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OBJECTIVE

OBJECTIVE. REVISION MOD 3. G. Check the side of the slide to see what level you are working at!. F. E. D. C. B. A. A*. INTEGERS. INTEGER is a whole number. HCF / LCM simple numbers – C HCF / LCM complex or more than two numbers – B Recognise prime numbers – C

archibald-richter
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OBJECTIVE

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  1. OBJECTIVE REVISION MOD 3

  2. G Check the side of the slide to see what level you are working at! F E D C B A A*

  3. INTEGERS • INTEGER is a whole number. • HCF / LCM simple numbers – C • HCF / LCM complex or more than two numbers – B • Recognise prime numbers – C • Write a number as product of its prime numbers – C • Find the reciprocal of a number - C

  4. Multiples G • These are all of the integers that appear in your number’s times table! • 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36. • 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84

  5. Factors G • These are all of the integers that will divide into your number and leave no remainder! • They are usually listed in pairs! e.g. the factors of 36 are: 1 & 36 2 & 18 3 & 12 4 & 9 6 & 6

  6. Prime Numbers & Prime Factors • APRIME NUMBERhasTWO DIFFERENT FACTORS 1 & ITSELF. The prime numbers less than 30 are …. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 • APRIME FACTOR, is a factor that is also a prime number. e.g. factors of12are1, 2 ,3 ,4 , 6 & 12of these2 & 3are prime factors. 12can be written as a product of prime factors… 12 = 2 x 2 x 3 in its INDEX FORM = 22 x 3 C

  7. Highest common Factor • Thehighest common factor (HCF)of two numbers, is the largest factor common to both. e.g.factors of 18 are 1,2,3,6,9,18 factors of 30 are 1,2,3,5,6,10,15,30 The highest factor common to both numbers is 6. We use HCF’s when cancelling fractions!!! C

  8. Lowest Common Multiple • The Lowest Common Multiple (LCM) of two numbers, is the smallest number that appears in both time tables. • The example below is for the 9 & 15 times table….. e.g. the multiples of 9 are 9,18,27,36,45,54,63,…. the multiples of 15 are 15,30,45,60,… 45 is the lowest common multiple of each sequence of numbers C

  9. Prime factor product trees • Products of prime numbers can be written as “trees”. 180 2 x 2 x 3 x 3 x 5 = 180 or; in INDEX FORM 22 x 32 x 5 = 180 90 x 2 45 x C 2 15 3 x 5 x x 3 x x x

  10. HCF and LCM • We can use prime factors to find the HCF and LCM… e.g. 504 = 2 x 2 x 2 x 3 x 3 x 7 700 = 2 x 2 x 5 x 5 x 7 HCF is 2 x 2 x 7 = 28 LCM is 2 x 2 x 2 x 3 x 3 x 5 x 5 x 7 = 12600 504 = 23 x 32 x 7 700 = 22 x 52 x 7 HCF is 22 x 7 LCM is 23 x 32 x 52 x 7 B This is what’s left from BOTH numbers when you take out the HCF

  11. Consecutive Numbers • A set of 5 consecutive numbers will increase by 5 each time, or are divisible by 5. e.g. 1+2+3+4+5 = 15 2+3+4+5+6 = 20 If n = starting number, then the next is (n+1), etc . n + (n+1) + (n+2) + (n+3) + (n+4) = 5n +10 = 5(n+2) C Thus5 is always factor of a series of five consecutive numbers

  12. INDICES • INDEXis another word forPOWER. • Recall integer squares / square roots to 15 –D • Recall integer cube / cube roots to 5 –D • Use index laws for positive powers –C • Use index laws for negative powers –B • Use index laws with simple fractional powers –A • Use index laws with complex fractional powers –A*

  13. Square Numbers & Cube Numbers • ASQUARE NUMBERis a NUMBER x ITSELF. 1 x 1 = 1, 2 x 2 = 4, 3 x 3 = 9, 4 x 4 = 16 and so on Remember the first 15 Square Numbers …. 1,4,9,16,25,36,49,64,81,100,121,144,169,196,225. • ACUBE NUMBERis a NUMBER x ITSELF x ITSELF. 1 x 1 x 1 = 1, 2 x 2 x 2 = 8, 3 x 3 x 3 = 27, and so on Remember the first 5 Cube Numbers …. 1, 8, 27,64,125. D

