LISTENING GAP FILL SHEET Nº1

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2 LISTENING GAP FILL SHEET Nº1 From: Just stop and think how important are to our lives. Numbers control us. In fact, we are numbers. We have passport numbers, security numbers, ID numbers, and more. We live in a house and a that has a number. We communicate with using numbers. We worry about how small or big the numbers are on our statements. We can t survive without numbers. Most of the world cannot function without the numbers and. These are the two numbers use to run their programmes. I don t think numbers were always so important. Maybe years ago, we only had to remember the number of sheep, goats or children we had. I wonder how numbers we need today!

3 CORRECT THE SPELLING From: Just stop and think how itrnmtoap numbers are to our lives. Numbers onlocrt us. In fact, we are numbers. We have passport numbers, social sieytcur numbers, ID numbers, and more. We live in a house and a street that has a number. We communicate with each other niusg telephone numbers. We yworr about how small or big the numbers are on our akbn statements. We can t rsviveu without numbers. Most of the world cannot unointfc without the numbers zero and one. These are the two numbers computers use to run their programmes. I don t think numbers were always so important. Maybe 500 years ago, we only had to rmemeber the number of sheep, goats or children we had. I worden how many numbers we need today! More free lessons at listenaminute.com - Copyright 010

4 UNJUMBLE THE WORDS From: stop Just important how think and numbers are to our lives. Numbers control us. numbers are we, fact In. We have passport numbers, social security numbers, ID numbers, and more. house a in live We street a and that has a number. We communicate with each other using telephone numbers. We or small how about worry big the numbers are on our bank statements. We can t survive without numbers. the of Most function cannot world without the numbers zero and one. These are the two numbers to run their programmes computers use. I don t important think numbers were always so. Maybe 500 years ago, remember the we only had to number of sheep, goats or children we had. I wonder how many numbers we need today! More free lessons at listenaminute.com - Copyright 010 3

5 DEPARTMENT OF MATHEMATICS I.E.S ZURBARÁN (NAVALMORAL) NAME DATE SHEET Nº WORDSEARCH: NUMBERS M T U O L Y P E X Z U G R M R U I U K V K Q L W R R J E L O L R R G P C F N U E I Q B T S T S M C Y Q K P A S K I M I I I E J A D D I T I O N M U L V P Y L C U F E B B V U R N E I L S F D F R V A G L E H N S D I N O I T C A R T B U S E S P E E V H W F O I M M Q S V T G D Y A X C T P U S Y V O E H T B N T C L L N M I N U S C A H Y A W A E D E D I V I D I N D Y V U H D D M P A W Q M X F F E Q R O M X B I P I O U U P S E X C X C X R Q B A T J Y D L ADDITION DIVIDED BY DIVISOR EQUAL TO EVEN NUMBER GREATER THAN LESS THAN MINUS MULTIPLE MULTIPLIED BY ODD NUMBER PLUS SUBTRACTION

6 DEPARTMENT OF MATHEMATICS I.E.S ZURBARÁN (NAVALMORAL) NAME DATE SHEET Nº I. Write the number names in words: WRITING BIG NUMBERS (i) 93 (ii) 4905 (iii) 39,001 (iv) 703,090 (v),193,015 (vi) 73,869,51 (vii) 337,16,647 (viii) 109,511,003 (ix),010,951,03 (x) 8,996,783,154 (xi),917,509,735 (xii) 91,330,000,501 (xiii) 609,071,008,09 (xiv) 909,999,909,990

7 DEPARTMENT OF MATHEMATICS I.E.S ZURBARÁN (NAVALMORAL) II. Write the numbers in numerals: (i) Ninety-nine (ii) Six hundred thirteen (iii) Nine thousand nine (iv) Sixty-one thousand seven hundred eight (v) Nine hundred eighty-one thousand five hundred sixty-one (vi) Seven million eight hundred six thousand five hundred eighty-five (vii) Nineteen million eight hundred sixty-nine thousand five hundred twenty-one (viii) Nine hundred million seven hundred ninety-three thousand fourteen (ix) Five hundred eleven million one hundred nine thousand nine hundred ninety-seven (x) Four billion seven hundred twenty-three million one hundred thirty-three thousand four hundred seventy-eight (xi) Three hundred seventy-four million five hundred twenty-six thousand four hundred thirty-eight (xii) Ten billion two hundred eighty-eight million three hundred forty-seven thousand eighty-two (xiii) Nineteen billion seven hundred ninety-two million six hundred sixteen thousand three hundred eleven (xiv) Two hundred fifty-nine billion one hundred eighty-six million seven hundred fifty-one thousand three hundred twenty-seven

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12 UNIT 0 SAYING NUMBERS REMEMBER THAT: IN DECIMAL NUMBERS: WE USE A COMMA ENGLISH PEOPLE USE A POINT Saying numbers : The figure 0 is usually called nought before a comma and oh after the comma. Example : 0,6004 :nought comma six oh oh two four and is used before the last two digits Example: 315 Three hundred and fifteen 803 Eight hundred and three CARDINALS ORDINALS 1 One First (1 st ) Two Second ( nd ) 3 Three Third ( 3 rd ) 4 Four Fourth (4 th ) 5 Five Fifth (5 th ) 6 Six Sixth (6 th ) 7 Seven Seventh (7 th ) 8 Eight Eighth (8 th ) 9 Nine Ninth (9 th ) 10 Ten Tenth (10 th ) 11 Eleven Eleventh ( 11 th ) 1 Twelve Twelfth (1 th ) 13 Thirteen Thirteenth ( 13 th ) 14 Fourteen Fourteenth ( 14 th ) 15 Fifteen Fifteenth ( 15 th ) 16 Sixteen Sixteenth ( 16 th ) 17 Seventeen Seventeenth ( 17 th ) 18 Eighteen Eighteenth ( 18 th ) 19 Nineteen Nineteenth ( 19 th ) 0 Twenty Twentieth (0 th ) 1 Twenty-one Twenty-first (1 th ) Twenty-two Twenty-second ( th ) 3 Twenty-three Twenty-third ( 3 th ) 4 Twenty-four Twenty-fourth ( 4 th ) 5 Twenty-five Twenty-fifth ( 5 th ) 6 Twenty-six Twenty-sixth ( 6 th ) 7 Twenty-seven Twenty-seventh ( 7 th ) 8 Twenty-eight Twenty-eighth ( 8 th ) 9 Twenty-nine Twenty-ninth (9 th ) 30 Thirty Thirtieth ( 30 th ) 40 Forty Fortieth ( 40 th ) 50 Fifty Fiftieth ( 50 th ) 60 Sixty Sixtieth (60 th ) 70 Seventy Seventieth (70 th ) 80 Eighty Eightieth ( 80 th ) 90 Ninety Ninetieth ( 90 th ) 100 One hundred Hundredth 1000 One thousand Thousandth One hundred thousand Hundred thousandth One million Millionth

