Working with whole numbers


 Charles Lloyd
 1 years ago
 Views:
Transcription
1 1 CHAPTER 1 Working with whole numbers In this chapter you will revise earlier work on: addition and subtraction without a calculator multiplication and division without a calculator using positive and negative whole numbers (integers) factors and multiples. You will learn how to: decompose integers into prime factors calculate Highest Common Factors (HCFs) and Lowest Common Multiples (LCMs) efficiently. You will also be challenged to: investigate primes. Starter: Four fours Using exactly four fours, and usual mathematical symbols, try to make each whole number from 1 to 100. Here are a few examples to start you off ( ) 5 6 You should try to stick to basic mathematical symbols such as,,, and brackets, wherever possible, but you may need to use more complicated symbols such as and! to make some of the higher numbers. Ask your teacher if you need some help with these symbols.
2 2 Chapter 1: Working with whole numbers 1.1 Addition and subtraction without a calculator You will sometimes need to carry out simple addition and subtraction problems in your head, without a calculator. These examples show you some useful shortcuts. Work out the value of When adding a string of numbers, look for combinations that add together to give a simple answer. Here, and 6 both give exact multiples of 10. Work out the value of Both these numbers are close to exact multiples of 100, so you can work out and then make a small adjustment. Work out is close to 100, so it is convenient to take away 100, then add 2 back on. Harder questions may require the use of pencil and paper methods, and you should already be familiar with these. Remember to make sure that the columns are lined up properly so that each figure takes its correct place value in the calculation.
3 1.1 Addition and subtraction without a calculator 3 Work out So Work from right to left. Add the units: Next, the 10s column: The digit 2 is entered, and the 1 is carried to the next column. Finally, the 100s column: Here are two slightly different ways of setting out a subtraction problem. You should use whichever of these methods you prefer. Work out Method For the units: 7 3 For the 10s: 2 5 cannot be done directly Exchange 10 from the 82 to give 70 and 12. Now and So Method So The first part is the same as method 1. Instead of dropping 82 down to 72, you can make 65 up to 75. Now and 8 7 1
4 Chapter 1: Working with whole numbers EXERCISE 1.1 Work out the answers to these problems in your head Use any written method to work out the answers to these problems. Show your working clearly An aircraft can carry 223 passengers when all the seats are full, but today 57 of the seats are empty. How many passengers are on the aircraft today? 20 The attendances at a theatre show were 75 (Thursday), 677 (Friday) and 723 (Saturday). How many people attended in total? 1.2 Multiplication without a calculator You will sometimes need to carry out simple multiplication problems in your head. This example shows one useful shortcut. Work out the value of is almost 50, so you can work out 50 3 then take off the extra So Harder questions will require pencil and paper methods. Here is a reminder of how short multiplication works. Work out the value of
5 1.2 Multiplication without a calculator So Begin with Enter as 8 with the 1 carried. Next, 7 6 2, plus the 1 carried, makes 3. Enter as 3 with the carried. Finally, , plus the carried, makes 16. Entered as 6 with the 1 carried; enter this 1 directly into the 1000s column. When working with bigger numbers, you will need to use long multiplication. There are two good ways of setting this out use whichever one you are most confident with. Work out the value of Method First, multiply 92 by to give Next, prepare to multiply 92 by 30, by writing a zero in the units column. This guarantees that you are multiplying by 30, not just times 3 gives 176. Finally, add 1968 and to give the answer So
6 6 Chapter 1: Working with whole numbers Method First, write 92 and 3 along the top and down the end of a rectangular grid Next, add diagonal lines, as shown Within each square of the grid, carry out a simple multiplication as shown. For example, 9 times 3 is Finally, add up the totals along each diagonal, starting at the right and working leftwards. EXERCISE 1.2 Use short multiplication to work out the answers to these calculations Use any written method to work out the answers to these problems. Show your working clearly
7 1.3 Division without a calculator 7 17 A company has 23 coaches and each coach can carry 55 passengers. What is the total number of passengers that the coaches can carry? 18 I have a set of 12 encyclopaedias. Each one has 199 pages. How many pages are there in the whole set? 19 Joni buys 16 stamps at 19 pence each and 13 stamps at 26 pence each. How much does she spend in total? 20 A small camera phone has a rectangular chip of pixels that collect and form the image. The chip size is 320 pixels long and 20 pixels across. Calculate the total number of pixels on the chip. 1.3 Division without a calculator Division is usually more awkward than multiplication, but this example shows a helpful method if the number you are dividing into (the dividend) is close to a convenient multiple of the number you are dividing by (the divisor). Work out the value of is So is almost 700, so you can work out then take off the extra 7 7 In most division questions you will need to use a formal written method. Here is an example of short division, with a remainder. Work out the value of remainder 1 So r 1 (or ) First, set the problem up using this division bracket notation. Divide into 6: it goes 1 time, with a remainder of 2. Next, divide into 27: it goes 6 times, with a remainder of 3. Finally, divide into 33: it goes 8 times, with a remainder of 1.
