Number Cardinal Ordinal

Transcription

1 UNIT 1: INTEGERS Remember how to read numbers: Complete this table: Number Cardinal Ordinal 1 One First (1 st ) 2 Two Second (2 nd ) 3 Three Third (3 rd ) 4 Fourth (4 th ) 5 Fifth 6 Sixth 7 Seventh 8 Eighth 9 Ninth 10 Tenth 11 Eleven Eleventh 12 Twelve Twelfth 13 Thirteen 14 Fourteen 15 Fifteen 16 Sixteen 17 Seventeen 18 Eighteen 19 Nineteen 20 Twenty 21 Twenty-one Twenty-first

2 30 Thirty Thirtieth 40 Forty One hundred Hundredth One thousand Hundred thousandth One million Millionth AND is use before the last to figures (tens and units) of a number. 103: a (or one) hundred and three 325: three hundred and twenty-five 2315: two thousand three hundred and fifteen The words hundred, thousand and million can be used in the singular form with a or one, but not alone. A is more common in an informal style; one is used when we were speaking more precisely. I want to live for a hundred years The journey took exactly one hundred light years I have a thousand euros A is also common in an informal style with measurement-words. A kilo of oranges costs a pound Mix one litre of milk with one kilo of flour... SINGULAR O PRURAL? Number are usually written in singular. Three hundred euros Several thousand light years The plural is only used with dozen, hundred, thousand, million, billion, if they are not modified by another number or expression (for example a few/several). Hundreds of pounds Thousands of light years PHONE NUMBERS Each figures is said separately (25 two five) The figure '0' is called oh (304 three oh four) Pause after groups of 3 or 4 figures two three four, seven eight oh nine two three four, five double seven eight (British English) or two three four, five seven seven eight (American English) 2

3 ZERO, NOUGHT, NIL, LOVE The figure 0 is usually called nought in British English and zero in American English In measurements, 0 is called zero: Water freezes at zero degrees Celsius In team games, zero scores are usually called nil in British English, and zero in American English. In tennis, the word love is used instead of zero (this is derived from French word l'oeuf, because zero can be egg-shaped): Spain three Germany nil (zero) Nadal is winning forty-love 1 BILLION In English, billion usually means a thousand million: Realize that in Spanish billion means PLACE VALUE Every digit in a number represents a different value depending on its position. For example: In 53, 5 represents fifty units In 5234, 5 represents five thousand units This is the place value table we need to write numbers, no matter how big they are: BILLION HUNDRED MILLION CALCULATIONS TEN MILLION MILLION HUNDRED THOUNSAND TEN THOUSAND THOUSAND HUNDRED TEN UNIT ADDITION 2+4=6 SUBTRACTION 8-5=3 MULTIPLICATION 6 5=30 DIVISION 12:3=4 Two and/plus four is/are/equals six Two added to four makes six What's two and four? It's six. Eight minus five is/are/equals three Eight take away five is three Five from eight leaves/is three Six times five is/equals thirty Six fives are thirty Six multiplied by five is/makes thirty (More formal way) Twelve divided by three is/are/equals four Three into twelve goes four times (for smaller calculations) POWERS 6 5 (6 is the base and 5 is the index or exponent) ROOTS 16=4 Six to the power of five Six to the fifth power Six raised to fifth The square root of sixteen is/equals four. SPECIAL POWERS 5 2 : Five squared 4 3 : Four cubed 3

4 ORDER OF OPERATIONS (PEMDAS) 1. Parenthesis 2. Exponents (Powers, Roots) 3. Multiplications and Divisions 4. Additions and Subtractions Remember the sentence: Please, Excuse Me Dear Aunt Sally Activities. 1. Write the following numbers in words as in the example: 3 456: Three thousand four hundred and fifty-six : 765: 237: : : : : 2. Write the following numbers in words as in the example: 3528: Three thousand five hundred and twenty eight : 987: 3 270: : : : : 3. Read the following number out loud:

5 4. Write the missing words. Then, write the answers in numbers and symbols: Ten plus three equals thirteen 10+3=13 Twelve minus six equals Seven times one equals Twenty-five divided by five equals Eight plus four minus nine equals 5. Write the missing numbers. Then, write the answers in words. 3+8 = 11. Three plus eight equals eleven. 3 = 30-5 =13 6: +2 = =30 (3 +3) : 10 =3 6. Write the missing symbols. Then, write the answers in words = = = = = = 9 7. Insert brackets to make the following calculations correct: 5+4 8= = :3= :3= = =37 24:4+2 7=28 240: =8 8. Calculate the following powers mentally and write them in words: 4 3 = 64 Four cubed equals sixty-four 5 4 = 11 2 = 2 5 = 5 3 = 10 3 = = 5

6 9. Calculate mentally and write in words as in the example: 16 = 4 The square root of sixteen is four 81 = 121 = 900 = 1600 = = 10. The words in all the statements are jumbled up. Rewrite them so that they make sense. 1. FIVE EQUALS PLUS FOUR NINE 2. FIFTEEN AND NINE MAKES SIX 3. TAKE THREE SEVEN TEN EQUALS 4. NINE FROM IS ELEVEN TWO 5. TWELVE TIMES THREE IS FOUR 6. TIMES TWO GOES NINE EIGHTEEN INTO 7. FIFTEEN MAKES NINE TO ADDED SIX 11. In each statements, the words are in the correct order, but the letters of each word have been jumbled up. Rewrite each sentence. 1. WOT SLUP THERE SKAME VIEF 2. EVENS DAD ENIN SLAQUE ENTEXIS 3. TIGHE KATE IXS IS WOT 4. VIEF FORM VETLEW IS VENES 5. WOT MITES THERE SQUEAL IXS 6. HERET MISTE OURF SKAME WELVET 7. ENTEROUF SUNIM NEVLEE SI REETH Integers: The first set of number we knew was the set of Natural Numbers (also called whole numbers): N = {0, 1, 2, 3, 4, 5, 6, 7, } There are many situations in which you need to use numbers below zero, one of these is temperature, others are money that you can deposit (positive) or withdraw (negative) in a bank, steps that you can take forwards (positive) or backwards (negative). Positive integers are all the whole numbers greater than zero: 1, 2, 3, 4, 5, Negative Integers are all the opposites of these whole numbers: -1, -2, -3, -4, -5, Integers allow us to count and order below and above zero. The set of all Integers is represented by the letter Z = { -4, -3, -2, -1, 0, 1, 2, 3, 4, } 6

7 The natural numbers are included in the set of Integers. This fact is represented by the symbol ' ' ' '. The Number Line: N Z is read N is a subset of Z. The number line is a line labelled with the integers in increasing order from left to right, that extends in both directions: For any two different places on the number line, the integer on the right is greater than the integer on the left. Examples: 4>-1 is read: four is greater than minus one -3<2 is read: minus three is less than two Opposite of an integer: The opposite of an integer is the same number with the other sign. The distance from a number to zero is the same as the distance from its opposite to zero. The opposite of +5 is -5 The opposite of -7 is +7 Absolute value of an integer: The absolute value of an integer is the number of units is from zero on the number line. If the number is positive, the absolute value is the same number. If the number is negative, the absolute value is the opposite. The absolute value of an integer is always a positive number (or zero). We specify the absolute value of a number n in between two vertical bars: n. Examples: 3 =3 5 =5 4 =4 7 =7 7

8 1. Plot on the number line and after order them from less to great Write the opposite and the absolute value of all these numbers. Adding and Subtracting Integers: Rules for Addition: When adding integers with the same sign: We add their absolute values, and give the result with the same sign. 4 6 = = = 7 When adding integers with the opposite signs: We subtract their absolute values (we subtract the smaller absolute value from the larger), and give the result with the sign of the integer with the larger absolute value. 7 9 = = = 5 Rules for Subtraction: Subtracting an integer is the same as adding the opposite. We convert the subtracted integer to its opposite, and add the two integers: The result of subtracting two integers could be positive or negative. 3 7 = 3 7 = = 2 8 = 6 8

