Teacher’s Guide 5
Series Editor: Peter Clarke Authors: Elizabeth Jurgensen, Jeanette Mumford, Sandra Roberts
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Number and place value
Contents
Counting and understanding numbers Page number
Introduction Key Principles of Busy Ant Maths How Busy Ant Maths supports the 2014 Primary National Curriculum for Mathematics The components of Busy Ant Maths Busy Ant Maths Teacher’s Guide Busy Ant Maths on Collins Connect Other components in Busy Ant Maths Busy Ant Maths Year 5 Medium-Term Plans
4 5 8 9 15 16 18
Getting Started 1: Number – Number and place value 2: Number – Addition and subtraction 3: Number – Multiplication and division 4: Number – Fractions 5: Number – Decimals 6: Number – Percentages (including fractions and decimals) 7: Measurement 8: Geometry – Properties of shapes 9: Geometry – Position and direction 10: Statistics
30 36 43 47 50 52 54 61 63 65
Teaching and Learning Units 1 – 12 Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7 Unit 8 Unit 9 Unit 10 Unit 11 Unit 12 Resource Sheets
68 102 136 170 204 238 272 306 340 374 408 442 478
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Introduction
Getting Started Introduction Key Principles of Busy Ant Maths Busy Ant Maths is a mathematics course that ensures complete coverage of the 2014 Primary National Curriculum for Mathematics.
The course has at its core the following seven key principles: 1 To inspire enjoyment of maths 2 To assist in developing children’s conceptual understanding of maths 3 To help raise levels of attainment for every child 4 To provide a rigorous and cohesive scope and sequence of the primary maths curriculum, while at the same time allowing for schools’ own curriculum design 5 To promote the most effective pedagogical methods in the teaching of mathematics 6 To offer manageable strategies for effective diagnostic, formative and summative assessment, to inform planning and teaching 7 To strengthen the home/school link. In addition to these seven key principles, Busy Ant Maths offers: • a straightforward yet flexible approach to the teaching of mathematics • lesson plans following a highly effective and proven lesson structure • a weekly bank of practical hands-on Learning activities • a detailed and systematic approach to the development of mental and written calculations • extensive teacher support through materials which: – are sufficiently detailed to aid confidence – are rich enough to be varied and developed – take into account issues of pace and classroom management – give careful consideration to the key skill of appropriate and effective questioning – provide a careful balance of teacher intervention and pupil participation – encourage communication of methods and foster mathematical rigor • controlled, manageable differentiation with activities and suggestions for at least three different ability groups in every lesson • a stand-alone resource aimed at developing children’s fluency in number facts containing hundreds of whole-class, group, paired and individual games and activities • pupil materials which are enjoyable and purposeful.
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Introduction
How Busy Ant Maths supports the 2014 Primary National Curriculum for Mathematics All of the components of Busy Ant Maths emphasise, and provide guidance on, the importance of the cyclical nature of teaching in order to best promote learning and to raise pupils’ attainment.
• Promote learning • Raise pupils’ attainment
Te a c h i
ng
sment s e ss
Planning
A Planning
Busy Ant Maths supports teachers in planning a successful mathematics programme for their unique teaching context and ensures: • a clear understanding of pupils’ pre-requisite skills before undertaking particular tasks and learning new concepts • considered progression from one lesson to another • regular revisiting and extension of previous learning • a judicious balance of objectives, and the time dedicated to each one • the use of a consistent format and structure. The elements of Busy Ant Maths that form the basis for planning can be summarised as follows:
Long-term plans The 2014 Primary National Curriculum for Mathematics constitutes the long-term plan for schools to follow. By closely reflecting the Programmes of Study, the Busy Ant Maths course embodies this long-term plan.
Medium-term plans The Busy Ant Maths Medium-term Plans show termly outlines of units of work with National Curriculum Attainment Target references, and specific lesson objectives. Using the Busy Ant Maths online Planning Tool via Collins Connect, these plans can be easily adapted to meet the specific needs of individual schools.
Short-term plans Individual Lesson Plans and accompanying Learning activities represent the majority of each yearly Teacher’s Guide. The lessons provide short-term plans that can easily be followed closely, or used as a ‘springboard’ and varied to suit specific needs of particular classes. An editable ‘Weekly Planning Grid’ is also provided on Collins Connect, which individual teachers can fully adapt.