8 8 Chapter 1: Working with whole numbers When dividing by a number bigger than 10, it is usually easier to set the working out as a long division instead. The next example reminds you how this is done. Work out the value of Begin by setting up the problem using division bracket notation. 13 will not divide into 3, so divide 13 into 33. This goes 2 times, with remainder Bring down the next digit, 0 in this case, to make the 7 up to divides into 70 five times, with remainder 5. Finally, bring down the digit 2 to make divides into 52 exactly times, with no remainder. So exactly
9 1. Positive and negative integers 9 EXERCISE 1.3 Use short division to work out the answers to these calculations. (Four of them should leave remainders.) Use long division to work out the answers to these problems. Show your working clearly. (Only the last two should leave remainders.) grams of chocolate is shared out equally between 6 people. How much does each one receive? 18 In a lottery draw the prize of 3250 is shared equally between 13 winners. How much does each receive? 19 Seven children share 100 sweets in as fair a way as possible. How many sweets does each child receive? 20 On a school trip there are 16 teachers and 180 children. The teachers divide the children up into equalsized groups, as nearly as is possible, with one group per teacher. How many children are in each group? 1. Positive and negative integers It is often convenient to visualise positive and negative whole numbers, or integers, placed in order along a number line. The positive integers run to the right of zero, and negative integers to the left: Smaller at this end larger at this end. Mathematicians describe numbers on the right of the number line as being larger than the numbers on the left. This makes sense for positive numbers, where 6 is obviously bigger than, for example, but care must be taken with negative numbers. is bigger than 6, for example, and 8 is smaller than 7. You need to be able to carry out basic arithmetic using positive and negative numbers, with and without a calculator. Many calculators carry two types of minus sign key: one for marking a number as negative, and another for the process of subtraction. So, in a calculation such as 6 5, you have to start with the quantity 6 and then subtract 5. Subtraction means moving to the left on the number line, so the answer is
10 10 Chapter 1: Working with whole numbers Take care when two minus signs are involved: the rule that two minuses make a plus is not always trustworthy. For example, (two minuses make even more minus!), whereas So two adjacent minus signs are equivalent to a single plus sign. If two adjacent signs are the same: or then the overall sign is positive. And if the signs are different: or then the overall sign is negative. Without using a calculator, work out the values of: a) 6 9 b) 5 c) 8 3 d) 5 6 a) b) 5 1 c) d) When multiplying or dividing with positive or negative numbers, it is usually simplest to ignore the minus signs while you work out the numerical value of the answer. Then restore the sign at the end. If an odd number of negative numbers is multiplied or divided, the answer will be negative. If an even number of minus signs is involved, the answer will be positive. Without using a calculator, work out the values of: a) ( 5) () b) ( ) ( 3) c) ( 8) ( 2) d) 5 ( ) ( 2) a) ( 5) () 20 b) ( ) ( 3) 12 c) ( 8) ( 2) d) 5 ( ) ( 2) and there are two minus signs, so the answer is positive. EXERCISE 1. Without using a calculator, work out the answers to the following: 1 ( 6) 2 6 ( 3) 3 3 ( 2) 2 ( 1) ( 5) (5) 5
11 1.5 Factors, multiples and primes Arrange these in order of size, smallest first: 8, 3, 5, 1, Arrange these in order of size, largest first: 12, 13, 5, 9,. 23 What number lies midway between and 12? 2 What number lies onethird of the way from 10 to 2? 1.5 Factors, multiples and primes You will remember these definitions from earlier work: A multiple of a number is the result of multiplying it by a whole number. The multiples of are, 8, 12, 16, A factor of a number is a whole number that divides exactly into it, with no remainder. The factors of 12 are 1, 2, 3,, 6, 12. A prime number is a whole number with exactly two factors, namely 1 and itself. The number 1 is not normally considered to be prime, so the prime numbers are 2, 3, 5, 7, 11, If a large number is not prime, it can be written as the product of a set of prime factors in a unique way. For example, 12 can be written as A factor tree is a good way of breaking a large number into its prime factors. The next example shows how this is done. Write the number 180 as a product of its prime factors Begin by splitting the 180 into a product of two parts. You could use 2 times 90, or times 5, or 9 times 20, for example. The result at the end will be the same in any case. Here we begin by using 18 times 10. Since neither 18 nor 10 is a prime number, repeat the factorising process.
12 12 Chapter 1: Working with whole numbers has been broken down into 9 times 2, and 10 into 2 times 5. The 2s and the 5 are prime, so they are circled and the tree stops there. The 9 is not prime, so the process can continue The factor tree stops growing when all the branches end in circled prime numbers. 3 3 Thus means the factor 2 is used twice (two squared). If it had been used three times, you would write 2 3 (two cubed). EXERCISE List all the prime numbers from 1 to 0 inclusive. You should find that there are 12 such prime numbers altogether. 2 Use your result from question 1 to help answer these questions: a) How many primes are there between 20 and 0 inclusive? b) What is the next prime number above 31? c) Find two prime numbers that multiply together to make 03. d) Write 91 as a product of two prime factors. 3 Use the factor tree method to obtain the prime factorisation of: a) 80 b) 90 c) 50 Use the factor tree method to obtain the prime factorisation of: a) 36 b) 81 c) 1 What do you notice about all three of your answers? 5 When 56 is written as a product of primes, the result is 2 a b where a and b are positive integers. Find the values of a and b. 1.6 Highest common factor, HCF Consider the numbers 12 and 20. The number 2 is a factor of 12, and 2 is also a factor of 20. Thus 2 is said to be a common factor of 12 and 20. Likewise, the number is also a factor of both 12 and 20, so is also a common factor of 12 and 20.