9 You can use a number line to help you to add or subtract integers: Calculate 4-6: Start at 4 and subtract 6 (move 6 units to the left): Calculate -3+7: The answer is -2. Start at -3 and add 7 (move 7 units to the right): The answer is Calculate: a) 5 2 b) 9 13 c) d) 7 13 e) 2 10 f) 4 5 g) 9 6 h) 8 3 i) Calculate using one of the methods of the example: Example: = 1 st Method (Doing the operations in order) = = = 3 5 = nd Method (Grouping positive and negative) = (8+2)+( ) = = - 2 a) b) c) d) e)

10 3. Calculate: (-6) + (5-2) - (-6+1) = 1 st Method (Removing first brackets) = = 2 2 nd Method ( Operating first the expressions into brackets) = (-5) = = 2 a) b) c) d) e) f) g) h) Calculate: a) b) c) d) e) Calculate: a) 6 [3 2 5 ] b) 4 [7 5 1 ] c) 5 [2 6 9 ] d) 3 [ ] 5 1 e) 5 [ ] 3 10 f) 1 2 [ ] 1 g) 1 [ ] Calculate: a) 7 2 [4 3 1 ] b) c) 2 [ ] d) [ ] e) [1 2 3 ] f) [4 3 5 ] [ 4 3 5] 10

11 Multiplying and Dividing Integers: Rules for Multiplication. To multiply a pair of integers: If both numbers have the same sign (positive or negative), their product is the product of their absolute values (their product is positive). If the numbers have opposite signs, their product is the opposite of the product of their absolute values (their product is negative). If a number is 0, the product is 0. Look at the chart below: PRODUCT = = = = 20 To multiply any numbers of integers: 1. Count the number of negative integers in the product. If this numbers is even, the product is positive, but if the number is odd, the product is negative. 2. Take the product of their absolute values. ( If any of the integers in the product is 0, the product is 0) = = = = = = 30 11

12 Rules for Division: To divide a pair of integers the rules are the same than for the product: If both numbers have the same sign (positive or negative), divide the absolute values of the first integer by the absolute value of the second integer (the result is positive). If the number have opposite signs, divide the absolute value of the first integer by the absolute value of the second integer, and give the result a negative sign. Look at the chart below: DIVISION : 3 = 4 12 : 3 = 4 12 : 3 = 4 12 : 3 = 4 1. Calculate: a) 3 4 b) 5 4 c) 10 3 d) 15 : 3 e) 40 : 8 f) 56 : 7 2. Calculate: a) b) c) 18 : 2 : 3 d) 20 : 2 : 1 e) f) g) 36 : 9 2 h) 15 3 : 5 12

13 Powers of Integers: Powers are products of equal factors: a n =a a... a, n times where a is the base and n is the exponent or index. Examples: 4 2 = 4 4 = = = = = = = 27 Sign of the power of an integer: If the base is positive, the sign will be always positive. If the base is negative, the sign will be positive if the exponent is even, and negative if it is odd. Example: 2 1 = = 2 2 = = = = = = = = = 64 Operations with powers: Multiplying powers: You can multiply powers with the same base by adding the exponents. a m a n =a m n Examples: =3 4 7 = = = 2 10 Dividing powers:you can divide powers with the same base by subtracting the exponents. a m :a n =a m n Examples: 3 12 : 3 7 = = :2=2 5 1 =2 4 13

14 Power of a power:you can simplify the power of a power by multiplying the exponents. a m n =a m n Examples: =4 2 5 =4 10 [ 3 4 ] 2 = = 3 8 Multiplying powers with the same exponent:you can multiply powers with the same exponent by multiplying the bases. a b n =a n b n Examples: = =15 4 [ 2 3] 5 = Dividing powers with the same exponent:you can divide powers with the same exponent by dividing the bases. a: b n =a n :b n Examples: 15 4 :3 4 = 15:3 4 =5 4 6:3 5 =6 5 : Express as just one power: a) b) c) d) 6 10 : 6 5 e) 7 6 : 7 2 f) g) [ 3 3 ] 5 h) [ 2 6 ] 2 2. Express as just one power: a) [ 2 2 ] 3 : 2 4 b) : 2 5 c) : d) 8 4 :8 2 :8 2 e) f) 6 4 :[ 2 8 :2 7 3] 3 14

