* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Mathematics
Law of large numbers wikipedia , lookup
Mathematics wikipedia , lookup
Infinitesimal wikipedia , lookup
Numbers (TV series) wikipedia , lookup
List of important publications in mathematics wikipedia , lookup
History of logarithms wikipedia , lookup
Approximations of π wikipedia , lookup
Georg Cantor's first set theory article wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
History of mathematics wikipedia , lookup
Collatz conjecture wikipedia , lookup
Location arithmetic wikipedia , lookup
Non-standard analysis wikipedia , lookup
Large numbers wikipedia , lookup
Foundations of mathematics wikipedia , lookup
Real number wikipedia , lookup
Positional notation wikipedia , lookup
Secondary School Mathematics Curriculum Improvement Study wikipedia , lookup
Organisation of content into year levels is advisory. Teachers will continue to make professional judgements about when to introduce content based on children’s prior learning and achievement. *National Consistency in Curriculum Outcomes, Statement of Learning – Mathematics Middle childhood: Mathematics/Number – Students use numbers and operations and the relationships between them efficiently and flexibly. Typical sequence of content: Year 4 Year 1 Year 2 Year 3 Year 5 Year 6 Year 7 Understand numbers Understand whole numbers and decimals Numbers to ten thousand Numbers to hundreds of thousands and decimal numbers to hundredths Numbers into the millions and decimal numbers to hundredths Numbers into the millions and beyond, decimal numbers to thousandths and negative numbers in real life contexts read, write and say whole numbers up to ten thousand, leaving a space between each set of three digits from right to left read, write and say whole numbers up to one million using the recurring pattern of H T O in the thousands and H T O in the ones use place value to read, write and say whole numbers into the millions and decimal numbers into the hundredths places (eg 3.4 is larger than 3.25 because there is a 4 in the tenths place) use place value to read, write and say whole numbers beyond the millions and decimal numbers into the thousandths (eg 3.4 is larger than 3.289 because 4 tenths equals to order and locate numbers on a suitably scaled number line (eg order and locate 2 250, 2 750 and 2 500 on a number line from 2 000 to 3 000) to order and locate numbers on a suitably scaled number line* (eg order and locate 30 000, 10 000 and 25 000 on a number line from 0 to 50 000) to order and locate numbers on a number line (eg locate 25, 50 and 75 on a number line from 0 to 100) to order and locate numbers on a suitably scaled number line (eg locate 200, 400 and 800 on a number line from 0 to 1 000) read, write and say correctly decimal numbers to two decimal places (eg 6.9 is ‘six point nine’ and 4.05 is ‘four point zero five’) 400 1000 , which is bigger than 289 ) 1000 investigate negative numbers in real life contexts* (eg temperature, bank balances, debit) © Department of Education and Training Western Australia, Middle childhood: Mathematics/Number scope and sequence, December 2007 1 Organisation of content into year levels is advisory. Teachers will continue to make professional judgements about when to introduce content based on children’s prior learning and achievement. *National Consistency in Curriculum Outcomes, Statement of Learning – Mathematics Typical sequence of content: Year 4 Year 1 Year 2 represent whole numbers in various Year 3 Year 5 Year 6 Year 7 Understand whole numbers and decimals (continued) represent whole numbers and decimal fractions in various ways* (eg represent decimals to two places, on an array, a 10 x 10 grid or a number line) represent whole numbers and decimal fractions in various ways (eg represent decimals to two places on an array, a 10 x 10 grid or a number line) the name and value of places in numbers up to ten thousand the name and value of places in numbers up to hundreds of thousands and decimals up to two decimal places the name and value of places into the millions and decimals up to two decimal places the name and value of places beyond the millions and decimals up to three decimal places the properties and patterns of odd and even numbers (eg even