Maths Unit 2 REvision
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The document defines different types of numbers including square numbers, cube numbers, powers, prime numbers, multiples, factors, prime factors, and recurring or terminating decimals. It also provides examples and explanations of concepts such as factor trees, LCM, HCF, fractions, algebraic fractions, and finding the midpoint of a line.
Maths Unit 2 REvision
- 1. Types of Numbers Square Numbers – any number times by itself. Examples: 4 (2x2),9 (3x3),16 (4x4),etc Cube Numbers – any numbers times by itself 3 times! (Like volume of a cube!) Examples: 8 (2x2x2), 27 (3x3x3), 64 (4x4x4),etc Powers – how many times a number multiplies itself by. Example: powers of 2: 2² = 2x2 = 4 2³ = 2x2x2= 8 2 = 2x2x2x2= 16 Prime Numbers – they only divide by themselves and one. If the number ends in 1,3,7 or 9, it means that there is a chance of it being a prime number – but it doesn‟t mean it is definitely a prime number! (e.g. 21,27 are NOT prime numbers though they end in 1/7 as they both divide by 3!) why
- 2. Multiples, Factors and Prime Factors Multiple = multiply = just keep multiplying! Multiples – the times table of that particular number. E.g. the multiples of 3 = 3,6,9,12,15,18…….. Factors of a number are pairs of numbers that you multiply to get another number. e.g. factors of 24 = 1x24,2x12,3x8,4x6 (multiply any of those pairs and you‟ll get 24!) – just start with 1 and keep going! why
- 3. Examples 1x64 Factor of 64 – 2x32 1,64,2,32,4,16,8 3x – 4x16 5x – 3,5,6,7 can’t be divided into 64. 6x – So leave it! 7x – Repeated 8x 8 numbers – so STOP! 1x24 Don’t write a number twice! 2x12 Factors of 24 – 1,24,2,12,3,8,4,6 3x8 Always start with ‘1’ times the 4x6 Repeated number! 5x – numbers – so 6x4 STOP! why
- 4. Factor Tree „Factor trees‟ are used to find prime factors (prime factors: a number broken down into a string of prime numbers all multiplied together) Factor tree of 420 – it A factor tree looks like this: will show you the prime factors of 420! Each ‘level’ of ‘Level’ 1 420 numbers multiplies to get the number above that level The numbers at the end ‘Level’ 2 42 42x10=420 10 (the ones in circles) are the prime factors! ‘Level’ 3 6 5 6x7=42 7 2 2x5=10 2 3 2x3=6 Finally! The prime factors of 420 =2,3,7,2 and 5. You would get 420 by multiplying those numbers. So it‟s usually written with times signs = 2x3x7x2x5 However – examiners like to see numbers in the simplest form, so we should write it like = 2²x3x7x5 (2²= 2x2) why
- 5. LCM LCM- Lowest Common Multiple – The smallest number that will divide by all of the numbers in the question. Once you know how to do the prime factors – LCM and HCF are fiddlesticks! LCM Example: Find the LCM of 42 and 36. 1) Find the prime factors of 42 and 36 2) 42=7x2x3 36=2x2x3x3 3) Circle the common prime factors 4) 42=7x2x3 36=2x2x3x3 5) Write down the ones you have circles ONCE =2x3 6) Then put the ones you haven‟t circled in the list= 2x3x2x3x7=252 This means if you wrote out all of the multiples in both numbers, 252 will be the first number that is common to both numbers. why
- 6. HCF HCF- Highest Common Factor – the biggest number that will divide into all the numbers in the question. Example of HCF: Let‟s use the same numbers: 42 and 36 We know that the prime factors of - 42=7x2x3 36=2x2x3x3 Now, we‟ve just got to circle the same things in both numbers! HCF of 42 and 36 = 2x3=6 This means the highest factor of both 42 and 36 is 6. why
- 7. Powers and Roots 1) When multiplying – you add the powers e.g. 3²x3²=3 (2+2=4) 8 is the index 2) When dividing – take away the powers number, I can‟t e.g. 3/3²=3² (4-2=2) get index numbers higher 3) When raising power to another, you multiply them. e.g. (5²)=5(8) than 5 on my laptop.. 4) Anything to the power of itself is just itself. e.g. 2=2 3=3 4=4 5) Anything to the power 0 is 1. e.g. 3˚=1 4˚=1 5˚=1 6) 1 to the power of anything is still 1. e.g. 1=1 1=1 1=1 7) Apply powers to top and bottom of fractions. e.g. (¾) = 9/16 why
- 8. Powers and Roots Continued.. 1) Negative powers – turn it upside down and make the power positive. 2) e.g. 7ˉ= 1/7=1/49 (3/5) ˉ = (5/3) 3) Fractional powers- ½= square root, 1/3= cube root, ¼ = fourth root,etc. e.g. 25½= square root of 25=5 4) Two-Stage Fractional Powers = 64⅜…..the top number of the fraction means the power, bottom number is the root.. So it means the eighth(8) root of 64, then the answer needs to be cubed(3). 5) Square roots can be positive or negative. e.g. 2² can be 2x2 = 4 but also (- 2)² = -2x-2=4 why
