Contents

IntroductionTeaching for mastery: an introduction 1Using this Teaching Pack 1

Ch 1 Whole numbers 1Topic overview (Book 2a pages 6–13) 3Place values (MNU 2-02a) 5Add/subtract whole numbers (MNU 2-03a) 7Rounding to the nearest 10 and 100 (MNU 2-01a) 9Using rounding to estimate answers (MNU 2-01a) 10

Ch 2 SymmetryTopic overview (Book 2a pages 14–19) 11Lines of symmetry (MTH 2-19a) 13Creating a symmetrical shape (MTH 2-19a) 15

Ch 3 Whole numbers 2Topic overview (Book 2a pages 20–27) 16Multiplication by a single digit (MNU 2-03a) 18Division by a single digit (MNU 2-03a) 20Multiplication by 10, 100, 1000 (MNU 2-03a) 21

Ch 4 Time 1Topic overview (Book 2a pages 28–36) 2312-hour time (MNU 2-10a) 2512-hour and 24-hour time (MNU 2-10a) 26Short time intervals (MNU 2-10c) 27Calendar and longer time intervals (MNU 2-10c) 28Minutes and seconds (MNU 2-10b) 29

Ch 5 Decimal numbers 1Topic overview (Book 2a pages 37–51) 30What is a decimal number? (MNU2-03b) 32Reading decimal scales (MNU 2-03b) 34Rounding to the nearest whole number (MNU 2-03b) 35Rounding to one decimal place number (MNU 2-03b) 36Adding or subtracting decimals (MNU 2-03b) 37

Ch 6 AnglesTopic overview (Book 2a pages 52–61) 38Types of angles (MTH 2-17a) 40Naming angles (MTH 2-17a) 41Measuring angles (MTH 2-17b) 42Drawing angles (MTH 2-17b) 43Compass points (MTH 2-17c) 44

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Ch 7 Decimal numbers 2Topic overview (Book 2a pages 62–8) 45Multiplying decimals by 10, 100, 1000 (MNU 2-03b) 47Multiplication by a single digit (MNU 2-03b) 49Mixed problems (MNU 2-03b) 51

Ch 8 MoneyTopic overview (Book 2a pages 69–79) 52Add, subtract, multiply and divide using money (MNU 2-09a) 54Money problems – bills (MNU 2-09a) 55Best buy – comparisons (MNU 2-09a) 56Mixed money problems (MNU 2-09a, MNU 2-09b) 57

Ch 9 Two dimensionsTopic overview (Book 2a pages 80–7) 582-dimensional shapes (MTH 2-16a, MTH 2-16c) 60Special triangles (MTH 2-16a) 61Describing a triangle fully (MTH 2-16a, MTH 2-16c) 62The circle (MTH 2-16a, MTH 2-16c) 63

Ch 10 AlgebraTopic overview (Book 2a pages 88–96) 64Basic equations (MTH 2-15a) 66Equations (MTH 2-15a) 68

Ch 11 FractionsTopic overview (Book 2a pages 97–106) 69Identifying fractions (MNU 2-07a) 71Equivalent fractions (MNU 2-07b) 72Fractions of a quantity (basic) (MNU 2-07b) 73

Ch 12 PercentagesTopic overview (Book 2a pages 107–112) 75What is a percentage? (MNU 2-07a) 77Linking fractions, decimals and percentages (MNU 2-07b) 78Finding a (simple) percentage of a quantity (MNU 2-07a) 79

Ch 13a LengthTopic overview (Book 2a pages 113–121) 80Measuring and drawing lengths (MNU 2-11a, MNU 2-11b) 82Units of length (MNU 2-11b) 83Perimeter (MNU 2-11b) 84

Ch 13b AreaTopic overview (Book 2a pages 122–9) 85Area of a shape (MNU 2-11b) 87Area of a rectangle (MNU 2-11c) 88Area of a right-angled triangle (MNU 2-11c) 89

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Ch 13c VolumeTopic overview (Book 2a pages 130–140) 90What is volume? (MNU 2-11a) 92Litres and millilitres (MNU 2-11b) 93Volumes by counting cubes (MNU 2-11c) 94Volume of a cuboid – a formula (MNU 2-11c) 95

Ch 13d WeightTopic overview (Book 2a pages 141–4) 96The kilogram and the gram (MNU 2-11b) 98

