Teacher’s Guide 6
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Contents Introduction
Collins International Primary Maths Stage 6 units
Key principles of Collins International Primary Maths
iv
How Collins International Primary Maths supports Cambridge Primary and the Cambridge Primary Mathematics Curriculum Framework
Teacher’s Guide Student’s Book Workbook Collins Connect DVD
viii xvi xvi xvii xvii
Collins International Primary Maths Stage 6 units and recommended teaching sequence
xviii
Collins International Primary Maths Stage 6 units link to Cambridge Primary Mathematics Curriculum Framework Cambridge Primary Mathematics Curriculum Framework Stage 6 link to Collins International Primary Maths units
xxi
xxxv
Refresh activities Number
Geometry Measure
Handling data
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2 Whole numbers 2
72
3 Decimals 1
78
4 Decimals 2
84
5 Fractions
90
iv
The components of Collins International Primary Maths: • • • • •
1 Whole numbers 1
Numbers and the number system 1 Calculation: Mental strategies, Addition and subtraction 20 Calculation: Mental strategies, Multiplication and division 28
6 Percentages
101
7 Ratio and proportion
107
8 Addition and subtraction 1
113
9 Addition and subtraction 2
119
10 Addition and subtraction 3
130
11 Multiplication and division 1
136
12 Multiplication and division 2
142
13 Multiplication and division 3
153
14 2D shape
164
15 3D shape
170
16 Angles
176
17 Position and movement
182
18 Length
193
19 Mass
199
20 Capacity
205
21 Time
211
22 Area and perimeter
217
Shapes and geometric reasoning Position and movement Length, mass and capacity Time Area and perimeter
38 44 48 54 56
23 Handling data
223
Resource sheets
234
Organising, categorising and representing data
58
Answers
xxx
Probability
60
Tracking back and forward through the Cambridge Primary Mathematics Curriculum Framework
xxx
Stage 6 Record-keeping charts
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Unit
15 3D shape
Refresh activities Drawing a dice Geometry – Shapes and geometric reasoning
Learning objective Code
Learning objective
6Gs2
Visualise and describe the properties of 3D shapes, e.g. faces, edges and vertices.
Resources dice (per class)
What to do • Display the Shape set tool showing a cube. • Hold up a dice. Show learners how the numbers on opposite sides of the dice add up to 7. • Ask learners to say which faces of the cube on display would have the numbers 1 to 6 so that they obey the rules. • Correct any misunderstandings.
Variation • Learners draw a sketch of a cube without labelling it.
Guess the 3D shape Learning objective Code
Learning objective
6Gs2
Visualise and describe the properties of 3D shapes, e.g. faces, edges and vertices.
Resources 3D shapes: cube, cuboid, sphere, cone, cylinder, one type of pyramid (per class); bag (per class)
What to do • Choose a 3D shape and place it inside a bag without learners seeing it. • Provide them with clues about the faces of the shape, for example: It has six faces all of which are oblongs. (cuboid) • Ask learners to guess the name of the shape and praise the learner who is first to identify the shape. • Repeat for different shapes.
Variation • Include different types of pyramids and prisms. Provide clues about the number of edges and vertices.
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Unit
15 3D shape
Mental shapes Code
Learning objective
6Gs4
Recognise and make 2D representations of 3D shapes including nets.
Geometry – Shapes and geometric reasoning
Learning objective
What to do • Describe the shape of the faces of a solid, but do not name it. Ask learners to visualise the shape that you describe. • Say: My shape has six faces that are all square. There are three pairs of parallel faces. (cube) • Invite learners to guess the shape described. • Repeat for different shapes, for example: My shape rests on a square face. Four triangles extend up from the square face and come together at a point. (square-based pyramid)
Variation • Extend the descriptions to include edges and vertices.
Sketch the shape Learning objective Code
Learning objective
6Gs4
Recognise and make 2D representations of 3D shapes including nets.