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Introduction
Teaching The most important role of teaching is to promote learning and to raise pupils’ attainment. To best achieve these goals Busy Ant Maths believes in the importance of teachers: • having high expectations for all pupils • systematically and effectively checking pupils’ understanding throughout lessons, anticipating where they may need to intervene, and doing so with notable impact on the quality of learning • generating high levels of engagement and commitment to learning • consistently providing high quality marking and constructive feedback to ensure that pupils make rapid gains • offering sharply focused and timely support and intervention that matches pupils’ individual needs.
To help teachers achieve these goals, Busy Ant Maths provides: • highly focused and clearly defined learning objectives • examples of targeted questioning, using appropriate mathematical vocabulary, that is aimed at both encouraging and checking pupil progress • a proven lesson structure that provides clear and accurate directions, instructions and explanations • meaningful and well-matched activities for pupils at all levels of understanding to practise and consolidate their learning • highly effective models and images to clearly illustrate mathematical concepts, including interactive digital resources. Each lesson in Busy Ant Maths has a specific learning objective derived from an Attainment Target from the 2014 Primary National Curriculum for Mathematics Programmes of Study, and follows the same teaching and learning sequence.
National Curriculum Attainment Target
Learning Objective
Teaching and Learning Sequence Getting Started
Teach
Individualised Learning
Plenary
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Introduction
Assessment Assessment, record-keeping and reporting continue the teaching and learning cycle and are used to form the basis for adjustments to the teaching programme. Busy Ant Maths offers manageable and meaningful assessment on four levels: • Diagnostic assessment The Assessment Tasks from the Busy Ant Maths Assessment Guide assist teachers in determining pupils’ readiness for a particular unit of work. They are designed to yield information that will directly support the teaching of individual pupils and whole-class teaching. • Short-term ‘on-going’ assessment Progress Check Questions ( ) are an important feature of every Busy Ant Maths lesson and are linked to specific learning objectives. They are designed to provide immediate feedback to pupils and to gauge pupil progress in order to adapt teaching. Shared Success Criteria are also provided in each lesson to assist pupils in identifying the steps required to achieve the learning objective. • Medium-term ’formative’ assessment As well as being used for diagnostic assessment, the Assessment Tasks, along with the Assessment Sheets, from the Busy Ant Maths Assessment Guide can be used to review and record the progress of both individual pupils and the class as a whole, in relation to the National Curriculum Attainment Targets. An End of Unit Test is provided for each of the 12 Busy Ant Maths units. Each test is designed to assess the mathematics covered during the three-week unit. The formative Assessment Tasks, Sheets and Tests provide individual and/or group opportunities to identify those pupils who are not yet reaching, or who are exceeding, national expectations. They can also be used to set individual targets for pupils. • Long-term ‘summative’ assessment The End of Year Class Evaluation document shows individual pupils’ attainment against national standards. It draws upon the data gathered throughout the year including results from Assessment Tasks and Sheets, performance in whole-class discussions, participation in group work, written evidence and any other supplementary notes. It is this document that forms the basis for reporting to parents and informing the next year’s teacher. Importantly, it also helps to determine whether pupils are on track to meet expectations at the end of the key stage. Collins Connect contains a powerful assessment tool for diagnostic, formative and summative assessments. Along with manageable and meaningful record-keeping formats it allows you to collect assessment data to store online and present digitally for class and whole school analysis. Refer to the Busy Ant Maths Assessment Guide or the Busy Ant Maths Getting Started Guide on Collins Connect for a more detailed explanation of the assessment, recordkeeping and reporting features of Busy Ant Maths.
