Maths Plans Year 5
Contents Introduction Introduction 1 Using the Plans 2 Autumn 1 7 Autumn 2 21 Spring 1 53 Spring 2 75 Summer 1 96 Basic Skills 115 Progression 125 The Liverpool Maths team have developed a medium term planning document to support effective implementation of the new National Curriculum. In order to develop fluency in mathematics, children need to secure a conceptual understanding and efficiency in procedural approaches. Our materials highlight the importance of making connections between concrete materials, models and images, mathematical language, symbolic representations and prior learning. There is a key focus on the teaching sequence to ensure that children have opportunities to practise the key skills whilst building the understanding and knowledge to apply these skills into more complex application activities. For each objective, there is a breakdown which explains the key components to be addressed in the teaching and alongside this there are a series of sample questions that are pitched at an appropriate level of challenge for each year group. An additional section (see appendix 1) provides a list of key, basic skills that children must continually practise as they form the building blocks of mathematical learning. 1
Using the plans This is not a scheme but it is more than a medium term plan The programme of study has been split into four domains: Number Measurement Geometry Statistics These allocations serve only as a guide for the organisation of the teaching. Other factors such as term length, organisation of the daily maths lesson, prior knowledge and cross-curricular links may determine the way in which mathematics is prioritised, taught and delivered in your school. As a starting point, we have taken these domains and allocated them into five half terms: Autumn 1 Autumn 2 Spring 1 Spring 2 Summer 1 Year 5 Number - number and place value - addition and subtraction Number - multiplication and division - fractions (including decimals and percentages) Measurement Geometry - properties of shapes - position and direction Statistics 2
Using the plans Within each half term, are some new objectives and some continuous objectives: Year 5 New objectives Continuous objectives Autumn 1 7 3 Autumn 2 18 8 Spring 1 5 10 Spring 2 7 10 Summer 1 2 10 The new objectives vary in length but cover the new learning for that half term, they will not appear again in their entirety. If the objective is in italics, it has been identified as an area that, once taught, should be re-visited and consolidated through basic skills sessions as these key skills form the building blocks of mathematical learning As before, the timings allocated and the organisation and frequency of delivery of these continuous objectives is flexible and will vary from school to school. Please note that Summer 2 has deliberately been left free for the testing period traditionally carried out at the end of summer 1. This also allows the flexibility to allocate time in Summer 2 to target specific areas identified through the assessment process as needing additional teaching time. There are 2 appendices attached: Appendix 1 - List of key basic skills with guidance notes Appendix 2 - Progression through the domains across the key stages The continuous objectives build up as you move through each half term. These objectives cover all the application aspects in mathematics. It is crucial that they are woven into the teaching continually during the year, so that once fluent in the fundamentals of mathematics, children can apply their knowledge rapidly and accurately to problem solving. 3
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YEAR 5 PROGRAMME OF STUDY DOMAIN 1 NUMBER NEW OBJECTIVES AUTUMN 1 NUMBER AND PLACE VALUE Objectives (statutory requirements) What does this mean? Example questions Notes and guidance (non-statutory) Read, write, order and compare numbers to at least 1 000 000 and determine the value of each digit Be able to recognise and record numbers in words and figures Be able to talk about the relative size of numbers, a number bigger than, less than, in between Write 453 002 in words, and vice versa Place 23 683 on a number line from 23 000 to 24 000 Think of a number that lies in between 23 490 and 23 890 Pupils identify the place value in large whole numbers. They continue to use number in context, including measurement. Pupils extend and apply their understanding of the number system to the decimal numbers and fractions that they have met so far. Order consecutive and non-consecutive numbers in ascending and descending order with particular focus on crossing boundaries and the use of zero as a place holder Present number lines in different ways and in different contexts (horizontal number line, vertical scale etc.) and place random numbers between two demarcations on a number line Order these numbers from smallest to largest and largest to smallest 23 542, 23 045, 23 005, 23 504 They should recognise and describe linear number sequences, including those involving fractions and decimals, and find the term-to-term rule. They should recognise and describe linear number sequences (for example, 1 1 3, 3, 4, 4...), including those 2 2 involving fractions and decimals, and find the term-to-term rule in words (for example, add 1). 7
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Count forwards or backwards in steps of powers of 10 for any given number up to 1 000 000 Interpret negative numbers in context, count forwards and backwards with positive and negative whole numbers, including through zero Count out loud, forwards and backwards following the sequence 10, 100, 1000 from different starting points Build on the counting skills identified previously to include bridging zero into negative numbers Using different starting points, count forwards and using multiples such as 6, 7, 9, 25 and 1000 bridging zero Starting at 10², children count in powers of 10 Starting from 54, count in multiples of 9 forwards and backwards Progression shown through starting the count from a number that is not a multiple of the step size Starting from 55, count in steps of 9 forwards and backwards Read and interpret negative numbers on a variety of scales Present scales both vertically and horizontally Round any number up to 1 000 000 to the nearest 10, 100, 1000. 10000, 100 000 Read Roman numerals to 1000 (M) and recognise years written in Roman numerals Using any number up to six digits, be able to round to one or more of the five criteria, 10, 100, 1000, 10000, 100 000 Ensure children can match Roman numerals to numbers Think of the number 78 456, round it to the range of criteria specified on the left Is 78 456 nearer to 70 000 or 80 000? Explain how you know CXII in Roman numerals represents which number? 9