- 9. Standard Index Form A number in standard index form must always be written in this way: A x 10ⁿ This number needs to be „n‟ is how many places the between 1 and 10. decimal moves. 2) “Express 0.0000623 in standard Examples: form.” 1) “Express 3560 in standard form.” The decimal point must move 5 -Move the decimal point until 3560 places to give 6.23, so the becomes 3.56 (A must be power of 10 is 5. between 1 and 10). - Since 0.0000623 is a small -The decimal point has moved 3 number, is must be 10ˉ, not 10 places, so „n‟ equals 3. Giving 10³ -3560 is a big number, so n is +3, not -3. -So, the answer is 3.56x10³ why
- 10. Recurring or Terminating There‟s one way which you can work out if the fraction is recurring (like 1/3=0.333333..) or terminating (like 1/20=0.2). It‟s very simple, as long as you know how to work out the prime factors of a number. If the denominator of the fraction has a prime factor other than 2 or 5, then it‟ll be a recurring decimal! e.g. 1/20 – denominator is 20 – prime factors of 20 are – 2x2x5,so 1/20 is a terminating decimal because the denominators‟ prime factors are either 2 or 5. e.g. 2 – 1/3 – denominator is 3, prime factor – 1x3…so..1/3 is a recurring decimal! why
- 11. Convert recurring decimals to fractions “convert 0.234234234… to fractions in its simplest form.” First – look at how many numbers are recurring, in this case – 3, because it‟s „234‟ that is just being repeated. Then, if one digit is recurring (like 0.3333333) then times it by 10;if two digits are recurring (like .23232323) then times it by 100; if three digits are recurring then times it by 1000 and so on. So, in this case it would by 0.234234234 – three digits are recurring, so times it by 1000. Then you get: 234.234234234234 Then, because you times it by Subtract the original number from it! a 1000, and took it away from its original (which is times 1), you get 1000-1=999. 234.234234234234234 So it becomes 234 - 0.234234234234234 999 234 When it is cancelled down, it becomes 26/11. why
- 12. Fractions Multiplying – just multiply across! e.g. ¾ x ⅜ = 9/32 Dividing – turn the 2nd fraction upside down, then multiply! e.g. ¾ ÷ ⅜ = ¾ x 8/3 =24/12=2 Adding/Subtracting- if the denominators are the same, just add/subtract the numerator. e.g. 3/8-1/8=2/8 However, if the denominators are NOT the same, then you have to MAKE THEM the same – it‟s called equalising. e.g. 4/7 + 5/3 = find the LCM of the denominators (7 and 3 in this case), but if you can‟t be bothered, then just times them. After that, times the numerators by the opposing denominator too! =12/21+35/21= 47/21=2 and 5/21 why
- 13. Continued… The example – solution! 4 5 Change their 4 4x3=12 5 5x7 =35 + = denominators first by = = 7 3 multiplying across. 7 7x3=21 3 3x7=21 12 35 47 DON‟T add the + = 21 21 21 denominator! Find a fraction of „something‟ – Cancelling down – dividing top and multiply the „something‟ by the TOP of bottom number by the same number the fraction, then divide by the bottom until they won‟t go any further. e.g. find 9/20 of £360 = (9x360) ÷ 20 = £162 e.g. “cancel 18 6 3 down = = 18/24” 24 8 4 Divide Divide by 3 by 2 why
- 14. Algebra Simplifying – collecting like terms e.g. “Simplify 2x-4+5x+6” 2x-4+5x+6 = put „x‟s together = 2x+5x=7x, then put numbers together = -4+6=+2 answer=7x + 2 Expanding brackets – the item outside the bracket multiplies each separate term inside the bracket. e.g. 1) 5(x+3) = 5x+15 2) 3x(x+4)=3x+12 Multiply ! Follow the arrows = 2px3p,2px1,- 4x3p,-4x+1 Double bracket = (2p-4)(3p+1)= Then tidy up = 6p +2p-12p-4 = 6p -10p -4 why
- 15. D.O.T.S –very important!! Difference of two squares. a -b = (a+b)(a-b) This is very important for factorising! You can easily pick up some marks for very hard algebraic fractions, by knowing this! You might get a question involving factorising „4x -36‟ Then, you will be surprised….but if you learn this – you will know the answer is – (2x+6)(2x-6) Trust me – it is very helpful. why
- 16. Factorising Putting brackets in! How to do this- 1) Take out the biggest number that goes into all numbers 2) Take each letter in turn and take out the highest power that will go into EVERY term 3) Open the brackets and fill in all the bits needed to reproduce each term. Example: Factorise 15xy+20xyz-35xyz Answer: 5xy(3x+4yz-7xz) Z is not in all three Biggest terms, so can‟t Highest power come out as a number that that x and y will go into common factor. would go into 15,20 and 35 all three terms why