numbers are always divisible by two, where odd numbers are not) identify factors and multiples of some two and three-digit numbers* (eg the factors of 80 are 1, 2, 4, 5, 8, 10, 16, 20, 40 and 80 and the first four multiples of 25 are 25, 50, 75, 100) properties of prime and composite numbers (eg use Eratosthenes’ Sieve to the find the prime and composite numbers between 1 and 100) properties of triangular numbers and square numbers (eg all square numbers end in 0, 1, 4, 5, 6 or 9) count in decimal amounts to two decimal places forwards and backwards from a given number (eg 2.34, 2.33, 2.32, 2.31) count forwards and backwards in decimal amounts to three decimal places, from a given number (eg 2.341, 2.342, 2.343) ways (eg symbols, words, concrete materials, calculators) patterns in the multiples of numbers* (eg multiples of 5 end in 5 or 0) and prime and composite numbers to 20, determining that a prime number has two factors, 1 and the number itself count in tenths* (eg 2.3, 2.4, 2.5) and hundredths for money and measures (eg use the scale on a tape measure to count forwards by 0.05 m from a given length [0.90 m, 0.95 m, 1.00 m, 1.05 m]) © Department of Education and Training Western Australia, Middle childhood: Mathematics/Number scope and sequence, December 2007 2 Organisation of content into year levels is advisory. Teachers will continue to make professional judgements about when to introduce content based on children’s prior learning and achievement. *National Consistency in Curriculum Outcomes, Statement of Learning – Mathematics Typical sequence of content: Year 4 Year 1 Year 2 Year 3 Year 5 Year 6 Year 7 Understand whole numbers and decimals (continued) recognise different representations of numbers involving decimal fractions* (eg 1.4 = 2+ 5 10 + 1 4 and 10 8 ) 100 2.58 = 2 + 58 100 or write decimals as fractions and fractions as decimals (eg 0.65 = 65 100 and 2 4 10 = 2.4) know decimal equivalents for 1 2 , 1 , 1 , 1 , 1 , 1 and use these 3 4 5 8 10 to find decimal and percentage equivalents* (eg 1 = 0.25 = 25% 4 and partition decimals in standard ways to two decimal places (eg 5.37 = 5 + 3 + 7 ) 10 100 1 8 = 0.125 = 12.5%) partition decimals in standard and non-standard ways (eg 2.46 = 2 + 0.46 as well as 2.2 + 0.26 as well as 2 + 4 + 6 ) 10 Money and measures using a decimal point Amounts of money and measures using a decimal point Connections between money, measures and decimals write money amounts with a decimal point between the whole number (dollar) part and the fractional (cents) part interpret a calculator display involving money (eg that, in a money context, 1.5 on the calculator screen translates to $1.50) read, write and order money and measures where a decimal point is used (eg 3.2 kg is heavier than 3.12 kg) write money with only one symbol (eg seventy five cents is written as 75c or $0.75 not $0.75c) round money to the nearest 5c (eg amounts ending in 1, 2, 6 or 7 round down, so 32c would be rounded to 30c and 37c rounded to 35c) interpret a calculator display involving money and measurements when the number has been truncated (eg 1.2 on the calculator might relate to 1.2 m, 120 cm or $1.20) © Department of Education and Training Western Australia, Middle childhood: Mathematics/Number scope and sequence, December 2007 100 3 Organisation of content into year levels is advisory. Teachers will continue to make professional judgements about when to introduce content based on children’s prior learning and achievement. *National Consistency in Curriculum Outcomes, Statement of Learning – Mathematics Typical sequence of content: Year 4 Year 1 Year 2 identify the whole part and the Year 3 fractional part in measurements (eg 6.3 m has 6 whole metres and 3 10 of a metre) Year 5 Year 6 Year 7 Understand whole numbers and decimals (continued) identify the whole part and the fractional part in measurements* (eg 5.34 m has 5 whole metres and 34 100 of a metre) Standard and non-standard partitions of two-digit and three-digit numbers Standard and non-standard placevalue partitions of numbers into the thousands Standard and non-standard partitions of numbers into the thousands and decimals to two decimal places Standard and non-standard partitions of whole numbers and decimals to three decimal places that