MATHS MILESTONES 3. Teacher guidance materials to develop good practice within Level 3 Number. Belfast Education and Library Board Numeracy Team

Transcription

1 MATHS MILESTONES 3 Teacher guidance materials to develop good practice within Level 3 Number Belfast Education and Library Board Numeracy Team

2 CONTENTS Page 1 Introduction 3 2 Assessments checks 8 3 Preparation activities 11 4 Appendices Language Suggested order for teaching multiplication tables Pupil Records Resources Bibliography Support materials Acknowledgements 70

3 Comments from teachers involved in consultation process Maths Milestones 3 was extremely user-friendly and easy to follow. Very useful for newly - qualified teachers clearly defined and shows a clear progression within Level 3. Commercial schemes usually jump from one Level to the next Level but pupils need the small steps inherent in the progression within Maths Milestones 3. The intervention strategies identified to use with pupils are clear, appropriate and concise. I found it useful as a baseline assessment of pupils for identification of their knowledge and skills to date. Using Maths Milestones 3 highlighted the areas that a pupil needed more reinforcement with or the skills that needed more explanation. It gave me confidence that I was helping to close the gap in the pupil s learning. I thought that some pupils were working at Level 3 but the Assessment Checks indicated that they had gaps in their understanding and I realised that these pupils would need more time to build a sound knowledge of Level 3. Maths Milestones 3 would be very useful for KS 1 teachers to use alongside the assessment of pupils they think may be Level 3. Introducing division alongside multiplication may initially appear a challenge since most schemes usually leave a gap before introducing division. But I understand that pupils working with multiplication and division together from the beginning highlights the links and builds strength in both understanding and problem solving. Flexibility in thinking is encouraged within multiplication and division by introducing the tables in a very structured order and linking new facts to tables pupils already know. 2

4 INTRODUCTION Numeracy is the development and application of mathematics across the curriculum and in real life situations Activities should be balanced between tasks which develop knowledge, skills and understanding, and those which develop the ability to apply mathematical learning and solve problems (The Northern Ireland Curriculum Primary, CCEA, 2007). The Numeracy Strategy identifies approaches that contribute towards improvements in teaching and learning of numeracy. During our work with schools teachers request additional guidance and support for those pupils, regardless of age, who are struggling mathematically. Many of these pupils require a little more time than their peers to grasp mathematical ideas, as well as a structured progression in small achievable steps. Two teacher guidance documents have been developed to support Number from level 1 level 3. Maths Milestones 1 and 2 1 was developed in 2005 and was shared with teachers through in-service training in October Maths Milestones 1 and 2 supported the development of mental approaches to calculation within 100 and Maths Milestones 3 has been developed for pupils who have successfully developed the skills for Assessment Check 10 within Maths Milestones 1 and 2 (i.e. those pupils who have a firm grasp of skills within level 2 Number). READINESS CHECK If pupils are still struggling with the following mathematical concepts and skills they will need to work with the practical activities outlined in the original Maths Milestones 1 and 2 document. Demonstrate 1:1 correspondence within 5 Partition sets within 5 confidently Demonstrate conservation of number within 10 Appreciate the inverse relationship between addition and subtraction Apply knowledge of addition and subtraction within 10 across a range of contexts Know, recognise and order numbers to 50, then 100 Understand composition of 2-digit numbers to 20, then 99 Partition numbers within 50 into tens and ones Understand our number system to 100 by describing the position of any number on the square Mentally add and subtract two 2-digit numbers within 50, then 100 If pupils have successfully gained these concept and skills they should be ready to work within level 3 Number using this Maths Milestones 3 document. 1 Maths Milestones 1 and 2: Teacher guidance materials to develop good practice within Number at levels 1 & 2 3

5 Pupils will become confident and competent in Number working within Level 3 by developing skills, knowledge and understanding in the maths milestones below: Have quick recall of number facts within 20, including doubles and halves Use grouping and regrouping / exchanging within our number system to support efficient mental and written calculations Use a range of efficient mental calculation strategies including approximation and estimation Use a range of strategies to solve problems and investigations across a variety of contexts within maths Know the order of our number system to at least 999 Know that the position of a digit indicates its value Understand and use both the face value and place value of numbers up to at least 999 Understand and use the concept of zero as a place holder with numbers up to 999 Understand and use the commutative aspects of addition and multiplication Recognise that subtraction and division are not commutative Understand and use multiplication and division as inverse operations Know and use key number facts involving all four operations within informal mental methods Understand and use efficient methods for multiplication and division in mental calculations Understand and use efficient written calculation and recording methods for addition and subtraction Recognise and use the concept of fractions as sharing of quantities and understand the link with division 4