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Year 5 Unit 2
Week 1: Number - Multiplication and division Week 2: Number - Fractions Week 3: Geometry - Position and direction
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Year 5, Unit 2
Week 1: Number - Multiplication and division National Curriculum attainment targets
Pupil targets
Pupils should be taught to:
• Multiply by 10/100 and adjust to find the answer when multiplying by 9/99
• multiply and divide numbers mentally drawing upon known facts • multiply and divide whole numbers by 10, 100 and 1000
• Recall all the multiplication facts for all multiplication tables up to 10 × 10 and multiply all by a multiple of 10, 100 or 1000 • Mentally multiply TO × O and then multiply by a multiple of 10, 100 or 1000 • Multiply a two-digit number by a multiple of 10 • Multiply by 5/50 using multiplication by 10/100 and halving • Multiply by 25 using multiplication by 100, halving and halving again
Weekly overview Pupils use their knowledge of the 10 multiplication table and adjusting to work out calculations that involve multiplying a two-digit number by 9, e.g. 67 × 9. They extend this understanding, along with multiplying by 100 and adjusting, to work out calculations that involve multiplying a two-digit number by 99, e.g. 34 × 99. Pupils multiply two-digit numbers by 10, 100 and 1000 and multiples of these numbers using knowledge of place value. They multiply two-digit near multiples of 10 using multiples of 10 and adjusting. Pupils develop their mental strategies to include multiplying by 5 using multiplying by 10 then halving; multiplying by 50 using multiplying by 100 then halving; and multiplying by 25 using multiplying by 100, then halving and halving again. Prerequisites for learning
Assessment
• Understand the effect of multiplying a number by 10, 100 and 1000
Use the following assessments from Busy Ant Maths Assessment Guide 5:
• Recall the multiplication tables up to 10 × 10
15: Multiply and divide numbers mentally drawing upon known facts
• Subtract mentally a two-digit number from a two- or threedigit multiple of 10 • Halve multiples of 10 and 100
17: Multiply and divide whole numbers and those involving decimals by 10, 100 and 1000
Tracking back and forward through the curriculum Year 4
Year 5
Year 6
• use place value, known and derived facts to multiply and divide mentally, including: multiplying by 0 and 1; dividing by 1; multiplying together three numbers
• multiply and divide numbers mentally drawing upon known facts
• find the effect of dividing a one- or two-digit number by 10 and 100, identifying the value of the digits in the answer as ones, tenths and hundredths [Domain: Number – Fractions (including decimals)]
• multiply and divide whole numbers and those involving decimals by 10, 100 and 1000
• perform mental calculations, including with mixed operations and large numbers
Related Busy Ant Maths Units Y4 Units Unit 4, Week 2 Unit 8, Week 2
Unit 11, Week 2 Unit 12, Week 1
Previous Y5 Units
Future Y5 Units Unit 4, Week 2 Unit 10, Week 1
Y6 Units Unit 2, Week 1 Unit 4, Week 1
Unit 6, Week 1 & 2 Unit 8, Week 1 & 2
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Year 5, Unit 2
Week 2: Number - Fractions National Curriculum attainment targets
Pupil targets
Pupils should be taught to:
• Divide a whole number by the denominator and multiply the answer by the numerator
• compare and order fractions whose denominators are all multiples of the same number
• Count forwards and backwards in simple fractions
• identify, name and write equivalent fractions of a given fraction, represented visually, including tenths and hundredths
• Recognise, describe and continue number sequences involving fractions
• develop their understanding of fractions as numbers, measures and operators by finding fractions of numbers and quantities *
• Write families of equivalent fractions • Compare and order fractions
• practise counting forwards and backwards in simple fractions * • recognise and describe linear number sequences, including those involving fractions and find the term-to-term rule * [Domain: Number – Number and place value] Weekly overview Pupils revise their understanding of fractions. They work out non-unit fractions of amounts and quantities using division and multiplication. They count forwards and backwards in simple fractions as a class and in pairs. Pupils look at sequences of fractions and find the term-to-term rule. They use a fraction wall to identify fractions that are equivalent to a half, a quarter and tenths. Pupils recognise the relationship between the denominators of equivalent fractions. They look at groups of fractions with different denominators and find the lowest common multiple so each fraction can be changed to an equivalent fraction all with the same denominator. Pupils use and apply this knowledge to compare and order fractions. Prerequisites for learning
Assessment
• Understand fractions as part of the number system
Use the following assessments from Busy Ant Maths Assessment Guide 5:
• Understand unit and non-unit fractions • Divide and multiply mentally • Have experience with fractions represented on number lines • Have some understanding of equivalent fractions
22: Compare and order fractions whose denominators are all multiples of the same number 23: Identify, name and write equivalent fractions of a given fraction, represented visually, including tenths and hundredths
Tracking back and forward through the curriculum Year 4