  14. Square Root TheNUMBERthat isSQUAREDto make9is3. 3is called theSQUARE ROOTof9and is written√9. Remember the square roots as the reverse of the square numbers. SO √1,√4,√9,√16,√25,√36,√49,√64,√81,√100,√121,√144,√169,√196,√225 are the numbers from 1 to 15. D

  15. What are Indices? • An Index is often referred to as a power For example: = 53 5 x 5 x 5 = 24 2 x 2 x 2 x 2 = 75 7 x 7 x 7x 7 x 7 5 is the INDEX 7 is the BASE NUMBER 75 & 24 are numbers in INDEX FORM

  16. Rule 1 : Multiplication 26 x 24 = 210 x 2x2x2x2x2x2 2x2x2x2 35 x 37 = 312 x 3x3x3x3x3 C 3x3x3x3x3x3x3 General Rule am x an = am+n

  17. Rule 2 : Division 44÷42 = 42 ÷ 4x4x4x4 4x4 26÷23 = 23 ÷ 2x2x2x2x2x2 C 2x2x2 General Rule am÷ an = am-n

  18. Rule 3 : Brackets (26)2 = 26 x 26 = 212 (35)3 = 35 x 35 x 35 = 315 C General Rule (am)n = am x n

  19. Rule 4 : Index of 0 How could you get an answer of 30? 35÷ 35 = 35-5 = 30 30 = 1 C General Rule a0 = 1

  20. Combining numbers x 2 x 2 x 2 x 2 5 x 5 x 5 = 53 x 24 We can not write this any more simply Can ONLY do that if BASE NUMBERS are the same

  21. Putting them together? 26 x 24 23 = 210 23 = 27 35 x 37 34 = 312 34 = 38 C 25 x 23 24 x 22 = 28 26 = 22

  22. ..and a mixture… 2a3 x 3a4 = 2 x 3 x a3 x a4 = 6a7 8a6÷ 4a4 = (8 ÷ 4) x (a6 ÷ a4) = 2a2 2 2 8a6 4a4 C

  23. Works with algebra too! a6 x a4 = a10 b5 x b7 = b12 c5 x c3 c4 = c8 c4 = c4 C a5 x a3 a4 x a6 = a8 a10 = a-2

  24. Summary of rules. 1. am x an = am+n 2. am÷ an = am-n 3. (am)n = am x n 4. a1 = a 5. a0 = 1

  25.  1 2 General Rule a-n = 1 an More rules….. Rule 6 negative indices 25 32 24 16 23 8 22 4 21 2 20 1 2-1  2-2  B

  26. General Rule a = √a n Rule 7 – Fractional Indices 9 x 9 = 91 =9 From Rule 1 & 4 So 9 = √9 A

  27. Rule 8 – Complex Fractional Indices 81= (4√81)³ = (3)³ = 27 General Rule Treat the bottom as a fractional index so find root, then use top part as a normal index. A*

  28. Standard Index Form • SIF is a way of writing big or small numbers using indices of 10. • Convert numbers to and from SIF – C • Use SIF in simple number problems – B • Use SIF in complex word problems – A

  29. Why is this number very difficult to use? 999,999,999,999,999,999,999,999,999,999 Too big to read Too large to comprehend Too large for calculator To get around using numbers this large, we use standard index form.

  30. But it still not any easier to handle!?! Look at this 100,000,000,000,000,000,000,000,000,000 At the very least we can describe it as 1 with 29 noughts.

  31. How could we turn the number 800,000,000,000 into standard index form? So, 800,000,000,000 = 8 x 1011 in standard index form Let’s investigate! Converting large numbers We can break numbers into parts to make it easier, e.g. 80 = 8 x 10 and 800 = 8 x 100 C 800,000,000,000 = 8 x 100,000,000,000 And 100, 000,000,000 = 1011

  32. Standard Form (Standard Index Form) 5.3 x 10n There will also be a power of 10 C The first part of the number is between 1 and 10 But NOT 10 itself!!

