Mental Maths Strategies for Multiplication and …

1 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 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).

2 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. 1 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.

3 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 2 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.

4 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 49 Think Multiplication for Division .. 51 Towards Fluency and Consolidation in Applying Strategies .. 53 Counting Activities using Multiplication and 58 Possible Sequence for Developing Multiplication Facts.

5 61 Further Examples for Exploring and Embedding Strategies .. 64 References .. 67 3 4 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.

6 Finally, a selection of engaging and enjoyable activities to consolidate learning and provide opportunities for pupils to master Multiplication and division facts is included. 5 Background Knowledge for Teachers Stages of Progression Arthur Baroody 1 identifies three stages through which children progress in order to acquire the basic facts of addition, subtraction, Multiplication and division: 1. Counting Strategies : using object counting (for example blocks or fingers) or verbal counting to determine the answer. For example, with 3 7 pupil starts with 7 and skip counts on verbally 7, 14, 21. 2. Reasoning Strategies : using known information to logically determine an unknown combination. For example, with 3 7 pupil knows that double 7 is 14 and one more set of 7 is 21.

7 3. Mastery: efficient (fast and accurate) production of answers. For example, with 3 7, pupil quickly responds, It s 21; I just know it. Multiplication Symbol 2 The use of the Multiplication sign can cause difficulties. Strictly, 3 4 means four threes or 3 + 3 + 3 + 3. Read correctly, it means 3 multiplied by 4. However, colloquially it is read as 3 times 4 , which is 4 + 4 + 4 or three fours. Fortunately, Multiplication is commutative: 3 4 is equal to 4 3, so the outcome is the same. It is also a good idea to encourage children to think of any product either way round, as 3 4 or as 4 3, as this reduces the facts that they need to remember by half. 1 Baroody, A (2006) Why Children Have Difficulties Mastering the Basic Number Combinations and how to help them 2 Crown (2010) Teaching Children to Calculate Mentally Three different ways of thinking about Multiplication are: As repeated addition, for example 3 + 3 + 3 + 3 As an array, for example four rows of three objects As a scaling factor, for example, making a line 3 cm long four times (Crown: 2010, ) 6 Known Facts 3 A useful link between Multiplication and addition allows children to work out new facts from facts that they already know.

8 For example, the child who can work out the answer to 8 6 (six eights) by recalling 8 5 (five eights) and then adding 8 will, through regular use of this strategy, become more familiar with the fact that 8 6 is 48. Teaching Multiplication and Division Multiplication and Division are often taught separately, with Multiplication preceding division. However division and Multiplication are inverse operations. Every Multiplication calculation can be replaced by equivalent division calculations and vice Therefore it is important to combine Multiplication and division soon after Multiplication has been introduced in order to help pupils see how they are related5 There are two concepts of division: The partition or fair-sharing idea, such as sharing 20 sweets among 4 children. The measurement or repeated subtraction concept.

9 If you have 80 and you spend 5 per day, how long will your money last?6 Dealing with Remainders7 Real life problems often result in remainders. A remainder can be dealt with in a number of ways. The remainder is discarded, leaving a smaller whole-number answer. The remainder can force the answer to the next highest whole number. The answer is rounded to the nearest whole number for an approximate result. Addressing what to do with remainders must be central to teaching about division. 3 Crown (2010) Teaching Children to Calculate Mentally 4 Crown (2010) Teaching Children to Calculate Mentally 5 Van de Walle, J., Karp, & Bay-Williams, (2010). Elementary and Middle School Mathematics Teaching Developmentally. 7th edn. Pearson 6 Van de Walle, J., Karp, & Bay-Williams, (2013).

10 Elementary and Middle School Mathematics Teaching Developmentally. 8th edn. Pearson: Allyn and Bacon p. 160 7 Van de Walle, J., Karp, & Bay-Williams, (2010). Elementary and Middle School Mathematics Teaching Developmentally. 7th edn. Pearson 7 Division using Arrays8 Using an array for a division problem such as 200 8 can help pupils think about the known and the unknown components in division. The array can be used to show how to build up to the dividend when solving a division problem with the Think Multiplication or Partial Quotient Strategies . Using 200 8 we can account for partial areas of (8 10) + (8 10) + (8 5), until we have a total area of 200 (Parrish 2010: ). 8 Parrish, S. (2010). Number Talks Helping Children Build Mental Math and Computation Strategies 8 10 = 80 8 10 = 80 8 5 = 40 8 10 10 5 An array model is as important to Multiplication and division as the number line model is to addition and subtraction.