Resources 3D shapes: cube, cuboid, sphere, cone, cylinder, one type of pyramid (per class)
What to do • Name a shape and ask learners to sketch it. • Reveal the actual shape and ask learners to comment on the accuracy of their sketches. • Repeat for different shapes.
Variation • Display the shape for learners to sketch.
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Unit overview
Unit 15: 3D shape Learning objectives
Geometry – Shapes and geometric reasoning
Code
Learning objective
6Gs2
Visualise and describe the properties of 3D shapes, e.g. faces, edges and vertices.
6Gs4
Recognise and make 2D representations of 3D shapes including nets.
Strand 5: Problem solving 6Pt4
Recognise 2D and 3D shapes and their relationships, e.g. a cuboid has a rectangular cross-section.
6Ps2
Deduce new information from existing information and realise the effect that one piece of information has on another.
Unit overview In this unit, learners extend their knowledge of properties of 3D shapes. They identify the shapes of faces of common solid shapes, as well as counting the number of faces, edges and vertices of cubes, cuboids, pyramids and prisms and investigating the relationship between them. Learners identify parallel and perpendicular faces of 3D shapes and use these properties when sorting and classifying. They describe the shape of the cross-section of a solid shape in a plane parallel to a base. From their experience of handling and describing 3D shapes, they visualise images of them. By unfolding cardboard containers, learners begin to understand how a net folds up to create the solid. Learners explore the nets of prisms and pyramids and determine whether a given net will fold to create a given solid.
Common difficulties and remediation Some learners confuse names of 2D shapes with names of 3D shapes, typically naming the 3D shape by its 2D face. Provide activities encouraging learners to name 3D shapes and learn their attributes, for example, sorting games involving asking questions, such as: What are the characteristics of this shape? Why did you sort the shapes into these groups?
Some learners find it difficult to represent a 3D object as a 2D drawing. Provide activities that involve visualising 3D shapes from 2D shapes, for example, recreating a 3D solid using multilink shapes from a photograph of the model.
Promoting and supporting language At every stage, learners require a mathematical vocabulary to access questions and problem-solving exercises. You may find it useful to refer to the online glossary at Collins Connect. If appropriate, when a new key word is introduced, ask learners to write a definition in their own words, for example: ‘A cube is a three-dimensional shape with six identical square faces joined at their edges.’ Encourage frequent usage of mathematical language to help embed vocabulary. For example, model the use of the word ‘vertex’ when learners refer to ‘corners of a shape’ as they make the transition to mathematically precise language. Also, help learners with pronunciation of tricky words, such as ‘prism’, pronounced priz-uh m. Throughout the unit it is important to encourage learners to seek clarification and confirmation of the mathematical language. This may involve prompting learners to call out and complete definitions, for example, following a demonstration of nets by saying: ‘A net is … what a 3D shape would look like if it were opened out flat.’
Look out for learners who find it difficult to identify nets that will form a cube. This is typical of learners who have had insufficient experience visualising 3D shapes, particularly how they can be opened up and folded again. Consequently, they find it difficult to ‘see’ how a net can be folded to make a 3D shape. Provide extended opportunities to draw and make nets of shapes, including those that do not always create a complete 3D shape. They should experience unfolding a range of packaging, such as tubes and prism-shaped packages.
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Student’s Book page 77
Unit
Workbook page 154
15 3D shape
Lesson 1: Visualising 3D shapes Code
Learning objective
6Gs2
Visualise and describe the properties of 3D shapes, e.g. faces, edges and vertices.
Strand 5: Problem solving 6Pt4
Recognise 2D and 3D shapes and their relationships, e.g. a cuboid has a rectangular cross-section.
6Ps2
Deduce new information from existing information and realise the effect that one piece of information has on another.
Prerequisites for learning
Vocabulary
Learners need to: • be able to visualise 3D shapes from 2D drawings • be able to visualise, from the front and from the side, 2D representations of 3D shapes made with interlocking cubes.
prism, pyramid, cube, edge, face, vertex
Success criteria Learners can: • visualise and describe the properties of a range 3D shapes.