  33. The first number must be a value between 1 and 10 One of the most important rules for writing numbers in standard index form is: But NOT 10 itself!! For example, 39 x 106 does have a value but it’s not written in standard index form. The first number, 39, is greater than 10. 3.9 x 107 is standard index form. C

  34. Indices of Ten Notice that the number of zeros matches the index number 2 100 10 10 10 10 3 1,000 4 10,000 5 100,000

  35. 10 So, 45,000,000,000 = 4.5 x 1010 Quick method of converting numbers to standard form For example, Converting 45,000,000,000 to standard form Place a decimal point after the first digit 4.5000000000 Count the number of digits after the decimal point. C This is our index number (our power of 10)

  36. And numbers less than 1? How can we convert 0.067 into standard index form? 0.067 = 6.7 x 0.01 0.01 = 10-2 C 0.067 = 6.7 x 10-2

  37. And numbers less than 1? How can we convert 0.000213 into standard index form? 0.000213 = 2.13 x 0.0001 0.0001 = 10-4 C 0.000213 = 2.13 x 10-4

  38. 56 567 5678 56789 0.56 0.056 0.0056 Write the following in standard form. 0.00056 23 234 4585 4.6 0.78 0.053 0.00123 How to write a number in standard form. Place the decimal point after the first non-zero digit then multiply or divide it by a power of 10 to give the same value. = 5.6 x 10 = 5.6 x 101 = 5.67 x 100 = 5.67 x 102 = 5.678 x 1000 = 5.678 x 103 = 5.6789 x 10 000 = 5.6789 x 104 = 5.6  10 = 5.6 x 10-1 C = 5.6  100 = 5.6 x 10-2 = 5.6  1000 = 5.6 x 10-3 = 5.6  10 000 = 5.6 x 10-4 2.3x 101 2.34x 102 4.585x 103 4.6x 100 7.8x 10-1 5.3x 10-2 1.23x 10-3

  39. Examples: Exp/EE? Calculate: 4.56 x 108x 3.7 x 105 +/- Sharp Standard Form on a Calculator You need to use the exponential key (EXP or EE) on a calculator when doing calculations in standard form. 4.56 Exp 8 x 3.7 Exp 5 = 1.6872 x 1014 1.7 x 1014(2sig fig) Calculate: 5.3 x 10-4 x 2.7 x 10-13 C 5.3 Exp - 4 x 2.7 Exp - 13 = 1.431 x 10-16 1.4 x 10-16 (2 sig fig) Calculate: 3.79 x 1018 9.1 x 10-5 3.79 Exp 18  9.1 Exp - 5 = 4.2 x 1022(2 sig fig)

  40. Calculations Using SIF B

  41. Multiply two numbers 4 x 1018 x 3 x 104 Numbers Powers of 10 4 x 3 x 1018 x 104 ADD powers = 12 x 1022 NOT Std Form! B = 1.2 x 101 x 1022 = 1.2 x 1023

  42. Complex word problems involving SIF The mass of the Earth is approximately 6 000 000 000 000 000 000 000 000 kg. Write this number in standard form. 6.0 x 1024 The mass of Jupiter is approximately 2 390 000 000 000 000 000 000 000 000 kg. Write this number in standard form. 2.39 x 1027 A How many times more massive is Jupiter than Earth? 398 2.39 x 1027 / 6.0 x 1024 =

  43. Complex word problems involving SIF The mass of a uranium atom is approximately 0. 000 000 000 000 000 000 000 395 g. Write this number in standard form. 3.95 x 10-22 The mass of a hydrogen atom is approximately 0. 000 000 000 000 000 000 000 001 67 g. Write this number in standard form. 1.67 x 10-24 How many times heavier is uranium than hydrogen? A 237 3.95 x 10-22/ 1.67 x 10-24 =

  44. Complex word problems involving SIF Writing Answers in Decimal Form (Non-calculator) Taking the distance to the moon is 2.45 x 105 miles and the average speed of a space ship as 5.0 x 103 mph, find the time taken for it to travel to the moon. Write your answer in decimal form. D 245 000 S 49 hours S = so T = = = T 5 000 D A

  45. Rounding to nearest integer (whole number). G.Rounding to nearest 10 or 100. G.Rounding to given number of decimal places. F.Rounding to given number of significant figures. E. Rounding.

  46. Roundingto the nearest whole number G • Is the arrow nearer to 6, 7 or 8? • If it is halfway between, then round UP 6 7 8

  47. Roundingto the nearest 10 G • Is the arrow nearer to 20, 30 or 40? • If it is halfway between, then round UP 20 30 40

  48. Roundingto the nearest 100 G • Is the arrow nearer to 400 or 500? • If it is halfway between, then round UP 400 500

  49. Decimal Places F

  50. Round the following number to 1dp 6.348 F If this number is a 0, 1, 2, 3 or 4 we don’t have to do anything else and we have our answer. Now look at the number immediately after where we stopped highlighting Firstly, highlight the number to the first number after the decimal point So we have 6.3 But is this the answer?

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