6 PURPOSE The purpose of these materials is to provide teachers with a series of Assessment Checks to enable them to identify pupil strengths, alongside the appropriate activities to develop the pupil s understanding of Number. These activities are designed to support the development of further mathematical skills and concepts (i.e. more maths milestones) rather than representing a series of tasks to be completed. The focus is on encouraging pupils to be confident in Level 3 aspects of Number, which should provide a secure foundation for work in other strands of mathematics as well as future number development. KEY PRINCIPLES Mental approaches Pupils should be encouraged to consider mental methods as a first resort in any calculation (Northern Ireland Strategy for Numeracy, Teaching and Learning file 2, 2000). Anghileri (2000) also notes that All pupils are expected ultimately to use efficient written methods for calculating but the only way such methods can be meaningful is if they are developed progressively to support and extend mental strategies. Borthwick and Harcourt-Heath s (2007) research looked at the range of calculation methods that pupils use when working within the four operations and revealed that pupils often find it difficult to choose the most efficient and effective strategy when faced with a written test situation. Their research showed that when pupils use a calculation strategy which is based on mental methods they usually reach the correct solution. This supports the emphasis placed within Maths Milestones 3 on pupils personal methods of recording prior to the introduction of more formal written recording. Mathematical language Use of language within these mathematical activities is fundamental to progression within pupils thinking. Modelling the use of specific mathematical terminology over a period of time is necessary before pupils can begin to use this language to explain their understanding. Opportunities for pupils to use language to express their thinking provide valuable assessment tools. Use of open and closed questioning techniques by both pupils and teachers encourages a rich language environment that challenges pupils thinking (Appendix 4.1) Pupil involvement Integrating thinking skills into mathematics means designing learning so that pupils will think more skilfully than they would otherwise do and as a result deepen their understanding across a range of contexts. Rich collaborative tasks allow learners to make decisions; involve learners in testing, proving, explaining, reflecting and interpreting; promote discussion and communication; encourage originality and invention; encourage what if? and what if not? questions; are enjoyable and contain the opportunity for surprise (Improving learning in mathematics: challenges and strategies, 2005, DFES). 2 Teaching and Learning file (T&L file) published by the Inter-Board Numeracy Group,

7 HOW TO USE The teaching approaches encouraged within these materials support the three core elements of Every School a Good School a strategy for raising achievement in literacy and numeracy (2008) Wave One: Quality whole class teaching Wave Two: Quality teaching plus additional support for identified pupils Wave Three: Quality teaching plus personalised support to meet the specific needs of individual pupils Structure Maths Milestones 3 provides support for pupils working towards Level 3 in Number. The materials are flexible and may be used to support whole class (Wave One), target groups (Wave Two) or individuals (Wave Three) within the numeracy lesson alongside appropriate classroom experiences. For example, within Wave Two support, an appropriate quality learning activity could be identified for the target group for that lesson and the role of the teacher is to actively teach / facilitate quality group interaction whilst the rest of the pupils work independently on appropriate activities to further develop their learning. Teacher guidance material is provided on Language (Appendix 4.1) and a Suggested Order for Teaching the Multiplication Tables (Appendix 4.2) as well as a Resources list and some Support Materials within Appendices 4.4 and 4.6 respectively. Appendix 4.3 provides a Pupil Record format to monitor skills progression as pupils work through the preparation activities and Assessment Checks and informs the next learning and teaching cycle. Maths Milestones 3 is structured as a series of Preparation Activities that develop a range of skills which are subsequently monitored through an appropriate Assessment Check. Preparation activities become progressively more challenging within and between each Assessment Check. This is not intended to restrict flexibility in use, but does indicate the structured progression inherent in mathematical conceptual development. 6

8 Assessment Maths Milestones 3 can be used to support Assessment for Learning. Elements within the Assessment Checks and the Preparation Activities can be used as targets for individual pupils or for groups of pupils. Targets and Learning Intentions can be shared with pupils so they can monitor their progress. Effective feedback tells learners where they have been successful, where they should focus their next improvement and precisely how to go about making that improvement happen. The oral introduction aspect of lessons could be linked to some of the preparation activities. Most of the tasks could form the basis of the main practical element of the lesson for a target group of pupils. It is anticipated that pupils would experience the whole range of preparation activities before moving on to the next Assessment Check. If pupils are still struggling with a particular preparation activity, teachers may need to review related activities in previous Assessment Checks. Some of the maths milestones may need to be more formally assessed as a pupil works through the Assessment Checks e.g. quick recall activities. An initial assessment of each pupil within a target group could involve identifying which Assessment Check and its Preparation Activities best describes the pupil s current level of understanding. For example, a pupil may be capable of achieving most of the statements within Assessment Check 13 and some of those within Assessment Check 14. This pupil would progress most by completing the full range of Preparation Activities supporting Assessment Check 13 before moving on to those related to Assessment Check 14. Assessment forces us to recognise that each learner is an individual with different learning needs, and to adapt the pace and content of teaching accordingly (Improving learning in mathematics: a professional development guide, 2005, DFES). The class teacher should liaise with the numeracy coordinator, SENCO and external educational partners to ensure an integrated approach within all of the numeracy provision for pupils. 7

