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OBJECTIVE REVISION MOD 3 G F Check the side of the slide to see E what level you are working at! 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 • Write a number as product of its prime numbers – C • Find the reciprocal of a number - C G Multiples • 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 G• Factors 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 Prime Numbers & Prime Factors • A PRIME NUMBER has TWO DIFFERENT FACTORS 1 & ITSELF. The prime numbers less than 30 are …. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 • A PRIME FACTOR, is a factor that is also a prime number. C e.g. factors of 12 are 1, 2 ,3 ,4 , 6 & 12 of these 2 & 3 are prime factors. 12 can be written as a product of prime factors… 12 = 2 x 2 x 3 in its INDEX FORM = 22 x 3 Highest common Factor • The highest common factor (HCF) of two numbers, is the largest factor common to both. 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. e.g. C We use HCF’s when cancelling fractions!!! 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….. C 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 Prime factor product trees • Products of prime numbers can be written as “trees”. 2 x 2 x 3 x 3 x 5 = 180 180 or; in INDEX FORM 90 C 2 x 2 45 x 3 x 15 x 3 x x 22 x 32 x 5 = 180 5 x x 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 B 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 This is what’s left from BOTH numbers when you take out the HCF Consecutive Numbers C • 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) Thus 5 is always factor of a series of five consecutive numbers INDICES • INDEX is another word for POWER. • Recall integer squares / square roots to 15 – D D Use index laws for positive powers – C Use index laws for negative powers – B • Recall integer cube / cube roots to 5 – • • A Use index laws with complex fractional powers – A* • Use index laws with simple fractional powers – • Square Numbers & Cube Numbers • A SQUARE NUMBER is a NUMBER x ITSELF. D 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. • A CUBE NUMBER is 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. Square Root The D NUMBER that is SQUARED to make 9 is 3. 3 is called the SQUARE ROOT of 9 and 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. What are Indices? • An Index is often referred to as a power For example: 5 x 5 x 5 = 53 2 x 2 x 2 x 2 = 24 7 x 7 x 7x 7 x 7 = 75 5 is the INDEX 7 is the BASE NUMBER 75 & 24 are numbers in INDEX FORM Rule 1 : Multiplication 26 x x 2x2x2x2x2x2 35 C 3x3x3x3x3 x x 24 = 210 2x2x2x2 37 = 312 3x3x3x3x3x3x3 General Rule am x an = am+n Rule 2 : Division 44 4x4x4x4 26 C 2x2x2x2x2x2 ÷ ÷ ÷ ÷ 42 = 42 4x4 23 = 23 2x2x2 General Rule am ÷ an = am-n 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 Rule 4 : Index of 0 How could you get an answer of 30? 35 ÷ 35 = 35-5 = 30 C 30 = 1 General Rule a0 = 1 Combining numbers 5x5x5 x2x2x2x2 = 53 x 24 We can not write this any more simply Can ONLY do that if BASE NUMBERS are the same Putting them together? 26 x 24 = 210 = 27 23 23 C 35 x 37 = 312 = 38 34 34 25 x 23 = 28 24 x 22 26 = 22 ..and a mixture… 2a3 x 3a4 = 2 x 3 x a3 x a4 = 6a7 8a6 ÷ 4a4 = (8 ÷ 4) x (a6 ÷ a4) = 2a2 C 2 6 28a 4 4a Works with algebra too! C a6 x a4 b5 x b7 c5 x c3 c4 a5 x a3 a4 x a6 = a10 = b12 = c8 = c4 c4 = a8 = a-2 a10 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 More rules….. Rule 6 negative indices 25 24 23 22 21 20 2-1 32 16 8 4 2 1 1 B 2-2 2 General Rule a-n = 1 an Rule 7 – Fractional Indices From Rule 1 & 4 1 9 x 9 = 9 =9 So 9 = √9 General Rule A a n = √a 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* 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 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. 