Powers and roots 5.1. Previous learning. Objectives based on NC levels and (mainly level ) Lessons 1 Working with integer powers of numbers


 Claude Townsend
 1 years ago
 Views:
Transcription
1 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 use squares, positive and negative square roots, cubes and cube roots, and index notation for small positive integer powers. Objectives based on NC levels and (mainly level ) In this unit, pupils learn to: represent problems and synthesise information in different forms use accurate notation calculate accurately, selecting mental methods or a calculator as appropriate estimate, approximate and check working record methods, solutions and conclusions make convincing arguments to justify generalisations or solutions review and refine own findings and approaches on the basis of discussions with others and to: use index notation for integer powers know and use the index laws for multiplication and division of positive integer powers extend mental methods of calculation with factors, powers and roots use the power and root keys of a calculator use ICT to estimate square roots and cube roots use the prime factor decomposition of a number. Lessons 1 Working with integer powers of numbers About this unit Assessment Estimating square roots 3 Prime factor decomposition Sound understanding of powers and roots of numbers helps pupils to generalise the principles in their work in algebra. It also helps pupils to be aware of the relationships between numbers and to know at a glance which properties they possess and which they do not. This unit includes: an optional mental test which could replace part of a lesson (p. 00); a selfassessment section (N5.1 How well are you doing? class book p. 00); a set of questions to replace or supplement questions in the exercises or homework tasks, or to use as an informal test (N5.1 Check up, CDROM). Common errors and misconception N5.1 Powers and roots Look out for pupils who: think that n means n, or that n means n ; wrongly apply the index laws, e.g , or ; think that 1 is a prime number; include 1 in the prime factor decomposition of a number; confuse the highest common factor (HCF) and lowest common multiple (LCM); assume that the lowest common multiple of a and b is always a b.
2 Key terms and notation Practical resources Exploring maths Useful websites problem, solution, method, pattern, relationship, expression, solve, explain, systematic, trial and improvement calculate, calculation, calculator, operation, multiply, divide, divisible, product, quotient positive, negative, integer factor, factor pair, prime, prime factor decomposition, power, root, square, cube, square root, cube root, notation n and n, n 3 and 3 n scientific calculators for pupils individual whiteboards Tier 5 teacher s book N5.1 Mental test, p. 00 Answers for Unit N5.1, pp Tier 5 CDROM PowerPoint files N5.1 Slides for lessons 1 to 3 Excel file N5.1 SquareRoot Tools and prepared toolsheets Calculator tool Tier 5 programs Multiples and factors quiz HCF and LCM Ladder method computers with spreadsheet software, e.g. Microsoft Excel, or graphics calculators Tier 5 class book N5.1, pp Tier 5 home book N5.1, pp Tier 5 CDROM N5.1 Check up Topic B: Indices: simplifying Factor tree nlvm.usu.edu/en/nav/category_g_3_t_1.html Grid game N5.1 Powers and roots
3 1 Working with integer powers of numbers Learning points A number a raised to the power 4 is a 4 or a a a a. The number that expresses the power is its index, so, 5 and 7 are the indices of a, a 5 and a 7. To multiply two numbers in index form, add the indices, so a m a n a m1n. To divide two numbers in index form, subtract the indices, so a m a n a m n. Starter Use slide 1.1 to discuss the objectives for the unit. This lesson is about finding positive and negative integer powers of numbers. Remind pupils that when a number is multiplied by itself the product is called a power of that number. So a a, or a squared, is the second power of a, and is written as a, a a a, or a cubed, is the third power of a and is written as a 3, and so on. For a 4 we say a to the power of 4, and similarly with higher powers. The number that expresses the power is its index. So 5 and 7 are the indices of a 5 and a 7. When the index is 1, it is usually omitted: we write a, rather than a 1. Ask pupils to calculate mentally powers of positive and negative integers, e.g. ( 8), 8, ( ) 5, 5, ( 3) 4, 3 4, ( 5) 3, 5 3 Record answers on the board, and ask pupils what they notice. Draw out that for negative numbers even powers are positive, and odd powers are negative. What is ( 1) 13? What is ( 1) 14? Extend to decimals, e.g. ask pupils to calculate mentally: (0.1) 3, (0.7), ( 0.) 