13 1. Write with Roman numbers: WORKSHEET L E S S O N a).345 = b) 749 = c) 5.78 =. Write with decimal numbers: a) MMCDXII = b) VICCCXXIV = c) DCCXLI = 3. How do you read these numbers? a) = b) = c) = 4. Partition these numbers: d) 1.35 = e) 49 = f) 1.449= d) XXIV = e) MMMXCV = f) CDXXXVIII = d) = e) 903 = f) = 1 a) = 3 Hundred Thousands + 4 Ten Thousands + 1 Thousands + Hundreds + 0 Tens + 8 Units = b) = c) 75.86= 5. Round to the nearest hundred: a).586 b) c) 7.47 d) Calculate: a) ( ) 19 = b) 3 : (16 4 3) + 1 = c) [ ( ) -1] + 19 = d) : 3= e) : + 76 = f) = 9

14 L E S S O N 1 WORD PROBLEMS 1. Peter buys a farm for $ and after selling it he makes a profit of $ How much does he sell it for?. With the money that I have now and $47 more, I can pay a debt of $55 and I still have $37. How much money do I have? 3. How many years are 6.05 days? 4. An airport has a plane landing every 10 minutes. How many planes land in one day? 5. There are 4500 inhabitants in a village, and there is a tree for every 90 inhabitants. How many trees are there in the village? How many trees have to be planted to have a tree for every 1 people? 6. Ann has 1187 in the bank, she spends 385 on a coat and 163 on a dress, how much money does she have in the bank now? Vocabulary Farm granja Sell vender Profit beneficio Debt deuda Land aterrizar Inhabitant habitante Buy comprar Spend gastar 10

15 UNIT 1 NATURAL NUMBERS 1.4. Decimal numeral system Several numeral systems have been developed over the course of history. Eventually we arrived at the Decimal system (its base is 10) which we use today. This system is a positional system based on ten digits: 0, 1,, 3, 4, 5, 6, 7, 8 and 9. It means that the position of each digit represents its value as the following example shows. Example 1: How do you write the number ?. Look at the following diagram: Million Hundred thousands Ten thousands Thousands Hundreds Tens Units Therefore the number is: six million eight hundred thousand thirty-one thousand seven hundred and forty-two. Example : Determine the value of each digit in the number ten thousands thousands 3 hundreds 0 tens. Representation and Order of the Natural numbers The Natural numbers can be represented by a line like that: 9 units Numbers are bigger when you move to the right on the number line. We can use symbols < and > to express which is the order relation between two numbers: less than and more than. Example: Order the following pairs of numbers: 3 and 9; 4 and 5; 9 and 8. We can write: 3 < 9 4 < 5 9 > 8 3 is smaller than 9 9 is greater than 8 One tip: If you have any doubts then check the correspondent position in the number line:

16 3. Operations with Natural numbers NATURAL NUMBERS UNIT 1 Many situations in real life involve operations with numbers so it is useful to practise them. We strongly recommend you practise them mentally Addition and subtraction The addition is an operation that combines numbers to get a total. The symbol is + Example: = 7 plus addends sum Remember the properties of the addition : Commutative: a + b = b + a Example: = Associative:( a + b )+ c = a +( b + c ) Example: ( +3) + 5 = + (3+5) Remember: solve the parenthesis first Example: Check the Associative property in the example above. ( +3) + 5 = + (3+5) = = 10 Then the property is correct The subtraction is an operation that takes one number away from another to get the difference. The symbol is minus Example: 4 17 = 5 minuend subtrahend difference The subtraction is NOT Commutative Example:

17 UNIT 1 NATURAL NUMBERS 3.. Multiplication and division Multiplying natural numbers is repeating additions. The symbol is You read five multiplied by four Example: 5 4 = = 0 factors product Remember the properties of the multiplication:: Commutative: a b = b a Example: 4 1 = 1 4 Associative:( a b ) c = a ( b c ) Example: ( 3) 5 = (3 5) Distributive: a (b + c )= a b + a c Example: (6 + 3)= a (b - c )= a b - a c (6-3)= 6-3 Dividing natural numbers is determining how many times one quantity is contained in another. We use the division to make groups or to share something. The symbol is : You read fifteen divided by three There are two kinds of divisions: Example: 15 : 3 = 5 dividend divisor quotient YOU CAN T DIVIDE BY ZERO EXACT DIVISION INEXACT DIVISION remainder REMEMBER THE PROOF OF DIVISIONS: 8 DIVIDEND = DIVISOR QUOTIENT + REMAINDER