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. But it still not any easier to handle!?! Let’s investigate! Converting large numbers How could we turn the number 800,000,000,000 into standard index form? We can break numbers into parts to make it easier, C e.g. 80 = 8 x 10 and 800 = 8 x 100 800,000,000,000 = 8 x 100,000,000,000 And 100, 000,000,000 = 1011 So, 800,000,000,000 = 8 x 1011 in standard index form Standard Form (Standard Index Form) 5.3 x 10 n There will also be a power of 10 C The first part of the number is between 1 and 10 But NOT 10 itself!! One of the most important rules for writing numbers in standard index form is: The first number must be a value between 1 and 10 But NOT 10 itself!! C 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. Indices of Ten Notice that the number of zeros matches the index number 2 10 3 10 4 10 5 10 100 1,000 10,000 100,000 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 C 4.5000000000 Count the number of digits after the decimal point. 10 This is our index number (our power of 10) So, 45,000,000,000 = 4.5 x 1010 And numbers less than 1? How can we convert 0.067 into standard index form? 0.067 = 6.7 x 0.01 C 0.067 = 6.7 x 10-2 0.01 = 10-2 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 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. 56 = 5.6 x 10 = 5.6 x 101 567 = 5.67 x 100 = 5.67 x 102 5678 = 5.678 x 1000 = 5.678 x 103 56789 = 5.6789 x 10 000 = 5.6789 x 104 0.56 = 5.6 10 = 5.6 x 10-1 C 0.056 = 5.6 100 = 5.6 x 10-2 0.0056 = 5.6 1000 = 5.6 x 10-3 0.00056 = 5.6 10 000 = 5.6 x 10-4 Write the following in standard form. 23 234 4585 4.6 0.78 0.053 0.00123 2.3x 101 2.34x 102 4.585x 103 4.6x 100 7.8x 10-1 5.3x 10-2 1.23x 10-3 Standard Form on a Calculator You need to use the exponential key (EXP or EE) on a calculator when doing calculations in standard form. Examples: Exp/EE? Calculate: 4.56 x 108 x 3.7 x 105 4.56 C 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 5.3 Exp Sharp - 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) Calculations Using SIF B Multiply two numbers 4 x 1018 x 3 x 104 Numbers 4x3 B NOT Std Form! Powers of 10 x 1018 x 104 = 12 x 1022 ADD powers = 1.2 x 101 x 1022 = 1.2 x 1023 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? 2.39 x 1027 / 6.0 x 1024 = 398 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 A How many times heavier is uranium than hydrogen? 3.95 x 10-22 / 1.67 x 10-24 = 237 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. A D S S = so T = T D 245 000 = = 49 hours 5 000 Rounding. 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. G Rounding to the nearest whole number • Is the arrow nearer to 6, 7 or 8? • If it is halfway between, then round UP 6 7 8 G Rounding to the nearest 10 • Is the arrow nearer to 20, 30 or 40? • If it is halfway between, then round UP 20 30 40 G Rounding to the nearest 100 • Is the arrow nearer to 400 or 500? • If it is halfway between, then round UP 400 500 F Decimal Places Round the following number to 1dp F 6.348 Firstly, highlight the number to the first number after the decimal point So we have 6.3 But is this the answer? If thisNow number look is at a 0, 1, the2,number 3 or 4 we don’t immediately have to doafter anything where else we andstopped we havehighlighting our answer. Round the following number to 1dp F 6.348 = 6.3 (1dp) What if the red number was a 5, 6, 7, 8 or 9? F Lets look at an example Round the following number to 1dp F If thisNow number look is at a 5, 6, the7,number 8 or 9 we increase immediately the last digit afterby where one. we stopped highlighting 9.2721 Firstly, highlight the number to the first number after the decimal point So we have 9.2 But is this the answer? So 9.2 becomes 9.3 Round the following number to 1dp F 9.2721 = 9.3 (1dp) Round the following number to 2dp F 7.456 If this number is a 0, 1, 2, 3 or 4 we don’t have to do anything Firstly, highlight Now look at else and we the number to the number have our the second immediately answer, but it is number after after where not, so we the decimal we stopped round up the point highlighting number in the second decimal place to give us our answer. 