13 1.6 Highest common factor, HCF 13 It turns out that 12 and 20 have no common factor larger than this, so is said to be the highest common factor (HCF) of 12 and 20. You can check that really is the highest common factor by writing 12 as 3 and 20 as 5; the 3 and 5 share no further factors. Find the highest common factor (HCF) of 30 and 80. By inspection, it looks as if the highest common factor may well be 10. Check: , and and clearly 3 and 8 have no further factors in common. So HCF of 30 and 80 is 10 By inspection means that you can just spot the answer by eye, without any formal working. There is an alternative, more formal, method for finding highest common factors. It requires the use of prime factorisation. Use prime factorisation to find the highest common factor of 30 and 80. By the factor tree method: Similarly, So HCF of 30 and Look at the 2 s: 30 has one of them, 80 has four. Pick the lower number: one 2 Look at the 3 s: 30 has one of them, but 80 has none. Pick the lower number: no 3s Look at the 5 s: 30 has one of them, and 80 has one. Pick the lower number: one 5 The prime factorisation method involves a lot of steps, but it is particularly effective when working with larger numbers, as in this next example. Use prime factorisation to find the highest common factor of 96 and 156. By the factor tree method: and HCF of 96 and
14 1 Chapter 1: Working with whole numbers It is important to be able to use the prime factorisation method in case it appears as an IGCSE examination question. You might like to try this ingenious alternative approach. A Greek mathematician named Euclid used it 3500 years ago, so it is often known as Euclid s method. Use Euclid s method to find the HCF of 96 and 156. [96, 156] [60, 96] [36, 60] [2, 36] [12, 2] [12, 12] Begin by writing the two numbers in a square bracket. Each new bracket contains the smaller of the two numbers, and their difference. Stop when both numbers are equal. So HCF (96, 156) 12 EXERCISE Use the method of inspection to write down the highest common factor of each pair of numbers. Check your result in each case. a) 12 and 18 b) 5 and 60 c) 22 and 33 d) 27 and 5 e) 8 and 27 f) 26 and Write each of the following numbers as the product of prime factors. Hence find the highest common factor of each pair of numbers. a) 20 and 32 b) 36 and 60 c) 80 and 180 d) 72 and 108 e) 120 and 195 f) 1 and Use Euclid s method to find the highest common factor of each pair of numbers. a) 12 and 30 b) 2 and 36 c) 96 and 120 d) 90 and 10 e) 78 and 102 f) 8 and Lowest common multiple (LCM) Consider the numbers 15 and 20. The multiples of 15 are 15, 30, 5, 60, 75, The multiples of 20 are 20, 0, 60, 80, Any multiple that occurs in both lists is called a common multiple. The smallest of these is the lowest common multiple (LCM). In this example, the LCM is 60. There are several methods for finding lowest common multiples. As with highest common factors, one of these methods is based on prime factorisation.
15 1.7 Lowest common multiple (LCM) 15 Find the lowest common multiple of 8 and 180. First, find the prime factors of each number using a factor tree if necessary Look at the powers of 2: Next, the powers of 3: Finally, the powers of 5: There are factors of 2 in 8, but only 2 in 180. Pick the higher of these: There is 1 factor of 3 in 8, but 2 in 180. Pick the higher of these: 2 There is no factor of 5 in 8, but 1 in 180. Pick the higher of these: 1 Putting all of this together: LCM of 8 and An alternative method is based on the fact that the product of the LCM and the HCF is the same as the product of the two original numbers. This gives the following result: a b LCM of a and b HCF of a and b This can be quite a quick method if the HCF is easy to spot. Find the lowest common multiple of 70 and 110. By inspection, HCF is 10 So: LCM
16 16 Chapter 1: Working with whole numbers It is also possible to find the HCF and LCM of three (or more) numbers. The prime factorisation method remains valid here, but other shortcut methods can fail. This example shows you how to adapt the factorisation method when there are three numbers. Find the HCF and LCM of 16, 2 and 28. Write these as products of prime factors: HCF of 16, 2 and 28 is 2 2 LCM of 16, 2 and 28 is The lowest number of 2s from 2 or 2 3 or 2 2 is 2 2 The highest number of 2s from 2 or 2 3 or 2 2 is 2 EXERCISE 1.7 Find the lowest common multiple (LCM) of each of these pairs of numbers. You may use whichever method you prefer and and and 5 25 and and and and and and and and and and and and and a) Write 60 and 8 as products of their prime factors. b) Hence find the LCM of 60 and a) Write 66 and 99 as products of their prime factors. b) Hence find the LCM of 66 and 99. c) Find also the HCF of 66 and a) Write 10, 36 and 56 as products of their prime factors. b) Work out the Highest Common Factor, HCF, of 10, 36 and 56. c) Work out the Lowest Common Multiple, LCM, of 10, 36 and a) Write 0, 8 and 600 as products of their prime factors. b) Work out the Highest Common Factor, HCF, of 0, 8 and 600. c) Work out the Lowest Common Multiple, LCM, of 0, 8 and 600.
17 Review exercise Virginia has two friends who regularly go round to her house to play. Joan goes round once every days and India goes round once every 5 days. How often are both friends at Virginia s house together? 22 Eddie owns three motorcycles. He cleans the Harley once every 8 days, the Honda once every 10 days and the Kawasaki once every 15 days. Today he cleaned all three motorcycles. When will he next clean all three motorcycles on the same day? REVIEW EXERCISE 1 Work out the answers to these arithmetic problems, using mental methods. Written working not allowed! Use pencil and paper methods (not a calculator) to work out the answers to these arithmetic problems Work out the answers to these problems using negative numbers. Do not use a calculator. 21 ( 7) ( 1) 22 6 ( 3) 23 ( 10) ( 13) ( 8) ( 3) 3 31 Use a factor tree to find the prime factorisation of: a) 70 b) 12 c) 96 d) a) Find the Highest Common Factor (HCF) of 2 and 56. b) Find the Lowest Common Multiple (LCM) of 2 and a) Write down the Highest Common Factor (HCF) of 20 and 22. b) Hence find the Lowest Common Multiple (LCM) of 20 and a) Write 360 in the form 2 a 3 b 5 c b) Write as an ordinary number.