18 NATURAL NUMBERS UNIT 1 5. Estimating Natural numbers Estimating a Natural number is substituting it by another natural number which is close to that number. There are many ways of estimating numbers. Rounding and Truncating are two of the ways of estimating numbers Rounding To Round a Natural number to a specific place value we look at the next digit to the right and: a) if the digit is 5 or more add one unit to the last digit and substitute the rest of the digits by zeros. For instance : Round off 475 to the nearest hundred 4800 b) if the digit is 4 or less simply write the number and add the correspondent zeros. For instance: Round off 7143 to the nearest thousand Example 1: Round off to the nearest ten thousand Example : Round off 8631 to the nearest hundred. 5.. Truncating To Truncate a number to a specific place value we simply write the number to the specified place value and drop all the remaining digits and substitute them by zeros. For instance: Truncate 56 to the ten 50 Example 1: Truncate to the nearest thousand Example 1: Truncate 8179 to the nearest ten

19 DEPARTMENT OF MATHEMATICS I.E.S. ZURBARÁN NAME: DATE: SHEET Nº: EXERCISES OF ESTIMATION 1. Round the following numbers to the specified place value: a. 134 to the nearest ten b to the nearest hundred c to the nearest ten thousand d. 400 to the nearest hundred e. 13 to the nearest ten f to the nearest thousand. Truncate the following numbers to the specified place value and compare the results with the previous exercise: a. 134 to the nearest ten b to the nearest hundred c to the nearest ten thousand d. 400 to the nearest hundred e. 13 to the nearest ten f to the nearest thousand.

20 DEPARTMENT OF MATHEMATICS I.E.S. ZURBARÁN NAME: DATE: SHEET Nº: LISTENING ABOUT BIG NUMBERS 1. Write IN NUMBERS the numbers you listen in the audio: a. b. c. d. e. f. g. h. i. k. l. m. n. o. p. q. r. s. j.. Now write IN WORDS the previous numbers.

21 I.E.S. Andrés de Vandelvira - Sección Europea Mathematics Powers 1 Power times It is a number obtained by multiplying a number by itself a certain number of For example 3 4 = 3x3x3x3 = 81 How to name powers. 5 6 Is read as - Six to the fifth power - Six to the power of five - Six powered to five. The most common is six to the power of five 6 is the base 5 is the index or exponent Especial cases: Squares and cubes (powers of two and three) 3 is read as: - Three to the second power - Three squared - Three to the power of two. - Three to the square power The most common is Three squared 3 5 Is read as: - Five cubed - Five to the third power - Five to the power of three. The most common is Five cubed 1

22 I.E.S. Andrés de Vandelvira - Sección Europea Mathematics Exercises 1. Calculate mentally and write in words the following powers: 3 a) 4 = 4 b) 5 = c) 11 = 5 d) = 3 e) 5 = f) 100 = 3 g) 10 =. Match the following numbers to their squares Describe the pattern formed by the last digit of any square number. - Are there any numbers that do not appear as the last digits? - Could 413 be a square number? 4. Match the following numbers to their cubes

23 I.E.S. Andrés de Vandelvira - Sección Europea Mathematics 5. Find and read the whole expression: 3 a) = 3 3 b) 6 = c) = 600 d) = e) 1000 = Fill in the missing numbers. a) [ ] 5 x 5x5x5xx5 = 5 b) [ ] 8 x 8x8x8 = 8 c) = [ ] 10 d) 81 = [ ] 4 e) 16 = [ ] f) [ ] 16 = 7. Write a list with the squares of all the whole numbers from 1 to 1 8. Write a list with the cubes of all the whole numbers from 1 to You can build up a pattern using square tiles. Shape 1 Shape Shape 3 a) Draw the next two shapes in the pattern. b) Count the numbers of tiles in each shape and put your results in a table 3

24 I.E.S. Andrés de Vandelvira - Sección Europea Mathematics Shape number Tiles c) How many tiles would be in: 1. Shape 6. Shape 9 3. Shape 15 d) Without drawing it, explain how to know the number of tiles when you know the number of the shape. 10. This pattern is built up using square tiles Shape 1 Shape Shape 3 e) Draw the next two shapes in the pattern. 4

25 I.E.S. Andrés de Vandelvira - Sección Europea Mathematics f) Count the numbers of tiles in each shape and put your results in a table Shape number Tiles g) How many tiles would be in: 1. Shape 5. Shape 7 3. Shape 10 h) Without drawing it, explain how to know the number of tiles when you know the number of the shape. 11. Some numbers are equal to the sum of two squares, for example = 34. Which numbers smaller than 100 are equal to the sum of two squares? way? How many of them are equal to the sum of two squares in more than one 4 Big numbers can be written using powers of 10, for example = 7x10, = 1.3x10 this form of writing numbers is called standard form. 5

26 I.E.S. Andrés de Vandelvira - Sección Europea Mathematics 1. Express in standard form the following numbers: a) 4,000,000,000 b) A billion c) 31,650,000 (round to the million) d) The length of the earth meridian in metres. e) The number of seconds in a year (round appropriately) 13. Write as ordinary numbers a) d) 5 3.4x 10 b) 7.6x 10 e) 0.05x 10 c) x 10 f).473x x 10 8 Operations with powers To manipulate expressions with powers we use some rules that are called laws of powers or laws of indices..1 Multiplication: When powers with the same base are multiplied, the base remains unchanged and the exponents are added Example: 8 ( 7x7x7x7x7) x( 7x7 7) x 7 = x = So x 7 = 7 8 Exercise 14. Fill in the missing numbers. 3 4 a) ( ) ( ) [ ] 3 3 = = = 3 b) [ ] 7 = 7 c) 6 7 [ ] 6 = 6 d) 5 [ ] = 6 6

