4 Show all working out. Show all working out. 0.7 518 7 5180 www.missbsresources.com Number Addition and Subtraction Addition Subtraction Work out the following. 4 +3 Addition + 3 1 6 8 Subtraction 7 5 β 5 3 8 2 4 7 2 6 5 6 8 5 7 2 4 7 Work out the following. 4 +3 1 www.missbsresources.com 2 6 4 7 5 5 7 2 4 7 Number Directed Numbers How much Factor 1 is this? + + + - 2 1 0 Factor 2 + + -- -- -- + + -- Product Overnight, the temperature dropped from 2 ºC to -4 ºC. By how many degrees did the temperature fall? -1 -2 +4 β +3 = +4 β β3 = +4 + β3 = β 4 β +3 = β 4 β β3 = β 4 + β3 = +5 × +3 = β5 × +3 = β5 × β3 = +6 ÷ β2 = β6 ÷ β2 = One day the level of the water in a river was 8cm above its average level. One week later it was 6cm below its average level. How far did the water level drop in the week? -3 -4 1 0 -1 -2 -3 -4 -5 The table shows the temperatures in four cities. Calculate the difference between he highest and lowest temperature. Highest: London 0π πΆ Lowest: π Moscow β9 πΆ Difference: Paris 6π πΆ Berlin At 7am, Joe recorded the temperature in his garden as being β4°C. He went back outside at 1pm and found that the temperature had increased by 12°C. What was the temperature at 1pm? β3π πΆ www.missbsresources.com Number Place Value Place the decimals in ascending order. Units . 1 10 1 100 1 1000 0 . 4 5 0 . 0 0 4 0 . 4 0 5 0 . 5 Place the decimals in ascending order. 0.602, 0.26, 6.02, 0.026, 0.06, 0.6 www.missbsresources.com Number Rounding 73 70 73 rounded to the nearest ten is 70, because 73 is closer to 70 than to 80. 76 80 1) Round 148 miles to the nearest hundred miles. 2) To the nearest 10p how much is in Sianβs purse? 76 goes up to 80, because 76 is closer to ____ than to _____. 3.9 + 4.1 7 1.14285714 Round to 2 significant figures. Calculate 11.7β3.1 9.6β2.4 Write down the full calculator display. Round to 3 significant figures. Place holder 9.83β1.622 Calculate 23.8β4.47×5.12 Calculate Work out the numerator: 8.95+ 7,84 2.03×1.49 Write down the full calculator display. Work out the denominator: Write down the full calculator display. Round the following numbers to 3 significant figures. Third 1) significant figure Place holder 2) 35260 2.347 Correct to 3 significant figures. Round the following numbers to 3 significant figures. 1) 2) 3) 4) 3568 42062 0.024537 0.0034078 www.missbsresources.com Number Fractions Simplify 14 21 Simplify Calculate the HCF of 14 and 21. Factors of 14 Factors of 21 1 x 14 2x7 HCF = To simplify divide the numerator and _____________ of the fraction by the HCF. Simplify 14 28 1 x 14 2x7 π) = To simplify divide the numerator and _____________ of the fraction by the HCF. Find the missing value πππ€ π·ππππππππ‘ππ πππ π·ππππππππ‘ππ ππππππ = 30 = Find the missing value ππππππ = πππ€ π·ππππππππ‘ππ πππ π·ππππππππ‘ππ ππππππ = 9 π) 15 Simplify Calculate the HCF of 14 and 28. Factors of 14 Factors of 28 HCF ππππππ = 12 π) 18 30 = × 24 π) 30 Find the missing value 2 ?? = 5 30 × 3 ?? π) = 5 15 π) 4 ?? = 5 60 Find the missing value × 2 ?? = 5 30 × 42 56 3 ?? π) = 8 48 4 ?? π) = 7 28 www.missbsresources.com Number Indices Fill in the missing gaps. 34 × 37 = 3 29 ÷ 25 = 2 2β1 1 = 2 =3 58 × 56 = _________ =2 79 × 7β3 = _________ 412 ÷ 43 = _________ 6β1 = _________ www.missbsresources.com Number Inequalities Match the inequality notation with the correct definition. < Less than or equal to β₯ β€ Greater than Greater than or