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Contents Introduction
Collins International Primary Maths Stage 3 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 3 units and recommended teaching sequence
xviii
Collins International Primary Maths Stage 3 units link to Cambridge Primary Mathematics Curriculum Framework Cambridge Primary Mathematics Curriculum Framework Stage 3 link to Collins International Primary Maths units
Geometry Measure
Handling data
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2 Whole numbers 2
48
3 Whole numbers 3
59
4 Fractions
65
5 Addition and subtraction 1
76
6 Addition and subtraction 2
87
7 Addition and subtraction 3
98
8 Multiplication and division 1
109
9 Multiplication and division 2
120
10 Multiplication and division 3
131
11 2D shape
142
12 3D shape
148
13 Position and movement
154
14 Money
165
15 Length
171
16 Mass
177
17 Capacity
183
18 Time
189
19 Handling data
195
Resource sheets
206
Answers
255
Tracking back and forward through the Cambridge Primary Mathematics Curriculum Framework
276
Stage 3 Record-keeping charts
295
xxi
xxxiii
Refresh activities Number
37
iv
The components of Collins International Primary Maths: • • • • •
1 Whole numbers 1
Numbers and the number system 1 Calculation: Addition and subtraction 8 Calculation: Multiplication and division 14 Shapes and geometric reasoning Position and movement Money Length, mass and capacity Time
19 23 25 27 33
Organising, categorising and representing data
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Introduction Key principles of Collins International Primary Maths Collins International Primary Maths is a mathematics course that ensures complete coverage of the Cambridge Primary Mathematics Curriculum Framework. The course offers: • a rigorous and cohesive scope and sequence of the Cambridge Primary Mathematics Curriculum Framework, while at the same time allowing for schools’ own curriculum design • a problem-solving and discovery approach to the teaching and learning of mathematics • lesson plans following a highly effective and proven lesson structure • a bank of practical hands-on learning activities • controlled, manageable differentiation with activities and suggestions for at least three different ability groups in every lesson
• extensive teacher support through materials which: – promote the most effective pedagogical methods in the teaching of mathematics – 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 learner participation – encourage communication of methods and foster mathematical rigor – are aimed at raising levels of attainment for every learner • manageable strategies for effective monitoring and record-keeping, to inform planning and teaching.
How Collins International Primary Maths supports Cambridge Primary and the Cambridge Primary Mathematics Curriculum Framework Cambridge Primary is typically for learners aged 5 to 11 years. It develops learner skills and understanding through the primary years in English, Mathematics and Science. It provides a flexible framework that can be used to tailor the curriculum to the needs of individual schools.
The Cambridge approach supports schools to develop learners who are:
Cambridge Primary Mathematics Curriculum Framework:
• reflective as learners, developing their ability to learn
• provides a comprehensive set of learning objectives in Mathematics for each year of primary education
• innovative and equipped for new and future challenges
• focuses on developing knowledge and skills which form an excellent foundation for future study
• engaged intellectually and socially, and ready to make a difference in the world.
• focuses on learners’ development in each year
The Cambridge Primary Mathematics Curriculum Framework is organised into six stages. Each stage reflects the teaching targets for a year group. Broadly speaking, Stage 1 covers the first year of primary teaching, when learners are approximately five years old. Stage 6 covers the final year of primary teaching when learners are approximately 11 years old.
• provides a natural progression throughout the years of primary education • is compatible with other curricula, internationally relevant and sensitive to different needs and cultures • is suitable for learners whose first language is not English
• confident in working with information and ideas – their own and those of others • responsible for themselves, responsive to and respectful of others
• provides schools with international benchmarks.
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Unit
1
Whole numbers 1
Refresh activities Clap counting Code
Learning objective
3Nn1
Recite numbers 100 to 200 and beyond.
What to do • Ask learners to sit in a circle. • Starting with a given number between 100 and 200, such as 154, learners count clockwise around the circle in ones, each saying one number. • At various points, clap. When the learners hear the clap, they change counting direction and begin to count backwards. They continue to count in a clockwise direction so that learners experience saying different numbers.
Variation • Learners arrange themselves in a line to represent counting forwards or backwards along a number line.
A counting wave Learning objectives Code
Learning objective
3Nn3
Count on and back in ones, tens and hundreds from two- and three-digit numbers.