15 Square root: The square root of a number a is another number b whose squared is a. a=b when b 2 =a The number a is called radicand, the symbol is called radical and b is called the squared root of a. The numbers with an exact square root are called perfect squares. Examples: 1=1 because 1 2 =1 4=2 because 2 2 =4 9=3 because 3 2 =9 16=4 because 4 2 =16 25=5 because 5 2 =25 36=6 because 6 2 =36 49=7 because 7 2 =49 64=8 because 8 2 =64 81=9 because 9 2 =81 100=10 because 10 2 = =11 because 11 2 = =12 because 12 2 = =13 because 13 2 = =14 because 14 2 = =15 because 15 2 = =16 because 16 2 = =17 because 17 2 = =18 because 18 2 = =19 because 19 2 = =20 because 20 2 =400 But, be careful! If we are working in the set of integers, a number can have two square roots: Example: 36=±6, because 6 2 =36 and 6 2 =36 100=±10, because 10 2 =100 and 10 2 =100 4, it does not exist because any squared integer is negative. 9, it does not exist. Integer square root: If a radicand is not a perfect square, the square root is not exact. In this case, we talk about integer square root. The integer square root of a number a is the greater number b whose squared is less than a. The remainder of the integer square root is the difference between the radicand a and the squared of the integer root b. Examples: 11 3 Remainder= =11 9= Remainder= =29 25= Remainder= =37 36=1 15

16 Order of operations: Do all operations in brackets (or square brackets) first. Then, do all the exponents (indexes and roots) Then, do multiplications and divisions in the order they appear. Finally, do additions and subtractions in the order they appear. Easy way to remember: Parenthesis Exponents Multiplications Divisions Additions Subtractions This gives you: PEMDAS: Please Excuse My Dear Aunt Sally. 1. Calculate: a) b) 15 : 5 22 : 2 c) d) 14 : 2 6 : 3 e) 50 :[ 6 4 ] f) 3.[ 56 : 8 ] 2. Calculate: a) 15: 3 1 b) 80 :[ 8 2] c) [ 80 : 8 ] 2 d) [9 8 ]:[ 3 4 ] e) : 2 f) 5 40 :

17 3. Calculate: a) b) 17 [ ] c) d) e) f) [ ]: Calculate: a) b) 2 [ ] 2 c) : 1 3 d) e) 2 [ ] f) 5 [ ] 5. Calculate: a) b) c) d) e) 25 9 : [2: 2 ] f)

18 6. Calculate: a) [ ]: 3 b) 3 : 3 12 : 6 3 c) 54 :[ ] d) :6 15: 3 e) 10: f) 1 [ 30 : 5 2] 35: 7 Activities. 1. Mount Everest is feet above sea level. The Dead Sea is feet below sea level. What is the difference of altitude between these two points? 2. The temperature in Chicago was 4º C at two in the afternoon. If the temperature dropped 12º C at midnight, what is the temperature now? 3. A submarine was situated feet below sea level. If it ascends feet, what is its new position? 4. Aristotle was born in 384 B.C. And died 322 B.C. How old was he when he died? 5. A submarine was situated feet below sea level. If it descends 125 feet, what is the new position? 18

19 6. This is the three-day forecast for Yellowknife (Canada) from the 24 th of November Today Tue Wed Nov 24 Nov 25 Nov 26 Snow Cloudy Snow -6º C -7ºC -6ºC -13º C -8ºC -14ºC What is the difference between the maximum and minimum temperatures each day? What are the maximum and minimum temperatures during these three days? 7. This is the three day forecast for Birmingham (UK) from the 24 th of November Today Tue Wed Nov 24 Nov 25 Nov 26 Rain Sunny Partly Cloudy 13º C 14º C 14º C 2º C -2º C -1º C What is the difference between the maximum and minimum temperatures each day? What are the maximum and minimum temperatures during these three days? 8. The Punic wars began in 264 B.C. and ended in 146 B.C. How long did the Punic Wars last? 19