18 18 Chapter 1: Working with whole numbers 35 Who is right? Explain carefully. If the HCF of two numbers is 1, then they must both be primes. Not necessarily true. Chuck Lilian 36 Pens cost 25p each. Mr Smith spends 120 on pens. Work out the number of pens he gets for The number 110 can be written as 3 2 c d, where c is a whole number and d is a prime number. Work out the value of c and the value of d. [Edexcel] [Edexcel] 38 a) Express 72 and 96 as products of their prime factors. b) Use your answer to part a) to work out the Highest Common Factor of 72 and 96. [Edexcel] Key points 1 Mental methods can be used for simple arithmetic problems. When adding up strings of whole numbers, look for combinations that add up to multiples of Harder addition and subtraction problems require formal pencil and paper methods. Make sure that you know how to perform these accurately. 3 Simple multiplication problems may be done mentally or by short multiplication. For harder problems, you need to be able to perform long multiplication reliably. If you find the traditional columns method awkward, consider using the box method instead both methods are acceptable to the IGCSE examiner. Long division is probably the hardest arithmetic process you will need to master. The traditional columns method is probably the best method there are alternatives, but they can be clumsy to use. If you have a long division by 23, say, then it may be helpful to write out the multiples 23, 6, 69,, 230 before you start. 5 Exam questions may require you to manipulate and order negative numbers. Remember to treat the two minuses make a plus rule with care, for example, 2 3 6, but Nonprime whole numbers may be written as a product of primes, using the factor tree method. This leads to a powerful method of working out the Highest Common Factor or Lowest Common Multiple of two numbers. 7 Sometimes you may be able to spot HCFs or LCMs by inspection. This result might help you to check them: a b LCM of a and b HCF of a and b
19 Internet Challenge 1 19 Internet Challenge 1 Prime time Here are some questions about prime numbers. You may use the internet to help you research some of the answers. 1 Find a list of all the prime numbers between 1 and 100, and print it out. How many prime numbers are there between 1 and 100? 2 Find a list of all the prime numbers between 1 and How many prime numbers are there between 1 and 1000? Compare your answers to questions 1 and 2. Does it appear that prime numbers occur less often as you go up to larger numbers? 3 Why is 1 not normally considered to be prime? How many British Prime Ministers had held office, up to the resignation of Gordon Brown in 2010? Is this a prime number? 5 Is there an infinite number of prime numbers? 6 What is the largest known prime number? 7 Is there a formula for finding prime numbers? 8 Where is the Prime Meridian? 9 Find out how the Sieve of Eratosthenes works, and use it to make your own list of all the primes up to 100. Check your list by comparing it with the list you found in question Some primes occur in adjacent pairs, which are consecutive odd integers, for example, 11 and 13, or 29 and 31. Find some higher examples of adjacent prime pairs. How many such pairs are there?
FACTORS, PRIME NUMBERS, H.C.F. AND L.C.M.
Mathematics Revision Guides Factors, Prime Numbers, H.C.F. and L.C.M. Page 1 of 16 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier FACTORS, PRIME NUMBERS, H.C.F. AND L.C.M. Version:
More informationCalculate Highest Common Factors(HCFs) & Least Common Multiples(LCMs) NA1
Calculate Highest Common Factors(HCFs) & Least Common Multiples(LCMs) NA1 What are the multiples of 5? The multiples are in the five times table What are the factors of 90? Each of these is a pair of factors.
More information1.16 Factors, Multiples, Prime Numbers and Divisibility
1.16 Factors, Multiples, Prime Numbers and Divisibility Factor an integer that goes into another integer exactly without any remainder. Need to be able to find them all for a particular integer it s usually
More informationCOMPASS Numerical Skills/PreAlgebra Preparation Guide. Introduction Operations with Integers Absolute Value of Numbers 13
COMPASS Numerical Skills/PreAlgebra Preparation Guide Please note that the guide is for reference only and that it does not represent an exact match with the assessment content. The Assessment Centre
More informationFACTORS, PRIME NUMBERS, H.C.F. AND L.C.M.
Mathematics Revision Guides Factors, Prime Numbers, H.C.F. and L.C.M. Page 1 of 10 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Foundation Tier FACTORS, PRIME NUMBERS, H.C.F. AND L.C.M. Version:
More informationChapter 11 Number Theory
Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications
More informationGrade 6 Math Circles March 10/11, 2015 Prime Time Solutions
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Lights, Camera, Primes! Grade 6 Math Circles March 10/11, 2015 Prime Time Solutions Today, we re going
More informationThe Crescent Primary School Calculation Policy
The Crescent Primary School Calculation Policy Examples of calculation methods for each year group and the progression between each method. January 2015 Our Calculation Policy This calculation policy has
More informationThere are 8000 registered voters in Brownsville, and 3 8. of these voters live in
Politics and the political process affect everyone in some way. In local, state or national elections, registered voters make decisions about who will represent them and make choices about various ballot
More informationHOSPITALITY Math Assessment Preparation Guide. Introduction Operations with Whole Numbers Operations with Integers 9
HOSPITALITY Math Assessment Preparation Guide Please note that the guide is for reference only and that it does not represent an exact match with the assessment content. The Assessment Centre at George
More informationMEP Y8 Practice Book A
2 Factors MEP Y8 Practice Book A 2.1 Factors and Prime Numbers A factor divides exactly into a number, leaving no remainder. For example, 13 is a factor of 26 because 26 13 = 2 leaving no remainder. A
More informationTYPES OF NUMBERS. Example 2. Example 1. Problems. Answers
TYPES OF NUMBERS When two or more integers are multiplied together, each number is a factor of the product. Nonnegative integers that have exactly two factors, namely, one and itself, are called prime
More informationYear 1. Use numbered number lines to add, by counting on in ones. Encourage children to start with the larger number and count on.
