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Contents Introduction
Collins International Primary Maths Stage 4 units
Key principles of Collins International Primary Maths
iv
How Collins International Primary Maths supports Cambridge Primary and the Cambridge Primary Mathematics Curriculum Framework
iv
The components of Collins International Primary Maths: • • • • •
Teacher’s Guide Student’s Book Workbook Collins Connect DVD
viii xvi xvi xvii xvii
Collins International Primary Maths Stage 4 units and recommended teaching sequence
xviii
Collins International Primary Maths Stage 4 units link to Cambridge Primary Mathematics Curriculum Framework Cambridge Primary Mathematics Curriculum Framework Stage 4 link to Collins International Primary Maths units
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xxxv
Refresh Number
Geometry Measure
Handling data
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Numbers and the number system 1 Calculation: Mental strategies, Addition and subtraction 13 Calculation: Mental strategies, Multiplication and division 22 Shapes and geometric reasoning Position and movement Length, mass and capacity Time Area and perimeter
32 36 38 42 44
Organising, categorising and representing data
46
1 Whole numbers 1
48
2 Whole numbers 2
54
3 Whole numbers 3
60
4 Decimals 1
66
5 Decimals 2
72
6 Fractions
78
7 Addition and subtraction 1
89
8 Addition and subtraction 2
100
9 Addition and subtraction 3
106
10 Multiplication and division 1
117
11 Multiplication and division 2
128
12 Multiplication and division 3
139
13 2D shape, including symmetry
150
14 3D shape
161
15 Position and movement
167
16 Length
173
17 Mass
179
18 Capacity
185
19 Measures
191
20 Time
197
21 Area and perimeter
203
22 Handling data 1
209
23 Handling data 2
215
Resource sheets
221
Answers
264
Tracking back and forward through the Cambridge Primary Mathematics Curriculum Framework
281
Stage 4 Record-keeping charts
298
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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
4Nn2
Count on and back in ones, tens, hundreds and thousands from four-digit numbers.
What to do • Learners sit in a circle. • Starting with a given four-digit number, learners count clockwise around the circle in ones, tens, hundreds or thousands, with each learner saying a single number. • At various points, clap. When the learners hear the clap, they change from counting forwards and begin to count backwards. They should continue to count around the circle in a clockwise direction so that learners experience saying different numbers. • Ask: What changes when you count in 1s / 10s / 100s / 1000s?
Variations 1 Use three-digit numbers as starting points. Learners practise counting in ones, tens and hundreds. 2 Give learners a starting four-digit number. Learners count in ones, tens, hundreds or thousands, but, instead of counting out aloud, learners clap when the count gets to them. At a certain point, stop the silent count and ask learners what number they think they reached.
Place value mastermind Learning objective Code
Learning objective
4Nn3
Understand what each digit represents in a three- or four-digit number and partition into thousands, hundreds, tens and units.
Resources
Number – Numbers and the number system
Learning objective
mini whiteboard and pen (per learner)
What to do • Choose a mystery four-digit number. • Ask learners to guess the number and to come up to the board and write it. • Draw a question mark above any digits of the number that are correct, but in the wrong place value position. Draw a tick above any digits of the number that are correct and in the correct place value position. For example, if the mystery number is 3521, the guess 6241 will result in a question mark above the 2 (the digit 2 is in the number, but is not in the hundreds place), and a tick above the 1 because the digit 1 is in the number and is in the correct place value position). • Continue to play until the mystery number is identified. • Leave guesses on the board so that learners can deduce the mystery number.
Variation Practise place value of three-digit numbers. When learners guess a digit in the correct place value position, tell them which digit it is.
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Unit
1
Whole numbers 1
Rounding around Learning objective Code
Learning objective
4Nn9
Round three- and four-digit numbers to the nearest 10 or 100.