3Nn4
Count on and back in steps of 2, 3, 4 and 5 to at least 50.
What to do • Arrange learners in either a circle or line. • Starting with a given two-digit or three-digit number, such as 346, they count around the circle (or along the line) in steps of 1, 10 or 100, each saying a different number. As they say their number, learners stand or jump up, raising their hands in the air to form a Mexican wave. • Encourage learners to consider the digits that change when counting in 1s, 10s or 100s.
Number – Numbers and the number system
Learning objective
Variation • Introduce further rules, for example: when ‘Change!’ is called out, learners count backwards or when a new number is called out, learners immediately start counting in steps of that number.
1
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Unit
1
Whole numbers 1
Place value bingo Learning objectives Code
Learning objective
3Nn5
Understand what each digit represents in three-digit numbers and partition into hundreds, tens and units.
Number – Numbers and the number system
Resources mini whiteboard and pen (per learner)
What to do • Learners each draw a 1 by 4 grid on their whiteboards. In each of the four sections they write a three-digit number, using any of the digits from 0 to 5. • Learners may write the same number more than once if they choose, or two numbers made from the same digits, for example, 431, 314. • Call out three random digits (for example: 3, 5 and 3). If a learner has written a number using these three digits (in this case: 533, 353 or 335) they should cross it out. They must only cross out one number each time. • The winner is the first to have all four numbers crossed out.
Variation • Learners draw a 3 by 2 grid and write six two-digit numbers.
Place that digit! Learning objectives Code
Learning objective
3Nn5
Understand what each digit represents in three-digit numbers and partition into hundreds, tens and units.
Resources mini whiteboard and pen (per learner)
What to do • Prior to the activity, inform learners of their target: to make the largest number they can. • Learners draw a 1 by 3 grid on a mini whiteboard. The sections of the grid represent the hundreds, tens and units of a three-digit number respectively. • Explain that you will call out four random digits. Learners decide where to write each digit as it is called out. One of the digits they must choose to discard. • Compare numbers across the class. Learners who have made the largest number win a point.
Variations • Learners draw two 1 by 3 grids with the aim of making the largest number in one and the smallest number in the other. Call out seven random digits. Learners decide where to write each digit as it is called out and should choose to discard one – as in the original activity. • Call out one digit at a time. Learners have to decide in which box on their grid to write the digit. Once a digit is written in a box it cannot be erased.
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Unit
1
Whole numbers 1
Human number line Code
Learning objective
3Nn9
Place a three-digit number on a number line marked off in multiples of 100.
Resources 9 chairs (per class); multiples of 100/10 labels – see Variations (per class)
What to do • Prior to the activity, spread nine chairs out in a line. Explain to the class that the chairs represent the multiples of 100 from 100 to 900. • Write four random digits on the board. • Learners use any three of the digits to each make a three-digit number. • Choose three learners to stand in the correct place (according to their chosen number) on the chair number line. • The rest of the class should attempt to guess their numbers. • For example: – the digits 5, 2, 8 and 0 are written on the board – learner stands between 500 and 600 – class guess numbers like 528, 502, 580, and so on, until they guess correctly.
Variations • Place multiples of 100 labels on the chairs. • Carry out the same activity, but with the chairs representing multiples of 10 and with learners choosing their own numbers that fit within the range of the number line.
Number – Numbers and the number system
Learning objective
3
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Unit
2
Whole numbers 2
Refresh activities Pat, click, tap Learning objective
Number – Numbers and the number system
Code
Learning objective
3Nn5 3Nn6* 3Nn7 3Nn9* 3Nn10*
Understand what each digit represents in three-digit numbers and partition into hundreds, tens and units. Find 1, 10, 100 more/less than two- and three-digit numbers. Multiply two-digit numbers by 10 and understand the effect. Place a three-digit number on a number line marked off in multiples of 100. Place a three-digit number on a number line marked off in multiples of 10.
What to do • Explain to learners that you will use gestures to signify digits within a number: – Hundreds digit = patting head – Tens digit = clicking fingers – Units digit = tapping knees • Pat, click and tap out a number. For example, pat your head twice (200), click your fingers eight times (80) and tap your knees five times (five). Ask learners to identify the number (285). • Repeat for different three-digit numbers. • Encourage learners to pat, click and tap out numbers of their own for the class to guess. Include numbers with no tens (no clicking fingers) and/or no units (no tapping knee).
