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Developing Mathematical Thinking In Number : Focus on Multiplication

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Aim of presentation To encourage staff reflection on approaches to teaching number. To stimulate professional dialogue. To use as a CPD activity for staff individually or collegiately.

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Experiences and Outcomes I can use addition, subtraction, multiplication and division when solving problems, making best use of the mental strategies and written skills I have developed. MNU 1-03a MNU 1-03a Having determined which calculations are needed, I can solve problems involving whole numbers using a range of methods, sharing my approaches and solutions with others. MNU 2-03a MNU 2-03a I can use a variety of methods to solve number problems in familiar contexts, clearly communicating my processes and solutions. MNU 3-03aMNU 3-03a Having recognised similarities between new problems and problems I have solved before, I can carry out the necessary calculations to solve problems set in unfamiliar contexts. MNU 4-03aMNU 4-03a

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Progression

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Building up times tables

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How many cubes? What would be efficient ways of finding out how many cubes there are?

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Group in 2s and Count in 2s?

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Group in 5s and Count in 5s?

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9 When children have mastered the facts of eg x2, x3, x4, x5, x10, children have only 10 more x facts to learn! Multiplication Facts Discuss!

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Multiplication Facts Using commutative property. The 10 more facts to learn are ie 6x6, 6x7,6x8,6x9 Why? 7x7, 7x8, 7x9 8x8,8x9 9x9 = How well do children calculate?

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6x6 Square numbers 5x54x43x32x2 Any other patterns? Why are they called square numbers? How do we encourage pupils to investigate?

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What is the most sensible order for teaching times tables? How can we help children see the links between the times tables?

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“I know the 2x and 3x table. My teacher tells me I know the rest.” Discuss !

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From x2 x4 and x8 (doubling) From x3 x6 (x2x3) and x9 (x3x3) From x2 and x3 x5 (x2+x3) From x3 and x4 x7 (x3+x4) Making the links between the tables What about x10? What tables does this help with? What about x10? What tables does this help with?

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From repeated addition to multiplication as array and as area 3+3+3+34+4+4 4 rows of 3 = 4 x 3 3 rows of 4 = 3 x 4 How do these images help children’s understanding?

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204 4 4 2444486872 3 x 24 = 24 + 24 + 24 Multiplication as repeated addition 20 444 4060646872 3 x 24 = (3 x 20) + (3 x 4) Using the distributive property of multiplication Progression 2 nd level – ‘ using their knowledge of commutative, associative and distributive properties to simplify calculations’

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24p Illustrating the distributive law using money 3 x 24p = (3x20p) + (3x4p) How do these images help children’s understanding? What might be an added challenge in this example? 24p

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14 30 Area = 30 x 14 Multiplication as area

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14 10 4 30 30 x 10 = 300 30 x 4 = 120 30 x 14 = (30 x 10) + (30 x 4) = 300 + 120 = 420 Area models for multiplication

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14 10 4 30 30 x 10 = 300 30 x 4 = 120 38 x 14 8 x 10 = 80 8 x 4 = 32 8 30 x 10 = 300 8 x 10 = 80 30 x 4 = 120 8 x 4 = 32 38 x 14 = 532 Area models for multiplication 38 X14 152 380 532 What is the explanation for the algorithm values ? Why include the zero?

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A challenge... Draw a similar diagram to explain what is happening in the calculation 48 x 34 ?

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Solution 34 30 4 40 40 x 30 = 1200 30 x 4 = 120 8 x 30 = 240 8 x 4 = 32 8

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2 x 2 x x 2 (x + 3) = 2x + 6 3 x 2 3 Area models for multiplication

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x 2 x X2X2 2x2x (x + 3) (x + 2)= x 2 + 3x + 2x + 3x2 = x 2 + 5x + 6 3x3x 3 x 2 3 Area models for multiplication

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y b x xy bxbx (x + a) (y + b) = xy + ay + bx + ab ayay ab a Area models for multiplication

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Further support for progression in mathematics http://www.ltscotland.org.uk/curriculumforexcellence/mathematics/outcomes/ moreinformation/developmentandprogression.asp

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Make the links 3x4=12 12÷3=4 12÷4=3 ¼ of 12 = 3 30 x 4= 120 30 x 40 = 1200 0.3x 4= 1. 2 0.4x 3= 1. 2 25% of 120 = 30

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Next steps What information will you share with colleagues? What might you or your staff do differently in the classroom? What else can you do as to improve learning and teaching about number What impact will this have on your practice?

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Developing Mathematical Thinking In Number : Focus on Multiplication

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0 x0 0. 0 x2 0 0 x1 0 0 x3 0 1 x7 7 2 x0 0 9 x0 0.

0 x0 0. 0 x2 0 0 x1 0 0 x3 0 1 x7 7 2 x0 0 9 x0 0.

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