numbers can be partitioned in standard ways (eg 376 = 300 + 70 + 6) or non-standard ways (eg 36 = 30 + 6 but also 18 + 18 or 20 + 16 and so on) that numbers can be partitioned in standard ways* (eg 536 + 428 is 500 + 400 and 30 + 20 and 6 + 8) and non-standard ways* (eg 376 = 200 + 176 or 150 + 150 + 76 and so on) ways to partition (eg 4.26 + 5.23 is 4 + 5 and 0.2 + 0.2 and 0.06 + 0.03 which is 9 + 0.4 + 0.09 = 9.49) ways to partition whole numbers (eg 15 x 6 can be 10 x 6 and 5 x 6 or 30 x 3) Compare and order numbers to ten thousand Compare and order numbers to hundreds of thousands, including money and measures Additively and multiplicatively compare and order whole numbers into the millions Additively and multiplicatively compare and order whole numbers and decimals to 3 decimal places order numbers using place value (eg 76 is bigger than 67 because there is a 7 in the tens place) use place value to compare and order numbers* (eg 7 896 is bigger than 7 096 because there is an 8 in the hundreds place and 3.4 m is longer than 3.25 m because there is a 4 in the tenths place) use place value to order larger whole numbers in ascending or descending order (eg 25 296 is bigger than 24 987 because there is a 5 in the ten thousands place) use place value to order numbers of any size in ascending or descending order (eg 235 500 is larger than 234 500 because there is a 5 in the thousands place) © Department of Education and Training Western Australia, Middle childhood: Mathematics/Number scope and sequence, December 2007 4 Organisation of content into year levels is advisory. Teachers will continue to make professional judgements about when to introduce content based on children’s prior learning and achievement. *National Consistency in Curriculum Outcomes, Statement of Learning – Mathematics Typical sequence of content: Year 4 Year 1 Year 2 the Year 3 relative size of numbers Year 5 Year 6 Year 7 Understand whole numbers and decimals (continued) the relative size of larger numbers (eg 1 000 is ten times as big as 100) the constant relationship between the places, and that the positions increase in powers of 10 from right to left (eg 100 is a hundred times smaller than 10 000) the constant relationship between the places, and that the positions increase in powers of 10 from right to left (eg 300 is a hundred times larger than 3 and 1 000 times larger than 0.3) count forwards and backwards in tenths from any number, including amounts of money (eg $2.70, $2.80, $2.90 and 78c, 68c, 58c) count forwards and backwards in tens, hundreds and thousands from any number (eg 4 500, 4 400, 4 300) count forwards and backwards in tens, hundreds, thousands and ten thousands from any number (eg 32 000, 31 000, 30 000) order and compare numbers and numerical statements using = and ≠, < and > symbols (eg 386 > 300 + 50 + 9) rounding up and down to the nearest 10 or 100 for practical purposes rounding up and down to the nearest 100, 1 000 or 10th for practical purposes rounding up and down to the nearest 100, 1 000 or 10th for practical purposes rounding up and down to the nearest 100, 1 000, 10th or 100th for practical purposes (eg 2 is quite small compared with 200) different ways of counting a collection (eg by twos, fives, tens) © Department of Education and Training Western Australia, Middle childhood: Mathematics/Number scope and sequence, December 2007 5 Organisation of content into year levels is advisory. Teachers will continue to make professional judgements about when to introduce content based on children’s prior learning and achievement. *National Consistency in Curriculum Outcomes, Statement of Learning – Mathematics Typical sequence of content: Year 4 Year 5 Year 1 Year 2 Partition Year 3 objects, collections and quantities into equal parts and read, write and count with unit fractions to represent common fractions (eg 1 3 , 1 2 , 1 4 ) orally, in drawings, (eg number lines, shapes), models (eg counters, objects) and in written forms Year 6 Year 7 Understand fractions Simple fraction equivalences and comparisons using a range of models, including number lines Compare and find equivalences for simple fractions and key percentages using a range of models, including number lines Compare and find equivalences for simple fractions