27 I.E.S. Andrés de Vandelvira - Sección Europea Mathematics 3 4 = f) e) [ ] [ ] 7 5 [ ] 6 = 7 g) [ ] [ ] 9 =. Division: When powers with the same base are divided, the base remains unchanged and the exponents are subtracted. Example: So : 6 = 6 3 Exercise 15. Fill in the missing numbers. a) 7 5 : 7 [ ] = 7 b) 1 13 :1 7 [ ] 8 5 = 1 c) 3 : [ ] = 3 3 d) [ ] [ ] 5 : [ ] = 5 e) 3 : [ ] = 3 f) [ ] : [ ] = 9.3 Power of a power: The exponents or indices must be multiplied Example: x5 15 ( ) = ( ) x ( ) x( ) x( ) x( ) = = = 5 = 3 15 So ( ) Exercise 16. Fill in the missing numbers. a) ( ) [ ] 4 5 = 4 b) ( ) [ ] = c) ( [ ] 8 3 ) = ( ) 5 d) [ ] = e) ( ) [ ] = [ ] 8.4 Powers with different base but the same exponent Multiplication: When powers with the same exponent are multiplied, multiply the bases and keep the same exponent. 7

28 I.E.S. Andrés de Vandelvira - Sección Europea Mathematics Example: 5 5 ( 7) x 7 = x = Division: When powers with the same exponent are divided, bases are divided and the exponent remains unchanged. Example: 3 3 ( 8 : ) : = =, we can also use fractions notation = 3 = 4 3 Exercise 17. Fill in the missing numbers. a) = [ ] 7 b) [ ] [ ] = 6 c) 5 [ ] = 15 [ ] 5 5 d) [ ] = 14 e) 8 : 4 = [ ] 5 f) [ ] 5 16 g) = [ ] 7 6 h) [ ] [ ] Exercise 18. Operate = = 5 a) 7 3 [ ] = 7 b) = 7 c) = 7 3 d) : 5 = e) ( ): = f) ( 14 ) = 7 3 g) 3 : ( 3 3 ) = h) ( 3 ) = [ ] 3 i) ( 1 1 ) = j) ( 6 : 3 ) = [ ] 8 3 [ ] k) [ 3 : ( 3 3 )] = 3 3 Square roots. The inverse operation of power is root l) = 4 5 The inverse of a square is a square root, that is: If we say that 9 = 3, that means that 3 = Calculate the following square roots: a) 81 = b) 11 = c) 100 = d) = e) 900 = f) 1600 = g) = 4 4 8

29 I.E.S. Andrés de Vandelvira - Sección Europea Mathematics There are numbers that are not squares of any other number, for example between 9 and 16 (squares of 3 and 4); there is not any whole square number. The square root of all numbers between 9 and 16 are between 3 and 4, for example 3 < 13 < 4, this is an estimation of the 13 value. Exercise 0. Estimate the value of the following square roots: a) < 57 < b) < 50 < c) < 700 < d) < 1500 < e) < 30 < f) < 057 < When we estimate a square root as 3 < 13 < 4 we can also say that 13 is 3 and the difference of 13 and 9 (square of 3), which is 4, is called the remainder. So we say that the square root of 13 is 9 and the remainder is 4. That means 13 = Exercise 1. Calculate the square roots and the remainders for the numbers of the previous exercise; write the meanings as in the example. 9

30 I.E.S. Andrés de Vandelvira - Sección Europea Mathematics Solutions 1. a) 4 3 = 64 four cubed is sixty four; b) 5 4 = 65 five to the power of four equals six hundred and twenty five; c) 11 = 11 eleven squared is one hundred and twenty one; d) 5 = 3 five to the power of five is/are thirty two; e) 5 3 = 15 five cubed is one hundred and twenty five; f) 100 = one hundred squared is ten thousand; g) 10 3 = 1000 ten cubed is one thousand. (169, 13 ); (196, 14 ); (5, 5 ); (81, 9 ); (400, 0 ); (10000, 100 ) 3. The last digit of any square number can be 0, 1,4, 5, 6 or 9; 413 can not be a square number 4. (15, 3 5 ); (8000, 3 0 ); (1, 3 1 ); (64, 3 4 );(1000, 3 10 ); (7, 3 3 ) 5. a) 3 3 = 7; b) 6 3 = 16; c) 600 = ; d) 7 = 18 ; e) = a) e) 16 = [ 4] ; f) [ 5] 5 x5x5x5xx5 = 5 ; b) [ 4] 16 = [ 4] [ 7] 8 x8x8x8 = 8 ; c) = 10 ; d) 81= [ 3] = 1; = 4 ; 3 = 9 ; 4 = 16 ; 5 = 5; 6 = 36; 7 = 49 ; 8 = 64; 9 = 81; 10 = 100 ; 11 = 11; 1 = = 1; 3 = 8 ; 3 3 = 7; 4 3 = 64 ; 5 3 = 15 ; 6 3 = 16; 7 3 = 343 ; 8 3 = 51 ; 93 = 79 ; 10 3 = a) b) Shape 4 Shape 5 Shape number Tiles Shape 6 = 36;. Shape 9 = 81; 3. Shape 15 = 5 c) The number of tiles is the square of the number of the shape 10