9 Assessment Check 11 ASSESSMENT CHECKS Have quick recall of number facts within 20 Apply quick recall of facts within 20 to support addition and subtraction within 100 Know half of 50 and 100 Understand and use a range of addition and subtraction vocabulary, including symbols Use commutative aspect of addition to mentally calculate within 100 Explain why subtraction is not commutative Assessment Check 12 Count in 100s, then 50s and finally in 1s bridging multiple of 100 using numbers up to 999 Order and sequence numbers within 999 Recognise and read numbers within 999 Approximate to the nearest 10 within 100, and then 999 Estimate first when mentally adding or subtracting within 999 Use easy fact within 10 to develop complements of 100 Use complements of 100 to support addition and subtraction within 999 Assessment Check 13 Group and regroup within 100 producing equal groups using cubes Group and regroup within 100 producing unequal groups using cubes Understand the relationship between the total number of objects, the number of groups, and the number within each group Group and regroup within 100 producing equal groups using a spike abacus Group and regroup within 100 producing unequal groups using a spike abacus Understand the repeated addition concept of multiplication Understand the concept of division through sharing activities Assessment Check 14 Appreciate the inverse relationship between multiplication and division Understand the concept of remainders within division using sharing activities Understand the repeated subtraction concept of division Understand the area model of multiplication Appreciate the commutative aspect of multiplication Understand and use a range of multiplication and division vocabulary, including symbols Assessment Check 15 Understand and use the link between doubling and multiplying by 2 Understand and use the link between halving and dividing by 2 Appreciate the inverse relationship between doubling and halving Have quick recall of 2, 5 and 10 times tables Apply knowledge of 2, 5 and 10 times tables across a range of contexts 8

10 Assessment Check 16 Use the commutative aspect of multiplication to derive unknown facts Use the inverse relationship between multiplication and division to derive division facts Explain why division is not commutative Apply quick recall of number facts within 20 to support mental addition and subtraction within 999 Use a range of mental strategies to add/subtract any numbers within 999 Understand the composition of number within 999 Assessment Check 17 Understand the position of each digit determines its value Appreciate the importance of zero as a place holder Understand and use both the face value and the place value aspects of numbers within 999 Recognise and write numbers up to 999 Appreciate that the digit on the left of a number has the greatest value and the digit on the right has the least value Assessment Check 18 Use the formal method for addition with no regrouping linking practical activities with horizontal and vertical recording Use the formal method for subtraction with no exchange linking practical activities with horizontal and vertical recording Use the formal method for addition with one regrouping linking practical activities with vertical recording Use the formal method for subtraction with one exchange linking practical activities with vertical recording Understand the process when handling zero during exchanging within subtraction and link practical activities with vertical recording Assessment Check 19 Use the formal recording method for addition requiring two regroupings Use the formal recording method for subtraction requiring two exchanges Use the inverse and commutative aspects of multiplication and division to derive the 4 times and 9 times tables and the ten remaining facts Understand the link between simple fractions of whole numbers and division facts based on the related multiplication facts Use a range of mental strategies within multiplication and division Have quick recall of all number facts based on the multiplication tables up to 10 x 10 Assessment Check 20 Apply knowledge of multiplication and division facts across a range of contexts Recognise whole numbers that are exactly divisible by 2, 5 and 10 Use partitioning strategy to support multiplication beyond the 10 x tables Use all four operations to solve problems across a range of contexts 9

11 Throughout the following preparation activities pupils should always have opportunities to use appropriate language in collaborative working situations. They should be encouraged to engage in effective questioning and to use practical materials as appropriate. How many cubes do you think Dienes man is made from? More importantly How could we begin to count the cubes? 10

12 Preparation Activities 11

13 Preparation activities to support Assessment Check 11 Exploring the components of numbers within 20 e.g , 5 + 7, 4 + 8, makes 12 (Exploring patterns using Cuisenaire or multilink) (Oral and written response) Developing quick recall of number facts, initially within 20, using the following progression: N + 1, 1 + N (e.g. 2+ 1, up to , ) N + 10, 10 + N N + N (e.g up to ) Near doubles (e.g is double 1 and then add 1, up to is double 10 then subtract 1) N + 2, 2 + N N + 3, 3 + N N + 4, 4 + N N + 5, 5 + N N + 6, 6 + N N + 7, 7 + N N + 8, 8 + N N + 9, 9 + N (e.g , up to , ) (Quick Recall Mental Strategy, Orange level, T & L file) (Towards mental response, with appropriate thinking time) Applying number facts in real-life contexts: e.g. John slept for 5 hours and then slept for 9 more hours. How long did John sleep for altogether? Susan sent 15 texts. She sent 9 more texts than John. How many texts did John send? (Solving oral and written word problems and responding mentally) Using the doubling strategy within 20 to develop an understanding of doubles up to 999 (e.g = 8 therefore = 80 and = 800) (Using Cuisenaire) (Oral response) Using the relationship between doubles and halves e.g. double 40 is 80 and therefore half of 80 is 40 (Using Cuisenaire) (Oral response) Knowing the half of 50 and 100 (Quick Recall Mental Strategy, Orange level, T & L file) (Oral response) 12

14 Applying quick recall of number facts within 20 to support addition and subtraction of numbers up to 100 e.g. using the easy fact strategy = 13 to solve = 53 (Ready For Calculating 3, Learning goal 3) (Oral and written response) Recognising, reading and understanding addition and subtraction vocabulary including symbols +, - and = (see Appendix 4.1: Language) (Using displays, word-banks, individual whiteboards, worksheets etc) (Oral and then written response) Exploring the range of vocabulary used in addition and subtraction word problems e.g. Tom gave 3 DVDs to his friend and was left with 4 DVDs. How many DVDS did Tom have in the beginning? (see Appendix 4.1: Language) (Using displays, word-banks, individual whiteboards, worksheets etc) (Oral and then written response) Using commutativity to add numbers in any order e.g is the same as (Using Dienes or grouping in tens where appropriate, using 100 square etc) (Oral response) Explaining why subtraction is not commutative e.g is not the same as 6 57 i.e. using a 100 square to try to count back 57 spaces from 6 which cannot be done (Oral response) Pupil should now have developed the skills for Assessment Check 11 3 Ready For Calculating: BELB,

