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Mental Maths Strategies for Multiplication and Division

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Mental Maths Strategies for Multiplication and Division Practical Approaches to Developing Mental Maths Strategies for Multiplication and Division × ÷ This manual has been designed by members of the Professional Development Service for Teachers. Its sole purpose is to enhance teaching and learning in Irish primary schools and it will be mediated to practising teachers in the professional development setting. Thereafter it will be available as a free downloadable resource on www.pdst.ie for use in the classroom. This resource is strictly the intellectual property of PDST and it is not intended that it be made commercially available through publishers. All ideas, suggestions and activities remain the intellectual property of PDST (all ideas and activities that were sourced elsewhere and are not those of the authors are acknowledged throughout the manual). It is not permitted to use this manual for any purpose other than as a resource to enhance teaching and learning. Any queries related to its usage should be sent in writing to: Professional Development Service for Teachers, 14, Joyce Way, Park West Business Park, Nangor Road, Dublin 12. Contents Introduction ............................................................................................................................................ 4 Background Knowledge for Teachers ..................................................................................................... 5 Stages of Progression .......................................................................................................................... 5 Multiplication Symbol ........................................................................................................................ 5 Known Facts ....................................................................................................................................... 6 Teaching Multiplication and Division .................................................................................................. 6 Dealing with Remainders .................................................................................................................... 6 Division using Arrays ........................................................................................................................... 7 Linear, Area and Set Model for Multiplication and Division ............................................................... 8 Possible Pupil Misconceptions ................................................................................................................ 9 Uses of Mental Calculation ............................................................................................................... 11 Fundamental Facts for Multiplication and Division .............................................................................. 12 The Commutative Property .............................................................................................................. 12 The Associative Property .................................................................................................................. 12 The Distributive Property .................................................................................................................. 12 Inverse Relationship .......................................................................................................................... 12 Teaching and Learning .......................................................................................................................... 13 Suggested Multiplication & Division Strategies ................................................................................ 13 Instructional Framework ................................................................................................................... 14 Classroom Culture ............................................................................................................................. 16 Problems and Solutions in Supporting and Developing Mathematical Thinking ............................. 17 Possible Resources ............................................................................................................................ 18 Key Teaching Principles for Mental Maths ....................................................................................... 18 Assessment ....................................................................................................................................... 19 Teacher Manuals for Supporting and Developing Mathematical Thinking ...................................... 20 Practical Strategies to Develop Multiplication and Division Properties ............................................... 21 Commutative Property ..................................................................................................................... 21 Distributive property of multiplication ............................................................................................. 22 1 Practical Activities for developing Multiplication & Division Strategies ............................................... 24 Skip counting ..................................................................................................................................... 24 Skip Counting for Division ................................................................................................................. 25 Multiplication Strategy: Repeated Addition ..................................................................................... 25 Repeated Subtraction for Division .................................................................................................... 27 Division as inverse multiplication with Cuisenaire Rods ................................................................... 30 Division as inverse multiplication- Area ............................................................................................ 31 Multiplication & Division Fact Families ............................................................................................. 32 Multiplication Strategy: Doubles ...................................................................................................... 33 Double and one more set ................................................................................................................. 33 Double and Double Again ................................................................................................................. 34 Multiplication Strategy: Known Facts ............................................................................................... 35 Multiples of 10 (links to facts for doubles, known facts and counting stick activities) .................... 35 Rounding and Compensating Strategy ............................................................................................. 35 Making Friendly Numbers Strategy .................................................................................................. 37 Partitioning Strategies ...................................................................................................................... 38 Partial Products ................................................................................................................................. 39 Division Strategy: Partial Quotients .................................................................................................. 42 Doubling/Halving .............................................................................................................................. 43 Pose the following problem to the children: .................................................................................... 47 Proportional Reasoning/Adjustment for division ............................................................................. 48 Breaking Factors into Smaller Factors and Factorisation.................................................................. 49 Think Multiplication for Division ....................................................................................................... 51 Towards Fluency and Consolidation in Applying Strategies ................................................................. 53 Counting Activities using Multiplication and Division........................................................................... 58 Possible Sequence for Developing Multiplication Facts ....................................................................... 61 Further Examples for Exploring and Embedding Strategies ................................................................. 64 References ............................................................................................................................................ 67 2 3 Introduction This booklet is intended to support teachers in developing multiplication and division mental maths strategies in their classrooms. It has been designed to accompany the PDST Mental Maths workshops. The booklet explores the key properties of number and number relationships relating to multiplication and division and outlines practical approaches to developing an understanding of these. It also explores background knowledge for teachers and fundamental facts in relation to mental maths. A range of concrete, practical activities that will support pupils in their development of multiplication and division mental maths strategies are also outlined. Also included is a suggested alternative strategic sequence through which pupils may acquire their multiplication facts. Finally,
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