equal to > Less than Match the inequality notation with the correct definition. β₯ Greater than < Less than or equal to β€ Less than > Greater than or equal to Find the maximum and minimum values for π₯ when 15 < 5π₯ < 20. What is the largest and smallest value π₯ can be? Minimum Maximum a) β5 < π₯ β€ 2 b)β3 β€ π₯ β€ 0 c) β9 < π₯ < β1 What is the largest and smallest value π₯ can be? Minimum Maximum a) β4 < π₯ β€ 3 b)β5 β€ π₯ < 0 c) β8 < π₯ < β1 Find the maximum and minimum values for π₯ when 20 < 4π₯ < 56. 15 < 5π₯ < 20 3< π₯ <4 Minimum: _______ Minimum: _______ Maximum: _______ Maximum: _______ www.missbsresources.com Number Bounds Lower Bound 21.5m Upper Bound 22.5m Amjadβs height is given as 162cm, correct to the nearest cm. Between which limits does Amjadβs height lie? A fence is m long to the nearest metre. Lower Bound £1.55 Upper Bound £1.65 A box is 8.5cm wide measured to the nearest tenth of a cm. What are the upper and lower bounds? A Krispy Crème doughnut costs £ to the nearest 10p. Lower Bound £1.45 Upper Bound £1.55 A box is 10.6cm wide measured to the nearest tenth of a cm. What are the upper and lower bounds? A Krispy Crème doughnut costs £ to the nearest 10p. A rectangle has a length of 20cm and height of 30cm to the nearest 10cm. What is the maximum and minimum area of the rectangle? Minimum Length Width Area The dimensions of a piece of carpet are given as 127cm x 68cm. Both lengths are correct to the nearest cm. Between what limits does the area of the carpet lie? Maximum 25cm 25cm 875ππ2 www.missbsresources.com Geometry Area and Perimeter Complete the sentences Area is the _____ of a shape. Find the area and perimeter of the shaded shape. Perimeter is the ________ of a shape. Complete the sentences Count the _________ to find the area. Find the area and perimeter of this shape. Count the _________ to find the perimeter. www.missbsresources.com Geometry Area and Perimeter Calculate the area of the triangle. Calculate the area of the triangle. 8 ππ 6 ππ π Area=______ππ 11 ππ 1 2 Area = × πππ π × βπππβπ‘ 5 ππ Area=__________ Calculate the area of the parallelogram. 8 ππ A right angled triangle is translated to the position shown to make a rectangles. The formula for the area of a parallelogram is the _________ as a rectangle. Half the _____ of the parallel sides. ________ the distance between them. That is how you calculate, area of a ___________. 11ππ Calculate the area of the trapezium. 6 ππ 5 ππ 10 ππ www.missbsresources.com Geometry Circles Match the key terms to the diagrams. Complete the sentences The radius is _______ the diameter. Diameter Circumference The diameter is _______ the radius. Radius Calculate the circumference of the circle. Calculate the circumference of the circle 10 ππ Radius= Diameter= 12 ππ Circumference= ππ Circumference= π × diameter Circumference=______πm Circumference=______ Calculate the area of the circle. Calculate the area of the circle. Radius= 3ππ Diameter= 4 ππ Area = ππ 2 Area = π × πππππ’π × πππππ’π Area = π × 3 × Area =______πππ Area =______ www.missbsresources.com Geometry Circles The formula for circumference in terms of radius is 2ππ Calculate the arc lengths. A The expression for arc length is B Calculate the arc length AB. 360 × 2 × π × 10 The formula for the area of a circle is Calculate the area of the sectors. A The formula for the area of a sector is B Calculate the area of the sector. 