15 Preparation activities to support Assessment Check 12 Exploring the effect of adding 9, 19, 29 etc or 11, 21, 31 etc to any number up to 100 (Rounding and Adjusting Mental Strategy, Orange level, T & L file) (Oral response) e.g. number add total Exploring the effect of subtracting 9, 19, 29 etc or 11, 21, 31 etc from a number up to 100 (Rounding and Adjusting Mental Strategy, Orange level, T & L file) (Oral response) e.g. number subtract total Solving a range of addition and subtraction problems involving number strings e.g. A 12-leafed plant loses 3 leaves and then 4 more leaves. How many leaves remain on the plant? (i.e = 5) (Oral and then written response) Developing the language of estimation using numbers within 100 (Approximation and Estimation, Orange level, T & L file) (Oral response e.g. it is more than but definitely not as many as. ) Exploring our number system by counting in 100s, then 50s and finally in 1s bridging the multiple of 100 (Counting on/back Mental Strategy Yellow Level, T & L file) (Counting orally forwards and backwards using different starts, from 90 to 999) Ordering a range of numbers within 999 with different starts, in particular bridging the multiple of 100 (Using a range of digit cards e.g. 485 to 515 alongside a number-line, washing line or applying their knowledge of ordering to identify a missing number) (Oral and written responses) Sequencing numbers within 999 (Identify the sequence in a given set of numbers e.g. 410, 420, 430 and the next number in the sequence) (Oral and written response) 14

16 Recognising and reading numbers up to 999 (When teacher displays a number, pupil can say number) (Oral response) Developing the skills of approximation to the nearest 10 within 100, and then 999 (Using counting stick and blank number line) (Approximation and Estimation, Yellow Level, T & L file) (Oral response and horizontal recording) Using mental strategies to develop estimation skills within 999 e.g is nearly = 330 (Approximation and Estimation, Yellow Level, T & L file) (Oral response) Exploring extended addition / subtraction patterns within 999 e.g. pupil could establish pattern and then apply to bigger numbers: (Understanding the Number System, orange level, T & L file) (Using Cuisenaire, number line, 100 square if needed, alongside horizontal written recording) Using the easy fact strategy to develop the complements of 100 (e.g = 10, = 100 or = 100) (Using Cuisenaire, hundred square and applying within the context of money) (Ready for calculating, Learning goal 2) (Oral response) Using complements of 100 to support addition / subtraction within 999 (e.g = 84 therefore = 684) (Using Cuisenaire, hundred square and applying within the context of money) (Oral response alongside horizontal recording) Pupil should now have developed the skills for Assessment Check 12 15

17 Preparation activities to support Assessment Check 13 Developing the concept of grouping and regrouping within 100 using the following progression: o Using same number + different group-size e.g groups of 2 producing equal groups - groups of 10 - groups of 5 - groups of 4 20 cubes 10 groups of 2 2 groups of 10 4 groups of 5 5 groups of 4 o Using different numbers + same group-size e.g groups of 5 producing equal groups 25 - groups of groups of groups of 5 Introducing the concept of unequal groups: o Using same number + different group-size e.g groups of 3 producing unequal groups - groups of 6 - groups of 9 - groups of 7 o Using different numbers + same group-size e.g groups of 6 producing unequal groups 27 - groups of groups of groups of 6 (Using multilink or unifix cubes) (Oral and written responses) 16

18 Exploring the relationship between: o the total number of objects o the number of groups o number within each equal group Investigate what happens if the total number stays the same but there is a different number within each group. (Using counters and hoops, multilink and Cuisenaire alongside oral + written response) Grouping and regrouping within 100 using an abacus representation along with the progression from the earlier examples in this check for same/different group-size and equal/unequal groups: 20 cubes Groups of 2 Groups of 10 Groups of 5 Groups of 4 Introducing the concept of multiplication through grouping i.e. a repeated addition strategy, explaining the process for example: (Understanding the number System, Yellow Level, T & L file) (Using counters, hoops, Cuisenaire, number-line, oral response alongside horizontal recording) = 6 sets of 2 = 12 Exploring the concept of division as grouping (using the same number and different group sizes) e.g. 24 can be equally grouped in 2s, 4s, 6s, 8s, 12s (Using counters and hoops, multilink and Cuisenaire alongside oral + written response) Exploring the concept of division as sharing e.g. 24 can be shared between 6 children using a 1 for you, 1 for me strategy (Using counters and hoops, multilink and Cuisenaire alongside oral + written response) Pupil should now have developed the skills for Assessment Check 13 17

