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2.5 X 3.2 = Aims of the Workshop Maths workshop Tuesday 27th January, 2015 2.5 x 3.2 = Aims of the workshop: • Understanding what underpins Maths teaching, the aims of the National Curriculum • What is place value and how to help our children understand it • The application of place value to the four operations • Demystifying decimals • Tables facts What underpins our Maths teaching? The responsibility of mathematics education is to enable all pupils to develop conceptual understanding of the mathematics they learn, its structures and relationships, and fluent recall of mathematical knowledge and skills to equip them to solve familiar problems as well as tackling creatively the more complex and unfamiliar ones that lie ahead. Ofsted: Mathematics Made to Measure May 2012 The Aims and Purpose of The National Curriculum (2014) for mathematics aims to ensure that all pupils: • become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils have conceptual understanding and are able to recall and apply their knowledge rapidly and accurately to problems • reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language • can solve problems by applying their mathematics to a variety of routine and non- routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions. What is place value and how can we help our children understand it? KEY FACTS • In our number system, all numbers can be represented using a finite set of digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). • The place the digit is written within a number gives us its value. • The system uses 10 as a base. • Larger whole numbers than 9 are constructed using powers of 10. 101 = 10 102 = 10x10 = 100 103 = 10x10x10 = 1000 • When 10 is accumulated in one place, this can be exchanged for one in the next place. What is place value and how can we help our children understand it? Place value grid H T U What is place value and how can we help our children understand it? Diennes and Coins www.topmarks.co.uk Order of operations: BIDMAS • Brackets • Indices • Division/Multiplication • Addition/Subtraction Use Bidmas to find what the difference is between: (4+5) x 2 and 4 + 5 x 2 Can you make statements equal to all the numbers 0 to 20 by using just four fours? You can use the four operations and brackets to do this. -4 -4 +4 +4 = 0 44÷44 = 1 (4÷4)+ (4÷4)= 2 (4+4+4) ÷4 = 3 (4-4) x 4 + 4 = 4 (4x4 + 4) ÷4 = 5 Addition and subtraction KEY FACTS • Addition and subtraction facts are inverse operations and can be grouped in families of facts eg: 3+4=7 and 4+3=7 and 7-4=3 and 7-3=4 • We can use our understanding of place value and these number facts to help us solve related calculations 30+40=70, 70-30=40, 0.3+0.4=0.7 and so on • Addition can be done in any order and still give the same answer (commutative law), this does not apply to subtraction • Any number from which 0 is subtracted leaves the number unchanged: 5-0=5 • Any number added to 0 leaves the number unchanged: 5+0=5 Addition and subtraction Progression in addition Progression in subtraction • Mental methods and key facts • Mental methods and key facts • Moving on from place value: • Moving on from place value: adding 1s, 10 and 100s subtracting 1s, 10 and 100s • Partitioning • Difference by counting on • Expanded column method • Expanded column method • Formal written method • Formal written method The word “sum” is only ever addition, not any other calculation, so avoid saying “this is a subtraction sum”. What is the sum of 19 and 17? It is 19+17. With subtraction, avoid saying, we always put the bigger or higher value number first, as later on, children will have to solve calculations such as 8-17 (temperature, bank balances etc!) When working on column addition or subtraction, avoid using the word “borrow”, who or what are we borrowing? It is better to say “exchanging” or “regrouping”. Times tables KEY FACTS • Multiplication and division facts are inverse operations and can be grouped in families of facts ie: 3x4=12 and 4x3=12 and 12÷4=3 and 12÷3=4 • We can use our understanding of place value and our times table facts to help us solve related calculations: 30x4=120 (effective 3x10x4=120) or 120÷40=3 (12x10÷4x10=3) or 30 x 0.4 = 12 • Multiplication can be done in any order and still give the same answer (commutative law), this does not apply to division • Any number multiplied by 1 stays the same: 1x2=2 1x56=56 • Any number or series of numbers multiplied by 0 gives an answer of 0: 5x0=0; 15x17x18x0=0 Times table facts What are all the factors of 24? Of 32? 24 32 1, 2, 3, 4, 6, 8, 12, 24 1, 2, 4, 8, 16, 32 What are the common factors? 1, 2, 4, 8 Can you solve the following? What would I multiply 0.6 What would I multiply 2 by by to get 24? to get 320? We never say that we add a zero when we multiply by 10 or take away a zero when we divide by 10. Avoid saying that we always put the bigger or higher value number first in division calculations. Times table facts 2.5 x 3.2 = 3.2 = 4 x 0.8 2.5 x 4 = 10 10 x 0.8 = 8 2.5 x 3.2 = 8 Times table facts (x only): 2x2=4 3x3=9 4x4=16 5x5=25 6x6=36 7x7=49 8x8=64 9x9=81 10x10=100 11x11=121 12x12=144 2x3=6 3x4=12 4x5=20 5x6=30 6x7=42 7x8=56 8x9=72 9x10=90 10x11=110 11x12=132 2x4=8 3x5=15 4x6=24 5x7=35 6x8=48 7x9=63 8x10=80 9x11=99 10x12=120 2x5=10 3x6=18 4x7=28 5x8=40 6x9=54 7x10=70 8x11=88 9x12=108 2x6=12 3x7=21 4x8=32 5x9=45 6x10=60 7x11=77 8x12=96 2x7=14 3x8=24 4x9=36 5x10=50 6x11=66 7x12=84 2x8=16 3x9=27 4x10=40 5x11=55 6x12=72 2x9=18 3x10=30 4x11=44 5x12=60 2x10=20 3x11=33 4x12=48 2x11=22 3x12=36 2x12=24 Demystifying decimals KEY FACTS AND IDEAS TO HELP • The decimal point is there the whole time even in integers (whole numbers) we just don’t always show it ie 3 is really 3.0 • The same principles apply when adding or subtracting with decimals of place value and exchange – it is still base 10. 1 = ten tenths • Use rounding to estimate answer in order to have a sense check eg: 1.9 x 2.7; this rounds to 2x3 so the answer should be about 6. • Use money to help explain decimals: let £1 be the unit and then 10p represent tenths and 1p represent hundredths • Convert a number to a whole number by multiplying by 10 or 100 and then divide by that number at the end: eg 1.19m x 8 can be multiplied by 100 to make 119cm x 8=952 and then divide it back by 100 to make 9.52m 2.5 x 4.04 x 0.9 = 9.09 2.5 x (4+0.04) x 0.9= ((2.5 x 4) + (2.5 x 0.04)) x 0.9 = (10 + 0.1) x 0.9 = 9 + 0.09 = 9.09 45 x 1.58 + 5.5 x 15.8 = 158 45 x 1.58 + 55 x 1.58 = (45 + 55) x 1.58 = 100 x 1.58 = 158 The decimal point does not move, the digits move and we put in or take away zero place holders. Websites: some suggestions Maths activities to support the new curriculum (other subjects too!): http://www.theschoolrun.com/ Various maths games: http://www.transum.org/Software/Game/ http://mathszone.co.uk/number-facts-xd/ http://www.primaryinteractive.co.uk/maths.htm Open-ended maths puzzles: http://nrich.maths.org/ Maths triangles: http://www.helpingwithmath.com/printables/others/fac0201fact_trian gle01.htm .
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