Year 1 Add with numbers up to 20 Use numbered number lines to add, by counting on in ones. Encourage children to start with the larger number and count on. +1 +1 +1 Children should: Have access to a wide
More informationCONTENTS. Please note:
CONTENTS Introduction...iv. Number Systems... 2. Algebraic Expressions.... Factorising...24 4. Solving Linear Equations...8. Solving Quadratic Equations...0 6. Simultaneous Equations.... Long Division
More informationMath 016. Materials With Exercises
Math 06 Materials With Exercises June 00, nd version TABLE OF CONTENTS Lesson Natural numbers; Operations on natural numbers: Multiplication by powers of 0; Opposite operations; Commutative Property of
More informationClass VI Chapter 3 Playing with Numbers Maths
Exercise 3. Question : Write all the factors of the following numbers: (a) 24 (b) 5 (c) 2 (d) 27 (e) 2 (f) 20 (g) 8 (h) 23 (i) 36 (a) 24 24 = 24 24 = 2 2 24 = 3 8 24 = 4 6 24 = 6 4 Factors of 24 are, 2,
More informationAn Introduction to Number Theory Prime Numbers and Their Applications.
East Tennessee State University Digital Commons @ East Tennessee State University Electronic Theses and Dissertations 82006 An Introduction to Number Theory Prime Numbers and Their Applications. Crystal
More informationSession 6 Number Theory
Key Terms in This Session Session 6 Number Theory Previously Introduced counting numbers factor factor tree prime number New in This Session composite number greatest common factor least common multiple
More informationUnit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.
Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L34) is a summary BLM for the material
More informationSquares and Square Roots
SQUARES AND SQUARE ROOTS 89 Squares and Square Roots CHAPTER 6 6.1 Introduction You know that the area of a square = side side (where side means the length of a side ). Study the following table. Side
More informationWelcome to Basic Math Skills!
Basic Math Skills Welcome to Basic Math Skills! Most students find the math sections to be the most difficult. Basic Math Skills was designed to give you a refresher on the basics of math. There are lots
More informationor (ⴚ), to enter negative numbers
. Using negative numbers Add, subtract, multiply and divide positive and negative integers Use the sign change key to input negative numbers into a calculator Why learn this? Manipulating negativ e numbers
More informationHigh Coniscliffe CE Primary School. Calculation Strategies Booklet. +  x. Guidance for Staff, Parents and Governors
High Coniscliffe CE Primary School Calculation Strategies Booklet +  x Guidance for Staff, Parents and Governors 20152016 Contents About this booklet.3 School Aims..3 Problem Solving..3 Reasons for using
More informationCubes and Cube Roots
CUBES AND CUBE ROOTS 109 Cubes and Cube Roots CHAPTER 7 7.1 Introduction This is a story about one of India s great mathematical geniuses, S. Ramanujan. Once another famous mathematician Prof. G.H. Hardy
More informationGrade 7 & 8 Math Circles October 19, 2011 Prime Numbers
1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Grade 7 & 8 Math Circles October 19, 2011 Prime Numbers Factors Definition: A factor of a number is a whole
More informationThe Relocation Property
Charles Ledger and Kim Langen, Spirit of Math Schools Introduction The Relocation Property When working in the intermediate level classroom in the North York School Board it became critically clear that
More informationCHAPTER 3 Numbers and Numeral Systems
CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,
More informationWhy Vedic Mathematics?
Why Vedic Mathematics? Many Indian Secondary School students consider Mathematics a very difficult subject. Some students encounter difficulty with basic arithmetical operations. Some students feel it
More informationSection 1.5 Arithmetic in Other Bases
Section Arithmetic in Other Bases Arithmetic in Other Bases The operations of addition, subtraction, multiplication and division are defined for counting numbers independent of the system of numeration
More information15 Prime and Composite Numbers
15 Prime and Composite Numbers Divides, Divisors, Factors, Multiples In section 13, we considered the division algorithm: If a and b are whole numbers with b 0 then there exist unique numbers q and r such
More informationAbercrombie Primary School Progression in Calculation 2014
Abercrombie Primary School Progression in Calculation 204 What you need to know about calculations Mathematics will be at the core of your child s schooling from the moment they start to the moment they
More information1 Number System. Introduction. Face Value and Place Value of a Digit. Categories of Numbers
RMAT Success Master 3 1 Number System Introduction In the decimal number system, numbers are expressed by means of symbols 0, 1,, 3, 4, 5, 6, 7, 8, 9, called digits. Here, 0 is called an insignificant
More informationPowers and roots 5.1. Previous learning. Objectives based on NC levels and (mainly level ) Lessons 1 Working with integer powers of numbers
N 5.1 Powers and roots Previous learning Before they start, pupils should be able to: recognise and use multiples, factors (divisors), common factor, highest common factor, lowest common multiple and primes
More informationMultiplication. Year 1 multiply with concrete objects, arrays and pictorial representations
Year 1 multiply with concrete objects, arrays and pictorial representations Children will experience equal groups of objects and will count in 2s and 10s and begin to count in 5s. They will work on practical
More informationSimply Math. Everyday Math Skills NWT Literacy Council
Simply Math Everyday Math Skills 2009 NWT Literacy Council Acknowledgement The NWT Literacy Council gratefully acknowledges the financial assistance for this project from the Department of Education, Culture
More informationChapter 1. Q3. Show that any positive integer is of the form 3q or, 3q + 1 or, 3q + 2 for some integer q.