Ch 14 CoordinatesTopic overview (Book 2a pages 145–151) 99Coordinates of a point (MTH 2-18a) 101More about the x-axis and the y-axis (MTH 2-18a) 102

Ch 15 PatternsTopic overview (Book 2a pages 152–9) 103Basic patterns (MTH 2-13a) 105Sequences (MTH 2-13a) 106Square and triangular numbers (MTH 2-13a) 107

Ch 16 Three dimensionsTopic overview (Book 2a pages 160–7) 1083-D shapes (MTH 2-16a) 1103-D shapes in the real world (MTH 2-16a) 111Drawing 3-D shapes (MTH 2-16c) 112

Ch 17 Multiples and factorsTopic overview (Book 2a pages 168–172) 113Multiples and factors (MNU 2-05a) 115

Ch 18 StatisticsTopic overview (Book 2a pages 173–187) 116Organising/interpreting Information from tables and graphs (MNU 2-20a) 118Interpreting pie charts (MNU 2-20a) 120Drawing graphs (MTH 2-21a) 121Conducting a survey (MNU 2-20b) 122

Glossary 123

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Introduction

Teaching for mastery: an introductionWhat is mastery learning?In a classroom where mastery learning is embedded, the learning outcomes are constant. A focus on aptitude, the time needed for learners to become proficient or competent, is key to the development of mastery. The expectation is that the vast majority of the class will move through content at roughly the same pace.

In a mastery classroom subject matter is broken down into blocks or units of learning with clearly delineated learning intentions. Learners work through a unit in a series of small, sequential steps. The majority of learners are expected to master the ‘fundamentals’ of the unit before moving on to the next block or unit. Working through each block sequentially ensures that learners are given opportunities to demonstrate a high level of understanding at a greater depth. Learners who struggle to reach the required level are supported through additional support, peer support or small group discussions to enable them to reach the anticipated level of understanding.

Why use this approach?By using key elements of collaborative learning and AiFL group work, for example peer tutoring or think-pair-share, mastery learning is a particularly effective vehicle for moving learners through the curriculum. Learners are encouraged to take responsibility for supporting each other’s progress. Taking a mastery approach demands a need for high expectations of all learners – a belief that all learners can and will be successful is key.

Recent analyses into mastery learning show that it may be a useful strategy for narrowing the attainment gap, particularly in relation to low attaining students.

Using this Teaching PackThis pack has been designed to support your daily teaching of mathematics. Included in the pack you will find several elements that are fundamental to learners’ understanding and progression in mathematics. They are explained below.

Topic overviewIn this section an overview of the entire chapter is provided so that it is possible for practitioners to see the ‘big picture’ of the learning over a sequence of lessons. Within this overview the big ideas – i.e. the key concepts to be taught and learned – have been identified to support teacher subject knowledge.

Potential misconceptions have also been included to support AiFL planning: it is useful to keep these barriers in mind during the daily lessons to explore misconceptions and therefore deepen understanding.

Core vocabulary that learners and teachers should be using to support the understanding of key concepts has also been identified. The vocabulary builds progressively across the full series, ensuring that learners are able to talk about their learning in the correct contexts.

Opportunities to develop non-cognitive skills have also been identified across a sequence of lessons. These skills support the development of a growth mindset and allow for opportunities to develop skills required for life, learning and work. Learning mathematics extends beyond learning concepts, procedures and their applications.

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2

Introduction

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We want to ensure that our learners develop a positive disposition toward mathematics and see mathematics as a powerful way for looking at situations.

Disposition refers not only to attitudes but to a way of thinking and acting in a positive manner. It is an expectation that mathematical dispositions manifest themselves in the way learners approach problems – whether with confidence, willingness to explore alternatives, perseverance and interest – and in their tendency to reflect on their own thinking. Evaluating these indicators and learners’ appreciation of the role and value of mathematics is central to the evaluation of mathematical knowledge as a whole.

Within each lesson: what could it look like in the classroom?Time to get started (anchor task to hook learners in) A problem given to the whole class that hooks the learners’ interest and gives them a purpose for learning. Learners should be allowed time to explore and reflect before feeding back their findings, ideas and thoughts to the class, peer groups or teacher.

Time to learn (main modelling by class teacher) This is the main modelling part of the lesson where learners are taught the skill guided by the shared understanding shown at the start of the lesson.