4 TO Main activity Use the Calculator tool to show pupils how to use the x y keys of their calculators. Use the key to explore raising a number to the power 0. Explain that this always has the answer 1. Discuss negative indices. Ask pupils to consider this pattern. How is each number found from the one above it? What is the pattern of the indices? What are the next few lines of the pattern? Establish that: Similarly 1 1, 1 4 and Show the table of powers of on slide 1.. Ask pupils in pairs to make up and record some multiplications using the table. Ask questions to help pupils to discover for themselves the rules for calculations with indices. N5.1 Powers and roots
4 What do you notice about the indices in these calculations? What is a quick way of multiplying powers of? Why does it work? What is this calculation in index form? [ ] [ ] [ ] What do you notice about the indices in these calculations? What is a quick way of dividing one power of by another? Why does it work? Repeat with the powers of 4 on slide 1.3. Now generalise. Write on the board m m 3. What will this simplify to? Explain why. [m m 3 (m m) (m m m) m m m m m m 5 ] Stress that the indices have been added, so that: m m 3 m 1 3 m 5 Repeat with m 5 m. Stress that for division the indices are subtracted, so that: m 5 m m 5 m 3 Discuss negative indices, e.g. m 3 m 7 m 3 7 m 4 = 1 m 4 m 5 m 3 m 5 3 m. Select individual work from N5.1 Exercise 1 in the class book (p. 00). Review Homework Show slide 1.4. Point to two different powers of 10. Ask pupils to multiply or divide them and to write the answer on their whiteboards. Stress that the rules for multiplying and dividing numbers in index form apply to both positive and negative indices. Ask pupils to remember the points on slide 1.5. Homework Ask pupils to do N5.1 Task 1 in the home book (p. 00). N5.1 Powers and roots
5 Estimating square roots Learning points n is the square root of n, e.g n is the cube root of n, e.g , Trial and improvement can be used to estimate square roots when a calculator is not available. Starter Tell pupils that in this lesson they will be estimating the value of square roots. Remind them that the square root of a is denoted by a, or more simply as a, and that a square root of a positive number can be positive or negative, e.g. if a 5 9, a Show the grid on slide.1. Write on the board: x 51, z 5 4. Point to an expression on the grid. Ask pupils to work out its value mentally and to write the answer on their whiteboards. Ask someone to explain how they calculated it. After a while, change the values for x and z to: x 5 9 and z 5 5. Main activity Discuss how to estimate the positive square root of a number that is not a perfect square, e.g. 70 must lie between 64 and 81, so Since 70 is closer to 64 than to 81, we expect 70 to be closer to 8 than to 9, perhaps about Show the class how they could find 7 if they had only a basic calculator with no squareroot key. Tell them that the process is called trial and improvement. Explain that 7 must lie between and 3, because 7 lies between and 3. Try Try.6 = Try.7 = 7.9. Try.65 = Try.64 = Try.645 = Too low Too low Too high. Very close but a little bit too high Very close but too low Still too low The answer lies between.645 and.65. All numbers between.645 and.65 round up to.65. So correct to two decimal places. Ask pupils to work in pairs and, using only the key on their calculator, to find 1 to two decimal places [answer: 3.46]. Establish first that it must lie between 3 and 4. N5.1 Powers and roots
6 Show the class how they could use a spreadsheet for this activity, without using the squareroot function, e.g. use the Excel file N5.1 SquareRoot. Point out that the strategy here is different. We work systematically in tenths from 3 to 4, then in hundredths from 3.4 to 3.5, then in thousandths from 3.46 to You can use this file to estimate other square roots by overtyping 3 and 3.4. If possible, pupils should develop similar spreadsheets, using either a computer or a graphics calculator. XL Select individual work from N5.1 Exercise in the class book (p. 00). Review Introduce root notation. Explain that if 79 is the cube of 9, then 9 is the cube root of 79, which is written as 3 _ The cube root, fourth root, fifth root, of a are denoted by 3 a, 4 a, 5 a, Use the Calculator tool to demonstrate how to find a cube root. You may need to explain that some calculators have a cube root key 3. Others have a key like x, or other variations. For example, to find the value of 3 _ 16, key in 1 6 x 3. [Answer: 6] Ask pupils to work out 3 64 and Explain that the cube root of a positive number is positive, and the cube root of a negative number is negative. TO Sum up the lesson by reminding pupils of the learning points. Homework Ask pupils to do N5.1 Task in the home book (p. 00). N5.1 Powers and roots
