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Workshop 4: Multiplication, Division and Fractions Presented by CW Building the knowledge foundations Factor of 10 between columns Groups to 20 Decade Pairs to 100 10’s in any number Name columns 10+ Counting Using decades 5+ Numerals X 2, x5, x10 Larger numbers Importance of 10 Doubles to 20 Groups to 5 Using 10 x Using facts to10 Count in 2’s 5’s 10’s Groups to 10 Sequence Doubles to 10 Language Sets Multiplication type problems I have 3 bags of lollies and there are 5 lollies in each bag. How many lollies have I got altogether? Can I solve the problem if I am stage 0 or stage 1? How would I solve the problem if I am stage 2 or stage 3? How could I solve the problem if I am at stage 4 and what knowledge MUST I have in my tool box? The key Idea to use Skip Counting Skip Counting like counting can be just rote learnt. Realisation that the skip counting sequence relates to putting the same sets of numbers together and the end count measures the set and that the rote count forwards gives the result of adding the set number of objects and backwards gives the result of subtracting the set number of objects. Stage 4 - 5 Key Idea: Children are learning to use addition strategies to solve problems that could be solved by multiplication. Multiplication as repeated addition Commutativity of multiplication Sharing into equal sets Grouping, how many sets can be made. Stage 5: Using known facts 2 x doubles 10 x place value 5 x place value and doubles (halves) X 2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10 4 6 8 10 12 14 16 18 20 6 8 12 14 16 18 15 20 30 35 40 45 30 40 10 15 20 25 30 35 40 45 50 60 70 80 90 20 30 40 50 60 70 80 90 100 Stage 5 - 6 Children are learning to derive further multiplication facts using addition and subtraction strategies from multiplication facts they already know. 4 x = 2 x + 2 x (doubling) 3x=2x+1x 6x=5x+1x ( or double 3 x when 3 x are known facts) X 2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10 4 6 8 10 12 14 16 18 20 6 9 12 15 18 21 24 27 30 8 12 16 20 24 28 32 36 40 10 15 20 25 30 35 40 45 50 12 18 24 30 36 42 48 54 60 14 21 28 35 42 16 24 32 40 48 18 27 36 45 54 70 80 90 20 30 40 50 60 70 80 90 100 Not many left to learn 7x =5x+2x 8 x = 5 x + 3 x or 10 x – 2 x 9 x = 10 x – 1 x Children need to understand and be able to use the relationships between the multiplication tables. When working with bigger numbers the “tables” become the knowledge tools for multiplicative thinking in the same way as basic addition facts are the knowledge tools for additive thinking. Tools in the tool box Counting Plastic tools, good to start with but not very efficient Addition and Subtraction Increasing range of more sophisticated tools Multiplication and Division The power tools. Increasingly efficient and sophisticated. Fractions at ENP Why? When ? How? Fractional Misconceptions No acceptance of the existence of numbers between numbers Share 5 biscuits between 4 children …. You can’t do it! Fractional Misconceptions The numerator relates to the whole ½ is 1 shared between 2 ¼ is 1 shared between 4 Fractional Misconceptions Half is a variable hence all fractions are variable ½ of 12 is 6 ½ of 8 is 4 ½ of 4 is 2……so the half keeps changing Fractional Misconceptions The symbol can be either way up You can’t have fractions greater than 1 4 = 6 6 4 never been challenged if fractions greater than haven’t been explored Fractional Framework Regions Knowing how many parts make a whole Number Sets Recognising the Using half as an symbol operator on Knowing where other numbers half lives on a number line Stage 2/3 Regions Know half is 2 equal parts. Know quarter is 4 equal parts. Number Recognise symbol ½ Recognise symbol ¼ Sets Doubles and halves Stage 4 Regions Number Sets 1 half, 2 halves, 3 halves 1 quarter, 2 quarters, 3 quarters 4 quarters 5 quarters Place of ½ on a number line. Place of 2 halves 3 halves etc. Find half and quarter of a set by equal sharing. Stage 5 Regions Number Sets Comparison and ordering of unit fractions by region. ½>¼>⅛ Extend knowledge to other fractions: Thirds, fifths, tenths etc. Meaning of – ths Identify symbols for fractions including those greater than 1 Place of ½ on a number line 2x½=1 4x½=2 Link to repeated addition for multiplication. Comparison and ordering of unit fractions by sets of ½ of 12 = 6 ¼ of 12 = 3 Link to repeated addition and commutativity of multiplication. ½ x 12 = 12 x ½ = 6