Strategies – Multiplication and Division

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Strategies – Multiplication and Division

4 x 3 does not represent the same as 3 x 4 but it has the same value.
From repeated addition to multiplication as an array 4+4+4 3 groups of 4 = 3 x 4 4 groups of 3 = 4 x 3 It is important that learners progress from understanding multiplication as ‘repeated addition’ to that of an area or an array. 4 x 3 does not represent the same as 3 x 4 but it has the same value.

Multiplication represented as an area
4 x 3 3 x 4 4 3 3 4 Representing multiplication as an area or array provides helpful imagery which is important in supporting learners conceptual understanding.

Multiplication represented as an area
4 x 3 = 12 3 x 4 = 12 4 3 3 4 An area or array model is particularly useful in supporting progression in multiplication as they can be used to represent: 1 digit number x 1 digit number (as above) or many digit number x many digit number as in the following slides

Multiplication represented as an area
Calculate 30 x 14 30 14 ie two digit number x two digit number

Multiplication as an area
14 10 4 30 30 x 10 = 300 30 x 4 = 120 two digit number x two digit number 30 x 14 = (30 x 10) + (30 x 4) = = 420

Modelling 38 x 14. Find ? 14 10 4 30 38 ? 30 x 10 ? x 10 30 x 4 ? x 4 two digit number x two digit number 38 x 14 = (30 x 10) + (? x 10) + (30 x 4) + (? x 4) = ? ? = ?

Area model for multiplication 38 x 14
30 8 10 8 x 10 = 80 30 x 10 = 300 Compare these models! 4 30 x 4 = 120 8 x 4 = 32 The area model is useful as it provides a pictorial image which highlights from where the numbers in the calculation come. 14 x 38 as an area 14 x 38 as a calculation 38 X 14 152 380 532 30 x 10 = 300 + 8 x 10 = 80 + 30 x 4 = 120 + 8 x 4 = 32 14 x 38 = 532 +

Modelling 38 x 142. 142 100 40 30 38 8 2 30 x 100 8 x 100 30 x 40 8 x 40 two digit number x three digit number 30 x 2 8 x 2

Progression in the area model for multiplication
2 (x + 3) = 2x + 6 x 3 2 2 x 3 2 x x An area or array model of multiplication also supports progression into algebraic methods, in this case the ‘distributive’ property.

Progression in the area model to multiplying decimal fractions
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Representation of 0∙3 An area model also supports understanding of the multiplication of decimal fractions.

Extending area model to multiplying decimal fractions
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0∙3 = 0.12 x 0∙4 Learners sometimes wrongly calculate 0.3 X 0.4 = instead of the correct solution Learners can confuse the rule for addition with that of multiplication of decimal fractions. An area model provides helpful imagery which can eliminate this confusion and support better understanding.

“I know the 2 times and 3 times table
“I know the 2 times and 3 times table. My teacher tells me I can work out the rest.” Discuss !

Multiplication facts from the 2 times and 3 times tables.
Using x2 table x4 and x8 (by doubling) you can derive the Using x x6 (x2 x3 or x3+x3) and x9 (x3 x3) Using x2 and x3 tables x5 (x2+x3) Using x3 and x x7 (x3+x4) x10 (x5+x5) deserves special attention. So, I can work out all the times tables up to x10???

Discuss! Learning multiplication facts
Hints and tips for helping with 7x8, 8x9, 9x9 etc When they have mastered the early multiplication facts eg x2, x3, x4, x5, x10, children have only 10 more x facts to learn! Discuss!

Learning multiplication facts
Use commutative properties, e.g. 3 x 4 = 4 x 3 So, when learning 6x table children already know facts up to 6 x 5 (from the x5 table) which leaves only four facts to learn: That leaves only six more facts to master on the x7, x8 and x9 tables, ie : = 6x6, 6x7, 6x8, 6x Why? Three facts of x7 table ie 7x7, 7x8, 7x9 Two facts of x8 table ie 8x8, 8x9 One fact of x9 table ie 9x9

The ‘distributive’ property for X and ÷
20 4 24 44 48 68 72 3 X = Multiplication as repeated addition 20 4 40 60 64 68 72 The use of ‘empty’ number lines is vitally important in supporting children’s early learning about the number system. Here, the number line can be used effectively to illustrate the relationship between two key concepts in multiplication ie ‘ repeated addition’ and the ‘distributive’ property. 3 X = 3 x x 4 Distributive property of multiplication Development & Progression 2nd level – ‘ using their knowledge of commutative, associative and distributive properties to simplify calculations’

Using money to illustrate the distributive property
3 x 21p = (3 x 20p) + (3 x 1p) Using money to illustrate the distributive property Progression – Level 2: ‘use their knowledge of ... distributive properties ...’ Multiplication 21p 21p 21p

Using money to illustrate the distributive property
Progression – Level 2: ‘use their knowledge of ... distributive properties ...’ Multiplication Show 3 x 24p = (3 x 20) + (3 x 4)

Using money to illustrate the distributive property
Progression – Level 2: ‘use their knowledge of ... distributive properties ...’ Division Show 87p ÷ 3

An effective strategy for multiplication
halving and doubling 8 x 15 example halving doubling = 4 x 30 = 2 x 60 = 1 x 120 = 120

An effective strategy for division
halving and halving example 204 ÷ 4 = 102 ÷ 2 halving doubling = 51 ÷ 1 = 51