Number – Numbers and the number system
Resources large ball (per class); mini whiteboard and pen (per learner)
What to do • Learners stand in a circle. Each learner writes a three-digit multiple of 10 or 100 on a piece of paper so that the numbers are in order around the circle and the rest of the class can see their number. For example: 570, 580, 590, 600, 610. • Give the ball to one of the learners. Call out a three-digit number, for example, 583. The learner who has the ball then rounds the number to the nearest 10 and throws the ball to the learner who has that number (580). • Repeat, ensuring that all learners get a chance to round the numbers. • Change the set of rounded numbers to allow learners to round to different numbers.
Variation Play the same game, but with four-digit numbers rounded to the nearest 100.
Rearrange those digits! Learning objective Code
Learning objective
4Nn12
Compare pairs of three-digit or four-digit numbers, using the > and < signs, and find a number in between each pair.
Resources mini whiteboard and pen (per learner)
What to do • Split the class into two teams. Write a four-digit number on the board. • Choose a player from Team A to write a symbol (> or <) on the board and then rearrange the four digits in the number so that the comparison makes sense. 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 same four digits a further time to make a number that goes in between. • For example: Write the number 6295 on the board. Team A player rearranges the digits and writes 6295 < 6925. Team B player rearranges the digits and writes 6592 in between.
Variation Play the same game, but using three-digit numbers.
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Unit
2
Whole numbers 2
Refresh activities Learning objective Code
Learning objective
4Nn7
Multiply and divide three-digit numbers by 10 (whole number answers) and understand the effect; begin to multiply numbers by 100 and perform related divisions.
Resources Resource sheet 1: 0–9 digit cards (per class), mini-whiteboard and pen
What to do • Shuffle the digit cards (do not use the 0 digit card) and ask a learner to take any five (for example: 5, 2, 1, 3 and 8). • Each learner should then make five different three-digit multiples of 10 by using the available digits combined with a 0 as the ones digit of each number (for example: 250, 810, 120 and so on). • Call out different × 10 calculations using the available digits (for example: What is 25 times 10?). • If learners have made the correct multiple, they draw a ring around that number. The first learner to draw a ring around all five multiples is the winner.
Variation Learners make two-digit numbers using the available cards. Ask division questions where multiples of 10 are divided by 10 to give two-digit answers (for example: What is 250 divided by 10?) Again, the first learner to draw a ring around all five answers is the winner.
Jump up, sit down Learning objective Code
Learning objective
4Nn15
Recognise odd and even numbers.
Number – Numbers and the number system
Make a multiple
What to do • Learners stand in a circle, facing outwards so they can’t see each other’s decisions. • Explain that learners who get the wrong answer will be out and the last learner to answer will also be out, so they need to answer correctly and quickly. • Call out numbers randomly (up to four-digit numbers). • If the number is odd, learners sit down on the floor. If it is even, they jump up. • Play the game as quickly as possible.
Variation Use numbers to 20 (progressing to 50) and play the game at a slower pace.
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Unit
3
Whole numbers 3
Refresh activities Negative numbers Number – Numbers and the number system
Learning objective Code
Learning objective
4Nn13
Use negative numbers in context, e.g. temperature.
Resources chalk / string (per class)
What to do • Chalk a number line (or use a piece of string) on the ground, from –10 through to 10. • Choose a learner to come to the front and stand on a certain number. They then shut their eyes. • Choose a second learner to guide them along the number line, either forwards or backwards, taking one step for each number. • Ask the first learner (and the class) what number they think they have ended up on.
Variation Play the game using positive numbers only and a number line from 0 to 20.
Sequence spotters Learning objective Code
Learning objective
4Nn14
Recognise and extend number sequences formed by counting in steps of constant size, extending beyond zero when counting back.
What to do • Split the class into two teams. • Choose six learners to come to the front and stand in a line. Four of these should be from Team A and two from Team B. • The four learners from Team A decide on a rule (for example: subtracting 3 each time) and a starting number and say the first four numbers in that rule (for example: 11, 8, 5, 2). • The two learners from Team B each get a point if they can continue the sequence for two more numbers (in this case, –1 and –4) and a third point if they can state the rule. • Repeat, swapping the learners each time.