20 9. This a table with the melting and boiling points of some metals Metal Melting Points ºC Boiling Points ºC Aluminium 660, Iron Gold 1064, Mercury -38,83 656,73 a) Calculate the difference between the melting and boiling point of each metal. b) How much warmer is the melting point of mercury than the melting point of iron. 10. On the 2 nd of January, the temperature dropped from 3º C at two o'clock in the afternoon to -11º C at 8 a.m. the next day. How many degrees did the temperature fall? 11. A Greek treasure was buried in the year 164 B.C. and found in 1843 A.D. How long was the treasure hidden? 12. On the 1 st of December, the level of the water in a reservoir was 130 cm above its average level. On the 1 st of July it was 110 cm below its average level. How many cm did the water level drop in this time? 20

21 Divisibility in the set of integers: The multiples of a number are obtained multiplying the number by each integer. Usually, the set of multiples of a number a is written ȧ. Example: Multiples of 2: 2={..., 6, 4, 2, 0, 2, 4, 6,...} The factors of a number are the numbers that divide exactly into it, with no remainder. Example: Factors of 20: {±1,±2,±4,±5,±10,±20} Factors and Multiples are linked: Prime Numbers: 12 is divisible by 3 12 is a multiple of 3 3 is a factor of 12. If a number has only two different factors, 1 and itself, then the number is said to be a prime number. Remember, we have already studied the Sieve of Erastothenes that gives us the list of the prime numbers. It starts as follows: Test of divisibility: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, Divisible by 2 A number is divisible by 2 if the last digit is 0, 2, 4, 6 or 8. Example: is divisible by 2 because the last digit is 6. Divisible by 3 A number is divisible by 3 if the sum of the digits is divisible by 3. Example: is divisible by 3 because the sum of the digits is 21 ( =21), and 21 is divisible by 3. Divisible by 4 A number is divisible by 4 if the number formed by the last two digits is either 00 or divisible by 4. Example: is divisible by 4 because 16 is divisible by 4. 21

22 Divisible by 5 A number is divisible by 5 if the last digit is either 0 or 5. Example: is divisible by 5 because the last digit is 5. Divisible by 6 A number is divisible by 6 if it is divisible by 2 (the last digit is 0, 2, 4, 6 or 8) and it is also divisible by 3 (the sum of the digits is divisible by 3) Example: 534 is divisible by 6 because is divisible by 2 (the last digit is 4) and it is divisible by 3 (the sum of the digits 5+3+4=12 is divisible by 3) Divisible by 10 A number is divisible by 10 if the last digit is 0. Example: is divisible by 10 because the last digit is 0. Divisible by 11 To check if a number is divisible by 11, sum the digits in the odd positions counting from the left (the first, the third, ) and then sum the remainder digits. If the difference between the sums is either 0 or divisible by 11, then so is the original number. Examples: Digits in odd positions: =24 Digits in odd positions: 9+8+9=26 Digits in even positions: =24 Digits in even positions: 1+2+1=4 The difference is 24-24=0 The difference: 26-4=22 So is divisible by 11. So is divisible by 11. There are a simple way of finding the prime factors of a number: = is the prime factorization of the number

23 Highest Common Factor (HCF) or Greatest Common Factor (GCF): Factors that are common to two or more numbers are said to be common factors. Example: Factors of 12 are: 1, 2, 3, 4, 6, 12. Factors of 18 are: 1, 2, 3, 6, 9, 18. So, common factors of 12 and 18 are 1, 2, 3, 6. The largest common factor of two or more numbers is called the highest common factor (HCF). In general, there are two methods for finding the Highest common factor of two or more numbers: Method I (for small numbers): List the factor of each number, and find the common factors. The largest of them is the highest common factor. Example: Calculate HCF (8,12): Factors of 8: 1, 2, 4, 8. Factors of 12: 1, 2, 3, 4, 6, 12. So, HCF(8,12)=4. Method II (general): To find the highest common factor of two or more numbers: Find the prime factorization of each number. Choose the common factor with the lowest exponents. Example: Find HCF (360,300): = = So, HCF (360,300) = =60. 23

