24 x 3 does not represent the same as 3 x 4 but it has the same value. From repeated addition to multiplication as an array4+4+43 groups of 4= 3 x 44 groups of 3= 4 x 3It 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.
3Multiplication represented as an area 4 x 33 x 44334Representing multiplication as an area or array provides helpful imagery which is important in supporting learners conceptual understanding.
4Multiplication represented as an area 4 x 3 = 123 x 4 = 124334An 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)ormany digit number x many digit number as in the following slides
5Multiplication represented as an area Calculate 30 x 143014ie two digit number x two digit number
6Multiplication as an area 141043030 x 10 = 30030 x 4 = 120two digit number x two digit number30 x 14 = (30 x 10) + (30 x 4)== 420
7Modelling 38 x 14. Find ?141043038?30 x 10? x 1030 x 4? x 4two digit number x two digit number38 x 14 = (30 x 10) + (? x 10) + (30 x 4) + (? x 4)= ? ?= ?
8Area model for multiplication 38 x 14 308108 x 10 = 8030 x 10 = 300Compare these models!430 x 4 = 1208 x 4 = 32The 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 area14 x 38 as a calculation38X 1415238053230 x 10 = 300+ 8 x 10 = 80+ 30 x 4 = 120+ 8 x 4 = 3214 x 38 = 532+
9Modelling 38 x 142.1421004030388230 x 1008 x 10030 x 408 x 40two digit number x three digit number30 x 28 x 2
10Progression in the area model for multiplication 2 (x + 3) = 2x + 6x322 x 32 x xAn area or array model of multiplication also supports progression into algebraic methods, in this case the ‘distributive’ property.
11Progression in the area model to multiplying decimal fractions 0.10.20.30.220.127.116.11.80.91.0Representation of 0∙3An area model also supports understanding of the multiplication of decimal fractions.
12Extending area model to multiplying decimal fractions 0.10.20.30.18.104.22.168.80.91.00∙3= 0.12x0∙4Learners 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.
13“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 !
14Multiplication facts from the 2 times and 3 times tables. Using x2 table x4 and x8 (by doubling)you can derive theUsing 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???
15Discuss! Learning multiplication facts Hints and tips for helping with 7x8, 8x9, 9x9 etcWhen they have mastered the early multiplication facts eg x2, x3, x4, x5, x10, children have only 10 more x facts to learn!Discuss!
16Learning multiplication facts Use commutative properties,e.g. 3 x 4 = 4 x 3So, when learning 6x table children already know facts up to6 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, 7x9Two facts of x8 table ie 8x8, 8x9One fact of x9 table ie 9x9
17The ‘distributive’ property for X and ÷ 20424444868723 X =Multiplication as repeated addition2044060646872The 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 4Distributive property of multiplicationDevelopment & Progression 2nd level – ‘ using their knowledge of commutative, associative and distributive properties to simplify calculations’
18Using money to illustrate the distributive property 3 x 21p= (3 x 20p)+ (3 x 1p)Using money to illustrate the distributive propertyProgression – Level 2: ‘use their knowledge of ... distributive properties ...’Multiplication21p21p21p
19Using money to illustrate the distributive property Progression – Level 2: ‘use their knowledge of ... distributive properties ...’MultiplicationShow 3 x 24p= (3 x 20) + (3 x 4)
20Using money to illustrate the distributive property Progression – Level 2: ‘use their knowledge of ... distributive properties ...’DivisionShow 87p ÷ 3
21An effective strategy for multiplication halving and doubling8 x 15examplehalvingdoubling= 4 x 30= 2 x 60= 1 x 120= 120
22An effective strategy for division halving and halvingexample204 ÷ 4= 102 ÷ 2halvingdoubling= 51 ÷ 1= 51