Variation Play the game as a whole-class activity. Split learners into pairs and write a sequence of numbers on the board (for Challenge 1 learners, use numbers greater than zero). Each pair gets a point if they can continue the sequence for two more numbers and a third point if they can state the rule.
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Unit
4
Decimals 1
Refresh activities First to four Code
Learning objective
4Nn4
Use decimal notation and place value for tenths and hundredths in context, e.g. order amounts of money; convert a sum of money such as $13.25 to cents, or a length such as 125 cm to metres; round a sum of money to the nearest pound.
4Nn5
Understand decimal notation for tenths and hundredths in context, e.g. length.
Number – Numbers and the number system
Learning objectives
Resources a 1–6 dice or Resource sheet 7: 1–6 Spinner (per class); 3 coloured markers (per class)
What to do • Split the class into three teams. • Display Slide R1. The teams take turns to spin the spinner and circle their position on the number line. • They start at zero and, for each number spun, they move a tenth (so an initial throw of five would mean moving five tenths from zero to 0·5). • After each round, the teams read the number they are on. The winning team is the first to reach the number 4. • If time allows, play the game in reverse, counting back in tenths from four to zero.
Variation Instead of a race between teams, learners spin the spinner and move one marker in different directions along the number line. After each movement, discuss the decimal they have ended on as a class.
Place your cards right! Learning objectives Code
Learning objective
4Nn4
Use decimal notation and place value for tenths and hundredths in context, e.g. order amounts of money; convert a sum of money such as $13.25 to cents, or a length such as 125 cm to metres; round a sum of money to the nearest pound.
4Nn5
Understand decimal notation for tenths and hundredths in context, e.g. length.
Resources Resource sheet 1: 0–9 large digit cards (per class)
What to do • Prior to each round of the activity, state what the objective is. This can be either to make the largest or the smallest decimal. Split the class into two teams and draw two sets of boxes on the whiteboard as templates · ). Shuffle a set of large digit cards. ready for two two-digit numbers with one decimal place (i.e. • Deal one card to the first team. They then decide in which one of their three boxes they should write it (tens, units or tenths). • Deal out the cards alternately until both teams’ boxes are full. The winning team is the one to have made the largest / smallest number depending on the original objective.
Variations For work with hundredths, learners place four cards onto the following template:
·
.
Deal out four cards to both teams at the same time. They then arrange their cards to make a TU.th number. Both teams then reveal their numbers and the largest / smallest number wins, depending on the original objective.
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Unit
5
Decimals 2
Refresh activities Decimal target board Number – Numbers and the number system
Learning objectives Code
Learning objective
4Nn4
Use decimal notation and place value for tenths and hundredths in context, e.g. order amounts of money; convert a sum of money such as $13·25 to cents, or a length such as 125 cm to metres; round a sum of money to the nearest pound.
4Nn5
Understand decimal notation for tenths and hundredths in context, e.g. length.
What to do • Display Slide R2. 12·1
13·5
11·9
12·4
14·7
13·2
15·9
14·2
10·8
11
14·3
12·5
16
12·2
16·1
14·8
• Ask learners to use the target board to answer the following challenges: ◊ Find numbers with an even number of tenths. ◊ Find pairs that differ by 0·1 (or 0·2). ◊ Find numbers that round to 12.
Variation Draw a target board featuring numbers to two decimal places. Learners find pairs that differ by 0·01 (or 0·02).
Conversions match Learning objectives Code
Learning objective
4Nn4
Use decimal notation and place value for tenths and hundredths in context, e.g. order amounts of money; convert a sum of money such as $13·25 to cents, or a length such as 125 cm to metres; round a sum of money to the nearest pound.
4Nn5
Understand decimal notation for tenths and hundredths in context, e.g. length.