Q1. State whether the following are true or not: Chapter 1 (i) 3 l 93 (ii) 6 l 28 (iii) 0 l 4 (iv) 5 l 0 (v) 2 l 8 (vi) 7 l 35 (vii) 6 l 6 (viii) 8 l 8 (ix) 13 l 25 (x) 1 l 1 Q2. Show that every
More informationIntroduction Number Systems and Conversion
UNIT 1 Introduction Number Systems and Conversion Objectives 1. Introduction The first part of this unit introduces the material to be studied later. In addition to getting an overview of the material
More informationSupplemental Worksheet Problems To Accompany: The PreAlgebra Tutor: Volume 1 Section 5 Subtracting Integers
Supplemental Worksheet Problems To Accompany: The PreAlgebra Tutor: Volume 1 Please watch Section 5 of this DVD before working these problems. The DVD is located at: http://www.mathtutordvd.com/products/item66.cfm
More informationCongruences. Robert Friedman
Congruences Robert Friedman Definition of congruence mod n Congruences are a very handy way to work with the information of divisibility and remainders, and their use permeates number theory. Definition
More informationNim Games Lesson Plan
Nim Games Lesson Plan Students play a variety of related counting games in order to discover and describe winning strategies. Students develop their number sense and problem solving skills as they investigate
More informationPrime Factorization 0.1. Overcoming Math Anxiety
0.1 Prime Factorization 0.1 OBJECTIVES 1. Find the factors of a natural number 2. Determine whether a number is prime, composite, or neither 3. Find the prime factorization for a number 4. Find the GCF
More informationProgression towards a standard written method of Calculation
Progression towards a standard written method of Calculation Introduction Children are introduced to the process of Calculation through practical, oral and mental activities. Through these activities they
More informationFACTORS AND MULTIPLES Answer Key
I. Find prime factors by factor tree method FACTORS AND MULTIPLES Answer Key a. 768 2 384 2 192 2 96 2 48 2 24 2 12 2 6 2 3 768 = 2*2*2*2*2*2*2*2 *3 b. 1608 3 536 2 268 2 134 2 67 1608 = 3*2*2*2*67 c.
More informationThe Teacher s Circle Number Theory, Part 1 Joshua Zucker, August 14, 2006
The Teacher s Circle Number Theory, Part 1 Joshua Zucker, August 14, 2006 joshua.zucker@stanfordalumni.org [A few words about MathCounts and its web site http://mathcounts.org at some point.] Number theory
More informationUnit 7 The Number System: Multiplying and Dividing Integers
Unit 7 The Number System: Multiplying and Dividing Integers Introduction In this unit, students will multiply and divide integers, and multiply positive and negative fractions by integers. Students will
More informationPreAlgebra Homework 2 Factors: Solutions
PreAlgebra Homework 2 Factors: Solutions 1. Find all of the primes between 70 and 100. : First, reject the obvious nonprime numbers. None of the even numbers can be a prime because they can be divided
More information47 Numerator Denominator
JH WEEKLIES ISSUE #22 20122013 Mathematics Fractions Mathematicians often have to deal with numbers that are not whole numbers (1, 2, 3 etc.). The preferred way to represent these partial numbers (rational
More informationIf A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?
Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question
More information54 Prime and Composite Numbers
54 Prime and Composite Numbers Prime and Composite Numbers Prime Factorization Number of Divisorss Determining if a Number is Prime More About Primes Prime and Composite Numbers Students should recognizee
More informationNumber: Multiplication and Division with Reasoning
count in multiples of twos, fives and tens (copied from Number and Number: Multiplication and Division with Reasoning MULTIPLICATION & DIVISION FACTS Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 count in
More informationI know when I have written a number backwards and can correct it when it is pointed out to me I can arrange numbers in order from 1 to 10
Mathematics Targets Moving from Level W and working towards level 1c I can count from 1 to 10 I know and write all my numbers to 10 I know when I have written a number backwards and can correct it when
More informationDecimals and other fractions
Chapter 2 Decimals and other fractions How to deal with the bits and pieces When drugs come from the manufacturer they are in doses to suit most adult patients. However, many of your patients will be very
More informationRepton Manor Primary School. Maths Targets
Repton Manor Primary School Maths Targets Which target is for my child? Every child at Repton Manor Primary School will have a Maths Target, which they will keep in their Maths Book. The teachers work
More informationIn order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature.
Indices or Powers A knowledge of powers, or indices as they are often called, is essential for an understanding of most algebraic processes. In this section of text you will learn about powers and rules
More informationMaths Module 6 Algebra Solving Equations This module covers concepts such as:
Maths Module 6 Algebra Solving Equations This module covers concepts such as: solving equations with variables on both sides multiples and factors factorising and expanding function notation www.jcu.edu.au/students/learningcentre
More informationDigital Electronics. 1.0 Introduction to Number Systems. Module
Module 1 www.learnaboutelectronics.org Digital Electronics 1.0 Introduction to What you ll learn in Module 1 Section 1.0. Recognise different number systems and their uses. Section 1.1 in Electronics.