19 Preparation activities to support Assessment Check 14 Exploring the concept of division within 25 through sharing activities and developing the link (i.e. the inverse relationship) between multiplication and division, e.g. given that 2 sets of 5 makes 10, how many twos are in 10? (Using Cuisenaire, counters and hoops) (Oral response alongside horizontal recording) Exploring the concept of division using sharing activities e.g. investigate a range of numbers using equal groups of 4. Discuss the numbers left over which are called remainders (Using counters and hoops, multilink and Cuisenaire alongside oral and written response) 15 is the same as 3 equal groups of 4 and 3 left over 28 is the same as 7 equal groups of 4 Exploring the concept of division using repeated subtraction of equal groups within 25, for example: (Understanding the number System, Yellow Level, T & L file) (Using counters and hoops, Cuisenaire and number-line alongside oral response and horizontal recording) 12 divided into groups of 2 makes 6 groups is the same as = 0 Exploring the effect of multiplying and dividing by zero e.g. 4 groups of zero = 0 or 6 sweets shared with zero groups = 0 since there are no groups to share (Using counters and hoops, Cuisenaire and number-line alongside oral response and horizontal recording) Exploring the number patterns generated by repeated addition / subtraction on a blank 100 square initially adding / subtracting 2, 5 or 10 to develop an understanding of multiplication / division (Understanding the number System, Yellow Level, T & L file) (Using blank 100 square alongside horizontal recording) Using repeated addition to introduce the area model of multiplication: (Using counters and hoops, multilink and Cuisenaire) (Oral and written recording) = 3 groups of 2 18

20 Exploring the area model of multiplication, highlighting the conservation of area (Using Cuisenaire and cm squared paper) (Oral and written recording) lots of 5 5 lots of 3 3 lots of 5 5 lots of 3 which can also be represented as: 3 x 5 = 15 or 5 x 3 = 15 Exploring the commutative aspect of multiplication using the area model e.g. 3 groups of 2 covers the same area when placed on top of 2 groups of 3 (Using counters and hoops, multilink and Cuisenaire) (Oral and written recording) Recognising, reading and understand multiplication and division vocabulary including symbols x, and = (see Appendix 4.1: Language) (Using displays, word-banks, individual whiteboards, worksheets etc) (Oral and then written response) Working towards quick recall of multiplication facts using the 1 and 10 times tables (Using Cuisenaire and 100 square) (Oral response alongside horizontal recording) Working towards quick recall of multiplication facts using the 2 and 5 times tables (Using Cuisenaire and 100 square) (Oral response alongside horizontal recording) Working towards quick recall of division facts based on the 2 and 5 times tables (Using Cuisenaire and 100 square) (Oral response alongside horizontal recording) Pupil should now have developed the skills for Assessment Check 14 19

21 Preparation activities to support Assessment Check 15 Exploring the effect of doubling and multiplying by 2 e.g. 2 x 3 = 6 therefore double 3 is 6 and 2 x 30 = 60 therefore double 30 is 60 (Using Cuisenaire and 100 square) (Understanding the Number System, Yellow Level, T & L file) (Oral response alongside horizontal recording) Exploring the effect of halving and dividing by 2 e.g. 6 2 = 3 therefore half of 6 is 3 and 60 2 = 30 therefore half of 60 is 30 (Using Cuisenaire and 100 square) (Understanding the Number System, Yellow Level, T & L file) (Oral response alongside horizontal recording) Exploring the inverse relationship between doubling and halving using appropriate language e.g. half of 60 is 30 and double 30 is 60 (Using Cuisenaire and 100 square) (Understanding the Number System, Yellow Level, T & L file) (Oral response alongside horizontal recording) Using the inverse relationship between doubling and halving to check the answers within simple word problems e.g. a petrol tank holds 44 litres and therefore half a tank holds 22 litres. True or false, does double 22 make 44? (Quick Recall, Yellow Level, T & L file) (Oral response alongside horizontal recording) Beginning to construct the multiplication square using squared paper and the multiplication facts for 2, 5 and 10 times tables (Using squared paper and a blank multiplication square) Developing quick recall of the multiplication facts for 2, 5 and 10 times tables and related division facts (Quick Recall, Yellow Level, T & L file) (Using 100 square, partially-completed multiplication square, counting stick and swinging apple) 20

22 Applying the knowledge of 2, 5 and 10 times tables to simple problem solving situations involving multiplication and division: (Using practical materials alongside oral and horizontal written responses) 5 children need 2 cubes each so that is = 10 or 5 x 2 cubes which makes 10 altogether or 10 cubes shared equally among 5 children gives 2 cubes each Calculating using addition, subtraction and simple multiplication and division e.g. shopping bills to 2 - ask pupils to generate their own simple shopping questions and discuss calculation strategies. (Using 5p, 10p and 20p coins) (Oral response and/or written recording) Investigating function machines for addition, subtraction, multiplication and division up to 200 using appropriate language input, function and output: (Oral and written response) Input: Function: Output: 110 add 25, subtract multiply by multiply by 5, divide by 2 10 Pupil should now have developed the skills for Assessment Check 15 21