Resources mini whiteboard and pen (per leaner)
What to do • Write ten different lengths on the board (in metres to two decimal places and in centimetres). For example: 640 cm, 6·24 m, 689 cm, 6·70 m, 620 cm, 6·55 m, 607 cm, 6·07 m, 602 cm, 6·04 m. • Ask learners to choose any five of these numbers and write them on a mini whiteboard. • Call out measurements equal to the lengths on the board. For example, 624 cm might be called as it is equal to 6·24 m, as well as random measurements which are not on the board to encourage learners to think carefully. • If learners hear a measurement that is equal to one of their numbers, they should circle it. For example, 6·40 m is equal to 640 cm, so they should circle 640 cm if it is one of their numbers. • The first learner to have circled all five of their numbers wins. • Repeat the game, using prices in dollars (as decimals) and cents, or a mixture of lengths and prices.
Variation Play the above game using measurements of length to one decimal place in order to practise conversion of measurements in tenths.
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Unit
6
Fractions
Refresh activities Learning objective Code
Learning objective
4Nn25
Find halves, quarters, thirds, fifths, eighths and tenths of shapes and numbers.
What to do • Write the following words in a column on the board: UP, WHY, TEAM, SPORT, QUARTERS and RECOGNISED. • Explain that each word represents a different type of fraction based on the number of letters it contains (WHY represents thirds because there are 3 letters, TEAM represents quarters and so on). • Explain that each letter within a word represents a fraction. For example, in TEAM 34 is the letter ‘A’ because it is the third quarter. 5 • Ask: Which letter represents four fifths? (R) Which fraction is the letter G? ( 10 ) • Give learners fractions in the form of a code. They should crack the code by finding the letters and spelling out a mystery word. For example: 5 4 3 1 6 ◊ 10 , 8, 5, 3, 10 = GROWN
◊ 18, 12, ◊ ◊
2 , 3 1 , 2
7 1 9 , , 10 4 10 3 2 2 3 , , , = 4 5 5 3 6 1 7 6 , , , 10 4 10 8
= QUITE HAPPY = UNTIE
Variation Encourage learners to create their own code and set clues for the rest of the class to identify.
Number – Numbers and the number system
Crack the fractions
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Unit
6
Fractions
Prove it! Learning objective Code
Learning objective
4Nn17
Order and compare two or more fractions with the same denominator (halves, quarters, thirds, fifths, eighths or tenths).
Number – Numbers and the number system
Resources mini whiteboard and pen (per learner); coloured interlocking cubes (per class); modelling clay (per class); strips of paper (per class)
What to do • • • •
Explain that in order for something to be proved true or false, there needs to be evidence. Give learners a statement of fact comparing fractions with the same denominator and ask them to prove it. In pairs, they discuss the statement and decide on a way to prove whether it is correct or not. Choose a pair to come to the front and share their evidence. Provide equipment so that learners can model the fractions they are referring to (coloured interlocking cubes, modelling clay, strips of paper, and so on). • Example challenges include: ◊ Prove that 45 is larger than 25. 6 2 ◊ Prove which is the smaller out of 10 and 10 . ◊ Prove that 38 is not larger than 58.
Variations 1 Give learners statements that refer to one fraction (without comparisons). For example: 2 ◊ Prove that 10 is 2 out of a possible 10. ◊ Prove what both numbers mean in the fraction 34. 2 Give learners statements to compare fractions with different denominators. For example: 2 . ◊ Prove that 35 is larger than 10 4 ◊ Prove that 8 is equivalent to 24. 3 Give learners statements to compare the place value of tenths in decimal numbers. For example: ◊ Prove that 0·5 is larger than 0·3. ◊ Prove that 4·46 is smaller than 4·77.
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Unit
6
Fractions
Is it equivalent? Learning objectives Code
Learning objective
4Nn19
Use equivalence to help order fractions, e.g. 7 and 3
4Nn20
Understand the equivalence between one-place decimals and fractions in tenths.
4Nn21
Understand that 1 is equivalent to 0·5 and also to 5 .