• recognise and show, using diagrams, families of common equivalent fractions • recognise and write decimal equivalents of any number of tenths or hundredths • recognise and write decimal equivalents to 1 1 3 4, 2 , 4
Year 5
Year 6
• compare and order fractions whose denominators are all multiples of the same number
• compare and order fractions, including fractions > 1
• identify, name and write equivalent fractions of a given fraction, represented visually, including tenths and hundredths
• recall and use equivalences between simple fractions, decimals and percentages, including in different contexts
• develop their understanding of fractions as • solve problems involving increasingly harder numbers, measures and operators by finding fractions to calculate quantities, and fractions fractions of numbers and quantities * to divide quantities, including non-unit fractions where the answer is a whole number • count up and down in hundredths; recognise that hundredths arise when dividing an object by one hundred and dividing tenths by ten
• multiply simple pairs of proper fractions, writing the answer in its simplest form [for example, 14 × 12 = 18 ] • divide proper fractions by whole numbers [for example, 13 ÷ 2 = 16 ]
• practise counting forwards and backwards in simple fractions *
• recognise and describe linear number sequences, including those involving fractions and find the term-to-term rule * [Domain: Number – Number and place value] Related Busy Ant Maths Units Y4 Units Unit 2, Week 2 Unit 6, Week 2
Previous Y5 Units
Future Y5 Units Unit 6, Week 2 Unit 10, Week 2
Y6 Units Unit 2, Week 2 Unit 7, Week 1
Unit 10, Week 2
* Notes and guidance (non-statutory)
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Year 5, Unit 2
Week 3: Geometry - Position and direction National Curriculum attainment target
Pupil targets
Pupils should be taught to:
• Translate a shape on a 2-D grid and know that the shape has not changed
• identify, describe and represent the position of a shape following a translation, using the appropriate language, and know that the shape has not changed
• Translate two or more shapes to make a tiling pattern on a 2-D grid • Create 2-D shapes and, by translating the shape in different directions, form a tiling pattern • Identify the position of a shape after a translation in the first quadrant of a coordinate grid
Weekly overview During this week, different translations of a shape are explored. A practical task is used to establish that a shape is unchanged after horizontal and vertical translations. Tiling patterns on game boards and grid papers are shown to be the repeated translation of a regular polygon. The horizontal translation of hexagons illustrates that the gaps are filled by equilateral triangles. Interlocking polygons are used to extend the pattern in a tiled floor in all directions. Pupils investigate the tiling patterns that can be made by translating different arrangements of four squares. They express translations to the left/right and up/down as a unit or units to be added to or subtracted from the x- and/or y-coordinate. This is then applied to the corresponding vertices of shapes to identify and describe the position of a shape after a translation in the first quadrant of a coordinate grid. Prerequisites for learning
Assessment
• Move a shape in directions to the left or right and up or down on a grid of squares
Use the following assessments from Busy Ant Maths Assessment Guide 5:
• Know that identical squares will fit together to form a grid of squares without any gaps or overlaps
47: Identify, describe and represent the position of a shape following a reflection or translation, using the appropriate language, and know that the shape has not changed
• Identify where a shape will be after a translation • Identify positions in the first quadrant by reading and plotting points Tracking back and forward through the curriculum Year 4 • describe positions on a 2-D grid as coordinates in the first quadrant • describe movements between positions as translations of a given unit to the left/right and up/down • plot specified points and draw sides to complete a given polygon
Year 5
Year 6
• identify, describe and represent the position of a shape following a reflection or translation, using the appropriate language, and know that the shape has not changed
• draw and translate simple shapes on the coordinate plane, and reflect them in the axes
Related Busy Ant Maths Units Y4 Units Unit 2, Week 3
Unit 11, Week 3
Previous Y5 Units
Future Y5 Units Unit 11, Week 3
Y6 Units Unit 2, Week 3
Unit 11, Week 3
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Year 5, Unit 2, Week 1, Lesson 1
Multiplying by 9, 99 using 10, 100 and ad justing National Curriculum attainment targets
Lesson objective
• Multiply numbers mentally drawing upon known facts
• Multiply numbers mentally drawing upon known facts
• Multiply whole numbers by 10, 100 Previous related lessons
Future related lessons
none
Unit 2, Week 1, Lesson 2; Unit 2, Week 1, Lesson 4
Prerequisites for learning
Success criteria
Pupils need to:
Pupils can:
• understand the effect of multiplying a number by 10
• multiply by 10 and adjust to find the answer when multiplying by 9
• subtract mentally a two-digit number from a two- or threedigit multiple of 10 Vocabulary
• multiply by 100 and adjust to find the answer when multiplying by 99
multiple, multiplied by, multiplication, subtract
Getting Started • Choose an activity from Number – Multiplication and division. • Choose a game or activity from Fluency in Number Facts: Y5/Y6 – Multiplication and division.