23 Preparation activities to support Assessment Check 16 Using commutativity to multiply numbers in any order e.g. 7 x 5 can be calculated as 5 x 7 using the 5 times tables (Using partially-completed multiplication square) (Understanding the Number System, Yellow Level, T & L file) Using the inverse relationship between multiplication and division to derive division facts: (Using partially-completed multiplication square alongside horizontal recording) 35 7 can be worked out using 5 x 7 = 35 Explaining why division is not commutative for example why 10 2 is not the same as 2 10 (Using Cuisenaire and squared paper alongside horizontal recording) Investigating numbers that cannot be divided equally by 2, 5 and 10 through grouping activities to introduce the concept of remainders, for example 24 can be shared as 4 equal groups of 5 with 4 left over (i.e. a remainder of 4) (Using counters and hoops, multilink alongside horizontal recording) Investigating the effect of multiplying a number by 10 or by 1 through oral discussion and practical exploration (Using counters and hoops alongside horizontal recording) (Understanding the Number System, Yellow Level, T & L file) Using all the number facts within the 2, 5 and 10 times tables to solve number problems e.g. e.g. ask pupils to generate their own simple problems and discuss calculation strategies. (Oral response) Using quick recall of addition and subtraction facts within 20 to support mental calculation within 999 without bridging, and then with bridging e.g = 12 therefore = 412 (Using Cuisenaire and oral response alongside horizontal written recording) 22

24 Using mental strategies to support addition/subtraction of any numbers within 999 e.g (Mental Strategies - Counting on/back, Re-ordering, Partitioning and Rounding and Adjusting, Yellow Level, T & L file) (Oral response alongside horizontal written recording) Using their knowledge of extended addition/subtraction patterns to solve real life problems e.g. players on a computer game are given 150 bonus points what would their new scores be? (Choosing and using addition/subtraction alongside efficient calculation strategies) (Oral explanations or using simple writing frames where appropriate) Exploring the composition of number within 999 e.g. 184 is the same as 1 hundred, 8 tens and 4 units, or 18 tens and 4 units, or 184 units (Using Dienes apparatus and Dienes board) (Oral response) Pupil should now have developed the skills for Assessment Check 16 23

25 Preparation activities to support Assessment Check 17 Investigating and explaining the position of each digit in terms of value using a spike abacus, e.g. 3 and 4 and 1 can be arranged on an abacus to give a range of numbers: (Oral response and digit cards) Explaining the importance of zero as a place holder through a range of activities Using place value arrow cards, calculators, place value flips, number fans or 0-9 digit cards to represent any number within 999 and explain the value and significance of each digit (Oral response) Linking the face value and place value aspects of numbers within 999 e.g. 435 is 400 and 30 and 5 as well as 4 hundreds, 3 tens and 5 units 502 is 500 and 2 as well as 5 hundreds and 2 units (Using Dienes apparatus alongside a spike abacus ) (Oral response alongside written recording) or 24

26 Recognising and writing numbers up to 999 (E.g. when teacher displays a number, pupil can say number name; and when teacher calls out a number-name pupil can use number fans, digit cards etc or use their individual whiteboard to write numerals) Applying knowledge of place value to play games, e.g Race to Place or calculator games (see Appendix 4.5) (Oral and written response) Beginning to appreciate that the digit on the left of a number has the greatest value and the digit on the right has the least value working with numbers up to 999, using structured materials (Using Dienes apparatus and Dienes board to represent the largest number or the smallest number possible when given any three digits e.g. given 5,6,4 which could represent 654 or 456 explaining the value of each digit) (Oral and written responses) Appreciating that the digit on the left of a number has the greatest value and the digit on the right has the least value using a range of materials (Using place value arrow cards, place value charts and place value flips to represent any number within 999 and explain the value of each digit) (Oral and written responses) Extending an understanding of place value up to 999 across a range of contexts including games, problems and investigations (Using a range of materials e.g. 0-9 dice, number fans or 0-9 digit cards, 0-9 spinners) (Oral and written responses) Pupil should now have developed the skills for Assessment Check 17 25

27 Pupils should have experience of addition, subtraction, composition of number and place value preparation activities before being introduced to written recording methods Assessment Checks 18 & 19 introduce the formal addition and subtraction written procedures through the following stages: practical activities horizontal recording vertical recording The language used during the exchanging aspect of subtraction should be carefully chosen to link pupils understanding of the mathematics from the practical activities to the vertical recording method. Teachers within a school will need to discuss and agree the appropriate language to bring mathematical meaning to the written procedure. 26

28 Preparation activities to support Assessment Check 18 o Addition with no regrouping (e.g = 567) add add 200 = add 20 = 60 4 add 3 = 7 4 add 3 = 7 40 add 20 = add 200 = 500 Pupils should use agreed language to explain and discuss their mathematical thinking at each stage within this process 27

29 o Subtraction with no exchange (e.g = 133) subtract 50 = 30 Recombining hundreds, tens and units = subtract 6 = 3 9 subtract 6 =3 80 subtract 50 = subtract 0 = 100 Pupils should use agreed language to explain and discuss their mathematical thinking at each stage within this process 28

30 o Addition with one regrouping (e.g = 562) 318 First, add the units Next, regroup the tens and units 362 add 244 add 200 Finally, add the hundreds add 4 = 12 units Next, regroup the tens and units 10 add 40 add 10 = add 200 = 500 Pupils should use agreed language to explain and discuss their mathematical thinking at each stage within this process 29

31 o Subtraction with one exchange (e.g = 416) 672 First, try to subtract the units Need to exchange 1 ten for 10 units 672 Now, subtract the units 666 Then, subtract the tens 616 Finally, subtract the hundreds subtract 6 Need to exchange 1 ten for 10 units 12 subtract 6 = 6 60 subtract 50 = subtract 200 = 400 Pupils should use agreed language to explain and discuss their mathematical thinking at each stage within this process 30