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NEW OBJECTIVES AUTUMN 1 ADDITION AND SUBTRACTION Add and subtract whole numbers with more than 4 digits, including using formal written methods (columnar addition and subtraction) Teaching to be in line with school Calculation Policy Methods: Expanded columnar Column Progression shown through: Start point THTU ± THTU (bridging 10 and 100) Expanded columnar Pupils practise using the formal written methods of columnar addition and subtraction with increasingly large numbers to aid fluency (see Mathematics Appendix 1). They practise mental calculations with increasingly large numbers to aid fluency (for example, 12 462-2300 = 10 162). End point THTU.t h ± THTU.t h (with multiple bridging) Column 11
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Add and subtract numbers mentally with increasingly large numbers Children need to be secure with the skills of bridging, partitioning, doubling and know their number bonds and pairs up to ten to add and subtract mentally Building on the skills introduced in Year 4, children add and subtract mentally THTU ± U, THTU ± T, THTU ± H, TU ± TU and HTU ±TU, increasing the complexity through the introduction of further bridging 1236 + 4, 1236 + 40, 1236 + 400, 36 + 57 1236 + 7, 1236 + 70, 1236 + 700, 136 + 57 13
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CONTINUOUS OBJECTIVES AUTUMN 1 Solve number problems and practical problems that relate to all of the above (number and place value) Be able to answer word and reasoning problems linked to place value Emma has used these digit cards to make the number 367.98 How many numbers with two decimal places can you make that round to 600? If you made the number that is seven tenths less than Emma s, which new digit card would you need? What is the smallest number with two decimal places that you can make? If you also had a zero digit card, how would this change your answer? Convince me that the number half way between 12.2 and 40.6 is 26.4 Find the numbers that could fit the following clues: Less than 100 and prime Not a multiple of 5 but a multiple of 3 Not odd but a square number Tens digit is double the hundredths digit 15
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Be able to use known facts in order to explore others. Include commutativity and inverse and other relationships between numbers: 42 x 8 is also 84 x 4 because one side of the multiplication is halved, the other side is doubled Are these statements true? If 32 x 8 = 256 then 256 8 = 32 If 32 x 8 = 256 then 256 32 = 8 If 32 x 8 = 256 then 8 256 = 32 If 32 x 8 = 256 then 320 x 80 = 2560 Starting with 42 x 8 = 336: 42 x 8 = 336 (and 336 = 42 x 8, 336 = 8 x 42) Understanding the inverse relationship between multiplication and division leads to equivalent statements, such as 42 = 336 8 Knowing division is not commutative, so 8 42 336 17
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Use rounding to check answers to calculations and determine, in the context of a problem, levels of accuracy Solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why Working with numbers up to THTU.t h digits, ensure that children have opportunities to: Estimate the answer Evidence the skill of addition and/or subtraction Prove the inverse using the skill of addition and/or subtraction Practice calculation skill including units of measure (km, m, cm, mm, kg, g, l, cl, ml, hours, minutes and seconds) Following the calculation sequence: Estimate 1245.85 + 1123.36 Calculate 1245.85 + 1123.36 Prove 2369.21 1123.36 = 1245.85 Calculate 2369.21m 1123.36m Solve missing box questions including those where missing box represents a digit or represents a number 2369.21cm - = 1245.85cm Solve problems including those with more than one step I have 1245.85 litres of water in one container and 1123.36 litres in another container, how much do I have altogether? I pour out 450 litres, how much is now left? Solve open-ended investigations Using the digit cards 1 to 9, make the smallest/biggest answer, an answer that is odd/even etc. 19
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YEAR 5 PROGRAMME OF STUDY DOMAIN 1 NUMBER NEW OBJECTIVES AUTUMN 2 MULTIPLICATION AND DIVISION Objectives (statutory requirements) What does this mean? Example questions Notes and guidance (non-statutory) Identify multiples and factors, including finding all factor pairs of a number, and common factors of two numbers Know and use the vocabulary of prime numbers, prime factors and composite (non-prime) numbers From a two-digit number, children can identify all factor pairs For all multiplication tables up to 12 x 12, children can identify multiples When given a pair of two-digit numbers, children can identify all factors that are common to both numbers A prime number is a number that can be divided evenly only by 1 or itself and it must be a whole number greater than one A composite (or non-prime) number is a whole number that can be divided evenly by numbers other than 1 and itself Prime factorisation is finding which prime numbers multiply together to make the original number List all the factor pairs of 24 Write all the two-digit multiples of 11 What are the common factors for the numbers 24 and 32? Circle the prime numbers in this list: 12, 3, 21, 23, 30 From a given set of numbers, identify which are prime and which are composite Find the prime factors of 12 (2 x 2 x 3) Pupils practise and extend their use of the formal written methods of short multiplication and short division (see Mathematics Appendix 1). They apply all the multiplication tables and related division facts frequently, commit them to memory and use them confidently to make larger calculations. They use and understand the terms factor, multiple and prime, square and cube numbers. Pupils interpret non-integer answers to division by expressing results in different ways according to the context, including with remainders, as fractions, as decimals or by rounding 8 (for example, 98 4 = 9 = 24 r 2 1 4 = 24 = 24.5 25). 2 Pupils use multiplication and division as inverses to support the introduction of ratio in year 6, for example, by multiplying and dividing by powers of 10 in scale drawings or by multiplying and dividing by powers of a 1000 in converting between units such as kilometres and metres. 21
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Establish whether a number up to 100 is prime and recall prime numbers up to 19 Multiply numbers up to 4 digits by a one- or two-digit number using a formal written method, including long multiplication for twodigit numbers Recall prime numbers up to 19 and derive prime numbers between 20 and 100 Teaching to be in line with school Calculation Policy Methods for X: Partitioning (grid) Short Long Progression shown through: Find all the prime numbers between 35 and 49 Partitioning (grid) Distributivity can be expressed as a(b + c) = ab + ac. They understand the terms factor, multiple and prime, square and cube numbers and use them to construct equivalence statements (for example, 4 x 35 = 2 x 2 x 35; 3 x 270 = 3 x 3 x 9 x 10 = 92 x 10). Pupils use and explain the equals sign to indicate equivalence, including in missing number problems (for example, 13 + 24 = 12 + 25; 33 = 5 x ). HTU x U THTU x U TU x TU Short Long 23