Refresh • Choose an activity from the Geometry – Shapes and geometric reasoning Refresh activities.
Discover • Discuss the image. Ask: How many 3D shapes can you see?
Resources 3D shapes: sphere, cylinder, cube, cuboid, range of pyramids including triangular- and square-based, range of prisms including triangular (include a mix of ‘teaching’ shapes and real world objects, such as containers and packaging) (per group)
of parallel edges does a square-based pyramid have? How many perpendicular edges? • Return the shapes to the table. Say: Look at these 3D shapes. Show me a face that is parallel to this one. Ask: Which face is perpendicular to this one?
Practise
Teach and Learn
• Workbook: Visualising 3D shapes page 154
• Discuss the text and images in the Student’s Book. Hold up 3D shapes in various orientations. Ask: What is the name of this shape? How many faces/vertices/edges does it have? • Display Slide 1. Say: When you look at the drawing, try to create a mental picture of the 3D shape. Remind learners of their experiences of handling 3D shapes and turning them to discover their properties. Ask: Which of the shapes has 0/8/12 vertices? Which of the shapes has 8/12 edges? Which of the shapes has faces that are squares/triangular/squares and triangular? Which 2D shapes would you need to make a hexagonal prism? (two hexagons, six rectangles) What shape could be made from a square and four triangles? (square-based pyramid) • Provide small groups of learners with 3D shapes. Ask them to classify the shapes using criteria such as, ‘has at least one pair of parallel faces’. Invite learners to share their classifications. • Remove the shapes and return to visualisation exercises. Say: Imagine a triangular prism. Ask: How many faces does it have? Are any of the faces parallel to each other? How many pairs
Apply • Display Slide 2 and ask pairs of learners to solve the problem. Share solutions and agree that the pyramids all have a pentagonal base and five triangular side faces.
Geometry – Shapes and geometric reasoning
Learning objective
Review • Say: Logan makes a lampshade from two octahedrons joined together. Tell your partner what Logan’s lampshade will look like and describe its properties.
Assessment for learning • I join straws to make three tetrahedrons. How many straws do I use? (18)
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Unit
Student’s Book page 78
15 3D shape
Workbook page 156
Lesson 2: Constructing 3D shapes Learning objective Code
Learning objective
6Gs2
Visualise and describe the properties of 3D shapes, e.g. faces, edges and vertices.
Geometry – Shapes and geometric reasoning
Strand 5: Problem solving 6Pt4
Recognise 2D and 3D shapes and their relationships, e.g. a cuboid has a rectangular cross-section.
6Ps2
Deduce new information from existing information and realise the effect that one piece of information has on another.
Prerequisites for learning
Vocabulary
Learners need to: • be able to visualise 3D shapes from 2D drawings • be able to visualise, from the front and from the side, 2D representations of 3D shapes made with interlocking cubes.
prism, pyramid, cuboid, tetrahedron, octahedron
Success criteria Learners can: • recognise, describe and build simple 3D shapes.
Refresh • Choose an activity from the Geometry – Shapes and geometric reasoning Refresh activities.
Discover • Discuss the text and image in the Student’s Book. Ask: How many straws you would need to build a triangular prism? How many joins would be needed? (9, 6)
Teach and Learn • Discuss the text and image in the Student’s Book. Arrange the learners into six groups and provide each group with shape-modelling materials and a picture of one of the following 3D shapes: cuboid, triangular-based pyramid (tetrahedron), squarebased pyramid, triangular prism, pentagonal prism, octahedron. Say: Using the materials, plan and then construct a 3D model of the shape you have been given. Share ideas. • Feed into the discussion by providing guidance on modelling techniques. Ask: Are there any groups that would benefit from working together? Elicit that the tetrahedron and octahedron groups could combine tetrahedrons to form octahedrons. Give learners time to construct their models. • Ask: Who can remember the definition of a prism? Praise learners who describe solid shapes that have identical parallel polygonal bases. Ask: What shape would be revealed if you cut a prism anywhere along its length parallel to its ends?