32 o Handling zero whilst exchanging during subtraction (e.g = b370) First, subtract the units Next, try to subtract the tens. Need to exchange 1 hundred for 10 tens Now, subtract the tens 570 Finally, subtract the hundreds subtract 5 = 0 0 subtract 3 Need to exchange 1 hundred for 10 tens 100 subtract 30 = subtract 200 = 300 Pupils should use agreed language to explain and discuss their mathematical thinking at each stage within this process Pupil should now have developed the skills for Assessment Check 18 31

33 Preparation activities to support Assessment Check 19 It is recommended that pupils should not move onto working with two regroupings or two exchanges until they have a secure understanding of one regrouping and one exchange. It is also critical that their understanding of the vertical recording method is based upon their mathematical understanding from the practical activities o Addition requiring two regroupings (e.g = 822) 9 add 3 = 12 units Next, regroup the tens and units 30 add 80 add 10 = 12 tens Next, regroup the hundreds and tens 600 add 100 add 100 = 800 o Subtraction requiring two exchanges (e.g = 256) 2 subtract 6 Need to exchange 1 ten for 10 units 12 subtract 6 = 6 0 subtract 50 Need to exchange 1 hundred for 10 tens 100 subtract 50 = subtract 100 = 200 Pupils should use agreed language to explain and discuss their mathematical thinking at each stage within this process 32

34 Reinforcing the inverse and the commutative aspects of multiplication and division through practical activities i.e. using Cuisenaire, 100 square (or a number-line) and a partially-completed multiplication square: o 4 times tables o 9 times tables o ten remaining facts 3 x 3 3 x 6 3 x 7 3 x 8 6 x 6 6 x 7 6 x 8 7 x 7 7 x 8 8 x 8 (Understanding the Number System, Yellow Level, T & L file) (Oral response alongside horizontal recording) Making the connections between simple fractions and multiplication and division using Cuisenaire, 100 square (or a number-line) and a partiallycompleted multiplication square within the context of: o half and the division facts based on the 2 times tables o quarter and the division facts based on the 4 times tables o 1/3 and the division facts based on the 3 times tables (Understanding the Number System, Yellow Level, T & L file) (Oral response alongside horizontal recording) Applying their understanding of commutative and inverse relationships within multiplication and division to support mental calculation (Using Cuisenaire and a fully-completed multiplication square) (Mental strategies - Re-ordering, Inverse Operation, Partitioning, Rounding and Adjusting and Factors, Yellow Level, T&L file) (Oral response alongside horizontal recording) 33

35 Developing quick recall of all of the number facts based on the multiplication tables up to 10 x 10 using a range of approaches including: o Commutative aspect (7 x 4 = 4 x 7) o Doubles (2 x tables) o Double two times (4 x tables) o Double three times tables (6 x tables) o Double four times tables (8 x tables) o Tens times tables take away 1 group (9 x tables) (Using paired or group work with games, number fans, digit cards, individual whiteboards) (Understanding the Number System, Yellow Level, T & L file) (Oral response alongside horizontal recording) Developing a reasonable approximation or estimation within a multiplication and division context using numbers up to 999 (Using Dienes, Cuisenaire, or children themselves e.g, estimate the number of children within a school) Pupil should now have developed the skills for Assessment Check 19 34

36 Preparation activities to support Assessment Check 20 Applying the knowledge of the multiplication and division facts to solve reallife problems, e.g. it takes 3 minutes to download 1 song onto your ipod, how long will it take to download 6 songs? (Oral response alongside horizontal recording) Recognising the whole numbers which are exactly divisible by 2, 5 and 10 (Using the fully-completed multiplication square alongside horizontal recording, number fans, individual whiteboards) Investigating the pattern of the products and quotients within all the number facts based on the multiplication tables, e.g. do the 3 times tables use every digit 0-9? (Using the fully-completed multiplication square alongside horizontal recording on individual whiteboards) Checking the results of calculations using the inverse relationship between multiplication and division (Oral response alongside horizontal recording) Investigating the effect of partitioning within multiplication: o Using the 1-10 x tables, for example 9 x 7 = 3 x 7 and 6 x 7, and add the products o Beyond 10 x tables, for example 13 x 7 can be calculated using their knowledge of 10 x 7 and 3 x 7 and add the products (Using the fully-completed multiplication square or a 100 square alongside horizontal recording or individual whiteboards) (Partitioning mental strategy, Yellow Level, T&L file) Investigating the effect of re-ordering when multiplying, for example 3 x 7 x 4 is the same as 3 x 4 x 7 (Oral response alongside horizontal recording) (Re-ordering mental strategy, Yellow Level, T&L file) Exploring the process of partitioning within division using the 1-10 tables mentally, for example 80 8 is the same as 40 8 and another group of 40 8 and add the quotients (Oral response alongside horizontal recording) 35