31 I.E.S. Andrés de Vandelvira - Sección Europea Mathematics 10. a) b) Shape number c) 1. Shape 5 = 15;. Shape 7 = 343; 3. Shape 10 = 1000 d) The number of tiles is the cube of the number of the shape 11. 1, 5, 8, 10, 13, 17, 18, 0, 5, 6, 9, 3, 34, 37, 40, 41, 45, 50, 5, 53, 58, 61, 65, 67, 7, 73, 74, 80, 8, 85, 89, 90, 9, 97, = = 50; = = 65 ; = + 9 = a) ; b) ; c) ; d) e) a) 3.4x10 5 = 340, 000 ; b) 0.05x10 = 5; c).473x10 8 = 47,300, 000 d) 7.6x10 = 76 ; e) 7.006x10 7 = 70,060, 000; f) 9x10 1 = 9,000,000,000, a) 15. a) [ 7] 3 ; b) [ 3] 7 ; b) Tiles [ 13] 7 ; c) [ 9 ] 6 ; d) [ 6] 1 ; c) [ 3 ] 3 ; d) 4 6 ; e) [ ] 3 [ ] 4 ; f) 7 [ 1] ; g) [ ] [ ] [ 9] 5 : [ 5] ; e) [ 3 ] [ 3] ; f) [ 9 ] : [ 9] = a) [ 10] 4 ; b) ( ) [ 4] 3 ; c) [ 4] ( ) ( ) 3 3 ; d) [ ] 5 ; e) ( ) [ ] = [ ] a) [ 4 ] 7 ; b) [ 3] [ ] ; c) 5 [ 3] [ ] ; d) [ 7] = 14 ; e) [ 3 ] 5 ; f) [ ] 5 g) [ ] 7 ; h) [ ] [ 1] 18. a) j) [ ] ; k) [ 8] 7 ; b) [ 6] 3 ; l) 1 4 ; c) ; d) 4 5 ; e) 5 ; f) 8 14 ; g) 19. a) 9 ; b) 11; c) 10 ; d) 100 ; e) 30 ; f) 40 ; g) ; h) [ 6 ] ; i) 0 1 ; 0. a) 7 < 57 < 8 ; b) 15 < 50 < 16 ; c) 6 < 700 < 7 ; d) 38 < 1500 < 39 ; e) 5 < 30 < 6 ; f) 45 < 057 < a) 57 = ; b) 50 = ; c) 700 = ; d) 1500 = ; e) 30 = ; f) 057 = ; 11

32 DEPARTMENT OF MATHEMATICS I.E.S. ZURBARÁN NAME: DATE: SHEET Nº: A READING ABOUT ROOTS The Rhind Mathematical Papyrus is a copy from 1650 BC of an even earlier work and shows how the Egyptians extracted square roots. In the Chinese mathematicas work Writings on Reckoning, written between 0 BC and 186 BC during the early Han Dynasty, the square root is approximated by using an excess and deficiency method [ ] Mahavira, a 9 th -century Indian mathematician, was the first to state that square roots of negative numbers do not exist. According to historian of mathematics D.E. Smith, Aryabhata s method for finding the square root was first introduced in Europe by Cataneo in A symbol for square roots, written as an elaborate R, was invented by Regiomontanus ( ). An R was also used for Radix to indicate square roots in Giralamo Cardano s Ars Magna. The symbor for the square root was first used in print in 155 in Cristoph Rudolff s Coss, which was also the first to use the thennew signs + and -.

33 DEPARTMENT OF MATHEMATICS I.E.S. ZURBARÁN Answer the following questions about the text: 1. Is the Rhind Mathematical Papyrus an original work? Why?. What does BC mean in the text? 3. Who gave a method to find a square root of large numbers? 4. What do you think that [ ] mean in the text? 5. Name two mathematicians whose studies about the square root were very important. 6. How old was Regiomontanus when he died? A LISTENING ABOUT POWERS Fill in the gaps:

34 DEPARTMENT OF MATHEMATICS I.E.S. ZURBARÁN NAME: DATE: SHEET Nº: ACTIVITY 1: RUNNING DICTATION DIVISIBILITY MULTIPLES AND FACTORS We say that a number A is a multiple of another number B if the division A : B is an exact division. That is, if B contains A a whole number of times. In this case we can also say that B is a factor or A. PRIME AND COMPOSITE NUMBERS A prime number only has two factors: the number one and itself. For example: 3, 5, 11, 17, etc. A composite number has more than two factors. For example: 4, 9, 15, 30, etc. A smart procedure to find the first prime numbers is the Sieve of Erathostenes. It consists of a table with the numbers from 1 to 100, and now do the following rules: Number is prime. Circle it, then cross out all the multiples of Circle the next number that is not crossed out (3) because it is prime too. And then, cross out all its multiples. Continue in this way, that is, circle the numbers which are not crossed out and cross out all its multiples until you finish with the table. Then you will have got the first prime numbers lower than 100.

35 DEPARTMENT OF MATHEMATICS I.E.S. ZURBARÁN ACTIVITY : READING Say if the following statements are TRUE or FALSE and explain why: 1. A is a multiple of B if you do the division and the remainder is not zero.. You can always say that if A is a multiple of B, then B is a multiple of A. 3. The opposite of a prime number is a composite number. 4. A prime number has got only two divisors: zero and one. 5. The number 73 is a prime number. 6. One way to find the first prime numbers is to use she Pythagoras Theorem. 7. The Sieve of Erathostenes consists of a table where you round all even numbers. 8. One of the rules of the Sieve of Erathostenes is to circle number 4 and cross out all its multiples. 9. Then you continue doing the same with 5, 6, 7 and so on, no matter if they are crossed out or not. 10. The goal of the Sieve of Erathostenes is to find the first prime number bigger than one hundred.

36 DEPARTMENT OF MATHEMATICS I.E.S. ZURBARÁN NAME: DATE: SHEET Nº: 11 DIVISIBILITY CRISS-CROSS Across 4. A number is divisible by 9 if the of its digits is a multiple of 9 7. A number is divisible by 3 when the of its digits is a multiple of 3 8. A number is divisible by 8 if its last digits are a multiple of 8 1. It is the relationship between two numbers when one is a factor of the other 13. A number is divisible by 6 when it is divisible by and A number is divisible by 5 when it ends in or The Common Multiple is the smallest number that is a multiple of both numbers Down 1. It is a number that results of multiplying one number by another one. The prime is the procedure to find all the prime numbers that are factors of a composite number 3. A number is divisible by 4 when the last two are a multiple of 4 5. It is a number that divides exactly another number 6. It is a number that has got more than two factors 9. A number is divisible by when it is an number 10. It is a number that only has two factors: 1 and itself 11. The Common Factor is the biggest number that is a factor of both numbers