Variation • Show how digits shift when multiplying by 10. Use actions to show a two-digit number, for example, 63 = six finger clicks and three knee taps. Learners then multiply by 10 and tap the answer (630 = six head pats and three finger clicks). This is a useful demonstration of how the digits do not change when multiplying by 10 – they just shift one place.
Rearrange those digits! Learning objectives Code
Learning objective
3Nn11
Compare three-digit numbers, use < and > signs, and find a number in between.
3Nn12
Order two- and three-digit numbers.
What to do • Split the class into two teams, A and B. • Write a three-digit number on the board. • Choose a player from Team A to write a symbol (> or <) on the board and then rearrange the digits in the number so that the comparison is true. • They get one point if they can do so correctly. • Choose a player from Team B. They also get one point if they are able to rearrange the digits a further time to make a number that goes in between the other two. • For example: – write the number 392 on the board – Team A player rearranges the digits and writes 392 < 932 – Team B player rearranges the digits and writes 923 in between, as 392 < 923 < 932.
Variation • Complete the activity without the symbols prior to the lesson introducing them. Player 1 can make a three-digit number from a given set of digits. Player 2 then says: ‘I can make a number more/less than this by rearranging the digits.’ The class should then guess Player 2’s number.
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Unit
3
Whole numbers 3
Refresh activities Challenge
Back and forth
1
Code
Number – Numbers and the number system
Learning objective Learning objective
3Nn7
Multiply two-digit numbers by 10 and understand the effect.
3Nn11*
Compare three-digit numbers, use < and > signs, and find a number in between.
What to do • Learners stand in a circle with you holding a ball in the centre. • Throw the ball quickly to one of the learners whilst calling out a two-digit number. • The learner should throw it back straight away and respond with the original number multiplied by 10; for example, if ‘39!’ is called, the learner should respond with ‘390!’ as they return the ball.
Variation • Decide beforehand whether learners are aiming to make a number larger or smaller than the given number and, consequently, the sign (< or >) that learners would use in between. Call out a three-digit number and throw the ball to a learner. They should rearrange the digits to make a larger/smaller three-digit number and call it out as they throw the ball back. For example, if ‘782!’ is called and the aim is to make a larger number, the learner could respond with ‘827!’
Mathletics
Challenge
1
Learning objectives Code
Learning objective
3Nn8*
Round two-digit numbers to the nearest 10 and round three-digit numbers to the nearest 100.
3Nn13*
Give a sensible estimate of a number as a range (e.g. 30 to 50) by grouping in tens.
3Nn12*
Order two- and three-digit numbers.
What to do • This activity is based on the idea of the high jump in an athletics competition. • Invite four learners to come to the front and give each of them two random digit cards. They should use these to make the largest two-digit number they can. • Set the high jump ‘bar height’ at 30 cm. If a learner’s number is more than 30, they clear the bar. If it is less, they should be given two new cards and two more attempts to clear the bar. • The ‘bar height’ should increase by 10 cm each time. • The winner is the player left at the end – the one with the largest two-digit number.
Variation • Learners should each be given three-digits to make the highest three-digit number they can from. They should then round their number to the nearest hundred. Go through each hundred from 100 to 1000. Learners should sit down as they fail to reach the new ‘bar height’. The winner is the player left at the end – the one with the largest three-digit number.
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Unit
4
Fractions
Refresh activities Challenge
Five-minute halving
2
Number – Numbers and the number system
Learning objectives Code
Learning objective
3Nn14
Find half of odd and even numbers to 40, using notation such as 13 1.
3Nn20
Find halves, thirds, quarters and tenths of shapes and numbers (whole number answers).
2
What to do • Learners each write down any five numbers from 5 to 20. They may be whole numbers or include halves, for example, 12 12. • Call out random numbers from 10 to 40. • If a learner has written down a number that is half of the number called out, they stand up and call out their number. • They should then cross out their answer. • The winner is the learner who crosses all five numbers first.
Variations • Learners complete the same activity but: – they each write down any five whole numbers from 5 to 20 – the teacher calls out random even numbers from 10 to 40.