37 Exploring the factors within 100 using the multiplication square to find the numbers that occur most frequently e.g. 24 occurs as the product of 6 and 4, 4 and 6, 8 and 3, 3 and 8, 2 and 12, 12 and 2 (Oral response alongside written recording) Using Game 24 to consolidate and apply knowledge of number facts for addition, subtraction, multiplication and division (Oral response within paired and group work) Investigating simple function machines for addition, subtraction, multiplication and division up to 999 using appropriate language input, function and output: (Oral and written response) Input: Function: Output: 410 add 200, subtract add 200, subtract add 200, subtract multiply by 5, divide by multiply by 5, divide by multiply by 5, divide by Calculating using addition, subtraction and simple multiplication and division to at least 500 e.g. ask pupils to generate their own shopping problems and discuss their calculation strategies. (Oral response and/or written recording) Pupil should now have developed the skills for Assessment Check 20 36

38 Appendices 37

39 APPENDIX 4.1: LANGUAGE Frequently pupils struggle with the language associated with problem solving and investigation activities. This needs to be considered carefully along with their knowledge of the appropriate mathematical language for pupils to be confident using the four operations. For example, there are many ways of asking simple addition and subtraction problems and pupils need experience of both responding and asking questions within a broad range of scenarios: Joe had 13 marbles. Then Tom gave him 5 more marbles. How many marbles does Joe have now? Joe had 13 marbles. Then Tom gave him some more marbles. Now Joe has 18 marbles. How many marbles did Tom give him? Joe has 18 marbles. Tom has 5 marbles less than Joe. How many marbles does Tom have? Joe has 18 marbles. He has 5 more marbles than Tom. How many marbles does Tom have? Joe had some marbles. Then Tom gave him 5 more marbles. Now Joe has 18 marbles. How many marbles did Joe have in the beginning? Joe and Tom have 18 marbles altogether. Joe has 13 marbles. How many marbles does Tom have? Joe has 18 marbles. Tom has 5 marbles. How many marbles does Joe have more than Tom? Joe has 18 marbles. Tom has 5 marbles. How many marbles does Tom have less than Joe? 38

40 Breadth of language for addition and subtraction (taken from The National Numeracy Strategy Mathematical Vocabulary booklet, DFES) Difference between Half, halve How many more/fewer is than? How much more/less is? Equals, sign, is the same as Tens, hundreds Inverse Add, addition, more, plus, increase Sum, total, altogether Score Double, near double How many more to make? Subtract, subtraction, take (away), minus, decrease Leave, how many are left/left over? Breadth of language for multiplication and division (taken from The National Numeracy Strategy Mathematical Vocabulary booklet, DFES) Lots of, groups of Times, multiply, multiplication, multiplied by Multiple of, product Once, twice, three times ten times Times as (big, long, wide and so on) Repeated addition Array Row, column Double, halve Share, share equally One each, two each, three each Group in pairs, threes tens Equal groups of Divide, division, divided by, divided into Remainder Factor, quotient, divisible by Inverse 39

41 Use of open-ended questions by pupils When Planning When Adapting When Evaluating How am I going to do it? Is it similar to anything I ve done before? Is it one of those? Do I understand it so far? Do I need to ask a question? Am I on the right track? Is there a better way? How did I do it? What method / strategy worked? What did I learn? Did my plan work out? Can I learn from my mistakes? Can I do better next time? (taken from the Thinking Skills and Personal Capabilities for Key Stage 1&2, Curriculum Support Implementation box, Northern Ireland Curriculum, CCEA) 40

42 APPENDIX 4.2: SUGGESTED ORDER FOR TEACHING THE MULTIPLICATION TABLES 1 times table 10 times table 2 times table 5 times table 4 times table 9 times table ten remaining facts 3 x 3 3 x 6 3 x 7 3 x 8 6 x 6 6 x 7 6 x 8 7 x 7 7 x 8 8 x 8 41

43 42 APPENDIX 4.3: PUPIL RECORD Pupil Names Assessment Check 11 Have quick recall of number facts within 20 Apply quick recall of facts within 20 to support addition and subtraction within 100 Know half of 50 and 100 Understand and use a range of addition and subtraction vocabulary, including symbols Use commutative aspect of addition to mentally calculate within 100 Explain why subtraction is not commutative

44 Pupil Names Assessment Check 12 Count in 100s, then 50s and finally in 1s bridging multiple of 100 using numbers up to 999 Order and sequence numbers within 999 Recognise and read numbers within 999 Approximate to the nearest 10 within 100, and then 999 Estimate first when mentally adding or subtracting within 999 Use easy fact within 10 to develop complements of 100 Use complements of 100 to support addition and subtraction within

45 44 Pupil Names Assessment Check 13 Group and regroup within 100 producing equal groups using cubes Group and regroup within 100 producing unequal groups using cubes Understand the relationship between the total number of objects, the number of groups, and the number within each group Group and regroup within 100 producing equal groups using a spike abacus Group and regroup within 100 producing unequal groups using a spike abacus Understand the repeated addition concept of multiplication Understand the concept of division through sharing activities

46 Pupil Names Assessment Check 14 Appreciate the inverse relationship between multiplication and division Understand the concept of remainders within division Understand the repeated subtraction concept of division Understand the area model of multiplication Appreciate the commutative aspect of multiplication Understand and use a range of multiplication and division vocabulary, including symbols 45

47 46 Pupil Names Assessment Check 15 Understand and use the link between doubling and multiplying by 2 Understand and use the link between halving and dividing by 2 Appreciate the inverse relationship between doubling and halving Have quick recall of 2, 5 and 10 times tables Apply knowledge of 2, 5 and 10 times tables across a range of contexts