7 3 Prime factor decomposition Learning points Writing a number as the product of its prime factors is called the prime factor decomposition of the number. You can use a tree method or a ladder method to find a number s prime factors. To find the highest common factor (HCF) of a pair of numbers, find the product of all the prime factors common to both numbers. To find the lowest common multiple (LCM) of a pair of numbers, find the smallest number that is a multiple of each of the numbers. QZ Starter Tell pupils that in this lesson they will be finding the prime factors of numbers and using them to find common factors and multiples of a pair of numbers. Remind them of the definitions of multiple, factor, factor pair and prime number. Launch Multiples and factors quiz. Ask pupils to answer on their whiteboards. Use Next and Back to move through the questions at a suitable pace. Main activity Write on the board three products such as: What do you notice about the numbers in these products? Establish that they are all prime numbers. Explain that when a number is expressed as the product of its prime factors it is called the prime factor decomposition of a number. Stress that because 1 is not a prime number it is not included in the decomposition. How can we find the prime factor decomposition of 80? First explain the tree method, i.e. split 80 into a product such as 0 3 4, then continue factorising any number in the product that is not a prime. Repeat with SIM Launch Ladder method. Use it to show the alternative method, where the number is repeatedly divided by any prime that will divide into it exactly. Demonstrate with 63, dragging numbers from the grid to the relevant positions. Continue to divide by prime numbers until the answer is 1. Express the answer as Repeat with SIM Show how to find the highest common factor (HCF) and lowest common multiple (LCM) of a pair of numbers. Launch HCF and LCM. N5.1 Powers and roots
8 Choose lowest common multiple. Select 8 and 6 using the arrows by the numbers. Drag multiples of 8 and multiples of 6 from the 100square to the answer boxes. (Numbers snap back to the 100square if dragged from answer box.) Numbers common to both boxes change colour to blue. Which numbers are both multiples of 8 and multiples of 6? Which is the lowest number that is both a multiple of 8 and 6? Drag the HCF into the box below the 100square. Repeat several times with different numbers, then change to highest common factor, which works similarly. Select 4 and 18, then drag factors to the answer boxes. Which numbers are both factors of 4 and factors of 18? Which is the highest number that is both a factor of 4 and 18? Repeat with different numbers. Show how to use a Venn diagram to find the HCF and LCM of a pair of numbers such as 36 and 30. Explain that: the overlapping prime factors give the HCF ( ); all the prime factors give the LCM ( ). Repeat with 18 and Select individual work from N5.1 Exercise 3 in the class book (p. 00). Review Sum up the lesson by stressing the points on slide 3.1. Round off the unit by referring again to the objectives. Suggest that pupils find time to try the selfassessment problems in N5.1 How well are you doing? in the class book (p. 00). Homework Ask pupils to do N5.1 Task 3 in the home book (p. 00). N5.1 Powers and roots
9 N5.1 Mental test Read each question aloud twice. Allow a suitable pause for pupils to write answers. 1 What number is five to the power three? Write all the prime factors of fortytwo. 3 Write down a factor of thirtysix that is greater than ten and less than twenty. 005 KS3 4 What is the next number in the sequence of square numbers? 004 KS3 One, four, nine, sixteen,... 5 Look at the numbers. Write down each number that is a factor of one hundred Y7 [Write on board ] 6 Write two factors of twentyfour which add to make eleven. 005 KS 7 What is the square root of eightyone? 001 KS3 8 What number is five cubed? 003 KS3 9 The volume of a cube is sixtyfour centimetres cubed. 00 KS3 What is the length of an edge of the cube? 10 What is the square of three thousand? 001 KS3 11 To the nearest whole number, what is the square root of 004 KS3 eightythree point nine? 1 I think of a number. I square my number and get the answer 007 KS3 one thousand six hundred. What could my number be? Key: KS3 Key Stage 3 test KS Key Stage test Y7 Year 7 optional test (1999) Questions 3 to 7 are at level 5; 8 to 11 are at level 6; 1 is at level 7 Answers 1 15, 3, or , 0, 5, and cm N5.1 Powers and roots