24 Lowest Common Multiple (LCM) or Least Common Multiple (LCM): Multiples that are common to two numbers are said to be common multiples. Example: Multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, Multiples of 3 are 3, 6, 9, 12, 15, 18, So, common multiples of 2 and 3 are 6, 12, 18, The smallest common multiple of two or more numbers is called the lowest common multiple (LCM). In general, there are two methods for finding the lowest common multiple of two or more numbers: Method I (for small numbers): List the multiple of the largest number and stop when you find a multiple of the other number. This is the LCM. Example: Calculate LCM (8,3): Multiples of 8 are: 8, 16, 24, 32, 40, Multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, So, LCM (8,3)=24 Method II (general): To find the lowest common multiple (LCM) of two or more numbers: Find the prime factorization of each number. Choose the non common factors and the common factors with the highest exponents. Example: Find LCM (18,24) = =2 3 3 So, LCM (18,24)= =72. 24

25 Activities. 1. Express these numbers as products of their prime factors: a) 48 b) 82 c) Calculate the highest common factor (HCF) of: a) 9 and 24 b) 12, 15 and 18 c) 96 and Find out the lowest common multiple (LCM) of: a) 9 and 24 b) 15 and 40 25

26 c) 20 and 30 d) 48, 54 and Factorise and then calculate the HCF and LCM of these groups of numbers: a) 168 and 490 b) 12, 100 and 6 c) 14 and 15 d) 1600 and 1200 e) 294, 1050 and 28 f) 14112, 1080 and

27 5. Sandra can pack her books in boxes of 5, 6 and 9, without having any book left. She has less than 100 books. How many books has she got? 6. Bus routes A and B start at seven o'clock in the morning from the same point. If the bus A passes the starting point every 24 minutes and the bus every 36 minutes, what time after seven do their departures coincide again? 7. We want to divide a rectangle of 600 cm by 90 cm into equal squares. Find out the length of the biggest square in cm. Calculate how many squares we get. 8. We want to cut two ropes that are 20 and 30 m long into pieces as big as possible and of the same length without wasting everything. What will each piece measure? 9. Iberia has a flight from Madrid to Ankara every 8 days, British Airways one every 12 days and Easy Jet one every 6 days. One day all three have a flight to Ankara. After how many days will the three flights coincide again? 27

28 10. In a cycling track one of the cyclists goes round the circuit every 54 seconds and the other every 72 seconds. They leave the starting line together. a) How long will it take them to meet again at the starting line for the first time? b) How many laps will each cyclist have done in that time? 11. What will the side of a square floor tile measure knowing that it has been used to pave the floor of a garage that is 123 dm long and 90 dm wide? (We have used an exact amount of floor tiles, without cutting any of them). 12. A baker needs to put 250 cakes and 75 biscuits in boxes as big as possible, with the same units per boxes but without missing both products in the same box. How many units will each box contain? How many boxes will he need? 13. A group of students can be organized in lines of 5, 4 and 3 students and there are less than 100. How many students are there? 14. On a Christmas tree, there are two strings of lights, red lights flash every 24 seconds and green lights every 36 seconds. They starts flashing simultaneously when connect the tree. When they flash together again? 28

29 Keywords: Natural Numbers= Números Naturales set= conjunto Integers= Números Enteros Positive Number= Número Positivo Negative Number= Número Negativo to be included= estar incluido (un conjunto en otro) subset= subconjunto Number Line= Recta Numérica opposite=opuesto absolute value=valor absoluto sign= signo addition= adición, suma to add= sumar subtraction= resta, substracción to subtract= restar multiplication= multiplicación to multiply= multiplicar division= división to divide= dividir even number= número par odd number= número impar power= potencia base= base exponent/index= exponente square root= raiz cuadrada radicand= radicando radical= radical (signo) perfect squares= cuadrados perfectos integer square root= raiz cuadrada entera remainder= resto Divisibility= Divisibilidad multiple= múltiplo factor= divisor to be divisible by= ser divisible por prime number= número primo Tests of divisibility= Criterios de divisibilidad Highest Common Factor (HCF) (UK) / Greatest Common Factor (GCF) (USA)= Máximo Común Divisor (m.c.d.) Lowest Common Multiple (LCM) (UK) / Least Common Multiple (LCM) (USA)= Mínimo Común Multiplo (m.c.m.) 29