360 × π × 10 2 The formula for the area of a circle is A The formula for the area of a sector is B Calculate the area of the sector. × π × 10 2 www.missbsresources.com Geometry Area and Perimeter Find the perimeter of the shape. Not to scale. b Not to scale. Length a = 15 -6 =____ Length b = 11 =____ Perimeter=_____cm Perimeter=______ 12 ππ 11 ππ 5 ππ Find the area of the shape. 14 ππ 15 ππ Not to scale. 6 ππ 6 ππ b Area a = 5 × =____ Area b = 6 × 11 =____ Find the area of the shape. 7 ππ Not to scale. a 6 ππ 6 ππ 15 ππ a Find the perimeter of the shape. 7 ππ 14 ππ 5 ππ Area=____πππ 11 ππ 12 ππ Calculate the total area. Calculate the total area. The formula for area of a circle is_______ Area A = ___ x ___ =______ππ2 B 20 cm 5 cm A π_____2 Diagram not drawn accurately. 10 cm ππ 2 Area B = = =_____ππ2 2 2 Total area= Area A + Area B = _____+______ =________ππ2 10 cm Diagram not drawn accurately. www.missbsresources.com Area=______ Geometry Volume Find the volume of the solid prism. Here is a solid prism made from centimetre cubes. Volume:_____πππ Find the volume of the solid prism. Calculate the volume of the cuboid. Calculate the volume of the cuboid. 4 ππ 4 ππ Volume = length × width × height 3ππ 7ππ Volume=______πππ Volume=______ CSA is an abbreviation for cross sectional ________. Calculate the volume of the cylinder. 5 cm CSA = 4 ×____ 2 4 cm 4 cm 6 cm 5 cm Volume = CSA x 5 cm Volume = ____ x 10 = _____ππ3 Find the volume of the cylinder. 4cm 10cm www.missbsresources.com Geometry Volume Calculate the surface area of the sphere. 2 SA=4ππ Calculate the surface area of the sphere hemisphere with a radius of 8cm. 8cm 10cm Calculate the volume of this strawberry ice cream and cone. Clearly show all working out. The formula for the volume of a sphere is The formula for the volume of a cone is Area is measured in units _______, because it has ____ dimensions. a, b and c are representations of lengths. Which expressions represent volume? π2 π Volume is measured in units _______, because it has ____ dimensions. www.missbsresources.com π(π2 + π) 2πππ 4ππ3 Geometry Surface Area Surface area is the area of each face added together. Sketch a net of the prism before calculating the surface area to help you visualise the faces, Include Write the dimensions. A 3cm B C C 4cm A B A dimensions on the net. 3cm Calculate the surface area of the cuboid. 4cm Diagram not drawn accurately. C B What is the completed formula for the surface area of a cylinder? Find the surface area of the cylinder. ππ 2 β Circumference = π × ππππππ‘ππ 4cm ππ· × β 10cm ππ 2 www.missbsresources.com Geometry Pythagoras Accurately label the hypotenuse with a C on each of these triangles. Pythagorasβ theorem is + = The hypotenuse is the ___________ side. It is also the side opposite the ________ angle. Pythagorasβ theorem is + = π = 4 2 +3 Pythagorasβ theorem is + = π = 4 2 +3 Calculate the missing lengths. (Clearly show all working out) 2 12cm π 2 = 16 + 9 = π = 25 β¦ = To find the length of a shorter side in a right-angled triangle we rearrange Pythagorasβ theorem. π Pythagorasβ theorem is = π = π 2 2 β 2 β 2 Calculate the missing lengths of the shorter sides. (Clearly show all working out) π§ π 4.2 cm 2 π€ 9cm π2 = 9 π2 = π₯ 12 cm 2 + π₯ 6cm π2 = 9 π2 = 16 π€ 8cm π 2 = 16 + 9 = π = 25 β¦ = π2 = 16 7 cm 2 10 cm 2 Calculate the missing hypotenuse lengths. (Clearly show all working out) π¦ 12 cm www.missbsresources.com Geometry Pythagoras and Trigonometry Problems Pythagorasβ theorem is Second diagonal + Calculate the length of the line segment AF. = 2cm Remember to find a logical order to answering the question. 