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Multiply and divide numbers mentally drawing upon known facts Using knowledge of multiplication tables to 12 x 12, children can recall and derive associated facts Include chanting of multiplication tables both consecutively and non-consecutively Recall of facts such as 6 x 8, 12 x 7, 40 5 Explore commutativity of multiplication Knowing that 0.8 x 7 is the same as 7 x 0.8 and that multiplication (without brackets) can be done in any order Recall related division facts and explore the inverse relationship of multiplication and division If 7 x 0.8 = 5.6, what are the related division facts? Using x and, 7, 0.8 and 5.6, write down some number sentences Sam multiplies two numbers together and gets the answer 3.6, what could his two numbers be? Know that to multiply by 12 is the same as multiplying by 3 then double and double again. Explore other similar patterns within multiplication tables 15 x 12 = 15 x 3 doubled and doubled again 25
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Divide numbers up to 4 digits by a one-digit number using the formal written method of short division and interpret remainders appropriately for the context Teaching to be in line with school Calculation Policy Methods for : Short Progression shown through: HTU U THTU U Expressing any remainders first using the notation r moving onto expression as a fraction then as a decimal To find a decimal remainder, use the skills of converting a fraction to a decimal, consolidating the links between fractions and decimals Short or 1441.67 correct to 2 decimal places Multiply and divide whole numbers and those involving decimals by 10, 100 and 1000 Use knowledge of place value columns when multiplying and dividing by 10, 100 and 1000 (i.e. when moving from right to left, each place value column is ten times bigger and vice versa) x 1000 = 28 300 5432 = 54.32 50.05 x 10 = 27
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Recognise and use square numbers and cube numbers, and the notation for squared ( 2 ) and cubed ( 3 ) A square number is formed by multiplying a digit by itself A cube number is formed by multiplying a digit by itself three times What is 7 squared? 55 is a square number, true or false? From the following numbers, which are squared, which are cubed which fit neither criteria? Ensure the correct notation is used and applied when teaching the objectives for area and volume in Spring 1 49, 13, 56, 81, 125, 343, 8, 104 29
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NEW OBJECTIVES AUTUMN 2 FRACTIONS (INCLUDING DECIMALS AND PERCENTAGES) Compare and order fractions whose denominators are all multiples of the same number Building on the work on fraction families in Year 4, children can order a set of fractions where the denominators are all multiples of the same number Start by using images to show how fractions, where denominators are multiples of the same number, can be compared When comparing fractions it is easier when the denominators are the same (that is, by finding a common denominator) Convert fractions using the skills of multiplication and the knowledge of fraction families, so that they have the same denominator and then be able to compare and order them Order this set of fractions,,, Use a fraction board to help initially and then progress into converting all fractions into twelfths 2 3 5 6 9 12 Pupils should be taught throughout that percentages, decimals and fractions are different ways of expressing proportions. They extend their knowledge of fractions to thousandths and connect to decimals and measures. Pupils connect equivalent fractions > 1 that simplify to integers with division and other fractions > 1 to division with remainders, using the number line and other models, and hence move from these to improper and mixed fractions. Pupils connect multiplication by a fraction to using fractions as operators (fractions of), and to division, building on work from previous years. This relates to scaling by simple fractions, including fractions > 1. Identify, name and write equivalent fractions of a given fraction, represented visually, including tenths and hundredths Equivalent fractions have the same value even though they may look different because when you multiply or divide both the numerator and denominator by the same number, the fraction keeps its value When given a fraction, children can derive other fractions that are equivalent to it using the skills of multiplication and division and the knowledge of fraction families Find two fractions that are equivalent to 3 8 Pupils practise adding and subtracting fractions to become fluent through a variety of increasingly complex problems. They extend their understanding of adding and subtracting fractions to calculations that exceed 1 as a mixed number. 31
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Recognise mixed numbers and improper fractions and convert from one form to the other and write mathematical statements > 1 as a mixed number [ for example, + = 1 = 1 ] 5 2 5 4 5 6 5 Add and subtract fractions with the same denominator and denominators that are multiples of the same number A proper fraction has a numerator smaller than the denominator An improper fraction has a numerator larger than (or equal to) the denominator Mixed numbers can also be called mixed fractions. A mixed fraction is a whole number and a proper fraction combined Children can convert mixed fractions to improper fractions and vice versa From a selection of mixed fractions and improper fractions, children can use the skills of conversion to place them in ascending and descending order Use denominators up to 10, ensure accurate notation used and calculations extend beyond one whole When given fractions where denominators are different but multiples of the same number, children can use skills of conversion so that the fractions have the same denominator and then are able to add and subtract Convert 2 to an improper fraction Order this set of fractions from smallest to 2 5 9 15 8 largest, 2,,, Use the skill of converting all fractions into twelfths 5 6 15 12 5 6 15 12 + _ + _ 5 6 9 12 2 3 2 3 3 3 8 6 12 12 3 Pupils continue to practise counting forwards and backwards in simple fractions. Pupils continue to develop their understanding of fractions as numbers, measures and operators by finding fractions of numbers and quantities. Pupils extend counting from year 4, using decimals and fractions including bridging zero, for example on a number line. Pupils say, read and write decimal fractions and related tenths, hundredths and thousandths accurately and are confident in checking the reasonableness of their answers to problems. They mentally add and subtract tenths, and one-digit whole numbers and tenths. They practise adding and subtracting decimals, including a mix of whole numbers and decimals, decimals with different numbers of decimal places, and complements of 1 (for example, 0.83 + 0.17 = 1). 33