10 N5.1 Check up and resource sheets Check up Answer these questions by writing in your book. Powers and roots (no calculator) N level 6 Which two of the numbers below are not square numbers? David says that What is 10? 3 To the nearest whole number, what is the square root of 93.7? _ 4 If 81 n 144, then n could be which of the following numbers? 5 Year 8 optional test level 6 Terry has 4 centimetre cubes. He uses them to make a cuboid that is one cube high. Tina has 4 centimetre cubes. She uses them to make a solid cuboid that is two cubes high What could the dimensions of her cuboid be? 1 cm high 4 cm wide 6 cm long 6 What is the biggest number that is a factor of both 105 and 135? 7 What is the smallest number that is a multiple of both 1 and 7? Powers and roots (calculator allowed) level 6 Mary thinks of a number. First I subtract 3.76 Then I find the square root of what I get My answer is 6.80 Which number did Mary think of? Pearson Education 008 Tier 5 resource sheets N5.1 Powers and roots N5.1 N5.1 Powers and roots 11
11 N5.1 Answers Class book Exercise 1 1 a 3 b 4 5 c 3 8 d ( 1) 4 e 5 f 6 1 a 64 b 43 c 56 d 18 e 1 f 1 g 1 h a 401 b c 1331 d e 104 f g h a 8 b 3 5 c d a 8 e 5 4 f 1 4 g 8 0 h b a b c d e f Rachel and Hannah are 14 and 11 years old. Extension problem 8 The smallest whole numbers are 6 and 10: Exercise 1 a x 3 b x 7 c x 1 d x 1 a 1.41 to d.p. b.15 to d.p. c 4 d 0. e 5 f 1. to d.p. g 1.73 to d.p. h 1 3 a b 7 c 11 d 8 4 Each answer is correct to 1 d.p. a.4 b 6.7 c 10.7 d Between 700 and 750 slabs will be used. 6 6 is 678, which is too few, and 8 8 is 784, which is too many. So the exact number of slabs is a a 9.7 to 1 d.p. b a 1.3 to 1 d.p. c a 0.4 to 1 d.p metres to d.p. Extension problem 8 98 = is not a perfect square. There is no whole number between the square root of 8880 and the square root of 8889 but 98 lies between the and Exercise 3 1 a 3 b 3 5 c 3 7 d 3 3 e 3 3 f 3 3 a 1,, 5, 10, 5, 50 b 5 3 a 1, 3, 5, 9, 15, 45 b e.g. 63 (with factors 1, 3, 7, 9, 1, 63) 5 a 7 and 30: HCF 6, LCM 360 b 50 and 80: HCF 10, LCM 400 c 48 and 84: HCF 1, LCM HCF 15, LCM HCF 10, LCM N5.1 Powers and roots
12 8 a and 3 b 6 c a 8 and 40: HCF 4, LCM 80 b 00 and 175: HCF 5, LCM 1400 c 36 and 64: HCF 4, LCM Extension problem 11 a 4 (with factors 1, and 4) b 4 16 (with factors 1,, 4, 8, 16) c 7 factors: factors: factors: factors: days from now, since 40 is the lowest common multiple of 1,, 3, 4, 5, 6 and 7. N5.1 How well are you doing? 1 a 3, 4, 5, 3 3 or 9, 16, 5, 7 b a is the largest b 5 and 7 are not square numbers. 3 a 3 9 b 7 18 c 3 = 18 4 a a 3 b b 5 a HCF is 1 b LCM is Suzy s number is Home book TASK 1 1 a 3 11 b 6 c 11 1 d x 6 e 4 3 f 10 4 g 8 10 h z a b c a (15) 5 (5) 65 b (11) 11 (19) 361 (1) 441 (9) 841 (31) 961 c (6) 3 16 d (13) TASK 1 a 19 b 9 c 9 d 5 e 4 f g 8.67 to d.p. h 4.4 to d.p cm to 1 d.p. 3 a 3 b 4 c 6 d 10 TASK 3 1 a b 3 5 a 3 5 b , 13 and 17 CDROM CHECK UP 1 5 and 7 are not square numbers B: 11 5 cm high by 1 cm wide by 1 cm long cm high by cm wide by 6 cm long cm high by 3 cm wide by 4 cm long N5.1 Powers and roots 13
Contents. Block A Understanding numbers 7. Block C Geometry 53. Block B Numerical operations 30. Page 4 Introduction Page 5 Oral and mental starters
Contents Page 4 Introduction Page 5 Oral and mental starters Block A Understanding numbers 7 Unit 1 Integers and decimals 8 Integers Rounding and approximation Large numbers Decimal numbers Adding and
More informationPROBLEM SOLVING, REASONING, FLUENCY. Year 6 Term 1 Term 2 Term 3 Term 4 Term 5 Term 6 Number and Place Value. Measurement Four operations
PROBLEM SOLVING, REASONING, FLUENCY Year 6 Term 1 Term 2 Term 3 Term 4 Term 5 Term 6 Number and Place Value Addition and subtraction Large numbers Fractions & decimals Mental and written Word problems,
More informationMaths Area Approximate Learning objectives. Additive Reasoning 3 weeks Addition and subtraction. Number Sense 2 weeks Multiplication and division
Maths Area Approximate Learning objectives weeks Additive Reasoning 3 weeks Addition and subtraction add and subtract whole numbers with more than 4 digits, including using formal written methods (columnar
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 informationYear 5. Pupils should identify the place value in large whole numbers.