Time to practise (practice guided by the teacher but learners working in pairs) Learners work in pairs to perform deliberate practice based on the problems modelled by the teacher.

Time to reflect (reflection/revisit success criteria) Learners reflect on the learning so far and identify the main success criteria for the lesson including non-cognitive attributes.

Time to work on our own (independent work) Learners work independently to practise the skill modelled with an increasing number of more difficult scenarios.

To end the lesson (overlearning) In some cases, usually with more challenging content, opportunities for overlearning have been referenced. This allows for additional content to be delivered, ensuring a deeper understanding.

Extension Advanced learners can undertake these tasks to deepen their understanding. Some extension tasks have been included in some lessons where it has been deemed appropriate.

Revision of First LevelChapter 0This chapter can be used to test and evidence learners’ understanding of First Level maths. The exercises in this chapter assess learners’ mathematical fluency and problem-solving skills. The exercises included in the textbooks follow the format of the other assessment opportunities seen throughout the books so that they feel familiar and provide the summative information used to inform teacher judgement.

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Potential misconceptions

The barriers to learning●● As the numbers increase, learners find it difficult to read numbers

aloud.

●● When counting in 1000s learners are sometimes unsure what comes after 9000 (10 000). They might also struggle with what comes before and after this number (9999, 10 001).

●● Learners do not make the link between these numbers and real-life contexts. (Could be linked to populations of countries, costs of items etc.)

●● Learners counting in powers of 10 are often ‘ill advisedly’ told that they need to ‘add a nought’. Learners must observe the transformation of numbers as the digits move into new places on the place-value chart.

Topic overview (Book 2a pages 6–13)

Chapter 1: Whole numbers 1

Resources

What you need from the maths cupboard●● Unifix® cubes

●● blank number tracks

●● tens frames

●● place-value arrow cards

●● base 10 materials

●● place-value charts

●● objects for counting

●● 1–100 number chart

●● place-value counters

Big ideas

The key concepts for this chapter●● Magnitude (the size of a number)

●● Cardinality (the amount in a set)

●● Need for organisation and keeping track

●● Thousands, hundreds, tens and units: understanding that columns can exist side by side.

●● Number names: knowing that instead of naming a number ‘two thousands, five hundreds, one ten and three units’, we give it a name, two thousand five hundred and thirteen.

●● Numbers in different forms: the number 4536 is usually thought of as four thousands, five hundreds, three tens and six units

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Topic overview: Whole numbers 1

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Revisit, review, revise!

Activities to consolidate learningPage 13 can be used as an assessment tool to support teacher judgement. Ideally it would be used a week after the content in the chapter has been taught.

Fundamentals

The basic skills to be mastered●● Learners can identify the place value of a digit in a five-digit number.

●● Learners can use concrete materials to represent numbers.

●● Learners can use concrete materials and/or pictorial representations to compare numbers.

●● Learners can represent numbers in the thousands on a number line.

●● Learners can round numbers to the nearest 10, 100, 1000.

Non-cognitive skills

The soft skills developed in each lessonCollaboration Resilience Flexibility Explanation Independence

Core vocabularynumeral place value order round stands for

represents regroup > greater than <

less than integer positive negative above

below zero minus next consecutive

sort classify property divisibility

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Lesson approach: Whole numbers 1

Time to get started (anchor task to hook learners in) Display the number 6942 with counters on a place-value chart. Allow learners time to discuss in pairs what each of the digits is worth. Then show the number 69 420 with counters on a place-value chart.

What is the same and what is different about the values shown in this number? Describe what has happened to the digits.

Time to learn (main modelling by class teacher)Gather feedback from the pairs and discuss findings. I have made this number (69 420) 10 times bigger than 6942. Did anyone notice that? Can anyone describe what has happened to the digits? Can you show me the process using the counters on the board? Show the information from the yellow box on page 6.

Model saying the value of each digit aloud as laid out on the place-value chart. Ask questions such as How many tens in 500? How many hundreds in 3500? How do you know? Can you prove it? These questions will help to extend the thinking of the advanced learners in the class. Repeat this process for a few different numbers.

Time to practise (practice guided by the teacher but learners working in pairs)

Pairs work together to answer questions 1–4. Place-value charts and counters should be available for all learners. Struggling learners could be supported through the use of counters. They could be encouraged to say the name and the value of all the digits. Advanced learners could be challenged by asking them to identify how many tens there are in thousands or in hundreds; how many hundreds and thousands there are in bigger numbers, etc.