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Multiply proper fractions and mixed numbers by whole numbers, supported by materials and diagrams Read and write decimal numbers as fractions 71 [for example, 0.71 = ] 100 Using images to support, children can multiply proper fractions by whole numbers Using images to support, children can multiply mixed numbers by whole numbers For any decimal number up to three decimal places but less than 1, children can express it as a fraction with a denominator of 10 and/or 100 and/or 1000 3 2 x = = 1 4 1 6 4 2 x 1 = 2 4 2 4 2 4 Express each of these numbers as a fraction: 0.8, 0.85, 0.857 Pupils should go beyond the measurement and money models of decimals, for example, by solving puzzles involving decimals. Pupils should make connections between percentages, fractions and decimals (for example, 100% represents a whole quantity and 1% 1 50 25 is, 50% is, 25% is ) and 100 100 100 relate this to finding fractions of. Recognise and use thousandths and relate them to tenths, hundredths and decimal equivalents Build on the knowledge of place value columns to include tenths and hundredths Reinforce the relationship between 1 the place value columns i.e. is ten 10 1 1 times bigger than, is ten times 1 100 100 bigger than 1000 Express 251 1000 as a decimal Link equivalent fractions to compare fractions. For example the knowledge 1 10 that = and to find decimal 10 100 equivalents Express 0.54 as a fraction 9 10 90 = = = 0.9 100 900 1000 35
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Round decimals with two decimal places to the nearest whole number and to one decimal place Read, write, order and compare numbers with up to three decimal places When rounding to the nearest whole number, children understand that the value of the tenth digit will determine whether they round up or down When rounding to one decimal place, children understand that the value of the hundredth digit will determine whether they round up or down Be able to recognise and record numbers in words and figures Order consecutive and non-consecutive numbers in ascending and descending order with particular focus on presenting sets of numbers that have a mix of one, two and three decimal places Repeat this with units of measure and money Be able to talk about the relative size of numbers, a number bigger than, less than, in between Round 15.47 to the nearest whole number Round 15.47 to one decimal place Three hundred and six point four seven nine Write this number in figures and then in words Order this set of numbers 54.673, 504.67, 54.67, 54.679, 54.03, 54.003 From a set of numbers with up to three decimal places, use the inequality symbols (< > ) to compare When presented with number lines place random numbers between two demarcations on a number line, working with numbers up to three decimal places From a number line with a start number of 54.3 and an end number of 54.5, place the number 54.38 37
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Recognise the per cent symbol (%) and understand that per cent relates to number of parts per hundred, and write percentages as a fraction with denominator 100, and as a decimal Per cent means per 100 Children understand the relationship between percentages, fractions and decimals When making these connections, children work with fractions with a denominator of 100 and convert these to decimals with up two decimal places Express the shaded area as a fraction and/or decimal and/or percentage Express 23% as both a fraction and a decimal Express 0.57 as both a fraction and a percentage Express 3 10 as both a decimal and a percentage 39
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CONTINUOUS OBJECTIVES AUTUMN 2 Solve number problems and practical problems that relate to all of the above (number and place value) Be able to answer word and reasoning problems linked to place value Emma has used these digit cards to make the number 367.98 How many numbers with two decimal places can you make that round to 600? If you made the number that is seven tenths less than Emma s, which new digit card would you need? What is the smallest number with two decimal places that you can make? If you also had a zero digit card, how would this change your answer? Convince me that the number half way between 12.2 and 40.6 is 26.4 Fill in the missing numbers: 0.6 x = 60 1000 = 1.6 6.03 x = 603 41
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Find the numbers that could fit the following clues: Less than 100 and prime Not a multiple of 5 but a multiple of 3 Not odd but a square number Tens digit is double the hundredths digit Be able to use known facts in order to explore others. Include commutativity and inverse and other relationships between numbers: 42 x 8 is also 84 x 4 because one side of the multiplication is halved, the other side is doubled Are these statements true? If 32 x 8 = 256 then 256 8 = 32 If 32 x 8 = 256 then 256 32 = 8 If 32 x 8 = 256 then 8 256 = 32 If 32 x 8 = 256 then 320 x 80 = 2560 Starting with 42 x 8 = 336: 42 x 8 = 336 (and 336 = 42 x 8, 336 = 8 x 42) Understanding the inverse relationship between multiplication and division leads to equivalent statements, such as 42 = 336 8 Knowing division is not commutative, so 8 42 336 43
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Use rounding to check answers to calculations and determine, in the context of a problem, levels of accuracy Solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why Solve problems involving number up to three decimal places Working with numbers up to THTU.t h digits, ensure that children have opportunities to: Estimate the answer Evidence the skill of multiplication and division Prove the inverse using the skill of multiplications and division Practice calculation skill including units of measure (km, m, cm, mm, kg, g, l, cl, ml, hours, minutes and seconds) Solve missing box questions including those where missing box represents a digit or represents a number Solve problems including those with more than one step Following the calculation sequence: Estimate 1245.85 + 1123.36 Calculate 1245.85 + 1123.36 Prove 2369.21 1123.36 = 1245.85 Calculate 2369.21m 1123.36m 2369.21cm - = 1245.85cm I have 1245.85 litres of water in one container and 1123.36 litres in another container, how much do I have altogether? I pour out 450 litres, how much is now left? Solve open-ended investigations Using the digit cards 1 to 9, make the smallest/biggest answer, an answer that is a multiple of 5 etc. 45