7.46 Round the following number to 2dp F 3.992 If this number is a 0, 1, 2, 3 or 4 we don’t have Firstly, highlight Now look at to do anything the number to the number else. In this the second immediately case it is so we number after after where have our the decimal we stopped answer point highlighting highlighted. 3.99 Round the following number to 1dp F 6.348 = 6.3 (1dp) What if the red number was a 5, 6, 7, 8 or 9? F Lets look at an example Round the following number to 1dp F If thisNow number look is at a 5, 6, the7,number 8 or 9 we increase immediately the last digit afterby where one. we stopped highlighting 9.2721 Firstly, highlight the number to the first number after the decimal point So we have 9.2 But is this the answer? So 9.2 becomes 9.3 Round the following number to 1dp F 9.2721 = 9.3 (1dp) Decimal Places (Rounding) Numbers can be rounded to 1,2, 3 or more decimal places. F Rounding to 1 d.p 4.8325 4. 8 4 2 5 4. 8 5 2 5 5 or bigger ? 5 or bigger ? 5 or bigger ? No No Yes 4.8 4.8 4.9 Decimal Places F It is often necessary/convenient/sensible to give approximations to real life situations or as answers to certain calculations. For example if a case of wine containing 6 bottles costs £25 then you could price a single bottle by calculating £25 6 = £4.166666667. It would be pointless to write out all the numbers on your calculator display. Since we are dealing with money (pounds and pence) we only need 2 decimal places (2 d.p.) So it would be much better to write down £4.17. Rounding to 1 d.p 4.8325 F 5 or bigger ? 4. 8 4 2 5 5 or bigger ? No 4. 8 5 2 5 5 or bigger ? No Yes 4.8 4.8 4.9 4. 8 6 2 5 4. 8 7 2 5 4. 8 9 2 5 5 or bigger ? 5 or bigger ? Yes 4.9 5 or bigger ? Yes 4.9 Yes 4.9 Rounding to 2 d.p 1. 4 2 6 1 5. 8 4 2 5 F 5 or bigger ? 0. 6 0 8 3 5 or bigger ? No Yes 1.43 5.84 0. 2 9 4 3 5 or bigger ? Yes 0.61 0. 5 5 5 0 0. 3 9 7 0 5 or bigger ? 5 or bigger ? 5 or bigger ? No Yes Yes 0.29 0.56 0.40 Rounding to 3 d.p 1. 4 2 6 1 8 5. 8 4 2 5 4 F 5 or bigger ? 0. 6 0 8 3 4 5 or bigger ? Yes 5 or bigger ? No 5.843 1.426 6. 2 9 4 7 1 5. 4 0 0 9 7 No 0.608 0. 3 9 9 7 7 5 or bigger ? 5 or bigger ? 5 or bigger ? Yes Yes Yes 6.295 5.401 0.400 Take Care! F • Round 3.48 to 1 d.p 3.5 • Round 3.48 to the nearest whole number 3 (not 4) E Significant Figures Example Round E 235440 To 2 significant figures 235440 E Underline the 1st 2 digits The 3 is changed to a 4 If this is 5 or more then you must round up Now look at the next digit 240000 All other digits are changed to zero • What are these numbers to 2 significant figures? 437900 69723 43490 2350 E 440000 70000 43000 2400 What about decimal numbers? E For example: Round 0.004367 to 2 significant figures 0.004367 E Underline the 1st 2 digits which are not zero Look at the next digit along You change the 3 to a 4 0.0044 If it is 5 or more you add 1 to the previous digit You can ignore any number after the 1st 2 digits which are not zeros Round the following to 2 significant figures E 0.05475 0.00475 0. 