More informationPrime Time: Homework Examples from ACE
Prime Time: Homework Examples from ACE Investigation 1: Building on Factors and Multiples, ACE #8, 28 Investigation 2: Common Multiples and Common Factors, ACE #11, 16, 17, 28 Investigation 3: Factorizations:
More informationNumber. ch?v=mquhqkknldk (maths is confusing funny)
Number http://www.youtube.com/watch?v =52CzD31SqaM&feature=related (maths is confusing II funny) http://www.youtube.com/wat ch?v=mquhqkknldk (maths is confusing funny) SLO To find multiples of a number
More informationALGEBRA. sequence, term, nth term, consecutive, rule, relationship, generate, predict, continue increase, decrease finite, infinite
ALGEBRA Pupils should be taught to: Generate and describe sequences As outcomes, Year 7 pupils should, for example: Use, read and write, spelling correctly: sequence, term, nth term, consecutive, rule,
More informationMathematics of Cryptography
CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter
More informationElementary Algebra. Section 0.4 Factors
Section 0.4 Contents: Definitions: Multiplication Primes and Composites Rules of Composite Prime Factorization Answers Focus Exercises THE MULTIPLICATION TABLE x 1 2 3 4 5 6 7 8 9 10 11 12 1 1 2 3 4 5
More informationGreatest Common Factor and Least Common Multiple
Greatest Common Factor and Least Common Multiple Intro In order to understand the concepts of Greatest Common Factor (GCF) and Least Common Multiple (LCM), we need to define two key terms: Multiple: Multiples
More informationTests for Divisibility, Theorems for Divisibility, the Prime Factor Test
1 Tests for Divisibility, Theorems for Divisibility, the Prime Factor Test Definition: Prime numbers are numbers with only two factors, one and itself. For example: 2, 3, and 5. Definition: Composite numbers
More informationIt is time to prove some theorems. There are various strategies for doing
CHAPTER 4 Direct Proof It is time to prove some theorems. There are various strategies for doing this; we now examine the most straightforward approach, a technique called direct proof. As we begin, it
More informationLesson 4. Factors and Multiples. Objectives
Student Name: Date: Contact Person Name: Phone Number: Lesson 4 Factors and Multiples Objectives Understand what factors and multiples are Write a number as a product of its prime factors Find the greatest
More information5544 = 2 2772 = 2 2 1386 = 2 2 2 693. Now we have to find a divisor of 693. We can try 3, and 693 = 3 231,and we keep dividing by 3 to get: 1
MATH 13150: Freshman Seminar Unit 8 1. Prime numbers 1.1. Primes. A number bigger than 1 is called prime if its only divisors are 1 and itself. For example, 3 is prime because the only numbers dividing
More informationWritten methods for addition of whole numbers
Stage 1: The empty number line Mathematics written methods at the Spinney Written methods for addition of whole numbers The mental methods that lead to column addition generally involve partitioning, e.g.
More informationUnit 2 Number and Operations in Base Ten: Place Value, Addition, and Subtraction
Unit 2 Number and Operations in Base Ten: Place Value, Addition, and Subtraction Introduction In this unit, students will review place value to 1,000,000 and understand the relative sizes of numbers in
More informationCONNECT: Powers and logs POWERS, INDICES, EXPONENTS, LOGARITHMS THEY ARE ALL THE SAME!
CONNECT: Powers and logs POWERS, INDICES, EXPONENTS, LOGARITHMS THEY ARE ALL THE SAME! You may have come across the terms powers, indices, exponents and logarithms. But what do they mean? The terms power(s),
More information1.6 The Order of Operations
1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative
More informationMathematics Calculation and Number Fluency Policy. Curriculum MMXIV. Chacewater School. +  x
Mathematics Calculation and Number Fluency Policy Curriculum MMXIV Chacewater School +  x Autumn 2014 Introduction The purpose of this document is to build on the successes of the Calculation Policy which
More informationAlgebra for Digital Communication
EPFL  Section de Mathématiques Algebra for Digital Communication Fall semester 2008 Solutions for exercise sheet 1 Exercise 1. i) We will do a proof by contradiction. Suppose 2 a 2 but 2 a. We will obtain
More informationTI83 Plus Graphing Calculator Keystroke Guide
TI83 Plus Graphing Calculator Keystroke Guide In your textbook you will notice that on some pages a keyshaped icon appears next to a brief description of a feature on your graphing calculator. In this
More informationChapter 4 Fractions and Mixed Numbers
Chapter 4 Fractions and Mixed Numbers 4.1 Introduction to Fractions and Mixed Numbers Parts of a Fraction Whole numbers are used to count whole things. To refer to a part of a whole, fractions are used.