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Solve problems involving multiplication and division including using their knowledge of factors and multiples, squares and cubes Working with numbers up HTU x U or THTU x U (where the answer is a 3 or 4 digit number) and HTU U or THTU U, ensure that children have opportunities to: Estimate the answer Following the calculation sequence: Estimate 214 x 7 = Solve problems involving addition, subtraction, multiplication and division and a combination of these, including understanding the meaning of the equals sign Evidence the skill of addition and/or subtraction Prove the inverse using the skill of addition and/or subtraction Practice calculation skill including units of measure (km, m, cm, mm, kg, g, l, cl, ml, hours, minutes and seconds) Calculate 214 x 7 = Prove 1498 7 = 214 Calculate 214 ml x 7 = Solve missing box questions including those where missing box represents a digit or represents a number 1498 = 214 Solve problems including those with more than one step One full barrel holds 214 litres and there are 7 full barrels, how much do I have altogether? I sell 2 barrels, how many litres do I have left? Solve open-ended investigations Using the digit cards 1 to 9, make the smallest/biggest answer, an answer that is odd/even etc. 47
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Solve problems involving multiplication and division, including scaling by simple fractions and problems involving simple rates. For multiplication and division, refer to Following the calculation sequence: above Scaling problems use the skills of multiplication and division for scaling up and down Link to work with measures by using recipes A recipe for 4 persons that must be scaled down to show quantities for 1 person using division skills A recipe for 2, that must be scaled up to feed 10 people, using the skills of multiplication Solve problems which require knowing percentage and decimal equivalents of 1 2 1,,,, and 4 1 5 those fractions with a denominator of a multiple of 10 or 25. 2 5 4 5 Children use the skills of converting between fractions, decimals and percentages and apply this in a problem solving context When making these connections, children work with fractions with a denominator of 100, 50, 25, 20 and 10 Which of the following discounts is the greatest and which is the least: 5 3, 0.25,, 0.3, 35%? 25 10 Here is a set of prices. All prices are to increase by 10%, calculate the new prices 450, 399, 505 If a television cost 300 and is reduced by 10%, what is the new price? A standard cereal box holds 500g. If you get 1 extra free, how many grams are in the box 4 now? There are 40 sweets in a packet. David eats some and there are now only 60% left. How many sweets has he eaten? 49
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YEAR 5 PROGRAMME OF STUDY DOMAIN 2 MEASUREMENT NEW OBJECTIVES - SPRING 1 Objectives What does this mean? Example questions Notes and guidance (statutory requirements) (non-statutory) Convert between different units of metric measure (for example, When converting, children will use decimal notation up to 3 decimal places Pupils use their knowledge of place value and multiplication and division to convert between standard units. kilometre and metre; centimetre and metre; centimetre and millimetre; gram and kilogram; litre and millilitre) Understand the explicit link with x and when converting, and build on the skills of multiplying and dividing by 10,100 and 1000 (e.g. there are 1000m in a km, therefore when converting km to m, multiply by 1000) Include lengths (km, m, cm, mm), mass (kg, g), volume/capacity (l, cl, ml) Using the full range of units of measure ask questions such as: Convert 3.7km into m If I was converting g to kg, would I multiply or divide by 10, 100 or 1000? True or false? 7539 cl = 7.539 l Pupils calculate the perimeter of rectangles and related composite shapes, including using the relations of perimeter or area to find unknown lengths. Missing measures questions such as these can be expressed algebraically, for example 4 + 2b = 20 for a rectangle of sides 2 cm and b cm and perimeter of 20cm. Pupils calculate the area from scale drawings using given measurements. Pupils use all four operations in problems involving time and money, including conversions (for example, days to weeks, expressing the answer as weeks and days). 53
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Understand and use approximate equivalences between metric units and common imperial units such as inches, pounds and pints Know which measures are imperial and which are metric Know common abbreviations for units Use approximate equivalents introducing sign meaning approximately equal to Length: Inches compared to centimetres using 1 2.5cm Miles compared to kilometre using 1 mile 1.5km Other commonly used equivalents for length include: 1cm = 2.5 inches 5 1km mile 8 400m = 1 mile 4 8km = 5 miles Answer questions such as: Approximately how many pints in 8 litres? 5 = cm (approximately) If I run 5 miles, approximately how many km is this? Mass: Pounds compared to kilograms using 2.2lb 1kg Volume/Capacity: Pints compared to litres using 2.2 pints 1litre 55
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Measure and calculate the perimeter of composite rectilinear shapes in centimetres and metres A composite shape is made up of two or more geometric shapes To find its area, it must be broken up into smaller shapes A rectilinear shape is one with right angles at all its vertices Calculate the perimeter of shapes by measuring the sides accurately with a ruler and /or calculating any unknown or missing lengths Calculate the perimeter of this shape 57
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Calculate and compare the area of rectangles (including squares), and including using standard units, square centimetres (cm 2 ) and square metres (m 2 ) and estimate the area of irregular shapes Building in the work on arrays in year 4, children understand that for a rectangle with length = a and width = b, area = ( a x b ) units ² Progression to be shown through: Starting with rectangles that are demarcated into square centimetres Calculate the area of this rectangle which has been drawn on cm² paper Build up to rectangles where lengths of sides are given I have a rectangle with a length of 13cm and a width of 6cm, calculate the area Finish with rectangles that require an accurate measurement of sides and then use of formula to find the area This is a scale diagram of a swimming pool with length of 15m and a width of 3m. What is the area of the swimming pool? Draw some rectangles with a perimeter of 20cm and then calculate their areas. Use knowledge of the area of a rectangle to make estimates of areas of irregular shapes Use a ruler to measure and then estimate the area of this shape in cm² 59