37 DEPARTMENT OF MATHEMATICS I.E.S. ZURBARÁN Try to fill in the missing numbers. Use the numbers 1 through 9 to complete the equations. Each number is only used once. Each row is a math equation. Each column is a math equation. Remember that multiplication and division are performed before addition and subtraction. CROSSWORD N Q L E W S W Q B B H G Q I W S G B L G P R O C E D U R E W X T K P R I M E C N I F B C S K Y T I L I B I S I V I D T L F A C T O R I Z A T I O N O H W B G L W C B Y I S S X N N M T T V U E T S E H G I H B Y S Q R C M S O L O V M B L P R B Q M C Y T A R Y R U L E M E X E A C O M P O S I T E A C Q B COMPOSITE DIVISIBILITY DIVISIBLE FACTOR FACTORIZATION HIGHEST LOWEST MULTIPLE PRIME PROCEDURE RULE

38 DEPARTMENT OF MATHEMATICS I.E.S. ZURBARÁN NAME: DATE: SHEET Nº: QUIZLET.COM: DIVISIBILITY RULES AND VOCABULARY 1. Open a google window and write: Click on. Write on your notebook what you read in the screen, until you finish the exercise with the 1 terms. 3. Click on. Complete the exercise on-line using the definitions you have in your notebook. 4. Write on your notebook what you hear when your teacher plays the audio. 5. Click on. Answer the test on line and check yourself. 6. Click on. Play the game by matching concepts and definitions to make them disappear. 7. Click on. Play the game by writing the definitions you have learnt.

39 DEPARTMENT OF MATHEMATICS I.E.S. ZURBARÁN NAME: DATE: SHEET Nº: GREATEST COMMON FACTOR AND LEAST COMMON MULTIPLE WORD PROBLEMS

40 DEPARTMENT OF MATHEMATICS I.E.S ZURBARÁN (NAVALMORAL) NAME...DATE...SHEET Nº... INTEGER NUMBERS (WHOLE NUMBERS) The INTEGERS are the natural numbers, their negatives, and zero:, 4, 3,, 1, 0, 1,, 3, 4, The positive integers are the natural numbers. The negative integers are the minus natural numbers: 1,, 3, 4, 5, Negative numbers Many years ago, you probably learnt to count by beginning at 1 and going on from there. When you were a little older, you probably learnt about the number zero. Now you are older still, you will need to know about numbers below zero. These are called negative numbers. Ordering negative numbers Any number above zero is a positive number. Positive numbers are written with no sign or a '+' sign in front of them and they are counted from zero to the right. Any number below zero is a negative number. Negative numbers are always written with a '-' sign in front of them and they are counted from zero to the left. Always look at the sign BEFORE the number to see if it is positive or negative. Adding and subtracting negative numbers To add and subtract numbers always begin counting from 0: When with positive numbers count to the right. When with negative numbers count to the left. EXAMPLE: Calculate Start from zero. Add 4. Then subtract 5. Then subtract 3. The answer is 4. Multiplying and dividing integers. Rules: Multiplying... The product of two numbers with the same signs is positive. The product of two numbers with different signs is negative. Dividing... The quotient of two numbers with the same sign is positive. The quotient of two numbers with different signs is negative = +. + = +. =. = + + : + = + : + = + : = : = +

41 DEPARTMENT OF MATHEMATICS I.E.S ZURBARÁN (NAVALMORAL) EXERCISE 1: Which word, 'bigger' or 'smaller', would fit correctly in each of these gaps? a) -7 is... than -1 b) + 6 is... than -71 c) -136 is... than -36 EXERCISE : Order the following numbers: EXERCISE 3: Work out the indicated operations (do them in columns!) a = b = c. 11 ( ) = d. ( 8) : (+4) = e. (+15) : (+3)= f. (-6) : ( )= g. (+0) : ( 10) = h. 4 ( ) + ( ) ( 3) (+ 5) ( 4) 8 ( 3) = i. ( 5) [(+ 5) + (+ ) ( )] = j. 6 8 k l. 160 : 40 m. 00 : 5 n. 7 [ ] o. 7 1 [ 5 3 ] =

42 DEPARTMENT OF MATHEMATICS I.E.S ZURBARÁN (NAVALMORAL) EXERCISE 4: Translate and solve the following word problems: 1. An elevator is on the 1 st floor. First, it goes up 6 floors, down, then down 5, up 8 and finally down 9. On which floor is the elevator now on? How can we interpret this result?. An iceberg floats on the maritime zone where the maximum depth is 157 metres and stands out of the water at 17 metres. The distance between the deepest point of the iceberg and the bottom of the sea is 6 metres. What is the total height of the iceberg? Solve in one combined operation with whole numbers. 3. Elvira spends euros every day from Monday to Friday, 5 euros every Saturday and 6 every Sunday. If she receives 5 euros every week, how much money does she save in a year? Consider that there are 5 weeks in a year. Use a combined expression to calculate it. 4. Jaime, Teresa, Lorenzo and Julia take a test of 10 questions and each question has 4 possible answers. For each correct question, they receive 4 points, for each mistake they lose 1 point, and for each question they leave blank, they do not receive a score. The following table shows the results for each person: RIGHT ANSWERS MISTAKES BLANK JAIME TERESA 4 4 LORENZO 3 5 JULIA Calculate their final scores. What is the highest and lowest score that you can get? 5. A swimming pool has 300,000 litres of water. Because of the cracks in the pool, it loses 5 litres of water every hour. After one week, they decide to refill it using a water tap which pours 1 litres of water per minute into the pool for one hour. What volume of water will the pool have now? Use a combined expression to calculate it. 6. A business sells kinds of products, A and B for 15 euros and 1 euros respectively and the cost for each is 5 and 4 euros respectively. The business has sold 50 units of A, and 300 of B. What is the profit of the sale? Express it in one combined operation and then calculate it. 7. According to the National Institute of Statistics, the average cost of a mortgage is 114,990 euros. If a person who has a mortgage of this cost and they pay 950 euros monthly, of which one half goes to interest and the other half to pay off the debt, how much money would they owe after one year? To calculate it, use one combined operation.