4Nn22
Recognise the equivalence between the decimal fraction and vulgar fraction forms of halves, quarters, tenths and hundredths.
2
4
Number – Numbers and the number system
10
10
Resources mini whiteboard and pen (per learner)
What to do • The aim of this activity is for a learner to try and guess the mystery fraction that they have been allocated, by using clues about equivalence to help. • Choose one learner to come to the front and stand facing the class with their back to the board. They are the ‘guesser’. • Write ten different fractions on the board. These should be fractions that are equivalent to 12, 14 or 15. • Ask learners to write down any one of these fractions on their whiteboards. • Point to one of the fractions without the guesser seeing. Explain that their job is to try and guess the mystery fraction. • The guesser then calls out various learners one by one. They bring their whiteboards, show the fraction and explain whether it is equivalent or not to the mystery fraction. • Continue comparing the fractions until the guesser discovers the mystery fraction. • For example: 2 is called to the front, they would say: ‘My fraction is not ◊ If the mystery fraction is 48 and a learner with 10 equivalent.’ ◊ If a learner with 24 is called to the front, they would say: ‘My fraction is equivalent.’ ◊ The guesser would then know that the mystery fraction must be equal to 24 (or 12).
Variations 1 Play with simple fractions, identifying equivalence between 12 and 24. 2 Play using decimal numbers. Write fractions (tenths, hundredths as well as 12, 14 and 34) on the board for learners to choose from. Choose a decimal number to act as the mystery number. The guesser then tries to guess the mystery number from clues given. For example: if it is not equivalent to 12, they know it is not 0·5.
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Unit
6
Fractions
Fraction and decimal loop cards Learning objectives
Number – Numbers and the number system
Code
Learning objective
4Nn20
Understand the equivalence between one-place decimals and fractions in tenths.
4Nn21
Understand that 1 is equivalent to 0·5 and also to 5 .
4Nn22
Recognise the equivalence between the decimal fraction and vulgar fraction forms of halves, quarters, tenths and hundredths.
2
10
Resources Resource sheet 12: Fraction and decimal loop cards (per class)
What to do • Distribute loop cards so each pair has at least one (if any cards are left over, some learners may have one each). 25 at the bottom, • Explain that each fraction has its pair in an equivalent decimal. For example, if a card says 100 there will be another card with 0·25 at the top. • Choose a pair to read out the fraction at the bottom of their card. • The pair with the equivalent decimal at the top of their card stands up, reads out the answer and then reads out the fraction at the bottom of 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.
Variations 1 Learners work in pairs to place the loop cards in a circle. 2 One pair reads out the question from their card. The rest of the class suggest any alternatives to the usual 3 30 6 = 0·3, 0·30, 100 , 20 , and so on). answer that are also equivalent (for example, 10
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Unit
6
Fractions
Fraction target boards Code
Learning objective
4Nn24
Relate finding fractions to division.
4Nn25
Find halves, quarters, thirds, fifths, eighths and tenths of shapes and numbers.
Resources mini whiteboard and pen (per learner)
What to do • • • •
Draw a 3 by 3 grid on the board to serve as a target board. In each cell, write a different multiple of 4. Ask learners to choose any five of the numbers on the target board and write them on their whiteboards. Call out ‘fractions of numbers’ questions related to each of the multiples. For example: One quarter of this number is 10. What is it? • If the answer to the question corresponds to one of their chosen multiples, learners should circle it. The first learner(s) to have circled all five of their multiples wins.
Variations 1 Write the numbers from 3 to 11 in the grid. Learners choose any five as before. Ask ‘What is one half / third / quarter / fifth / eighth / tenth of [x]’ questions that relate to the numbers in the grid. For example: What is one fifth of 20? 2 Write even numbers in the grid. Ask quickfire ‘What is half of ...’ questions that relate to the numbers in the grid. Choose learners to come and point to the answers.
Number – Numbers and the number system
Learning objectives
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