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Estimate volume [for example, using 1 cm 3 blocks to build cuboids (including cubes)] and capacity [for example, using water] Volume is a measure of the space taken up by something that is either a liquid or a solid Capacity is the amount that a given container can hold Units of measure are: Volume of liquid is measured in litres (l), centilitres(cl), and millilitres (ml) Capacity is measured in litres (l), centilitres (cl) and millilitres (ml) The capacity of this measuring cylinder is 300ml. the volume of liquid in the jug is 150ml. Show similar images with a variety of scales, including partially demarcated scales where estimates must be made Volume of a solid is measured in cubic metres (m³) or cubic centimetres (cm³) Using cm³ build a cuboid with a volume of 24cm³ Each edge of this cube measures 5cm. What is the volume of the cube? 61
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CONTINUOUS OBJECTIVES SPRING 1 Solve number problems and practical problems that relate to all of the above (number and place value) Be able to answer word and reasoning problems linked to place value Emma has used these digit cards to make the number 367.98 How many numbers with two decimal places can you make that round to 600? If you made the number that is seven tenths less than Emma s, which new digit card would you need? What is the smallest number with two decimal places that you can make? If you also had a zero digit card, how would this change your answer? Convince me that the number half way between 12.2 and 40.6 is 26.4 Fill in the missing numbers: 0.6 x = 60 1000 = 1.6 6.03 x = 603 63
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Find the numbers that could fit the following clues: Less than 100 and prime Not a multiple of 5 but a multiple of 3 Not odd but a square number Tens digit is double the hundredths digit Be able to use known facts in order to explore others. Include commutativity and inverse and other relationships between numbers: 42 x 8 is also 84 x 4 because one side of the multiplication is halved, the other side is doubled Are these statements true? If 32 x 8 = 256 then 256 8 = 32 If 32 x 8 = 256 then 256 32 = 8 If 32 x 8 = 256 then 8 256 = 32 If 32 x 8 = 256 then 320 x 80 = 2560 Starting with 42 x 8 = 336: 42 x 8 = 336 (and 336 = 42 x 8, 336 = 8 x 42) Understanding the inverse relationship between multiplication and division leads to equivalent statements, such as 42 = 336 8 Knowing division is not commutative, so 8 42 336 65
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Use rounding to check answers to calculations and determine, in the context of a problem, levels of accuracy Solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why Solve problems involving number up to three decimal places Working with numbers up to THTU.t ht, ensure that children have opportunities to: Estimate the answer Evidence the skill of multiplication and division Prove the inverse using the skill of multiplications and division Practice calculation skill including units of measure (km, m, cm, mm, kg, g, l, cl, ml, hours, minutes and seconds) Solve missing box questions including those where missing box represents a digit or represents a number Solve problems including those with more than one step Solve open-ended investigations Following the calculation sequence: Estimate 1245.85 + 1123.36 Calculate 1245.85 + 1123.36 Prove 2369.21 1123.36 = 1245.85 Calculate 2369.21m 1123.36m 2369.21cm - = 1245.85cm I have 1245.85 litres of water in one container and 1123.36 litres in another container, how much do I have altogether? I pour out 450 litres, how much is now left? Using the digit cards 1 to 9, make the smallest/biggest answer, an answer that is odd/even etc. 67
Notes 68
Solve problems involving multiplication and division including using their knowledge of factors and multiples, squares and cubes Working with numbers up HTU x U or THTU x U (where the answer is a 3 or 4 digit number) and HTU U or THTU U, ensure that children have opportunities to: Estimate the answer Following the calculation sequence: Estimate 214 x 7 = Solve problems involving addition, subtraction, multiplication and division and a combination of these, including understanding the meaning of the equals sign Evidence the skill of multiplication and division Prove the inverse using the skill of multiplications and division Practice calculation skill including units of measure (km, m, cm, mm, kg, g, l, cl, ml, hours, minutes and seconds) Calculate 214 x 7 = Prove 1498 7 = 214 Calculate 214 ml x 7 = Solve missing box questions including those where missing box represents a digit or represents a number Solve problems including those with more than one step Solve open-ended investigations 1498 = 214 One full barrel holds 214 litres and there are 7 full barrels, how much do I have altogether? I sell 2 barrels, how many litres do I have left? Using the digit cards 1 to 9, make the smallest/biggest answer, an answer that is a multiple of 5 etc. 69
Notes 70
Solve problems involving multiplication and division, including scaling by simple fractions and problems involving simple rates. For multiplication and division, refer to Following the calculation sequence: above Scaling problems use the skills of multiplication and division for scaling up and down Link to work with measures by using recipes A recipe for 4 persons that must be scaled down to show quantities for 1 person using division skills A recipe for 2, that must be scaled up to feed 10 people, using the skills of multiplication Solve problems which require knowing percentage and decimal equivalents of 1 2 1,,,, and 4 1 5 those fractions with a denominator of a multiple of 10 or 25. 2 5 4 5 Use the skills of converting between fractions, decimals and percentages and apply this in a problem solving context When making these connections, children work with fractions with a denominator of 100, 50, 25, 20, 10, 5 and 2 Which of the following discounts is the greatest and which is the least: 5 3, 0.25,, 0.3, 35%? 25 10 Here is a set of prices. All prices are to increase by 10%, calculate the new prices 450, 399, 505 If a television cost 300 and is reduced by 10%, what is the new price? A standard cereal box holds 500g. If you get 1 extra free, how many grams are in the box 4 now? There are 40 sweets in a packet. David eats some and there are now only 60% left. How many sweets has he eaten? 71
Notes 72
