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Study the NC2014 approach to multiplication and division, develop fluency of times tables, and explore different representations and written methods. Consider the role of calculators and links to measurement.
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SCITT Day 4 2018 Multiplication (and links to Division)
DAY 4 - CALCULATION – MULTIPLICATION with some links to division • Study the NC2014 approach to multiplication and division including using manipulatives and/or recording mental methods, informal jottings and formal written methods; • Consider how the use and complexity of arrays progress through the primary years • Explore the importance of developing fluency of times tables and deriving related division facts Associated Issues for Teaching • The role of calculators in the classroom • Making links to Measurement • Multiplication – facts, representations, flexible, connections, grid, formal • Division – making connections
My role: Provide time to think and consider You: Make sense of the ideas and activities Stop and ask questions…
Words/ phrases x and ÷ What different words/ phrases can you list… ?
Day 4 - Multiplication and Division Multiplication as repeated addition Arrays grid method Standard written methods Arrays Multiplication as scaling Build it Language Models and images White Rose Calculation guidance Understanding x/÷ Fluency Learning x tables Division as grouping Division as sharing Links between x/÷ Understandings and connections Number facts Written methods
Representations… “4 children each have a bag of sweets, there are 3 sweets in each bag.” • Represent MANY ways [&…? &…?] • Unstructured v structured • Language/ vocabulary Cubes Counters Tens Frames Numicon Number rods Dienes …….. Number lines X grids Other…? “Show me. Convince me.” Cbuild it, make it Pdraw or sketch it Avisualise, write it • Key Messages: • Identifying a group • Orientation • Making connections • Commutativity • Division • Representation matching language!
Two types of multiplication • Repeated addition • Scaling Tracking National Curriculum Expectations • What are the expectations for x and ÷ in your year group? • Are both multiplication types mentioned? • Any ‘unclear’ language/ any examples required? 2 2 + 2 + 2 2 2 2 ‘3 times longer’
WRMH guidance • Make sense of the images • Are there important messages? • Compare with practice in your classroom What other CPA images could you add to this guidance?
3 x 5 = How could you work it out? 2 x sentences 2 ÷ sentences
Secrets for Cheating at x tables Secret 1: If you know one fact, then you know a second. Secret 2 X tables always give you a buy one, get three free offer. Secret 3: If you can double, then you can multiply by 2, 4 and 8. Secret 4: If you can multiply by 1, then you can multiply by 10. Secret 5: If you can halve, then you can multiply by 5. Secret 6: If you use fingers, then you can multiply by 9 Nines in your head: Multiply by 10 and then adjust Secret 7: If you know the 3 times table, then you know the 6 times table.
If I know that 5 x 7 = 35 Repeated addition Place Value 50 x 7 = 500 x 7 = 0.5 x 7 = 5 + 5 + 5 + 5 + 5 + 5 + 5= 7 + 7 + 7 + 7 + 7 = Any order and inverse 7 x 5 = 35 35 ÷ 7 = 5 35 ÷ 5 = 7 Fractions One fifth of 35 = 7 Two fifths… Three fifths… One seventh of 35 = 5 Two sevenths… 6 x 7 = Or 4 x 7 = Derive new facts from known facts Repeated subtraction 5 x 6 = Or 5 x 8 = 35 – 7 – 7 – 7 – 7 – 7 = 0 35 – 5 – 5 – 5 – 5 – 5 – 5 – 5 = 0
H T U? 0 2 4 0
Making your own ‘fact families’ x and ÷ cards 3 x 4 =12 4 x 3=12 12÷3=4 12 shared between 3 equals 4 One third of 12 = 4 ¼ of 12 = 3
Formal methods… 1st Learners UNDERSTAND what x/÷ is 2nd x/÷ facts are fluent 3rd Learners apply their learning to solve problems Then ready to explore a progression to formal methods
‘Grid’ or ‘Area’ multiplication Multiplication 13 x 6 = 3 10 60 18 60 + 18 = 78 6 Can this be built more efficiently? so 13 x 6 = 78
Multiplication 13 x 6 = NB from reading 78 ÷ 6 = (60 + 18) ÷ 6 = 10 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10 10 10 10 10 Now build and record: 24 x 3 = 125 x 4 =
Just use grid multiplication… 64 x 8 = 234 x 6 = 34 x 78 =