Time to reflect (reflection/revisit success criteria)Bring learners back together and discuss challenges and successes. Generate success criteria for identifying values of different digits. Display the success criteria on the working wall. Display question 5.

How would we order these numbers? How can we do this by using what we already know? Model ordering them based on learner input. Show the first number line in question 7.

Discuss what is shown. How could you determine what A is? Discuss feedback.

Place values

This section covers pages 6–7 of the TeeJay CfE Book 2a.

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Lesson approach: Whole numbers 1

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Time to work on our own (independent work)Learners solve questions 5–9 independently. Question 10 can be used as an extension activity. Advanced learners could be encouraged to create similar problems for peers to solve. Struggling learners may benefit from support from the class teacher.

NoteFurther lessons should be taught using the above structure to consolidate the concept.

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Lesson approach: Whole numbers 1

Time to get started (anchor task to hook learners in)Remind learners of previous learning around addition and subtraction. What strategies do we have to solve 390 add 540? Allow pairs to discuss and take feedback. Circulate, notice and encourage mental methods.

Time to learn (main modelling by class teacher)Take feedback and allow learners to share some ideas. Model using the two mental methods shown in the yellow box on page 8. Think aloud when carrying out each method.

For method one – I can see that 390 is close to 400. Multiples of 10 are much easier to add so I’m going to turn 390 into 400 by adding a 10. 400 add 540 equals 940. Now I need to take away 10, so my answer is 930. Would this calculation work if I took 10 away from the 540 and added it onto the 390 to make 400? Why does it work? 530 + 400 = 930.For method two – I’m going to take the 40 off 500 and deal with it later. So now I have 390 + 500 = 890. Now I need to add 40 so I’ll add 10 + 30. 890 add 10 is 900 and 900 add 30 equals 930. Which method would you choose and why? Does anyone do it a different way? Ask learners to discuss in pairs and feed back.

Model each method with counters to support struggling learners.

Time to practise (practice guided by the teacher but learners working in pairs)

Ask learners to answer question 1 in pairs. Place-value charts, number lines or hundred squares can be used to support struggling learners. Advanced learners could be encouraged to find at least two methods for each calculation.

Time to reflect (reflection/revisit success criteria)Discuss the successes and challenges. Re-teach if necessary. Show some calculations exhibiting common errors – allow learners time to discuss what’s gone wrong and provide feedback with advice. Show question 2a.

This time we are subtracting. What mental strategies have we learned previously to help solve this? Model partitioning and subtracting.

Show question 2b.

Would partitioning be a helpful strategy for this calculation? Why or why not? Model solving the calculation by partitioning the 39 into 30 and 4 and 5. Model subtracting each part of the number to find the answer. What other methods do you know?

Add/subtract whole numbers

This section covers pages 8–10 of the TeeJay CfE Book 2a.

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Lesson approach: Whole numbers 1

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Time to work on our own (independent work)Learners solve questions 2 and 3 independently. Advanced learners should use at least two methods. Struggling learners could use apparatus to support them, such as place-value counters or base 10 materials.

An additional sequence of lessons could be taught using the word problems in question 4. Model question 4a using a bar model to visualise what the problem is asking us to do. Think aloud – what do we know about the problem? What happens? And so on. Learners can solve the problems in groups, pairs or independently, as required. Extra practice (pages 9–10) is there to provide structure for additional lessons. The questions in the red box on page 10 can be used as an extension activity for advanced learners.

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Lesson approach: Whole numbers 1

Time to get started (anchor task to hook learners in)Recap on prior learning around rounding numbers.

My friend says that he has a rule for rounding to the nearest 10 or 100. Display the information from the yellow box on page 11.

Allow learners time to discuss this and demonstrate understanding by trying out different examples.

Time to learn (main modelling by class teacher)Take feedback. Allow learners to demonstrate understanding on whiteboards to show how they have calculated the answer. Display further examples of rounding to 10 or 100. Ensure learners understand the thought process. For example, rounding 47 to the nearest 10:

I need to draw a number line and mark on the two multiples of 10 that 47 lies between: 40 and 50. Now I need to mark the place for 47. Is it closer to 50 or 40? I can see that it is closer to 50. So I round the tens digit up to 5 (50). Does this follow the rule in the yellow box? Repeat for rounding to 100. Include rounding three-digit numbers to the nearest 10 with examples such as 342.