45475 0.055 0.0048 0.45 Significant figures E • When first identifying significant numbers, zeros at the beginning or end don’t usually count, but zeros ‘inside’ the number do. • Digits of a number kept in place by zeros where necessary. • The rounded answer should be a suitable reflection of the original number e.g. 24,579 to 1 s.f could not possibly be 2 24,579 to 1 s.f is 20,000 Write questions and answers in your books? E 49382.95 to 2 s.f. and 1dp = 49000 49383.0 0.05961 to 1 s.f. and 2dp = 0.06 0.06 374.582 to 3 s.f. and 1dp = 375 374.6 0.0009317 to 2 s.f. and 3dp = 0.00093 0.001 Objective: • Share a quantity into a given ratio. C. • Find an unknown number that fits a given ratio. C. Sharing a quantity into a given ratio For example, share 36 into the ratio 2:7 First ADD the ratio 2 + 7 = 9 Second DIVIDE this answer into the quantity to be shared 36 ÷ 9 = 4 C Third MULTIPLY the ratio by this answer 2 X 4 : 7 X 4 This is the answer 8 : 28 When sharing into a given ratio, the name to remember is: ADaM C A D + ÷ and M X Share 32 into the the ratio 3 : 5 3+5=8 32 ÷ 8 = 4 3 X 4 = 12 : 5 X 4 = 20 Answer 12 : 20 Share these into the given ratio C Ratio 2:3 2:5 3:7 4:5 1:2:5 Quantity 50 28 20 360 32 Share Share these into the given ratio C Ratio 2:3 2:5 3:7 4:5 1:2:5 Quantity Share 20 : 30 50 8 : 20 28 6 : 14 20 160 : 200 360 4 : 8 : 20 32 Finding an unknown number that fits a given ratio Example – If the ratio of red beads black beads is 3 : 5, how many black beads will I need for 21 red beads? C Red : Black 3:5 21 : ? First find the number that you multiply 3 by to get 21 Red : Black C 3:5 X7 21 : ? Red : Black C 3:5 X7 X7 21 : 35 Another Example Red : Black 2:7 C X6 12 : ? 2 X 6 is 12 so you multiply 7 by 6 to get the ? Red : Black 2: 7 C X6 12 : 42 X6 FRACTIONS Top number is the NUMERATOR, bottom number is the DENOMINATOR Find equivalent fractions. F Simplify a fraction to its lowest form. E Add and subtract fractions with common denominator. D Multiply and divide fractions. D Add and subtract fractions with different denominator. C Convert to and from fractions, decimals and percentages. D Be able to convert a recurring decimal to a fraction. C What makes a fraction? Part Umerator D enominator o N Whole n Something we do with Fractions. find EQUIVALENT F ent iv ♫ EQU = AL = ♫ Something we do with Fractions. SIMPLIFY E S im = 8 16 p L = i 4 = 8 = = f y 2 4 = Adding Fractions D 1 4 + 2 4 = 3 4 Subtracting Fractions D 5 2 7 7 3 7 What Happens if • The two bottom numbers are different C 1 1 + 3 4 Find LCM (Lowest Common Multiple) 1 1 + 3 4 Find multiples of 3 and 4 X table shows the multiples of 3 3,6,9,12,15,18,21,…….. C X table shows the multiples of 4 4,8,12,16,20,24,………… 1 1 + 3 4 Change the denominators into 12 4 C 4 1 1 3 4 + 3 + 3 12 12 3 4 4+3 7 12 12 + 1/4 1/3 1/4 + = ? 1/3 1/3 1/4 C + = 3/12 + 4/12 = 7/12 Another example but TAKING AWAY • The two bottom numbers are different C 2 1 3 6 Find LCM (Lowest Common Multiple) 2 1 3 6 Find multiples of 3 and 6 X table shows the multiples of 3 3,6,…….. C X table shows the multiples of 6 6,………… 2 C 2 21 4 1 3 6 6 6 3 6 1 2 1/3 1/3 1/6 1/6 4/6 C 2/ 3 2X2 2X3 - 1/6 = 3/6 = 1/2 1/ - = ? 6 1 6 = 4 6 + 1 6 3 1 = = 6 2 Multiplying Fractions - Method 5 1 4 5 4 54 D 12 12 12 12 3 4 Dividing Fractions - Method • Invert the second fraction and then multiply D 3 5 3 8 24 6 4 8 4 5 20 5 1 1 5 If it’s fractions you’ve got to sum, the first thing to do is check its bum. Add tops together if bums are the same, if they’re not, then it’s a pain. Equal bums is what you need, use times tables, your bums to feed. Take away is the same as add, times and divide are not so bad. For times do the bottom and then the top, divide do the same with the 2nd bottom up C NEILSEAL METHOD FOR COVERTING FPD OUT OF THE RED AND INTO BLACK FRACTION DECIMAL Add up to last place fraction and cancel. FRACTION PERCENTAGE % 100 (cancel if possible) DECIMAL PERCENTAGE Numerator ÷ Denominator (Numerator x 100) Decimal x 100 ÷ Denominator % ÷ 100 . How can we write 0.3 as a fraction. . Let n = 0.3 . So 10n = 3.3 . . So 10n - n = 3.3 – 0.3 B So 9n = 3 So n = 3 = 1 9 = 3 . . How can we write 0.3451 as a fraction. . . Let n = 0.3451 . . So 10000n = 3451.451 .. .. So 10000n - 10n = 3451.451 – 3.451 B So 9990n = 3448 So n = 3448 9990 = 1724 4995