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© PDST 2014
This work is made available under the terms of the Creative Commons Attribution Share Alike 3.0 Licence http://creativecommons.org/licenses/by-sa/3.0/ie/. You may use and re-use this material (not including images and logos) free of charge in any format or medium, under the terms of the Creative Commons Attribution Share Alike Licence.
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Mental Maths Strategies Workshop 2:Multiplication &
Division
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PDST Workshop Booklet
Support Materials
PDST Mental Maths Handbook
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• To provide participants with opportunities to reflect on current practice
• To develop participants’ facilitation of learning skills
• To actively engage participants in identifying, naming and applying mental maths strategies for multiplication and division
Objectives
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• The teacher’s role is to facilitate mathematical thinking
• Concrete and pictorial exploration of number is the foundation of mental maths strategy development
• Pupils should devise, communicate and reflect on various mental maths strategies
Key Messages
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• What are the main challenges pupils face when calculating mentally?
• What is your current practice in relation to mental calculation?
Reflection
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“Rapid recall of number facts is one aspect of mental calculation but there are others. This involves presenting children with calculations in which they have to work out the answer using known facts and not just recall it from a bank of number facts that are committed to memory.”
(Crown:2010, p.12)
Number Facts or Mental Maths
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• Commutative Property
• Associative Property
• Distributive Property
• Inverse Relationship
Fundamental Facts
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• Professional Reading Time-handbook p.8
Linear, Area and Set Models
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• Starts by assessing the knowledge of the group
• Information flows in many different directions
• Facilitators elicit a variety of methods from pupils
• Facilitators encourage pupils to critically evaluate solution methods
Teacher as Facilitator
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Practical Task
In groups, use Cuisenaire to show the distributive property of 5 ×14•How many ways can you show this? •Represent some of these ways in the pictorial mode using squared paper.
•Make up a multiplication problem and use Cuisenaire to demonstrate the associative property of the problem.
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• Identify good practice from the clip
• Describe the role of the teacher
• What skills are the pupils displaying?
Video Footage
12Sherry Parrish, Number Talks Clip 3.7 Array Discussion
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Eliciting
Supporting
Extending
Instructional Framework
Child-centred Teacher as Facilitator
Language-based
Revoicing
Strategy-sharing
Higher-order
skillsTeacher not sole
validator of
mathematical
knowledge
Pupil self-
reflection
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“To help children to learn and draw on a range of mental methods, you need to raise their awareness and understanding of the range of possible strategies, develop their confidence and fluency by practising using the strategies, and help them to choose from the range the most efficient method for a given calculation.”
(Crown:2010, p.15)
A Range of Methods…
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I just know it
• Counting Strategies – using object counting (for example blocks or fingers) or verbal counting to determine the answer
• Reasoning Strategies – using known information to logically determine an unknown combination
• Mastery – efficient (fast and accurate) production of answers
Stages of Progression
15Arthur Baroody (2006)
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COUNTING
Mental Strategies for × and ÷
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Name the strategy…
• 4.3 ×20 • 188 ÷4
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Handbook p.33
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Name the strategy…
• 5 × 13 • 8 ×25
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Handbook p.35
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Name the strategy…
• 15 ×16
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Handbook p.43
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Name the strategy…
• 720 ÷ 36
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Handbook p.48
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Name the strategy…
• 38 × 7 • 860 ÷ 4
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Handbook p.38
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Name the strategy…
• 32 ×15 • 384 ÷16
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Handbook p.39 , 42
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Name the strategy…
• 398 ×6 • 1764 ÷18
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Handbook p.35
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Name the strategy…
• 12 × 15 • 348 ÷12
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Handbook p.49, 50
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Name the strategy…
• 15 ÷3
25
Handbook p.51
•550 ÷15
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Skip Counting Doubles 2s, 4s, 8s Doubling & Halving
Repeated Addition/Subtraction
Partitioning by Place Value
Break Down Factors/ Factorisation
Rounding & Compensating
Using Known Facts
Division
•Think multiplication
•Proportional Adjustment/Reasoning
Multiplication and Division Strategies
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Partial Products/ Quotients
Multiplying/Dividing by multiples of 10
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No pressure
You can work with a partner to figure them out
The Test !