Fraction doodles
Challenge
or
2
Learning objectives Code
Learning objective
3Nn15
Understand and use fraction notation recognising that fractions are several parts of one whole, e.g. 3 is three quarters and 4
2 is two thirds. 3
3Nn16
Recognise equivalence between 1, 2, 4 and 5 using diagrams.
3Nn17
Recognise simple mixed fractions, e.g. 11 and 21.
3Nn20
Find halves, thirds, quarters and tenths of shapes and numbers (whole number answers).
2 4 8
10
2
4
Resources large piece of paper or whiteboard (per class/group)
What to do • Choose a learner to come to the front and draw a simple doodle of an object in 20 seconds. The subject can be anything they wish (for example, car, lemon, planet). • They then split it into equal parts and shade some of them. • The class should identify the object and the fraction shaded.
Variation • Learners should draw and shade fractions of circles, squares or rectangles instead of objects.
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Unit
Rounding mixed numbers
4
Fractions
Challenge
2
Learning objective Code
Learning objective
3Nn18
Order simple or mixed fractions on a number line, e.g. using the knowledge that 1 comes half way between 1 and 3, and 2
4
4
Number – Numbers and the number system
that 11 comes half way between 1 and 2. 2
What to do • • • • •
Learners each write down any five whole numbers from 1 to 10. Call random mixed numbers. Learners should round the mixed number to the nearest whole number and circle it, if they have it. Remind learners that halves should be rounded upwards (4 12 becomes 5). The first learner to circle all five whole numbers is the winner.
Variations • Call out more difficult mixed fractions, for example, 3 58. • Learners have to decide whether the fraction is more or less than a half before rounding it.
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Unit
5
Addition and subtraction 1
Refresh activities Number bond tennis
Challenge
2
Number – Calculation: Mental strategies, Addition and subtraction
Learning objective Code
Learning objective
3Nc1
Know addition and subtraction facts for all numbers to 20.
3Nc2
Know the following addition and subtraction facts: – multiples of 100 with a total of 1000 – multiples of 5 with a total of 100.
Resources ball or bean bag (per class); piece of string (per class)
What to do • • • • •
Split the class into two teams, in an open space, standing either side of a piece of string (the net). Choose a number bond to practise, for example, number bonds to 20. Player 1 in Team A takes the ball or bean bag and passes it over the net whilst calling out a number. Player 1 in Team B responds by calling out the corresponding number to make 20. Play moves to the second players in each team, and so on.
Variation • Practise number bonds in pairs, with two players either side of the net.
Fizz buzz
Challenge
or
2
Learning objectives Code
Learning objective
3Nc2
Know the following addition and subtraction facts: – multiples of 100 with a total of 1000 – multiples of 5 with a total of 100.
What to do • • • • •
Learners sit in a circle and begin counting in ones. Each learner says a new number. If a learner’s number is a multiple of 5, they should say ‘Fizz!’ instead of the number. For example, learners should count: ‘1, 2, 3, 4, Fizz! 6, 7, 8, 9, Fizz!’ If a learner makes a mistake, they are out.
Variations • To extend the activity, introduce a second multiple. For example, if a number is a multiple of 5, they should say ‘Fizz!’ If it is a multiple of 3, learners should say ‘Buzz!’ If it is a multiple of both 5 and 3, they should say ‘Fizz Buzz!’ • This could be extended further with the introduction of a third multiple and the word ‘Whizz!’
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Unit
Addition and subtraction loop
5
Addition and subtraction 1
Challenge
2
Code
Learning objective
3Nc1
Know addition and subtraction facts for all numbers to 20.
3Nc12
Add several small numbers.
Number – Calculation: Mental strategies, Addition and subtraction
Learning objectives
Resources Resource sheet 8: Addition and subtraction loop cards (per pair)
What to do • Distribute loop cards so each pair has at least one. • Choose a pair to read out the question from their card. • The pair with the correct answer stands up, reads out the answer and then reads out the question on their loop card. • Time the activity to see how quickly the class can get back to the starting card. • Shuffle the cards and encourage learners to try to beat their time.
Variation • Learners work in pairs to place the loop cards in a circle. • One pair reads out the question from their card. • The rest of the class suggest alternatives to the usual answer that could be used with the = sign, for example, 14 + 4 equals 18, but also 20 – 2 and 10 + 8.
Agree or disagree?
Challenge
or
2
Learning objectives Code
Learning objective
3Nc11
Use the = sign to represent equality, e.g. 75 + 25 = 95 + 5.