Solve problems involving converting between units of time Be able to convert: hours to minutes minutes to seconds years to months weeks to days and vice versa, applying this skill when solving problems Give problems that include mixed units and that specify how an answer is expressed so that a conversion is required 3.5 years = months = days 3 runners ran a marathon and their times were recorded as such: Runner A = 4 hours 12 minutes 3 seconds Runner B = 254 minutes 25 seconds Runner C = 17 200 seconds Place the runners in order of fastest to slowest Use all four operations to solve problems involving measure [for example, length, mass, volume, money] using decimal notation, including scaling. For problem solving all four operations refer to Following the calculation sequence: above Scaling problems involve changing the quantities for groups of different size. Scaling down to decrease quantities and scale up to increase quantities Include decimal notation in measures Here is the recipe to make 25 cookies. If you need to make 100 cookies, calculate the new quantities you would need Here is another recipe but I only need Calculate the new quantities 1 3 of it. 73
Notes 74
YEAR 5 PROGRAMME OF STUDY DOMAIN 3 GEOMETRY NEW OBJECTIVES SPRING 2 PROPERTIES OF SHAPES Objectives What does this mean? Example questions Notes and guidance (statutory requirements) (non-statutory) Identify 3-D shapes, including cubes and other cuboids, from 2-D representations Know angles are measured in degrees: estimate and compare acute, obtuse and reflex angles Name a 3-D shape and describe its properties based on a 2-D representation Include shapes such as cube, cuboid, pyramids, prisms, spheres When given a set of angles, children can classify as acute, obtuse or reflex Using knowledge that a right angle = 90 and that a full turn = 360, children can estimate the size of an angle to a reasonable degree of accuracy Name this shape and describe its properties Include number and shapes of the faces and number of edges and vertices Name each angle and estimate its size in degrees Pupils become accurate in drawing lines with a ruler to the nearest millimetre, and measuring with a protractor. They use conventional markings for parallel lines and right angles. Pupils use the term diagonal and make conjectures about the angles formed between sides, and between diagonals and parallel sides, and other properties of quadrilaterals, for example using dynamic geometry ICT tools. Pupils use angle sum facts and other properties to make deductions about missing angles and relate these to missing number problems. Draw given angles, and measure them in degrees ( ) Using a ruler and protractor, children can draw angles with a good degree of accuracy Draw and label the following angles: an acute angle measuring 55 an obtuse angle measuring 130 a reflex angle measuring 280 75
Notes 76
Identify: -angles at a point and one whole turn (total 360 ) angles at a point on a 1 straight line and a turn 2 (total 180 ) -other multiples of 90 Children understand the relationship between a right angle, a straight angle and a whole turn and the associated measurements in degrees Answer questions such as: If I face west and then turn clockwise through 3 right angles, what direction am I facing now? If I complete 4.5 turns, how many right angles have I turned through? Use the properties of rectangles to deduce related facts and find missing lengths and angles Distinguish between regular and irregular polygons based on reasoning about equal sides and angles Know that the interior angles of a rectangle comprise of 4 right angles and add to 360 Know that parallel sides in a rectangle are equal in length Know that the perimeter of a rectangle is calculated by adding the length of all 4 sides or by using the formula perimeter = 2 ( a + b ) units Know that the area of a rectangle is calculated by multiplying the length by the width and can use the formula, area = ( a x b ) units² A polygon is a 2-dimensional shape made up of straight lines If all the angles and sides are equal it is regular, otherwise it is irregular Polygons include shapes such as triangles, quadrilaterals, pentagons, hexagons, heptagons and octagons When reasoning about shapes include reference to regular / irregular, number and properties of sides, number and size of angles c 8cm What is the size of angle c? What is the length of b? b If the area is 32cm², what is the length of a? What is the perimeter of the rectangle? Name these polygons and describe their properties a 77
Notes 78
NEW OBJECTIVES SPRING 2 POSITION AND DIRECTION identify, describe and represent the position of a shape following a reflection or translation, using the appropriate language, and know that the shape has not changed. When a shape is reflected, it ends up facing the opposite direction, appearing to be reflected as in a mirror. The movement is a flip With the mirror line vertical or horizontal, children can reflect a given shape accurately Reflect these shapes in the given mirror line Pupils recognise and use reflection and translation in a variety of diagrams, including continuing to use a 2-D grid and coordinates in the first quadrant. Reflection should be in lines that are parallel to the axes. This should include examples where the shape touches the mirror line as well as examples where it does not When a shape is translated, it moves from one place to another. The movement is a slide Every point of the shape must move the same distance in the same direction Translation will be described using the vocabulary of left, right, up and down Translate this shape by moving it down 4 and 2 to the left 79
Notes 80
CONTINUOUS OBJECTIVES SPRING 2 Solve number problems and practical problems that relate to all of the above (number and place value) Be able to answer word and reasoning problems linked to place value Emma has used these digit cards to make the number 367.98 How many numbers with two decimal places can you make that round to 600? If you made the number that is seven tenths less than Emma s, which new digit card would you need? What is the smallest number with two decimal places that you can make? If you also had a zero digit card, how would this change your answer? Convince me that the number half way between 12.2 and 40.6 is 26.4 Fill in the missing numbers: 0.6 x = 60 1000 = 1.6 6.03 x = 603 Find the numbers that could fit the following clues: Less than 100 and prime Not a multiple of 5 but a multiple of 3 Not odd but a square number Tens digit is double the hundredths digit 81
Notes 82
Be able to use known facts in order to explore others. Include commutativity and inverse and other relationships between numbers: 42 x 8 is also 84 x 4 because one side of the multiplication is halved, the other side is doubled Are these statements true? If 32 x 8 = 256 then 256 8 = 32 If 32 x 8 = 256 then 256 32 = 8 If 32 x 8 = 256 then 8 256 = 32 If 32 x 8 = 256 then 320 x 80 = 2560 Starting with 42 x 8 = 336: 42 x 8 = 336 (and 336 = 42 x 8, 336 = 8 x 42) Understanding the inverse relationship between multiplication and division leads to equivalent statements, such as 42 = 336 8 Knowing division is not commutative, so 8 42 336 83
Notes 84