43 DEPARTMENT OF MATHEMATICS I.E.S. ZURBARÁN NAME: DATE: SHEET Nº: 15 LISTENING ABOUT FRACTIONS Let's learn fractions! Fill in the gaps with some words you hear from the video: 1. It's easy to count things, like a pizza, a pie, or a donut.. The bottom number is called the. 3. The number is called the numerator. 4. That means that, in a, the denominator would be eight. 5. If you normally eat two of pizza, then two would be the numerator. 6. That would mean that you eat two of a pizza. 7. This pie is divided into pieces, and you eat pieces of it. 8. Let's try one example. 9. When the fraction is one over two, we call that one.

44 DEPARTMENT OF MATHEMATICS I.E.S. ZURBARÁN READING ABOUT FRACTIONS QUESTIONS: 1. When can we add fractions?. Once they are prepared to be added, what do we have to do? 3. What is the meaning of the word of when we are talking about fractions? 4. To multiply fractions, what do we have to do? 5. What is the main difference between adding and multiplying fractions?

45 DEPARTMENT OF MATHEMATICS I.E.S ZURBARÁN (NAVALMORAL) NAME... DATE...SHEET Nº OPERATIONS WITH FRACTIONS a) 1 b) c) : d) e) f) Answer these questions: a. Draw the fractions: OTHER EXERCISES WITH FRACTIONS 3 4 and b. Write as a mixed number. 4 4 c. Represent in the real line: 6 1, 9 and d. Write an equivalent fraction to 7 8 whose numerator is e. Simplify until you have an irreducible fraction: 36 f. Say if these pairs of fractions are or not equivalent and explain why: i. and ii. and g. Order from lowest to highest: 17. A square has a side of m. Find out its area and the perimeter. 6 3, , 4 3. Alberto has got 40 at home. He spends 8 to go to Cirque Du Soleil. How much money has he left? What is the fraction of the money he hasn t spent yet? students passed a Maths test in a classroom, they represent three quarters of the students. How many students are there in the classroom? 5. Laura has spent 1 5 of her money with her friends. She comes back home with 8 euros. a. How much money did she have when she went out home? b. How much money has she spent? 6. A farmer owns 360 hectares of land. He plants potatoes on 10 3 of his land and beans on 6 1 of the remainder. How many hectares are planted with potatoes? How many hectares are planted with beans? How many hectares are left?

46 DEPARTMENT OF MATHEMATICS I.E.S. ZURBARÁN NAME: DATE: Fibonacci Sequence The Fibonacci Sequence is the series of numbers: 0, 1, 1,, 3, 5, 8, 13, 1, 34,... The next number is found by adding up the two numbers before it. The is found by adding the two numbers before it (1+1) Similarly, the 3 is found by adding the two numbers before it (1+), And the 5 is (+3), and so on! Example: the next number in the sequence above is 1+34 = 55 Makes A Spiral When we make squares with those widths, we get a nice spiral: Do you see how the squares fit together? For example 5 and 8 make 13, 8 and 13 make 1, and so on. History Fibonacci was not the first to know about the sequence, it was known in India hundreds of years before! About Fibonacci The Man His real name was Leonardo Pisano Bogollo, and he lived between 1170 and 150 in Italy. "Fibonacci" was his nickname, which roughly means "Son of Bonacci". As well as being famous for the Fibonacci Sequence, he helped spread Hindu-Arabic Numerals (like our present numbers 0,1,,3,4,5,6,7,8,9) through Europe in place of Roman Numerals (I, II, III, IV, V, etc). That has saved us all a lot of trouble! Thank you Leonardo. Fibonacci Day Fibonacci Day is November 3rd, as it has the digits "1, 1,, 3" which is part of the sequence. So next Nov 3 let everyone know!

47 DEPARTMENT OF MATHEMATICS I.E.S. ZURBARÁN Read carefully the previous text and answer the following questions: 1. After 1, 34, what are the next three numbers in Fibonacci Sequence?. What polygon do you need to draw to get this beautiful spiral? 3. How many years did Fibonacci live? 4. What is a nickname? 5. What are Hindu-Arabic Numerals? 6. Why is November, the 3 rd Fibonacci s Day?

48 Key stage mathematics 007 mental mathematics test First name Last name School Total marks Practice question 10 cm % 30% 40% 11 Time: 5 seconds 50% 60% p g m Time: 15 seconds Time: 10 seconds p remainder

49 DEPARTMENT OF MATHEMATICS I.E.S ZURBARÁN (NAVALMORAL) NAME... DATE...SHEET Nº 19 Directly Proportional and Inversely Proportional

50 DEPARTMENT OF MATHEMATICS I.E.S ZURBARÁN (NAVALMORAL) TRANSLATE AND SOLVE THESE PROBLEMS: 1. Share 70 in parts directly proportional to 3, 5 and 6. Share 1080 in parts inversely proportional to 1, 3 and 6 3. Find the missing term in the following proportions: 4. Anne buys 5 pounds of potatoes at the market. If pounds cost $0.80, how much has to pay Anne? 5. A car travels 81 km with 4.5 litres. How far can it go with 0 litres of petrol if it travels at the same speed? 6. You can empty the water in a tank in 00 times with a bucket of 15 litres. How many times can you empty it with a bucket of 5 litres? 7. Thirty-five workers can build a house in 16 days. How many days will 8 workers take to build the same house? 8. If we open 4 taps in a water tank, the tank is empty in 8 minutes. How long does it take you to empty it with only 3 taps? 9. A grandfather shares 450 between his three grandchildren who are 8, 1 and 16 years of age. If he distributes the money in proportion to their ages, how much will each one receive? 10. A gold ingot of 0,340 kg. is worth 14. What is the value of a 30 grams portion?