More informationCISC  Curriculum & Instruction Steering Committee. California County Superintendents Educational Services Association
CISC  Curriculum & Instruction Steering Committee California County Superintendents Educational Services Association Primary Content Module IV The Winning EQUATION NUMBER SENSE: Factors of Whole Numbers
More information3 PLAYING WITH NUMBERS
3 PLAYING WITH NUMBERS Exercise 3.1 Q.1. Write all the factors of the following numbers : (a) 24 (b) 15 (c) 21 (d) 27 (e) 12 (f) 20 (g) 18 (h) 23 (i) 36 Ans. (a) 24 = 1 24 = 2 12 = 3 8 = 4 6 Hence, factors
More informationDividing Whole Numbers
Dividing Whole Numbers INTRODUCTION Consider the following situations: 1. Edgar brought home 72 donut holes for the 9 children at his daughter s slumber party. How many will each child receive if the donut
More informationGrade 7/8 Math Circles Fall 2012 Factors and Primes
1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Fall 2012 Factors and Primes Factors Definition: A factor of a number is a whole
More information9. The Pails of Water Problem
9. The Pails of Water Problem You have a 5 and a 7 quart pail. How can you measure exactly 1 quart of water, by pouring water back and forth between the two pails? You are allowed to fill and empty each
More informationPrimes in Sequences. Lee 1. By: Jae Young Lee. Project for MA 341 (Number Theory) Boston University Summer Term I 2009 Instructor: Kalin Kostadinov
Lee 1 Primes in Sequences By: Jae Young Lee Project for MA 341 (Number Theory) Boston University Summer Term I 2009 Instructor: Kalin Kostadinov Lee 2 Jae Young Lee MA341 Number Theory PRIMES IN SEQUENCES
More informationFACTORISATION YEARS. A guide for teachers  Years 9 10 June 2011. The Improving Mathematics Education in Schools (TIMES) Project
9 10 YEARS The Improving Mathematics Education in Schools (TIMES) Project FACTORISATION NUMBER AND ALGEBRA Module 33 A guide for teachers  Years 9 10 June 2011 Factorisation (Number and Algebra : Module
More informationWe now explore a third method of proof: proof by contradiction.
CHAPTER 6 Proof by Contradiction We now explore a third method of proof: proof by contradiction. This method is not limited to proving just conditional statements it can be used to prove any kind of statement
More informationClock Arithmetic and Modular Systems Clock Arithmetic The introduction to Chapter 4 described a mathematical system
CHAPTER Number Theory FIGURE FIGURE FIGURE Plus hours Plus hours Plus hours + = + = + = FIGURE. Clock Arithmetic and Modular Systems Clock Arithmetic The introduction to Chapter described a mathematical
More informationPrimes. Name Period Number Theory
Primes Name Period A Prime Number is a whole number whose only factors are 1 and itself. To find all of the prime numbers between 1 and 100, complete the following exercise: 1. Cross out 1 by Shading in
More informationSection P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities
Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.
More informationNumber Systems Richard E. Haskell
NUMBER SYSTEMS D Number Systems Richard E. Haskell Data inside a computer are represented by binary digits or bits. The logical values of these binary digits are denoted by and, while the corresponding
More informationWootton Wawen C.E. Primary School. A Guide to Mental Maths
Wootton Wawen C.E. Primary School A Guide to Mental Maths Why is it important that children recall maths facts? Calculators should not be used as a substitute for good written and mental arithmetic. (National
More informationFractions and Decimals
Fractions and Decimals Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles December 1, 2005 1 Introduction If you divide 1 by 81, you will find that 1/81 =.012345679012345679... The first
More informationPigeonhole Principle Solutions
Pigeonhole Principle Solutions 1. Show that if we take n + 1 numbers from the set {1, 2,..., 2n}, then some pair of numbers will have no factors in common. Solution: Note that consecutive numbers (such
More informationMaths Calculation Booklet for parents
Maths Calculation Booklet for parents INTRODUCTION This booklet is intended to explain the ways in which your children are taught to write down their calculations. We have revised the school calculation
More informationOur Lady of the Visitation Catholic Primary School. CalculationDivision Policy Years 16
Our Lady of the Visitation Catholic Primary School CalculationDivision Policy Years 16 Date of Ratification: Signed: Miss K. Coll (HEAD TEACHER) Review date: Mrs H.McKenzie (CHAIR OF GOVERNORS) Signed:
More information8 Primes and Modular Arithmetic
8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.
More informationQM0113 BASIC MATHEMATICS I (ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION)
SUBCOURSE QM0113 EDITION A BASIC MATHEMATICS I (ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION) BASIC MATHEMATICS I (ADDITION, SUBTRACTION, MULTIPLICATION AND DIVISION) Subcourse Number QM 0113 EDITION
More informationQuick Reference ebook
This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed
More informationNUMBERING SYSTEMS C HAPTER 1.0 INTRODUCTION 1.1 A REVIEW OF THE DECIMAL SYSTEM 1.2 BINARY NUMBERING SYSTEM
12 Digital Principles Switching Theory C HAPTER 1 NUMBERING SYSTEMS 1.0 INTRODUCTION Inside today s computers, data is represented as 1 s and 0 s. These 1 s and 0 s might be stored magnetically on a disk,
More informationNow that we have a handle on the integers, we will turn our attention to other types of numbers.
1.2 Rational Numbers Now that we have a handle on the integers, we will turn our attention to other types of numbers. We start with the following definitions. Definition: Rational Number any number that
More informationReal Numbers. Learning Outcomes. chapter. In this chapter you will learn:
chapter Learning Outcomes In this chapter you will learn: ÂÂ ÂÂ ÂÂ ÂÂ ÂÂ ÂÂ ÂÂ ÂÂ How to define the real numbers About factors, multiples and prime factors How to write a whole number as a product of prime
More informationPROGRESSION THROUGH CALCULATIONS FOR MULTIPLICATION
PROGRESSION THROUGH CALCULATIONS FOR MULTIPLICATION THE FOLLOWING ARE STANDARDS THAT WE EXPECT THE MAJORITY OF CHILDREN TO ACHIEVE BY THE END OF THE YEAR. YR Related objectives: Count repeated groups of
More informationFactors, powers and roots
10 Factors, powers and roots This chapter is about multiples, factors, powers and roots. In the hinese calendar two separate cycles interact. There are 10 heavenly stems and 1 zodiac animals. You can use
More informationIn order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature.
Percentages In this unit we shall look at the meaning of percentages and carry out calculations involving percentages. We will also look at the use of the percentage button on calculators. In order to
More information