3cm 6cm First diagonal What is the formula for area of a triangle? Calculate the are of the triangle. 16 cm πππ π = π‘πππ = Calculate the angle between the length AE and the base ABCD in the pyramid pictured below, giving your answer to 1 decimal place. Opposite Remember πππ π πππ = βπ¦π π Adjacent www.missbsresources.com Geometry Missing Angles Find the missing angle The angles in a triangle add up to _________. Find the missing angle The angles in a quadrilateral add up to ________. The angles on a straight line add up to_________. The angles around a point add up to _________. Find the missing angle Find the missing angle www.missbsresources.com Algebra Sequences Match the sequence to its rule Write down the next three terms Fill in the blanks Fill in the blanks Mark each sequence with A (arithmetic) or G (Geometric) Write down the formula for the nth term of these sequences www.missbsresources.com Algebra Notation & Expressions Fill in the blanks from the word bank below to complete the sentences. Six Add Match each statement to the correct expression. π+3 Subtract a from 3 πβ3 Subtract The expression π + 5 means we five to π. Add 3 to a The expression π β 5 means we five to π. The expression 6 β π means we subtract a from . Fill in the blanks from the expression bank below to complete the sentences. c÷7 π×π 7×π The expression π means . The expression 7π means . π The expression means . 7 When simplifying expressions we collect like terms together. For example 1) π‘ + π‘ + + + = 5t = 5π‘ Multiply b by 3 Divide b by 3 Multiply b by itself Simplify the following 1) π + π + π + π = 2) 7π₯ β 2π₯ = 3) 3 × 5π = 4) 3π₯ + 4π¦ β π₯ + 2π¦ = 3) 8 × 3βπ Match each statement to the correct expression. π2 2 2) 2π‘+ Subtract 3 from a 3 π = 16π www.missbsresources.com 3 π 3π π 3 Algebra Expansion Expand the following brackets. Expand means you need to ___________ out the brackets. 1) 3 π₯ + 5 πΈπ₯ππππ 2(π₯ + 3) Use the method your teacher has taught. π 2 π₯+3 = 2π₯ +π 2) 2 3π¦ β 4 2 = 3) π₯ π₯ + 2 Expand and simplify Expand and simplify 1) 2 π₯ β 5 + 4 π₯ + 2 4 π₯β5 β3 π₯+7 Expand each bracket first. = 4π₯ β 20 β 3π₯ =π₯ 2) 5 π₯ + 3 β 2(π₯ β 4) Simplify by collecting like terms together. What does the word expand in mathematics mean you need to do? Expand 1) 3 π₯ + 2 2) 4(π₯ β 3) www.missbsresources.com Algebra Expansion Expand and simplify: Expand π₯ + 4 π₯ β 7 × π₯ π₯ β7 a) ( x + 4 )( x + 9 ) β7π₯ b) ( x + 3 )( x β 5 ) +4 = π₯2 c) ( 3x + 4 )( x β-2 ) β7π₯ = First Outside Inside Last Expand and simplify: (2π₯ + 5)(4π₯ β 3) a) ( 2x + 1 )( x + 3 ) β6π₯ b) ( 5x β 3 )( x β 5 ) = = 8π₯ 2 c) ( 3x + 5 ) www.missbsresources.com 2 Algebra Factorisation Factorise the following expression: 12 a + 8 Find the numerical highest common factor between each term(12 and 8) = 4 4(3a + 2) 12 a + 8 4 goes outside the bracket 4 x 3a = 12a and 4 x 2 = 8 Factorise the following expression: 12 a + 18 Find the numerical highest common factor between each term(12 and 18) = Factorise fully: a) 6b β 9 b) 36 β 6c c) 27a β 18b β 9c Factorise fully: a) 4b β 12 b) 24 β 6c 6 12π + 18 c) 14a β 21b +7c So ( + ) Factorise the following expression: 4π2 + 8π Factorise fully: a) 3π2 + 2π ×π× + ×2× b) 10π + 15π 2 So 4a( + ) c) 12ππ 2 + 18π www.missbsresources.com Algebra Factorisation Factorise 2 π₯ + π₯ β 12 What factor pairs Times to make multiply to make -12 End = ( π₯+ )( π₯β ) Add to make Middle Be careful of the signs. Factorise 1) π₯ 2 β 8π₯ + 15 2) π₯ 2 β 3π₯ β 18 3) 2π₯ 2 + 10π₯ + 8 www.missbsresources.com Algebra Algebraic Fractions Express 3 π₯+4 × 2π₯+5 2 as a single fraction. 