Year 5 Year 5 programme of study (statutory requirements) Number, place value, approximation and estimation Number, place value, approximation and estimation Pupils should identify the place value in large
More informationSolution: There are TWO square roots of 196, a positive number and a negative number. So, since and 14 2
5.7 Introduction to Square Roots The Square of a Number The number x is called the square of the number x. EX) 9 9 9 81, the number 81 is the square of the number 9. 4 4 4 16, the number 16 is the square
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 informationCALCULATIONS. Pupils should be taught to: As outcomes, Year 7 pupils should, for example:
CALCULATIONS Pupils should be taught to: Consolidate understanding of the operations of multiplication and division, their relationship to each other and to addition and subtraction; know how to use the
More informationMATHS LEVEL DESCRIPTORS
MATHS LEVEL DESCRIPTORS Number Level 3 Understand the place value of numbers up to thousands. Order numbers up to 9999. Round numbers to the nearest 10 or 100. Understand the number line below zero, and
More information2. Perform the division as if the numbers were whole numbers. You may need to add zeros to the back of the dividend to complete the division
Math Section 5. Dividing Decimals 5. Dividing Decimals Review from Section.: Quotients, Dividends, and Divisors. In the expression,, the number is called the dividend, is called the divisor, and is called
More informationMaths Level Targets. This booklet outlines the maths targets for each sublevel in maths from Level 1 to Level 5.
Maths Level Targets This booklet outlines the maths targets for each sublevel in maths from Level 1 to Level 5. Expected National Curriculum levels for the end of each year group are: Year 1 Year 2 Year
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 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 informationread, write, order and compare numbers to at least and determine the value of each digit
YEAR 5 National Curriculum attainment targets Pupils should be taught to: Number Number and place value read, write, order and compare numbers to at least 1 000000 and determine the value of each digit
More informationYear 8  Maths Autumn Term
Year 8  Maths Autumn Term Whole Numbers and Decimals Order, add and subtract negative numbers. Recognise and use multiples and factors. Use divisibility tests. Recognise prime numbers. Find square numbers
More informationYear 5 Mathematics Programme of Study Maths worksheets from mathsphere.co.uk MATHEMATICS. Programme of Study. Year 5 Number and Place Value
MATHEMATICS Programme of Study Year 5 Number and Place Value Here are the statutory requirements: Number and place value read, write, order and compare numbers to at least 1 000 000 and determine the value
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 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 informationGrade 7  Chapter 1 Recall Prior Knowledge
MATH IN FOCUS Grade 7  Chapter 1 Recall Prior Knowledge REFRESH YOUR MEMORY! CHAPTER 1 Recall Prior Knowledge In order to be successful with the new information in Chapter 1, it is necessary to remember
More informationy x x 2 Squares, square roots, cubes and cube roots TOPIC 2 4 x 2 2ndF 2ndF Oral activity Discuss squares, square roots, cubes and cube roots
TOPIC Squares, square roots, cubes and cube roots By the end of this topic, you should be able to: ü Find squares, square roots, cubes and cube roots of positive whole numbers, decimals and common fractions
More informationYear 1 Maths Expectations
Times Tables I can count in 2 s, 5 s and 10 s from zero. Year 1 Maths Expectations Addition I know my number facts to 20. I can add in tens and ones using a structured number line. Subtraction I know all
More informationSupporting your child with maths
Granby Primary School Year 5 & 6 Supporting your child with maths A handbook for year 5 & 6 parents H M Hopps 2016 G r a n b y P r i m a r y S c h o o l 1 P a g e Many parents want to help their children
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 informationA fairly quick tempo of solutions discussions can be kept during the arithmetic problems.
Distributivity and related number tricks Notes: No calculators are to be used Each group of exercises is preceded by a short discussion of the concepts involved and one or two examples to be worked out
More informationYear Five Maths Notes
Year Five Maths Notes NUMBER AND PLACE VALUE I can count forwards in steps of powers of 10 for any given number up to 1,000,000. I can count backwards insteps of powers of 10 for any given number up to
More informationReteaching. Properties of Operations
 Properties of Operations The commutative properties state that changing the order of addends or factors in a multiplication or addition expression does not change the sum or the product. Examples: 5
More informationThere are eight lessons in this unit, Number 4. The objectives covered in this unit are:
Number 4 contents There are eight lessons in this unit, Number 4. N4.1 Multiplying and dividing decimals by 10 or 100 3 N4.2 Equivalence of fractions 6 N4.3 Comparing fractions 9 N4.4 Fractions and percentages
More informationMathematics standards
Mathematics standards Grade 6 Summary of students performance by the end of Grade 6 Reasoning and problem solving Students represent and interpret routine and nonroutine mathematical problems in a range
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 informationTopic Skill Homework Title Count to and across 100, forwards and backwards, beginning with 0 or 1, or from any given number.