What digit should I look at when rounding this to the nearest 10? Why? What digit would I look at when rounding the same number to the nearest 100? Repeat for a few more examples.

Time to practise (practice guided by the teacher but learners working in pairs)

Pairs work together to answer questions 1 and 2. Struggling learners could use a number line to support understanding. Advanced learners could be encouraged to round three- or four-digit numbers to the nearest 10 and 100 each time.

Time to reflect (reflection/revisit success criteria)Reflect on the learning so far. Recap on rounding rules and allow learners to explain these in their own words, using examples to demonstrate their understanding. Model solving question 3 with input from the learners.

Time to work on our own (independent work)Learners solve questions 3–5 independently. Struggling learners may still use a number line to solve the questions. Advanced learners could provide further examples for the working wall. They could also suggest how to round numbers to the nearest thousand using prior knowledge.

Rounding to the nearest 10 and 100

This section covers page 11 of the TeeJay CfE Book 2a.

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Lesson approach: Whole numbers 1

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Time to get started (anchor task to hook learners in)Recap on learning from previous lessons. Allow partners time to recap on rounding rules that they recall from previous learning. When do you think it would be helpful to write numbers? Display the information from the yellow box on page 12.

Time to learn (main modelling by class teacher)Gather feedback from the first part of the lesson. Model rounding to help estimate the answers to the calculation shown. Think aloud.

I will round 51 to 50, the tens digit stays the same because 51 is closer to 50. 78 is closer to 80 than 70 so the tens digit rounds up. When we approximate an answer we use the approximation . Having an approximate answer helps us to know if the actual answer we calculate is correct. 50 + 80 is approximately 130. Now I can calculate 51 + 78 = 50 + 70 + 1 + 8 = 129. Repeat the process for the other calculations, inviting learners to offer ideas for rounding.

Time to practise (practice guided by the teacher but learners working in pairs)

Pairs work together to solve questions 1–3. Struggling learners could use a number line to support the rounding. Advanced learners could provide a range of different estimates that would result in the same answer to the calculation.

Time to reflect (reflection/revisit success criteria)Bring the class back together. Show this calculation: 242 add 364 is approximately 500.

242 add 360 equals 200 add 340 add 62 equals 602. Ask learners what’s gone wrong. Allow time for learners to talk to their partners before gathering feedback. What advice would they give to help this learner calculate accurately?

Time to work on our own (independent work)Learners solve questions 4 and 5 independently. Struggling learners could use a number line to help with rounding. Advanced learners could write word problems to match the calculations..

Using rounding to estimate answers

This section covers page 12 of the TeeJay CfE Book 2a.

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Topic overview (Book 2a pages 14–19)

Chapter 2: Symmetry

Resources

What you need from the maths cupboard●● 2-D shape sets

●● 2-D paper shapes

●● mirrors

Big ideas

The key concepts for this chapter●● The idea that figures and shapes can be made up of exactly similar parts

facing each other or around an axis.

●● Some shapes have more than one line of symmetry.

●● Geometrical designs from different cultural traditions provide rich experiences of transformations and symmetry.

Potential misconceptions

The barriers to learning●● Learners are able to discuss shapes when handling them but cannot

visualise the shape.

●● Learners think that all 2-D shapes only have one line of symmetry.

●● Learners believe that if a figure is made up of symmetrical basic shapes, it will be symmetrical.

●● Learners think that all polygons have the same number of lines of symmetry as they do numbers of sides/angles.

Core vocabularyline symmetry reflect translation congruent octahedron

scalene triangle axis of symmetry reflective symmetry

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Topic overview: Symmetry

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Revisit, review, revise!

Activities to consolidate learningPage 19 can be used as an assessment tool to support teacher judgement. Ideally it would be used a week after the content in the chapter has been taught.

Fundamentals

The basic skills to be mastered●● Learners can determine if a 2-D shape is symmetrical by folding it.

●● Learners can identify a line of symmetry in a figure.

●● Learners can draw a symmetrical figure.

Non-cognitive skills

The soft skills developed in each lessonCollaboration Resilience Flexibility Explanation Independence

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