3 2x + 5 = 2 x+4 = Express 2 π₯β7 × 3π₯+6 4 as a single fraction. When multiplying expressions, watch out for expanding brackets. 3 2x + 5 2 x+4 Express 2 π₯β2 + 3 π₯+5 as a single fraction. Express 3 π₯+6 + 4 π₯β8 as a single fraction. 2 π₯+5 +3 β + π₯β2 Expand Simplify Simplify fully π₯+ = π₯+ π₯+ = π₯β π₯ 2 +3π₯+2 π₯ 2 βπ₯β6 π₯+ π₯β Simplify fully Factorise the numerator and denominator. Simplify www.missbsresources.com π₯ 2 β7π₯β8 π₯ 2 +5π₯+4 Algebra Substitution Evaluate the following expressions when π = π πππ π = π 1) 2π₯ + π¦ = π2 = π × 3π 2 = × × If π = 3 what is the value of 4π2 = 4 × 3 × = 2) 2π¦ 2 = 3) 2π₯ 2 β 2π¦ = Use this rule to work out the total cost of hiring a bouncy castle. πππ‘ππ πππ π‘ = £10 πππ βππ’π πππ’π £15 Michelle hires a bouncy castle for 6 hours. πππ‘ππ πππ π‘ = £10 × πππ‘ππ πππ π‘ = Use this rule to work out the total cost of a taxis journey. πππ‘ππ πππ π‘ = £3 πππ ππππ πππ’π £2 Maria travels 5 miles in a taxis. Work out the total cost. + £15 + £15 πππ‘ππ πππ π‘ = The total cost is £125, how many hours did Adam hire the bouncy castle for?. πππ‘ππ πππ π‘ = £10 πππ βππ’π πππ’π £15 Cost × 10 +15 Hire Cost ÷ 10 β15 Hire The total cost of a taxis journey is £23. How many miles did Abdul drive? πππ‘ππ πππ π‘ = £3 πππ ππππ πππ’π £2 www.missbsresources.com Algebra Formulae When we follow a rule we substitute the terms with the corresponding values. The rule is Power = Voltage × Current If the voltage is 12 volts and the current is 5 amperes. What the is the power in watts? Aiman wants to calculate the momentum of a curling stone. The stone had a mass of 8kg and velocity of 3 m/s. Use the rule below to calculate the momentum. Momentum = Mass × Velocity Power = 12 × = watts π + π = 15 πβπ =8 c & d could be π = ππ and π = π because ππ + π = ππ Write down two different pairs of answers for c and d. π = ______ and d = ______ Write down two different pairs of answers for a and b. π = ______ and π = ______ π = ______ and π = ______ π = ______ and π = ______ A taxis cost £5 for each mile that it travels. Write this as a formula. Cost = It costs £10 per hour to rent a bouncy castle. Express this as a formula. × www.missbsresources.com Algebra Formulae Find the value of s when π = 3, π‘ = 2 πππ π’ = 4 1 π = π’π‘ + ππ‘ 2 2 1 π =π’× + × × π‘2 2 1 π = 4× + × × 22 2 Find the value of s when π = 2, π‘ = 5 πππ π’ = 3 1 π = π’π‘ + ππ‘ 2 2 www.missbsresources.com Algebra Forming Equations Write down an expression for the perimeter of this shape. Write down an expression for the perimeter of this shape. 2π₯ Add the edges together and then collect like terms. 