Year 1 (Age 56) Number and Place Value Count to and across 100, forwards and backwards, beginning with 0 or 1, or from any given number. Count up to 10 and back (Age 56) Count up to 20 objects (Age 56)
More informationLevel Descriptors Maths Level 15
Level Descriptors Maths Level 15 What is APP? Student Attainment Level Descriptors APP means Assessing Pupil Progress. What are the APP sheets? We assess the children in Reading, Writing, Speaking & Listening,
More informationPossible Stage Two Mathematics Test Topics
Possible Stage Two Mathematics Test Topics The Stage Two Mathematics Test questions are designed to be answerable by a good problemsolver with a strong mathematics background. It is based mainly on material
More informationFive daily lessons. Page 23. Page 25. Page 29. Pages 31
Unit 4 Fractions and decimals Five daily lessons Year 5 Spring term Unit Objectives Year 5 Order a set of fractions, such as 2, 2¾, 1¾, 1½, and position them on a number line. Relate fractions to division
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 informationSwavesey Primary School Calculation Policy. Addition and Subtraction
Addition and Subtraction Key Objectives KS1 Foundation Stage Say and use number names in order in familiar contexts Know that a number identifies how many objects in a set Count reliably up to 10 everyday
More informationSQUARESQUARE ROOT AND CUBECUBE ROOT
UNIT 3 SQUAREQUARE AND CUBEUBE (A) Main Concepts and Results A natural number is called a perfect square if it is the square of some natural number. i.e., if m = n 2, then m is a perfect square where m
More informationTeaching Guidance. For. Counting and Partitioning Numbers
Teaching Guidance For Counting and Partitioning Numbers (Overcoming Barriers Moving Levels 1 to 2, 2 to 3, 3 to 4, 4 to 5) 1 of 2 The National Strategies Primary Overcoming barriers in mathematics helping
More informationDECIMAL MODULE. I. Adding Decimals. II. Subtracting Decimals. III. Multiplying Decimals. IV. Dividing Decimals. BMR.
DECIMAL MODULE I. Adding Decimals II. Subtracting Decimals III. Multiplying Decimals IV. Dividing Decimals BMR.Decimals Page 1 I. Adding Decimals. Introduction: This is the first of four parts on working
More informationMathematics Success Grade 8
T92 Mathematics Success Grade 8 [OBJECTIVE] The student will create rational approximations of irrational numbers in order to compare and order them on a number line. [PREREQUISITE SKILLS] rational numbers,
More informationSection R.2. Fractions
Section R.2 Fractions Learning objectives Fraction properties of 0 and 1 Writing equivalent fractions Writing fractions in simplest form Multiplying and dividing fractions Adding and subtracting fractions
More informationMaths Targets Year 1 Addition and Subtraction Measures. N / A in year 1.
Number and place value Maths Targets Year 1 Addition and Subtraction Count to and across 100, forwards and backwards beginning with 0 or 1 or from any given number. Count, read and write numbers to 100
More informationArithmetic 1 Progress Ladder
Arithmetic 1 Progress Ladder Maths Makes Sense Foundation Endofyear objectives page 2 Maths Makes Sense 1 2 Endofblock objectives page 3 Maths Makes Sense 3 4 Endofblock objectives page 4 Maths Makes
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 informationOral and Mental calculation
Oral and Mental calculation Read and write any integer and know what each digit represents. Read and write decimal notation for tenths and hundredths and know what each digit represents. Order and compare
More informationNumber: Multiplication and Division
MULTIPLICATION & DIVISION FACTS count in steps of 2, 3, and 5 count from 0 in multiples of 4, 8, 50 count in multiples of 6, count forwards or backwards from 0, and in tens from any and 100 7, 9, 25 and
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 informationPythagorean Theorem. Inquiry Based Unit Plan
Pythagorean Theorem Inquiry Based Unit Plan By: Renee Carey Grade: 8 Time: 5 days Tools: Geoboards, Calculators, Computers (Geometer s Sketchpad), Overhead projector, Pythagorean squares and triangle manipulatives,
More information36 The National Strategies Secondary Mathematics exemplification: Y7 NUMBER. Place value, ordering and rounding
36 The National Strategies Secondary Mathematics exemplification: Y7 Pupils should learn to: Understand and use decimal notation and place value; multiply and divide integers and decimals by powers of
More informationMultiples and factors quiz
Level A 1. 18, 27 and 36 are all multiples of nine. 2. 500 is a multiple of 200. 3. Which list is made up of multiples of 12? A) 1, 12, 48 B) 12, 24, 36 C) 12, 22, 32 4. The digit sums (the one digit answer
More informationMaths Progressions Number and algebra
This document was created by Clevedon School staff using the NZC, Maths Standards and Numeracy Framework, with support from Cognition Education consultants. It is indicative of the maths knowledge and
More informationNUMBERS AND THE NUMBER SYSTEM
NUMBERS AND THE NUMBER SYSTEM Pupils should be taught to: Understand and use decimal notation and place value; multiply and divide integers and decimals by powers of 0 As outcomes, Year 7 pupils should,
More informationLesson 3 Compare and order 5digit numbers; Use < and > signs to compare 5digit numbers (S: Bonds to 100)
Abacus Year 5 Teaching Overview Autumn 1 Week Strands Weekly summary 1 Number and placevalue (NPV); and order 5digit Read, write, compare Written addition numbers, understanding and subtraction the placevalue
More informationYear 6 Maths Objectives
Year 6 Maths Objectives Place Value COUNTING COMPARING NUMBERS IDENTIFYING, REPRESENTING & ESTIMATING NUMBERS READING & WRITING NUMBERS UNDERSTANDING PLACE VALUE ROUNDING PROBLEM SOLVING use negative numbers
More information10410 Year 9 mathematics: holiday revision. 2 How many nines are there in fiftyfour?