Use rounding to check answers to calculations and determine, in the context of a problem, levels of accuracy Solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why Solve problems involving number up to three decimal places Working with numbers up to four digits, ensure that children have opportunities to: Estimate the answer Evidence the skill of addition and/or subtraction Prove the inverse using the skill of multiplications and division Practice calculation skill including units of measure (km, m, cm, mm, kg, g, l, cl, ml, hours, minutes and seconds) Solve missing box questions including those where missing box represents a digit or represents a number Solve problems including those with more than one step Solve open-ended investigations Following the calculation sequence: Estimate 1245.85 + 1123.36 Calculate 1245.85 + 1123.36 Prove 2369.21 1123.36 = 1245.85 Calculate 2369.21m 1123.36m 2369.21cm - = 1245.85cm I have 1245.85 litres of water in one container and 1123.36 litres in another container, how much do I have altogether? I pour out 450 litres, how much is now left? Using the digit cards 1 to 9, make the smallest/biggest answer, an answer that is odd/even etc. 85
Notes 86
Solve problems involving multiplication and division including using their knowledge of factors and multiples, squares and cubes Working with numbers up HTU x U or THTU x U (where the answer is a 3 or 4 digit number) and HTU U or THTU U, ensure that children have opportunities to: Estimate the answer Following the calculation sequence: Estimate 214 x 7 = Solve problems involving addition, subtraction, multiplication and division and a combination of these, including understanding the meaning of the equals sign Evidence the skill of multiplication and division Prove the inverse using the skill of multiplications and division Practice calculation skill including units of measure (km, m, cm, mm, kg, g, l, cl, ml, hours, minutes and seconds) Calculate 214 x 7 = Prove 1498 7 = 214 Calculate 214 ml x 7 = Solve missing box questions including those where missing box represents a digit or represents a number Solve problems including those with more than one step Solve open-ended investigations 1498 = 214 One full barrel holds 214 litres and there are 7 full barrels, how much do I have altogether? I sell 2 barrels, how many litres do I have left? Using the digit cards 1 to 9, make the smallest/biggest answer, an answer that is a multiple of 5 etc. 87
Notes 88
Solve problems involving multiplication and division, including scaling by simple fractions and problems involving simple rates. Solve problems which require knowing percentage and decimal equivalents of 1 2 1,,,, and 4 1 5 those fractions with a denominator of a multiple of 10 or 25. 2 5 4 5 For multiplication and division, refer to Following the calculation sequence: above Scaling problems use the skills of multiplication and division for scaling up and down Use the skills of converting between fractions, decimals and percentages and apply this in a problem solving context When making these connections, children work with fractions with a denominator of 100, 50, 25, 20, 10, 5 and 2 Link to work with measures by using recipes A recipe for 4 persons that must be scaled down to show quantities for 1 person using division skills A recipe for 2, that must be scaled up to feed 10 people, using the skills of multiplication Which of the following discounts is the greatest and which is the least: 5 3, 0.25,, 0.3, 35%? 25 10 Here is a set of prices. All prices are to increase by 10%, calculate the new prices 450, 399, 505 If a television cost 300 and is reduced by 10%, what is the new price? A standard cereal box holds 500g. If you get 1 extra free, how many grams are in the box 4 now? There are 40 sweets in a packet. David eats some and there are now only 60% left. How many sweets has he eaten? 89
Notes 90
Solve problems involving converting between units of time Be able to convert: hours to minutes minutes to seconds years to months weeks to days and vice versa, applying this skill when solving problems Give problems that include mixed units and that specify how an answer is expressed so that a conversion is required 3.5 years = months = days 3 runners ran a marathon and their times were recorded as such: Runner A = 4 hours 12 minutes 3 seconds Runner B = 254 minutes 25 seconds Runner C = 17 200 seconds Place the runners in order of fastest to slowest Use all four operations to solve problems involving measure [for example, length, mass, volume, money] using decimal notation, including scaling. For problem solving all four operations refer to Following the calculation sequence: above Scaling problems involve changing the quantities for groups of different size. Scaling down to decrease quantities and scale up to increase quantities Include decimal notation in measures Here is the recipe to make 25 cookies. If you need to make 100 cookies, calculate the new quantities you would need Here is another recipe but I only need Calculate the new quantities 1 3 of it. 91
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Summer
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YEAR 5 PROGRAMME OF STUDY DOMAIN 3 STATISTICS NEW OBJECTIVES - SUMMER 1 Objectives What does this mean? Example questions Notes and guidance (statutory requirements) (non-statutory) Solve comparison, sum and difference problems using information A line graph shows information that is connected in some way (such as a change over time) This graph shows the cost of phone calls in the daytime and in the evening Pupils connect their work on coordinates and scales to their interpretation of time graphs. presented in a line graph Children should be able to read and interpret information on such graphs in order to answer simple questions They begin to decide which representations of data are most appropriate and why. How much does it cost to make a 9 minute call in the daytime? How much more does it cost to make a 6 minute call in the daytime than in the evening? 95
Notes 96
Carol went on a 40-kilometre cycle ride. This is a graph of how far she had gone at different times. Complete, read and interpret information in tables, including timetables. Where information is presented in a table, children can read and interpret the information in order to answer simple questions How many minutes did Carol take to travel the last 10 kilometres of the ride? Use the graph to estimate the distance travelled in the first 20 minutes of the ride. Carol says, 'I travelled further in the first hour then in the second hour'. Explain how the graph shows this. This table shows the distances in kilometres between five towns. Use the table to find the distance from London to Manchester. James goes from Newcastle to Birmingham, and then on to Cardiff. How many kilometres does he travel? 97
Notes 98
Here is part of a train timetable How long does the first train from Edinburgh take to travel to Inverness? Ellen is at Glasgow station at 1.30pm. She wants to travel to Perth. She catches the next train. At what time will she arrive in Perth? 99