Crucial Knowledge – Stage 1 - Number BIDMAS The order you do calculations in: B rackets I ndices D ivision M ultiplication A ddition 4 Operators Negative Numbers S ubtraction • Addition (or Sum) + • Adding or subtracting – USE A • Subtraction (or Difference) – NUMBER LINE • Multiplication (or Product) x • Multiplying or dividing use the rules • Division ÷ + x + = + + ÷ + = + + x - = - + ÷ - = - Fractions Decimals and Percentages - x + = - - ÷ + = - • Different ways of saying part of a whole number - x - = + - ÷ - = + • You can change from one to the other Prime Numbers Rounding • Have exactly two factors • Decimal places (column after decimal point) • No other whole numbers, except 1 and itself divide into them • Significant Figures (highest value column) Crucial Knowledge – Stage 1 - Number Highest Common Factor (HCF) Lowest Common Multiple (LCM) Standard Form • Write down all the factors from • A way of writing very BIG or very SMALL numbers the numbers and find the biggest • Think BIG numbers – distance between planets and SMALL numbers value on both lists – This is the – sizes of atoms. Highest Common Factor • BIG numbers have POSITIVE powers and SMALL number have NEGATIVE powers. • Write down all the multiples of Always a multiply Always the number 10 Always a number Positive for very large numbers the two numbers and find the between 1 and 10 smallest on both lists – This is the 4 x 106 Negative for very small numbers Lowest Common Multiple Percentages Fractions • An amount out of 100 • Multiplying – Multiply top by top and • To Calculate a percentage of an amount (What is 40% of £50) bottom by bottom. Percentage ÷ 100 x amount (40 ÷ 100 x 50 = £20) • Dividing - ‘Keep Change Flip’. • To change to a percentage (you score 4 out of 5 in a test, what • Addition or Subtraction – You need same percentage is this?) bottom number (denominator). Amount you got ÷ what it is out of x 100 (4 ÷ 5 x 100 = 80%) Crucial Knowledge – Stage 2 - Number Percentage Change Inequalities • If a value goes up, it’s a percentage increase. • Understand inequality symbols < > ≤ ≥ • If a value goes down, it’s a percentage decrease. • List values that satisfy a inequality. • We work out percentage of amount and either add it on • Show by drawing on a number line values that or subtract it from our starting value satisfy inequality. • Or we work out the percentage change by working out the difference in values and dividing by our original value Estimation and then multiplying by 100. • An answer close to the exact answer. • All values are rounded to 1 significant figure. Powers • Follow BIDMAS to get your estimation. • If we multiply powers we add. y3 x y4 = y(3 + 4) = y7 • If we divide powers we subtract. y10 ÷ y6 = y(10 - 6) = y4 Use of Calculator • Anything to the power zero is always 1 • Must be able to use brackets ( ) on calculator to get an answer to multi stage calculations. Product of Primes • Must be able to use powers on calculator. • Any value split into prime numbers MULTIPLIED together. • Must be able to use for Standard Form calculations . • First 5 prime numbers are 2, 3, 5, 7 and 11. • Must be able to use fraction button for all multi tier • Sometimes we put into a VENN diagram to calculate LCM calculations. and HCF. • Must be able to use calculator for percentage calculations. Crucial Knowledge – Stage 3 - Number Recurring Decimals Advanced Powers Upper and Lower Bounds • A decimal with repeating values • A negative power means • Upper is slightly above your values • We indicate the repeating reciprocal (”1 over”) • Lower is slightly below your values % % numbers with a dot above • 4"# = = • Using bounds affects calculations – you ̇ &' %( 0. 6 = 0.666666 … • A fractional powers means must find bounds before any calculations ̇ ̇ • Example: 0. 656 = 0.656656656 … find a root ̇ ̇ * * 0.716 = 0.7161616 … + • Must be able to convert recurring • �' = � �+ = � Q: A field measures 34m x 28m both decimals to fractions • More complicated fractions measured to the nearest metre. What is the require using powers and minimum and maximum area the field Fractions roots could have? + . • Mixed number to improper • 16' = 16 = 4 = 64 Upper Bound Lower Bound 1 13 4 = • Evaluate a negative fractional A: Bounds Length (34m) 34.5m 33.5m 3 3 power in this order Width (28m) 28.5m 27.5m 4 � 3 + 1 = 13 3rd 2nd . 2 • Improper to mixed number " Maximum Area = 34.5 x 28.5 = 983.25m 13 1 16 & 2 = 4 Minimum Area = 33.5 x 27.5 = 921.25m 3 3 1st % • = (0 16)".