DAY 1 Mental questions 1 Multiply seven by seven. 49 2 How many nines are there in fiftyfour? 54 9 = 6 6 3 What number should you add to negative three to get the answer five? 8 4 Add two point five to
More informationFactors and Products
CHAPTER 3 Factors and Products What You ll Learn use different strategies to find factors and multiples of whole numbers identify prime factors and write the prime factorization of a number find square
More informationRoots of Real Numbers
Roots of Real Numbers Math 97 Supplement LEARNING OBJECTIVES. Calculate the exact and approximate value of the square root of a real number.. Calculate the exact and approximate value of the cube root
More informationHow do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.
The verbal answers to all of the following questions should be memorized before completion of prealgebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics
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 informationKnowing and Using Number Facts
Knowing and Using Number Facts Use knowledge of place value and Use knowledge of place value and addition and subtraction of twodigit multiplication facts to 10 10 to numbers to derive sums and derive
More informationUnit 5 Length. Year 4. Five daily lessons. Autumn term Unit Objectives. Link Objectives
Unit 5 Length Five daily lessons Year 4 Autumn term Unit Objectives Year 4 Suggest suitable units and measuring equipment to Page 92 estimate or measure length. Use read and write standard metric units
More informationNumber: Fractions (including Decimals and Percentages) Reasoning
Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 COUNTING IN FRACTIONAL STEPS Pupils should count in fractions up to 10, starting from any number and using the1/2 and 2/4 equivalence on the number line (Non Statutory
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 informationNumber & Place Value. Addition & Subtraction. Digit Value: determine the value of each digit. determine the value of each digit
Number & Place Value Addition & Subtraction UKS2 The principal focus of mathematics teaching in upper key stage 2 is to ensure that pupils extend their understanding of the number system and place value
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 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 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 informationIntroduction 6 National curriculum objectives 7
Contents Introduction 6 National curriculum objectives 7 Number Add/subtract Lesson Plan 1 Place value Code book 8 Lesson Plan 2 Counting Stepping stones 12 Lesson Plan 3 Negative numbers Fairground ride
More informationPaper 1. Calculator not allowed. Mathematics test. First name. Last name. School. Remember KEY STAGE 3 TIER 5 7
Ma KEY STAGE 3 Mathematics test TIER 5 7 Paper 1 Calculator not allowed First name Last name School 2009 Remember The test is 1 hour long. You must not use a calculator for any question in this test. You
More informationWorking with whole numbers
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
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 informationOral and mental starter
Lesson Objectives Order fractions and position them on a number line (Y6) Vocabulary gauge, litre numerator, denominator order Resources OHT. individual whiteboards (optional) Using fractions Oral and
More informationIndices and primes. 3.1 Overview TOPIC 3. Why learn this? What do you know? Learning sequence. number and algebra
number and algebra TOPIC 3 N LY Indices and primes 3.1 Overview N AL U AT IO Indices are very useful in everyday life because they allow us to write very large and very small numbers more simply. For calculations
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 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 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 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 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 informationA summary of maths in year 5
Handale Primary School Maths Curriculum Year 5 (910 years old) A summary of maths in year 5 In Year 5 pupils extend their understanding of the number system and place value to include larger numbers,
More informationChapter 4  Decimals
Chapter 4  Decimals $34.99 decimal notation ex. The cost of an object. ex. The balance of your bank account ex The amount owed ex. The tax on a purchase. Just like Whole Numbers Place Value  1.23456789
More informationMathematics Success Grade 6
T276 Mathematics Success Grade 6 [OBJECTIVE] The student will add and subtract with decimals to the thousandths place in mathematical and realworld situations. [PREREQUISITE SKILLS] addition and subtraction
More informationWithin each area, these outcomes are broken down into more detailed stepbystep learning stages for each of the three terms.