- 17. Algebraic Fractions Multiplying – it‟s the same, just multiply across. st 35stw 35stw 7st x = = 10w 12w 6 60w Dividing – turn the second fraction upside down, then multiply across…it‟s the same rule…again.. st 6 st 35stw 7st ÷ = x = 10w 10w 12w 35stw 60w Adding/Subtracting – get the same denominator by multiplying across. As long as you can do fractions, this is a piece of cake! why
- 18. Midpoint of a line To find the midpoint of a line – just divide! “Find the midpoint of the line AB. A has the co-ordinates of (4,5), B has the co-ordinates of (6,3)” Add the Xs and the Ys together separately, then divide by two. 4+6/2=5 The first 5+3/2=4 number is x, second Then put the Xs and Ys back. (5,4) number is y, always. why
- 19. y=mx+c m- the gradient of the line c- the y-intercept (where the graph hits the y axis) The gradient of a line = y divided by x. m is always the number with x, c is always the number on its own, so don’t get it mixed up when they give you ‘y=2+4x’ – put it back – ‘y=4x+2’ why
- 20. Distance-time graph Gradient=speed The steeper the line, the faster the speed. Straight lines means it has stopped. why
- 21. Parallel and perpendicular lines Parallel lines have the same gradient (m), but different y intercept. So the lines y=2x+3,y=2x-1,y=2x+6 are all parallel. The gradient of two perpendicular lines multiply to give -1. If the gradient of the first line is m, the gradient of the other line will be -1/m, because mx-1/m=-1. why
- 22. The gradients of these lines are 2 and -1/2. The product of the gradients is 2 x -1/2= -1. why
- 23. Geometry Angles in a triangle adds up to 180. Angles around a point adds up Angles on to 360. a straight line adds up to 180. Angle d = a+b Angles in Isosceles quadrilateral (4- triangles – 2 sides sided shapes) adds the same, two up to 360. angles the same. why
- 24. Continued… a a a a b a a and a are a and b add a and a are the same up to 180 the same These are These are called These are called alternate called corresponding angles supplementary angles angles why
- 25. Polygons Interior angle Exterior Angle Exterior angle= 360/number of sides. Interior angle = Sum of interior 180-exterior angles = (number angle of sides - 2) x 180 why
- 26. Symmetry Line Symmetry- draw one or more mirror lines across a picture and both sides will fold together exactly. Rotational Symmetry- rotate the shape into different positions and it will look exactly the same. why
- 27. Circles Tangent A tangent is a straight line that just touches the outside of the circle. A chord is a line drawn across the inside of a circle. Chord An arc is just part of the circumference of the circle. Same length tangents The angle between a tangent and the radius is always 90 degrees!! why
- 28. Areas why
- 29. Net, Volume of prism Surface area of the solid = area of net why
- 30. Volume of Prism A prism is a solid object which is the same shape all the way through (such as a triangular prism and a cube). Volume of prism = cross =area of cross-section (triangle) section x length x length =(4x3)/2x10 =60cm cubed 4cm 3cm 10cm why
- 31. Plan and Elevation You can look at an object from different points of view – plan (from the top), front (from the front!) and side (obviously, from the side XD). In your exam, you would then have to draw an image of the different points of views. So – get practising~ why
- 32. Converting Units Metric Units Imperial Units 1 (kg) 2.2 pounds (lb) Jayden drives at an average 1 litre 1¾ pints speed of 60mph. How long will it take him to drive 120 km? 4.5 litres 1 gallon 8km=5 miles 8 km 5 miles 60/5=12 8x12=96 30 cm 1 foot (ft) 60mph=96km/h Time = Speed/Distance These are the ones you need =120/96=1¼ hours you know for your exam. why
- 33. Speed, Distance and Time This is used to find out the above…but you would‟ve been using it for a long time, so I will not ramble~ You can think of this equation as SDT – SoDiT- SOD IT! “A car travels 90 miles at 36 miles per hour. How long does it take?” T=D/S = 90/36=2.5 hours. But- Remember to get the units right! “A boy walks 800m in 10 minutes. Find his speed in km/h” 800m=0.8km 10minutes =0.1667 hours Then divide 0.8km by 0.1667hours to get 4.8km/h why