Resources 3D shape-modelling materials: straws or sticks, sticky tape or sticky putty (per group); pictures of 3D shapes: cuboid, triangular-based pyramid (tetrahedron), square-based pyramid, triangular prism, pentagonal prism, octahedron (per class); ruler (per learner); 3D shapes: cuboid, tetrahedron, triangular prism, hexagonal pyramid, hexagonal prism (per group) Establish that the cross-section would be the same shape as the end faces. • Ask learners who made prisms to hold up their shapes. Prompt and include the cuboid group. • Say: Who can remember the definition of a pyramid? Praise learners who describe solid shapes with a flat base and all other faces triangles that meet at a point. Say: The point is actually a common vertex and is given a special name: an apex.
Practise • Workbook: Constructing 3D shapes page 156 • Refer to Activity 1 from the Additional practice activities.
Apply • Display Slide 1 and ask groups of learners to solve the problem, then share their solutions. (14 rectangles, 2 octagons, 1 hexagon, 6 triangles)
Review • Say: Melinda uses straws to construct a solid shape. The shape has two pentagonal bases joined by five rectangular faces. Ask: How many straws did she use?
Assessment for learning • I join straws to make three hexagonal pyramids. How many joins did I make in all?
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Student’s Book page 79
Unit
Workbook page 158
15 3D shape
Lesson 3: Nets (1) Code
Learning objective
6Gs4
Recognise and make 2D representations of 3D shapes including nets.
Geometry – Shapes and geometric reasoning
Learning objective
Strand 5: Problem solving 6Pt4
Recognise 2D and 3D shapes and their relationships, e.g. a cuboid has a rectangular cross-section.
6Ps2
Deduce new information from existing information and realise the effect that one piece of information has on another.
Prerequisites for learning
Vocabulary
Learners need to: • be able to visualise 3D shapes from 2D drawings.
cube, square, net, base, side, vertex
Success criteria
cube-shaped packaging (per class); Resource sheet 15: Net for a closed cube (per learner); scissors (per learner); Resource sheet 16: squared paper (per pair); coloured pencils (per learner)
Learners can: • use knowledge of the properties of cubes to identify and draw different nets of cubes.
Refresh • Choose an activity from the Geometry – Shapes and geometric reasoning Refresh activities.
Discover • Discuss the text and image. Show learners how the opened up box in the image clearly shows the arrangement of the six faces and how they come together to form a cuboid. Say: Visualise opening up a cube-shaped box. Ask: What would the net look like?
Teach and Learn • Display the Nets tool and play the cube video. Ask: What is the definition of a net? Establish a net is a flat representation of a solid shape that can be folded to form the shape that it represents. Say: A net is what the shape would look like if it was opened out flat. The easiest solid shape to represent as a net is a cube, because all the faces are the same. • Demonstrate carefully unfolding a cardboard box by pulling back the tabs. Fold back the faces and flatten out the card to form a net. • Provide learners with Resource sheet 15: Net for a closed cube and scissors. Ask learners to cut out their nets. • Say: Visualise the net folded. Ask: Which part would be the base? Which part would be the top of the cube? Learners label what would be the edges and vertices of the cube, then make the cube. • Discuss the images in the Learn section of the Student’s Book then say: There are 11 possible nets that can make up a cube. Can you visualise how to fold them to make cubes?
Resources
• Discuss the Example and ask learners to decide what shapes net A and C would make. (triangular pyramid and cube)
Practise • Workbook: Nets (1) page 158
Apply • Distribute squared paper and scissors. Learners work in pairs to draw three nets for a closed cube, then cut them out and make the cubes.
Review • Ask two or three volunteers to come to the front to show their Apply activity nets, then discuss the similarities and differences between the different nets as a class.
Assessment for learning • Ignoring any tabs, how many folds do I need to assemble a cube from its net? (5)
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Unit
Student’s Book page 80
15 3D shape
Workbook page 160
Lesson 4: Nets (2) Learning objective
Geometry – Shapes and geometric reasoning
Code
Learning objective
6Gs4
Recognise and make 2D representations of 3D shapes including nets.