MATHEMATICS PROGRAMME OF STUDY COVERAGE all topics are revisited several times during each academic year. Existing learning is consolidated and then built upon and extended. Listed below are the end of
More informationProgression in written calculations in response to the New Maths Curriculum. September 2014
Progression in written calculations in response to the New Maths Curriculum This policy has been written in response to the New National Curriculum, and aims to ensure consistency in the mathematical written
More informationNumber: Fractions (including Decimals and Percentages) COUNTING IN FRACTIONAL STEPS Year 1 Year 2 Year 3 Year 4 Year 5 Year 6
Pupils should begin to count in halves, using practical resources to support Number: Fractions (including Decimals and Percentages COUNTING IN FRACTIONAL STEPS Pupils should count in count up and down
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 informationNUMBERS AND THE NUMBER SYSTEM
NUMBERS AND THE NUMBER SYSTEM Pupils should be taught to: Know the number names and recite them in order, from and back to zero As outcomes, Year 1 pupils should, for example: Join in rhymes like: One,
More informationStage 5 PROMPT sheet. 5/3 Negative numbers 4 7 = 3. l l l l l l l l l /1 Place value in numbers to 1million = 4
Stage PROMPT sheet / Place value in numbers to million The position of the digit gives its size / Negative numbers A number line is very useful for negative numbers. The number line below shows: 7  l
More informationThis is Radical Expressions and Equations, chapter 8 from the book Beginning Algebra (index.html) (v. 1.0).
This is Radical Expressions and Equations, chapter 8 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons byncsa 3.0 (http://creativecommons.org/licenses/byncsa/
More informationAssociative Property The property that states that the way addends are grouped or factors are grouped does not change the sum or the product.
addend A number that is added to another in an addition problem. 2 + 3 = 5 The addends are 2 and 3. area The number of square units needed to cover a surface. area = 9 square units array An arrangement
More informationMULTIPLICATION. Present practical problem solving activities involving counting equal sets or groups, as above.
MULTIPLICATION Stage 1 Multiply with concrete objects, arrays and pictorial representations How many legs will 3 teddies have? 2 + 2 + 2 = 6 There are 3 sweets in one bag. How many sweets are in 5 bags
More informationMath Help and Additional Practice Websites
Name: Math Help and Additional Practice Websites http://www.coolmath.com www.aplusmath.com/ http://www.mathplayground.com/games.html http://www.ixl.com/math/grade7 http://www.softschools.com/grades/6th_and_7th.jsp
More informationExponents, Factors, and Fractions. Chapter 3
Exponents, Factors, and Fractions Chapter 3 Exponents and Order of Operations Lesson 31 Terms An exponent tells you how many times a number is used as a factor A base is the number that is multiplied
More informationDECIMALS. Rounding Decimals Review and Multiplying Decimals. copyright amberpasillas2010. Rounding Review. Decimals are read as and
DECIMALS Rounding Decimals Review and Rounding Review Decimals are read as and 5 1 8 2 1, Thousands Hundreds Tens Ones Tenths Hundredths Read as 518 and 21 hundredths Thousandths Ten Thousandths 1 Rounding
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 informationCharlesworth School Year Group Maths Targets
Charlesworth School Year Group Maths Targets Year One Maths Target Sheet Key Statement KS1 Maths Targets (Expected) These skills must be secure to move beyond expected. I can compare, describe and solve
More informationUnit 8 Angles, 2D and 3D shapes, perimeter and area
Unit 8 Angles, 2D and 3D shapes, perimeter and area Five daily lessons Year 6 Spring term Recognise and estimate angles. Use a protractor to measure and draw acute and obtuse angles to Page 111 the nearest
More informationDecimal Notations for Fractions Number and Operations Fractions /4.NF
Decimal Notations for Fractions Number and Operations Fractions /4.NF Domain: Cluster: Standard: 4.NF Number and Operations Fractions Understand decimal notation for fractions, and compare decimal fractions.
More informationSubtraction. Fractions
Year 1 and across 100, forwards and backwards, beginning with 0 or 1, or from any given number write numbers to 100 in numerals; count in multiples of twos, fives and tens Children continue to combine
More informationMaths Targets  Year 5
Maths Targets  Year 5 By the end of this year most children should be able to Multiply and divide any whole number up to 10000 by 10 or 100. Know what the digits in a decimal number stand for, e.g. the
More informationCalculations Policy. A shared document between parents and teachers.
Calculations Policy A shared document between parents and teachers. R. Fairgrieve November 2015 1 2 Introduction This calculations policy takes into account the changes in the curriculum since September
More information