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Do:
Mastering the Basic Facts
1. Ask students to self-monitor
2. Focus on self improvement
3. Drill in short time segments
4. Work on facts over time
5. Involve families
6. Make drill enjoyable
7. Use technology
8. Emphasise the importance of quick recall of facts
Avoid:
1. Using lengthy timed tests
2. Using public comparisons of mastery
3. Proceeding through facts in order from 0 to 9
4. Moving to memorisation too soon
5. Using facts as a barrier to good mathematics
28Van de Walle, p.188
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1. Recognise that more drill will not work
2. Provide hope
3. Inventory – the known and unknown facts for each student in need
4. Identify strengths and weaknesses
5. Focus on reasoning strategies
6. Build in success
7. Provide engaging activities for drill
Steps to Success for Pupils with SEN…
29Van de Walle (p.186/187)
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• Anxiety
• Use the Multiplication Table
• Identify the known facts
• One times, times one
• Doubles, fives and tens
• Squares
• Commutative property
• Only 15 facts remaining!
Struggling Pupils
30First Steps in Mathematics, Book 2 p.193
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• Build up Facts to 5
• Ones and Twos
• Tens
• Squares
• Fives
• Fours and Eights
• Threes and Sixes
• Nines
Possible Sequence for Fact Mastery
31First Steps in Mathematics,: Number, Book 2
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5 5 10 15 20 25
4 4 8 12 16 20
3 3 6 9 12 15
2 2 4 6 8 10
1 1 2 3 4 5
× 1 2 3 4 5
Build up facts to 5
32First Steps in Mathematics,: Number, Book 2
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5 5 10 15 20 25
4 4 8 12 16 20
3 3 6 9 12 15
2 2 4 6 8 10
1 1 2 3 4 5
× 1 2 3 4 5
Ones and Twos
33First Steps in Mathematics,: Number, Book 2
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10 10 20 30 40 50 60 70 80 90 100
9 9 18 81
8 8 16 64
7 7 14 49
6 6 12 36
5 5 10 15 20 25 30 35 40 45 50
4 4 8 12 16 20 24 28 32 36 40
3 3 6 9 12 15 18 21 24 27 30
2 2 4 6 8 10 12 14 16 18 20
1 1 2 3 4 5 6 7 8 9 10
× 1 2 3 4 5 6 7 8 9 10
Tens & Squares
34First Steps in Mathematics,: Number, Book 2
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10 10 20 30 40 50 60 70 80 90 100
9 9 18 45 81
8 8 16 40 64
7 7 14 35 49
6 6 12 30 36
5 5 10 15 20 25 30 35 40 45 50
4 4 8 12 16 20 24 28 32 36 40
3 3 6 9 12 15 18 21 24 27 30
2 2 4 6 8 10 12 14 16 18 20
1 1 2 3 4 5 6 7 8 9 10
× 1 2 3 4 5 6 7 8 9 10
Remaining Fives
35First Steps in Mathematics,: Number, Book 2
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10 10 20 30 40 50 60 70 80 90 100
9 9 18 36 45 72 81 90
8 8 16 24 32 40 48 56 64 72 80
7 7 14 28 35 49 56 70
6 6 12 24 30 36 48 60
5 5 10 15 20 25 30 35 40 45 50
4 4 8 12 16 20 24 28 32 36 40
3 3 6 9 12 15 18 21 24 27 30
2 2 4 6 8 10 12 14 16 18 20
1 1 2 3 4 5 6 7 8 9 10
× 1 2 3 4 5 6 7 8 9 10
Fours & Eights
36First Steps in Mathematics,: Number, Book 2
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10 10 20 30 40 50 60 70 80 90 100
9 9 18 27 36 45 54 72 81 100
8 8 16 24 32 40 48 56 64 80
7 7 14 21 28 35 42 49 56 70
6 6 12 18 24 30 36 42 48 54 60
5 5 10 15 20 25 30 35 40 45 50
4 4 8 12 16 20 24 28 32 36 40
3 3 6 9 12 15 18 21 24 27 30
2 2 4 6 8 10 12 14 16 18 20
1 1 2 3 4 5 6 7 8 9 10
× 1 2 3 4 5 6 7 8 9 10
Threes & Sixes
37First Steps in Mathematics,: Number, Book 2
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10 10 20 30 40 50 60 70 80 90 100
9 9 18 27 36 45 54 63 72 81 100
8 8 16 24 32 40 48 56 64 72 80
7 7 14 21 28 35 42 49 56 63 70
6 6 12 18 24 30 36 42 48 54 60
5 5 10 15 20 25 30 35 40 45 50
4 4 8 12 16 20 24 28 32 36 40
3 3 6 9 12 15 18 21 24 27 30
2 2 4 6 8 10 12 14 16 18 20
1 1 2 3 4 5 6 7 8 9 10
× 1 2 3 4 5 6 7 8 9 10
Nines
38First Steps in Mathematics,: Number, Book 2
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• How will your practice in relation to mental maths change now?