= 2". = 13 ÷ 3 = 4 ��������� 1 3 Crucial Knowledge – Stage 1 – Ratio and Proportion Ratio as a measure Equivalent ratios Dividing a given ratio • A ratio is a comparison of parts • Same values but different • The question matches the order of items to • Use a colon (:) to separate parts numbers the order of parts in the ratio. The first of a ratio • Values used can get larger, as thing mentioned gets the first part of the • A colon is read as ‘to’ well as smaller ratio • 2 or 3 parts • Do same to all parts • Find the total number of parts in the ratio • Understand the parts add up 3:6 (+) and stay in proportion x4 x4 • Divide the amount to be shared by the total 12:24 parts (÷) Cancelling ratios • Basic unit conversions Multiply by each part of the ratio (x • Like simplifying fractions • Convert units of length (mm, Example • Look for common factors cm, m, km) Q: Adam and Ben share £45 in the ratio 1:2. • Do the same to both parts of • Be able to convert to common Who gets how much? the ratio unit before calculating A: 1 + 2 = 3 parts in total 3:6 • Convert units of time £45 ÷ 3 = £15 per part ÷3 ÷3 1:2 1:2 • Convert units of measure (ml, l) • Convert units of mass (g, kg, t) x15 x15 15:30 Adam gets £15 and Ben gets £30 Crucial Knowledge – Stage 2 – Ratio and Proportion Unit conversions Ratio calculations Recipe Scaling • Area conversions • Use a ratio to scale • Work out we have enough to complete Use the same conversions as measurements up and • How much of something do we need for length, but squared down • Volume conversions • Examples include Example: Use the same conversions as using maps and scale Q: A recipe uses 300g of flour and 150g of butter to for length, but cubed drawings make a cake for 4 people. How much of each FGHIJKLM • ����� = • Size calculations ingredient is needed to bake a cake for 6 people. IGNM • Units for speed include relative to scale and metres per second (m/s) real life A: 6 ÷ 4 = 1.5 (scale factor). and kilometres per hour 300g x 1.5 = 450g flour (kmph) 150 x 1.5 = 225g butter Crucial Knowledge – Stage 3 – Ratio and Proportion Interest Calculations Proportionality Compound Measures _GHIJKLM • Compound Interest is an • Values that have a relationship with each other, • ����� = accumulating interest, as one changes, so does the other one `GNM changing over time, as a • Y = kx dJHH • ������� = growth • y is directly proportional to x. efghNM • Depreciation is a reduction Q: If y=24, then x=8 jfkLM • Compound Interest Work out the value of y when x=2. • �������� = Always multiply Always 1 Number of lkMJ years A: y = kx 24 = kx8 k=3 and so y = 3x Starting K amount a x (1 ± �) So when x = 2 y = 3 x 2 = 6 + for growth Interest rate as - for depreciation a decimal Inverse Proportionality • A reverse percentage is • Values that have a relationship with each other, as one changes, so does finding the original value the other, but inverse ����� ����� ] �������� = �100 • � = 100 − % �ℎ���� ^ • y is inversely proportional to x. Q: When y=2, x=3. Work out the value of y when x=18 ] ] ( A: � = 2 = k = 6 and so � = ^ . ] When x = 18 y = 6 ÷ 18 = 1/3 Crucial Knowledge – Stage 1 – Geometry and Measures Coordinates Terminology Shape • Remember “along the corridor • Edge – Where 2 faces meet then up the stairs” Use of Protractor • Vertices – Where 3 faces meet • X and y values written on the • Measure angles accurately • Face – side of a 3d shape axes • Draw bearings • Quadrilateral – a 4 sided polygon • 4 quadrants • Polygon – a 2d shape with straight Angle Reasoning sides • Angles on straight line = 180o Area and perimeter • Acute – an angle less than 90o • Angles in a triangle = 180o • Perimeter is distance around shape • Obtuse – an angle between 90o • Vertically opposite angles are • Area is space inside a shape (2D), and 180o always equal measure in square units • Reflex – an angle more than 180o • Rectangle ���� = �����ℎ × ����ℎ • Angles in quadrilateral = 360o % o • ���� = ���� × ℎ���ℎ� • Angles at a point = 360 Triangle # Only use diagonals for perimeter Types of Triangles % • ���� = � + � × ℎ���ℎ� • Trapezium # Scalene – all sides and angles are different Only use diagonals for perimeter • Isosceles – 2 sides and angles are the same • Circle ���� = � × ������# • Equilateral – 3 sides and angles are the same ������������� = 2� × ������ • Right – contains a right angle Circumference is the perimeter of a circle Crucial Knowledge – Stage 2 – Geometry and Measures Bearings Pythagoras • 3 digit format • Measure clockwise from Polygons c North, 000o a • A shape with 3 or more straight sides • Be able to draw and add onto • Total Interior Angles = (n-2) x 180 b a diagram • Interior + Exterior = 180o • �# + �# = �# • Measure reflex angles using a • Sum of Exterior = 360o Square root c2 to find Hypotenuse compass # # # • � + � = � Basic transformations • Calculations using North for Square root b2 to find shorter side • Reflections parallel lines Over straight lines (y=, x= ) including Plans and Elevations Angles with parallel lines diagonals (y=x) • Images from 3 different directions • F – Corresponding • Rotations • Front, side and plan Always equal Direction, Distance and Centre • Work out size or volume • Z – Alternate • Translation • Draw 3 images from a 3D drawing ���ℎ� + ���� − Always equal • Draw a 3D image from 3 plans and • C – Co-Interior elevations �� + ���� − Always add to 180o • Enlargement Scale factor and Centre Crucial Knowledge – Stage 3 – Geometry and Measures Loci and Constructions Advanced Volumes & • Perpendicular line bisector • ������ = ��.