Strand 5: Problem solving 6Pt4
Recognise 2D and 3D shapes and their relationships, e.g. a cuboid has a rectangular cross-section.
6Ps2
Deduce new information from existing information and realise the effect that one piece of information has on another.
Prerequisites for learning
Vocabulary
Learners need to: • be able to visualise 3D shapes from 2D drawings.
prism, pyramid, net, base, side, vertex
Success criteria
Resource sheet 17: Nets of pyramids, preferably enlarged to A3 (per group); Resource sheet 18: Nets of prisms, preferably enlarged to A3 (per group); ruler (per pair); paper (per pair)
Learners can: • use knowledge of the properties of prisms and pyramids to identify and draw different nets of these shapes.
Refresh • Choose an activity from the Geometry – Shapes and geometric reasoning Refresh activities.
Discover • Discuss the image of prisms and pyramids. Say: Prisms have two parallel bases and sides that are parallelograms. Pyramids have polygonal bases and triangular side faces that connect to an apex.
Teach and Learn • Following a discussion about the text and examples in the Student’s Book, organise learners into four groups and provide them with paper, rulers, and Resource sheet 17: Nets of pyramids. Say: We are going to make nets for shapes that have one or more triangular faces. Ask: What shapes could these be? Elicit descriptions for square- and hexagonal-based pyramids, a tetrahedron and an octahedron. Ask: What do these nets have in common? (triangles) Which face will form the base of the shape when assembled? (the shape that is not a triangle). What shapes will they make? Ask each group to assemble a different solid shape from one of the nets on the resource sheet (so each group makes a different shape). Say: Starting at a vertex, open up your shape and lay it flat on your table. Ask: What happens if you cut/take one of the side faces of your net and reattach it at some other place? Does it form a net that will make the same shape? • Repeat the activity for prisms using Resource sheet 18: Nets of prisms. Ask: What kind of information can be obtained from a net of a
Resources
prism about the solid it creates? What is the relationship between the number of sides on the base and the number of faces on the prism? How does the net of a prism differ from the net of a pyramid?
Practise • Workbook: Nets (2) page 160 • Refer to Activity 2 from the Additional practice activities.
Apply • Display Slide 1 and ask pairs of learners to work together to answer the questions about nets of shapes.
Review • Display the Nets tool. Review the construction of nets for prisms and pyramids. Ask: How are they different?
Assessment for learning • To build a triangular prism using two equilateral triangles and six rectangles, what dimensions must be equal? (the length of sides of each triangle must equal the length of the shorter side of the rectangle)
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Unit
15 3D shape
Additional practice activities Activity 1
Challenge
2
• Visualise different pyramids in order to determine the relationship between the number of faces and the number of vertices.
Resources
Variation
interlocking triangles, squares and hexagons, or any other shape-building equipment, such as straws and sticky putty (per pair); Resource sheet 19: Pyramid properties (per pair)
Challenge
3
Learners explore the relationship between faces and edges.
What to do • Ask learners to investigate the relationship between the number of faces and vertices of any pyramid and complete the table by visualising or constructing each shape. • They use the numbers in the table to find the relationship between the faces and the vertices and then share their results and conclusions. (The number of faces and the number of vertices are the same.)
Activity 2
Challenge
2
Learning objective • Identify and draw the nets of a cube.
Resources
Variation
Resource sheet 14: Square dot paper (per learner); ruler (per learner)
Learners draw symbols on the faces of a net and assemble the cube. They swap with other pairs to determine the original net. Challenge
3
Geometry – Shapes and geometric reasoning
Learning objective
What to do • Prior to the activity, remind learners that an ordinary 1–6 dice is a cube and that opposite faces add up to 7. • Prior to the activity write on the board: Draw the net of a cube. Write the numbers 1 to 6 on the faces of your net so that if it was folded into a cube the opposite faces would add up to 7. How many different arrangements of the numbers will work? • Learners investigate and share their arrangements.
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