51 DEPARTMENT OF MATHEMATICS I.E.S ZURBARÁN (NAVALMORAL) NAME...DATE...SHEET Nº PROBLEMS WITH PERCENTAGES

52 DEPARTMENT OF MATHEMATICS I.E.S ZURBARÁN (NAVALMORAL)

53 DEPARTMENT OF MATHEMATICS I.E.S ZURBARÁN (NAVALMORAL) NAME...DATE...SHEET Nº... Polynomials A polynomial looks like this: Polynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term")... so it says "many terms". A polynomial can have: constants (like 3, 0, or ½) variables (like x and y) exponents (like the in y ), but only 0, 1,, 3,... etc are allowed

54 DEPARTMENT OF MATHEMATICS I.E.S ZURBARÁN (NAVALMORAL) What is Special About Polynomials? Because of the strict definition, polynomials are easy to work with. For example we know that: If you add polynomials you get a polynomial If you multiply polynomials you get a polynomial So you can do lots of additions and multiplications, and still have a polynomial as the result. Also, polynomials of one variable are easy to graph, as they have smooth and continuous lines. Degree The degree of a polynomial with only one variable is the largest exponent of that variable. Standard Form The Standard Form for writing a polynomial is to put the terms with the highest degree first. EXERCISES:

55 FUNCTIONS (TABLES AND GRAPHS) SHEET Nº Name: 1. Name the point that has the coordinates. a. (, ) b(-6, ) c.(1,-4) d.(0,-6) e (-4,-). Write the coordinates of each point. a. B b. G c. E d. N e. H 3. In what quadrant is each point located? a. C b. J c. L d. M e. K 4. The following table shows electricity usage of a household at various times. a. At what time was the highest usage? b. Are there times when the usage doesn t vary? What times? c. Between which times does the usage increase the most? 5. Study the following graph and answer the questions: Hours Usage (kwh) a. What was the temperature on the first day? And on the 5 th day? b. On what day was the temperature the highest? c. How much did the temperate rise between the first and the third days?

56 6. Determine whether the graph is a function and explain your reasoning. 7. Suppose a notebook costs 1.50 a. State the formula that shows the relationship between the number of notebooks we buy (x) and the total cost that we pay for (y). Which ones are independent and dependent variables? b. Fill in the table: x (number of notebooks) y (total cost) 1.5 c. Sketch the graph. How is the relationship between the two magnitudes? 8. The following table represents the surface of a square for a given side: l (cm) A(cm ) 4 9 a) Complete the table. b) Which ones are independent and dependent variables. c) Write the formula that represents the relationship between the two variables. d) Sketch the graph. 9. When taking a taxi, we have to pay a fixed fee of.50 and 1 per kilometer. a. State the formula that represents the relationship between the number of kilometers and the final cost. b. If we go over 5 kilometers, how does it cost? c. Plot the graph.

57 DEPARTMENT OF MATHEMATICS I.E.S. ZURBARÁN NAME: DATE: SHEET Nº: LISTENING There are lies, damned lies, and. I like this saying. It means statistics lie even more than the lies. It s true. Statistics can be used to dress up any. Politicians always use statistics to let us know things are better than they really are. Or to show things are much worse. Most governments have that pump out statistics. I actually like statistics. It s interesting to look at the and make comparisons. I also like trying to find in the numbers. My favourite statistics are about. I can look at football stats for hours. The statistics I don t like so much are those about my. There s lots of red ink with these numbers and lots of signs.

58 DEPARTMENT OF MATHEMATICS I.E.S. ZURBARÁN READING Fill in the gaps with the following words: average; axis; coordinate; decreasing; frequency; graphs; minimum; mode; proportional; related; relationships; representation; rows; sectors; times; variable

59 DEPARTMENT OF MATHEMATICS I.E.S. ZURBARÁN NAME: DATE: SHEET Nº: STATISTICAL WORD PROBLEMS 1. The math marks of a class of 5 students in the final exam are following: a. Group the data in a frequency table, relative and in percentage. b. Calculate the mean, the mode, the range and the median. c. Make a pie chart with two data: fail and pass. The weights of eight children are the following: 14, 3, 18, 5, 40, 4, 35 and x kg. If the average weight is 9,5 kg, what is the weight of the missing child? 3. From the study of a group of forty marriage couples by the number of siblings has the next result: a. Group the data in a frequency table. Calculate the relative frequencies and the percentage frequency. b. Make a bar chart. c. Calculate the mean, the mode, the range and the median.

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61 What s the Probabilty? Answer the questions below using a number and a fraction. Circle your answer. A bag contains 4 blue marbles, 3 red marbles, and 1 orange marble. If a marble is taken from the bag at random, what is the probability that the marble will be red? There are 5 girls and 8 boys in a class. If a male student closes his eyes and randomly selects another student to join his group, who is he more likely to select, a girl or a boy? What is the probability that the marble will be blue? What are the chances he will select a girl? Ken is bobbing for apples. The tub contains 8 red, 6 green, and 4 yellow apples. If Ken closes his eyes and bites into the first apple he touches, what color apple is he least likely to bite into? Ming has a pack of gumballs. The pack contains 4 pink, 8 white, green, and 3 blue gumballs. If she draws one at random, what is the probability it will be pink? What is the probability the apple will be green? What is the probability she will pick a blue gumball? Carl has 13 toy cars: 5 are red, are silver, 4 are black, and are blue. If his brother selects 1 car at random, what is the chance it will be silver? Davie has 5 flags in a bag. The bag contains 5 red flags, 10 yellow flags, 7 blue flags, and 3 white flags. If he grabs a flag from the bag at random,what is the probability he will grab a blue flag? What is the probability that his brother will randomly select a black car? What is the probability he will grab a yellow flag? Copyright 013 Education.com LLC All Rights Reserved More worksheets at