What to do • Learners stand up and all face forwards. • Write on the board addition and subtraction statements, using a combination of calculations that are true and those that are not, for example: 36 – 12 = 24 or 45 + 3 = 47. • If learners agree with the statement, they make a tick symbol (✓) with their arms (one arm vertical, one arm diagonal). If they disagree with the statement, they make a cross symbol (✗) with their arms (arms crossed diagonally). • Repeat, varying the statement and discussing learner answers each time.
Variations • This activity can be extended once learners are familiar with the way that the equals sign can be used to show equivalence between statements, with statements such as 4 + 7 = 14 – 3. • Use statements that are a matter of opinion, such as: The quickest way to add two numbers is to look for numbers that make ten. • Encourage learners to justify their choice of agreeing or disagreeing.
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Unit
6
Addition and subtraction 2
Refresh activities Jumps, steps and hops
Challenge
2
Number – Calculation: Mental strategies, Addition and subtraction
Learning objective Code
Learning objective
3Nc14
Add and subtract pairs of two-digit numbers.
3Nc15
Add three-digit and two-digit numbers using notes to support.
What to do • Learners stand in a line. Tell them that they are standing on an invisible number line. Every 100 is a jump, every ten is a step and every unit is a small hop. • Depending on the objective, call out a two- or three-digit addition or subtraction. Learners model the calculation by saying the first number and then jumping/stepping/hopping forwards (addition) or backwards (subtraction) to find the answer. For example, if the calculation is 62 + 26, learners say ‘62’ and then take two steps forwards, counting in tens (72, 82). They then take six hops forwards, counting in units (83, 84, 85, 86, 87, 88).
Variations • Practise partitioning of numbers by calling out a two- or three-digit number. • Learners model the number by jumping/stepping/hopping to show the numbers of hundreds, tens and units.
Step around the circle
Challenge
2
Learning objectives Code
Learning objective
3Nc17
Add/subtract single-digit numbers to/from three-digit numbers.
3Nc18
Find 20, 30, … 90, 100, 200, 300 more/less than three-digit numbers.
Resources beanbag (per class)
What to do • Learners sit in a circle. Each chooses a three-digit number for themselves. • Stand in the centre of the circle and pass a beanbag to one of the learners. At the same time, call out the number by which learners are to step-count, either ‘10s!’ or ‘100s!’ • The learner with the beanbag stands up, calls out their three-digit number and begins stepping around the circle. either forwards or backwards. • The rest of the class step-count in 10s or 100s, starting with the three-digit number. If the learner decides to walk forwards, the class step-count forwards and vice versa. • The class count aloud in time with as the learner’s steps and until they arrive back to sit in their original place and pass the beanbag back.
Variation • Learners choose a two-digit number each. They then step-count in either 1s or 10s.
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Unit
6
Addition and subtraction 2
Challenge
Spot the mistake
2
Code
Learning objective
3Nc18
Find 20, 30, … 90, 100, 200, 300 more/less than three-digit numbers.
Number – Calculation: Mental strategies, Addition and subtraction
Learning objectives
What to do • Write on the board a HTU + TU calculation with an incorrect answer. The mistake should be based on common errors learners make, for example, when adding across hundreds boundaries. Include the notes used to work out the answer, for example, a number line showing the error. • Learners identify the mistake and demonstrate how to answer the question correctly.
Variations • Provide learners with a calculation for which they should devise their own wrong answers for the rest of the class to spot. Encourage learners to think carefully about small mistakes that are commonly made and how these might affect the answer. • Learners discuss the steps they could take to ensure they don’t make similar mistakes. Challenge
Number triangles
or
or
2
Learning objectives Code
Learning objective
3Nc18
Find 20, 30, … 90, 100, 200, 300 more/less than three-digit numbers.
What to do • Draw a triangle with a number at each vertex. There should be a ‘rule’ to the positioning of each number, for example, if learners add the two numbers along the base, it makes the total at the apex. • Learners identify the rule and then solve various problems where only two numbers are given. For example: 312
362
592
50
?
702
30
302
?
Variation • Draw squares instead of triangles and write numbers that correspond to a two-step calculation (numbers in each corner and in the centre). • Learners identify the rule. For example: 592
40 632
562
30
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