3π₯ + 2 π₯+4 The perimeter=26cm what is the value of π₯? = 26 The perimeter=54cm what is the value of π₯? Solve for π₯. Iqbal has 17 pencils all together. He has a box of pencils and 5 extra pencils. Emily has a 20 sweets all together. She has two bags of sweets. Write this as an equation. Write this as an equation. How many pencils are in the box? I think of a number I multiply it by 5 and then subtract 6. n 5n 5n How many sweets are in a bag? Julie thinks of a number. She multiplies it by 4 and adds 16. The result is also six times as big as the number. What is the number? The result is also two times the number Iβm thinking. The equation is What is the number? π₯+5 6 Both rectangles have the same area. What is the value of the missing length? 5 2π₯ β 4 www.missbsresources.com Algebra Solving Equations Solve the equations Solve the equations π+5=9 π‘ + 12 = 15 Solve the equations 5π¦ = 15 2π€ = 14 πβ8=7 Solve the equations π =4 5 π€ =6 2 Solve 2π₯ + 8 = 22 Solve 3π₯ + 5 = 23 π β 11 = 14 5π₯ β 3 = 17 3π₯ = π₯= Solve 2 π₯ + 5 = 24 + Solve 3 π₯ + 4 = 27 = 24 2π₯ = π₯= www.missbsresources.com Algebra Solving Equations Solve the equation 3π‘ β 9 = 5π‘ β 17 β3π‘ β3π‘ β9 = 2π‘ + 17 Solve the equation 5π‘ + 12 = 7π‘ + 8 Solve the following equation Solve the following equation 3π‘ β 8 = 5π‘ + 20 β3π‘ β3π‘ 2π‘ β 6 = 6π‘ + 12 β8 = 2π‘ + 20 Solve the following equation Solve the following equation 2 π₯ + 2 = 6π₯ β 8 3 π₯ + 2 = 4π₯ β 7 Expand 3π₯ + 6 = Solve the following equation Solve the following equation 3 π₯ β 4 = 5(π₯ + 7) Expand 3π₯ β 12 = www.missbsresources.com 5 π₯ β 7 = 4(2π₯ β 3) Algebra Solving Inequalities Solve 6π₯ + 5 β₯ 4π₯ β 11 Solve 4π₯ + 15 < 8π₯ β 16 β₯ β₯ β₯ Represent the following solution sets on a number line. π₯>5 2π₯ + 4 < 10 π₯ β€ β4 Expand 1) 3 π₯ + 5 Solve the following and represent the solution set on a number line. 4 3π₯ β 5 > 2π₯ β 35 2) 2(3π₯ β 5) www.missbsresources.com Algebra Solving Quadratic Equations Factorise π₯ 2 β 2π₯ β 15 (π₯ )(π₯ Solve π₯ 2 + 4π₯ β 21 = 0 ) www.missbsresources.com Algebra Simultaneous Equations Solve 3π₯ + 2π¦ = 4 4π₯ + 5π¦ = 17 Aim to get one coefficient value the same. 1000 tickets were sold. Adult tickets cost £8.50, children's cost £4.50, and a total of £7300 was collected. How many tickets of each kind were sold? www.missbsresources.com Algebra Trial and Improvement The equation π₯ 3 + 3π₯ = 4 Has solutions between 3 and 4. π₯ Output The equation π₯ 3 β 6π₯ = 72 Has solutions between 4 and 5. Big/Small www.missbsresources.com Algebra Change the Subject To rearrange equations you follow the reverse order of operations. Rearrange these formulae to make m the subject. Make π₯ the subject 1) π = π β 4 π§ = 5π₯ β 3 2) q = 7π + 5 +3 π¦=π₯+4 β4 π§ + 3 = 5π₯ ÷5 β4 π§+3 π¦ =π₯ So π₯ = =π₯ Make π₯ the subject of the formula. Make π₯ the subject of the formula. π¦ = 4π₯ + π§ π¦= The opposite to the square root is to _____________ a number. 4π₯ 5 π¦ 2 = 4π₯ + z Rearrange to make π₯ the subject of the formula Rearrange to make π₯ the subject of the formula 3 π₯ β π‘ = π§(π β π₯) = π§π β π§π₯ Factorise 3π₯ + π§π₯ = π§π + 3π‘ p π¦ + π₯ = π‘(π€ β ππ₯) Expand Rearrange (3 + π§) = π§π + 3π‘ Rearrange π₯= www.missbsresources.com Algebra Coordinates Plot the missing coordinate to make a square. What is the coordinate? _______ When plotting coordinates we need to remember the rhyme . Plot the missing coordinate to make a rectangle. What is the coordinate? _______ Plot the coordinates A (3,4) B (2,5) C (0, 3) D (4,0) Along the ____________ and up the ___________. www.missbsresources.com Algebra Coordinates Write down the coordinates of the points 1) A _________ B 2) B _________ (-3, 2) 3) C _________ A Plot and label the following coordinates 4) D (0, 5) 5) E (β2, 4) 6) F (β3, β4) 7) G (5, β6) C (3, -3) (-4, -6) What is the midpoint between these pairs of numbers? Here are a set of coordinates A (0, 4) B (4, 8) C (-2, 1) D (3, -4) (Midpoint means halfway between) 4 8 5 10 -6 What is the midpoint of the coordinates a) A and B b) C and B c) A and D -2 www.missbsresources.com Algebra Line Graphs B What do you notice about the 3 points? What is the equation of the line? π₯ = C D A What are the equations of the lines a) c) b) d) Plot and connect the following coordinates on the grid. (-2, 5) (1, 5) (3, 5) (6, 5) What is the equation of the line they make? Draw and label the lines of 1) π¦ = 4 2) π₯ = 2 3) π¦ = β5 www.missbsresources.com Algebra Straight Line Graphs π = ππ Complete the table using the graph to help you. π = ππ π β2 β1 0 1 2 π β2 4 Complete the table, draw and label the graph. π = ππ β π π β2 β1 π 0 1 β3 Circle the graphs with a positive gradient. π¦ π¦ a 2 Match the equation with the correct sketches. π¦ π¦ π¦ = 3π₯ +4 c π₯ π₯ b π¦ Circle the graphs with the intercept of 4. π¦ π¦ a π₯ 2 π₯ π¦ = βπ₯ β1 π¦ c 4 -2 π₯ π₯ 4 π₯ b -4 4 Identify the gradient (m) and intercept (c) of these equations. 1) π¦ = 3π₯ β 5 Gradient = Intercept = 2) π¦ = β2π₯ + 6 Gradient = Intercept = π₯ π¦ = 2π₯ β4 π¦ π₯ π¦ π₯ www.missbsresources.com π¦ = β2π₯ +5 Algebra Graphs Complete the table, draw and label the graphs π = ππ π β2 β1 π 0 1 1 2 3 4 π = ππ β π π β2 β1 π 0 1 β5 2 β1 π = ππ β ππ β π π β2 π 5 www.missbsresources.com β1 0 β3 1 2 β3 3 Algebra Equations of a Line The equation of a straight line is normally in the form. π¦= What is the gradient of the equations? a) 2π¦ = 8π₯ + 10 π₯+ If I have 2 lots of an equation I need to divide the equation by 2. b) 4π¦ = 12π₯ β 20 2π¦ = 6π₯ β 8 ÷π ÷π π¦= π₯β4 What is the same when lines are parallel? Which of these lines are parallel? A C π¦ = 3π₯ + 2 B 4π¦ = 12π₯ β 20 3π¦ = 12π₯ β 15 Gradient = E D π¦ = 2π₯ β 3 πβππππ ππ π¦ πβππππ ππ π₯ What is the intercept of the graph? Calculate the gradient of this line. What is the equation of the line? www.missbsresources.com 5π¦ β 10π₯ = 20 Algebra Equation of a Line When lines are parallel they have the same ________________. Calculate the equation of a line parallel to π¦ = 3π₯ +5, passing through the point (2, 6). E.g. Calculate the equation of a line parallel to π¦ = 2π₯ + 4, passing through the point (1,3). So, π¦ = 2π₯ + π Substitute in the 3= 2×1+π coordinate values to find c. 3=2+π 1=π So, y = 2π₯ + 1 When line graphs are perpendicular to each other the gradients should multiply together to make βπ. Are these pairs of lines perpendicular to each other? (True or False) π¦ = 5π₯ + 8 What are the missing numbers? 1) 4 × 1 2) 3 × = β1 = β1 π¦ = β3π₯ + 6 1 π¦= π₯+3 2 www.missbsresources.com 1 π¦=β π₯β3 5 1 π¦ = 3π₯ β 3 π¦ = β2π₯ + 4 T F Algebra Graphs Match the graphs to the equations. π¦=3 π Given that π π₯ is the graph above math the following graphs with the functions. π¦=π₯ π¦ = π₯3 2 π π₯ +3 π π₯ β3 www.missbsresources.com π¦= π π₯+3 1 π₯ π π₯β3 Algebra Exponential and Reciprocal Graphs Complete the table of values for π¦ = 0.6 π₯ 0 1 π¦ 1 0.8 2 3 π₯ Note β the variable that changes is the power 4 0.13 1 Draw the graph π¦ = 0.6 0.8 Use your graph to solve the equation 0.6 0.6 π₯ = 0.4 Answer _________ 0.4 Draw the line π¦ = 0.4, because π¦ has been replaced with 0.4 so π¦ = 0.4 0.2 0 π₯ 1 2 3 4 5 www.missbsresources.com