2100 280 = 2380 240 32 = 272 Grid method 3 4 7 8 x 3 2 2 4 0 2 8 0 2 1 0 0 + 2 6 5 2 3 4 8 x 2 7 2 3 4 7 8 x 2 7 2 2 3 8 0+ 2 6 5 2 2 3 8 x 4 8 x 30 70 x 4 70 x 30 3 4 7 0 x 2 3 8 0 1 1 Long multiplication Expanded method 2 x Short multiplication
White Rose Materialswww.whiterosemaths.com Planning Task: Time to explore the materials and engage with the mathematics activities and ideas… What are the age-related expectations? What ideas are offered to cement fluency? How do the R &PS activities offer challenge? (Through breadth…? depth…?) OTHER SLIDES FOLLOW…
Two structures of division • Sharing (links with ‘finding fractions of amounts’) • Grouping (repeated subtraction) Draw a picture… • There are 18 sweets share equally between 3… • There are 18 sweets, divide them into 3s… Other models and images…? Counters/ arrays/ number lines/ number rods/ Numicon …
The language of division Should always use the language ‘divided’ 18 ÷ 3 = “18 dividedby 3” Sharing – “18 dividedbetween 3” Grouping – “18 divided into 3’s” How would you calculate 18 ÷ 3 = ? Also a fraction: “Each equal part is a third” Language must match the model
Division 18 ÷ 3 = 18 grouped into 3’s How many 3’s in 18? 18 shared equally between 3 How many each? “One for you, one for you…” Answer 6 6 each 6 groups
Division 18 ÷ 3 = 18 grouped into 3’s How many 3’s in 18? What would this look like with Numicon? With Number rods? Answer 6 Equal in area Equal in length
Remainders 20 ÷ 3 = 6 r2 20 grouped into 3’s How many 3’s in 20? 20 shared equally between 3 “One for you, one for you…” ‘Remainder 2’ or 2 ‘left over’ 2 2
Calculate… 21 ÷ 3 = 24 ÷ 6 = 15 ÷ 5 = 20 ÷ 4 = 24 ÷ 8 = 35 ÷ 7 = Remainders 26 ÷ 5 = 25 ÷ 4 = Going over ‘10x’ - mental 38 ÷ 2 = 39 ÷ 3 = 44 ÷ 4 = 48 ÷ 6 = Calculate then discuss ‘how to teach’ • Numicon • Number rods • Counters/ cubes arrays • Dienes
Splitting 81 ÷ 3 = Moving beyond the 10x table… How can I split this number using knowledge of the 3 x table? Do it Sharing or grouping? 10 1 10 10 10 10 10 10 10
Splitting 81 ÷ 3 = Moving beyond the 10x table… How can I split this number using knowledge of the 3 x table? This could be sharing between 3 people (horizontal) “10 for you, 10 for you…” OR as a group/ column of 3 tens Do it 1 10 10 10 10 10 10 10 10
Splitting 81 ÷ 3 = How can I split this number using knowledge of the 3 x table? 3 X ? ? ? 30 30 21 10 + 10 + 7 = 27 Practice: 84 ÷ 7 = / 72 ÷ 3 = / 65 ÷ 5 =
Division – Written MethodsWhat happens when the dividend goes beyond 10 x ? Use sharing or grouping to build an array for: 10 1 10 1 52 ÷ 4 = 10 10 10
Division – Written Methods with Base 10 52 ÷ 4 = 3 1 1 1 Grouping in 4s – REMOVING THE PLACE VALUE “How many groups of 4 (in each column)?”
Division – Written Methods with Base 10 52 ÷ 4 = 1 1 10 1 C build it, make it P draw or sketch it A visualise, write it 1 1 1 10 1 1 1 10 1 1 1 10 Making links back to grid method
Division – Written Methods with Base 10 72 ÷ 3 = 10 10 10 10 1 1 C build it, make it P draw or sketch it A visualise, write it 10 10 10 Making links back to grid method
94 ÷ 4 = 94 ÷ 4 = 2 3 r2 Chunking Short Division 4 9 4 9 4 4 0 – 5 4 4 0 – 1 4 1 2 – 2 10 x 10 x 3 x 23 r2 1 How many groups of 4 are in this column? 2 r1 How many groups of 4 are in this column? 3 r2 )
98 ÷ 7 = Long division How many 7s in this column? Short division 98 - 70 28 - 28 0 10x 4 x 1 4 2 How many 7s in this column?
Your turn! Use chunking & short division to work out 144 ÷ 6 =
TASK 2a and 2b[EITHER +/- or x/ ÷] HAND IN DAY 5 • TASK 2b: Observe a lesson where a model/ image is used by the teacher and/or learners. After the lesson interview the children and critically evaluate the impact on their understanding. • LEARNING OUTCOME: To be more aware of the important role of models and images when teaching for mastery understanding • Maths TASK 2a – Briefly review the use models and images used in your classroom (CPA) for Place Value. Photo/ sketch/ draw the models to discuss how they are used. • LEARNING OUTCOME: To be more aware of the important role of models and images when teaching for understanding
GROUP READING FOR DAY 5: • D Haylock ‘Mathematics Explained for Primary Teachers’ Third Edition • Chapter 14 ‘Fractions and Ratios’