• List the resources you may need to acquire to support pupils’ learning
Reflect Again
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• Can you identify any of today’s strategies in the clip?
• How would you rate the communication and expression skills of the pupils?
• Are they problem-solving? How?
Video Footage
40Sherry Parrish, Number Talks Clip 5.2
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School-Self EvaluationTeaching & Learning Framework; 6 Step SSE Process; gathering, collating & analysing relevant data; implementing the SIP for
literacy, numeracy and any other area of teaching & learning.
Health and Wellbeing SPHE including Child Protection and Stay Safe Relationships and Sexuality Education (RSE) Active Learning Methodologies Anti Bullying-Procedures and Policy, Awareness
Raising, Intervention, Prevention and Intervention Strategies
PE Wellbeing for teachers
PDST Websites
www.pdst.iepdsttechnologyineducation.iescoilnet.ie (portal for resources)teachercpd.ie (on-line courses)
Overview of PDST Primary Supports for Leading Learning in the 21st Century
www.pdst.ie/schoolsupport
Models of support:in-school support (in-class modeling, meeting principal/groups of teachers, Croke Park hrs), seminars, workshops, clusters
Literacy SSE & strategies for improving
oral language, writing, reading and the use of broadcast /digital media.
NumeracySSE & strategies for improving
numeracy. Supporting the development of Mathematical
thinking.
GaeilgeTacaíocht chun Curaclam na Gaeilge a chur i
bhfeidhm i Scoileanna T1&T2, Féinmheastóireacht Scoile, chomh maith le
tacaíocht lán Ghaeilge a sholáthar do Ghaelscoileanna agus scoileanna sa
Ghaeltacht.
Integrating ICT - PrimaryActive Learning Methodologies- Key methodologies of the primary curriculum through the effective use of ICT.Literacy & Numeracy – Creating, developing and using ICT to create teaching resources in all areas of primary literacy and numeracy.SSE – Online tools for gathering, collating & analysing relevant data.eAssessment & ePortfolios – Assessment of / for learning with ICT.ePlanning & Collaboration – Google Apps for Education.Tablet Technology Integration – Pedagogy, Curriculum & SEN Virtual Learning Environments (VLEs) – Google Classroom, Edmodo
PDST Leadership ProgrammesMisneach.....New PrincipalsTánaiste.......New Deputy PrincipalsTóraíocht.....Aspiring Leaders accredited by
Maynooth UniversityForbairt .......Experienced Principals & ALNsSpreagadh...NAPD & PDST collaboration
www.pdst.ie
The PDST is funded by the Teacher Education Section (TES)
of the Department of Education and Skills (DES) and is
managed
by Dublin West Education Centre
Fís
Fog
hlai
mF
orba
irt
This work is made available under the terms of the Creative Commons Attribution Share Alike 3.0 Licence http://creativecommons.org/licenses/by-sa/3.0/ie/. You may use and re-use this material (not including images and logos) free of charge in any format or medium, under the terms of the Creative Commons Attribution Share Alike Licence.
© PDST 2014