Teach • Review multiplying by 10. Ask, whilst writing the calculation on the board: What is 23 x 10? (230); What is 65 x 10? (650)
Year 5, Unit 2, Week 1
• Ask: What happens when a number is multiplied by 10? (the number becomes 10 times larger as digits move one place to the left) • Write 34 × 9 = on the board. • Say: Explain to your partner how you would work out the answer to this calculation. • Ask children to share their explanations with the class. (Children have learned various written strategies for this type of calculation, although most children should be able to work this out mentally using partitioning.) • Say: We are going to learn a quick mental strategy to help work out the answer to calculations that involve multiplying by 9. When we multiply by 9, we can multiply by 10 first then subtract the number we are multiplying by 9 to find the answer. If children are confident at calculating mentally, it is not necessary to record all of the steps above. You may wish to leave out the first line of the recording or just allow children to record the calculation and answer.
1
• Display: Slide 1 (34 × 9). Show the 34 rows of 10. Ask: What is 34 × 10? (340) Show how 34 × 10 becomes 34 × 9 by removing one row of 34. Ask: What is 340 – 34? (306) • Say: 34 × 9 equals 306.
1–3
• Display: Slides 1–3. • Show children how to record this strategy, i.e. 34 × 9 = (34 × 10) – 34 = 340 – 34 = 306 • Repeat with other examples on the slides, i.e. 28 x 9; 46 x 9. • Ask: What is the answer to 68 × 99? Explain how you worked out the answer. • Write 46 × 9 = on the board. • Say: Explain to your partner how you would work out the answer to this calculation. • Ask children to share their explanations with the class.
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Unit 2: Number - Multiplication and division If children are confident at calculating mentally it is not necessary to record all of the steps above. You may wish to leave out the first line of the recording or just allow children to record the calculation and answer.
• Say: We can use a mental strategy to work out the answer to this calculation. When we multiply by 99, we can multiply by 100 first then subtract the number we are multiplying by 99 to find the answer. 4
• Display: Slide 4 (46 × 99). Show the 46 rows of 100. Ask: What is 46 × 100? (4600) Show how 46 × 100 becomes 46 × 99 by removing one row of 46. Ask: What is 4600 – 46? (4554) • Say: 46 × 99 equals 4554.
4–6
• Display: Slides 4–6. • Show children how to record this strategy, i.e. 46 × 99 = (46 × 100) – 46 = 4600 – 46 = 4554 • Repeat with other examples on the digital resource, i.e. 38 x 99; 57 x 99. • Ask: What is the answer to 78 x 99? Explain how you worked out the answer.
Individualised Learning Refer to Activity 1 from the Learning activities on page 114.
Pupil Book 5A: – Page 28: Multiplying by 9, 99 using 10, 100 and adjusting Progress Guide 5: – Support, Year 5, Unit 2, Week 1, Lesson 1: Multiplying by 9 using 10 and adjusting – Extension, Year 5, Unit 2, Week 1, Lesson 1: Multiplying by 9, 99 using 10, 100 and adjusting Resources: 30 counters (per pair)
Plenary Resources mini whiteboard, pen and eraser (per child) • Review using the 10 multiplication table and adjusting as a strategy for multiplying by 9. Write multiplication calculations on the board one at a time, for example, 75 × 9 =. • Say: Explain to your partner how you would work out the answer to this calculation using your mini whiteboard. • Ask the children to explain how to use this strategy to work out the answer. Repeat with other examples. • Review multiplying by 100 and adjusting as a strategy for multiplying by 99. Write multiplication calculations on the board one at a time, for example, 37 × 99 =. • Say: Explain to your partner how you would work out the answer to this question using your mini whiteboard. • Ask the children to explain how to use this strategy to work out the answer. Repeat with other examples.
Overcoming Barriers • Some children may find these methods difficult if they are unable to subtract mentally from multiples of 10 and 100. Ensure they are able to subtract mentally as using the written methods of subtraction are inefficient and incompatible with these mental strategies.
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