and key percentages using a range of models, including number lines read, write and say unit fractions using drawings, oral and written forms, in everyday contexts* (eg a quarter of an hour is fifteen minutes. This can be shown on the clock face) compare fractions using models, drawings and number lines (eg use read, write and order proper fraction strips to show 1 3 = 2 6 = 3 9 fractions (eg 1 ) improper fractions 3 ) (eg 7 5 ) and mixed numerals* (eg 3 3 ) 4 partition an object or collection into equal parts (eg cut a piece of ribbon into 12 pieces and give three people a quarter of the original length) partition an object or collection in a variety of ways to show equal parts* (eg to find a quarter of 20 cards, deal out 20 cards to 4 students) partition an object or collection in a variety of ways to show equal parts (eg three fifths of the class is girls, that is 0.6 or 60% of the class) find fractional amounts of a thing, quantity or collection of things (eg find 3 of a class of 28 students, 4 2 3 of a bag of marbles, 4 of a glass 10 of water) recognise simple equivalences of fractions through diagrams and models (eg fold a page in half and half again to see 2 = 1 ) 4 2 recognise simple equivalences of fractions to tenths through diagrams and models* (eg fold a page in half and half again to see 2 = 1 ) 4 2 recognise simple equivalences of fractions including mixed numerals and improper fractions through diagrams and models (eg colour 1 6 grids to see it equals 1 3 or 7 ) 8 4 4 recognise simple equivalences of fractions including mixed numerals and improper fractions through diagrams and models* (eg colour 2 3 grids to see it equals 11 ) 4 4 use terminology of unit fractions in context (eg a pie cut into six equal parts has one-sixth as its unit fraction) © Department of Education and Training Western Australia, Middle childhood: Mathematics/Number scope and sequence, December 2007 6 Organisation of content into year levels is advisory. Teachers will continue to make professional judgements about when to introduce content based on children’s prior learning and achievement. *National Consistency in Curriculum Outcomes, Statement of Learning – Mathematics Typical sequence of content: Year 4 Year 1 Year 2 half Year 3 of a half is one-quarter (eg show by cutting, folding, measuring, dealing out) order unit fractions using diagrams and models (eg ‘1 2 more than 1 of it’) 3 of this snake is Year 5 Year 6 Year 7 Understand fractions (continued) that parts of a whole can look different but still be equal in size (eg two halves do not have to be symmetrical to be half, but must have the same quantity) the size of fractions are relative to the whole (eg ‘ 1 of your pocket 3 money might be more or less than 1 3 of mine’) count in fractional amounts (eg one-third, two-thirds, one whole, one and one-third) count in fractional amounts (eg four-fifths, one whole, one and one-fifth or four-fifths, five-fifths, six fifths) count in fractional amounts (eg one and one-third, one and twothirds, two, two and one-thirds) order unit fractions using diagrams, order easily visualised fractions, with like and unlike denominators, by placing them on a number line, using order easily visualised fractions, with like and unlike denominators, by placing them on a number line, using number lines and models* (eg 1 3 of twelve counters is more than 1 ; on 4 a grid of two rows of five, show which is bigger, 7 or 3 ; order and 10 5 reference points such as 0, 1 and 1 2 (eg place 3 4 and and compare) 1 2 on a number line reference points such as 0, 1 and 1* 2 (eg place 3 4 and 6 10 on a number line and compare) locate mixed numbers such as 2, 2 1 , 3, 3 1 , 4 on a number line) 2 2 compare simple fractions and percentages based on multiples of 10% and 25% from a model or diagram (eg 25% of 2 litres equals 50% of 1 litre) © Department of Education and Training Western Australia, Middle childhood: Mathematics/Number scope and sequence, December 2007 7 Organisation of content into year levels is advisory. Teachers will continue to make professional judgements about when to introduce content based on children’s prior learning and achievement. *National Consistency in Curriculum Outcomes, Statement of Learning – Mathematics Typical sequence of content: Year 4 Year 1 Year 2 Year 3 Year 5 Year 6 Year 7 Understand fractions (continued) Simple rates rates are used in everyday situations (eg price per kilogram, cost per litre of petrol, cost per text message) Simple ratios Simple ratios ratios are used to describe straightforward scales and concentrations (eg cordial is made up of one part cordial to four parts water. The ratio is 1:4) ratios are used to describe straightforward scales and concentrations (eg in a Year 7 class of 14 girls and 8 boys, the girl/boy ratio is 14:8 or 7:4) Understand operations Represent number stories involving change, combine and compare situations, using diagrams and addition and subtraction number sentences Represent the full range of additive problems using diagrams as needed and addition and subtraction number sentences (equations) identifying the part or the whole as the unknown Represent the full range of additive problems using diagrams as needed and addition and subtraction number sentences (equations) identifying unknowns Use addition and subtraction equations (and diagrams as needed) to represent the full range of additive situations and situations that involve more than one problem type write number sentences using the terms change, combine and compare (eg ‘Combine Tim’s 14 marbles with Lisa’s 36 marbles and find the total’ is written as ‘14 + 36 = ’; ‘Lisa has 12 party hats and 15 friends, how many friends will not get a hat?’ is written as ’15 – 12 = ’) write number sentences* (eg ‘Ted began with 26 marbles and now has 12, how many did he lose?’ is written as ‘26 – = 12’; ‘Compare the height of two rose bushes, one is 95 cm tall, the other is 87 cm tall’ is written as ‘95 – 87 = cm’) identify and write number sentences for all types of problems (change, combine, equalise or compare situations), including those with larger whole numbers and where money, measurements and simple fractions are involved identify and write number sentences for problems, including contexts where the clues to assist in choosing the operation are not obvious, and where larger whole numbers, money, measurements and simple fractions are involved* © Department of Education and Training Western Australia, Middle childhood: Mathematics/Number scope and sequence, December 2007 8 Organisation of content into year levels is advisory. Teachers will continue to make professional judgements about when to introduce content based on children’s prior learning and achievement. *National Consistency in Curriculum Outcomes, Statement of Learning – Mathematics Typical sequence of content: Year 4 Year 1 Year 2 write Year 3 number stories to match given number sentences write (eg for ‘25 – 12 = ’ write ‘A farmer had 25 sheep and sold 12, how many are left?’) Year 5 Year 6 Year 7 Understand operations (continued) write number stories to match given number sentences write (eg for ‘58 + 12 – 7 = ’ write ‘Tom had 58 football cards and bought another packet of 12. He then gave 7 of his cards to his cousin, how many cards were left?’) Match the semantics (meaning) of addition and subtraction problem types with the numbers and symbols in their number sentences, identifying the unknowns Use the whole amount and the parts within number stories and in open number sentences to link and choose between addition and subtraction for two and three-digit numbers Use the inverse relationship between addition and subtraction for large whole numbers, money and familiar measures Use the inverse relationship between addition and subtraction for large whole numbers and decimals to two decimal places and readily visualised fractions Use the inverse relationship between addition and subtraction for large whole numbers and decimals to three decimal places and readily visualised fractions state subtraction facts from known addition facts (eg 6 + 8 = 14 therefore 14 – 8 = 6 and 14 – 6 = 8) use the inverse relationship to restate the problem* (eg 36 + 24 = 60 therefore 60 – 24 = 36 and 60 – 36 = 24) use the inverse relationship to restate the problem (eg 3.8 + 4.6 = 8.4 therefore 8.4 – 4.6 = 3.8 and 8.4 – 4.6 = 3.8) use the inverse relationship to restate and simplify the problem* (eg for 13.06 – 9.82 = ‘I know 9.82 + 3.24 = 13.06 so 13.06 – 9.82 = 3.24’) solve missing number problems (eg 15 + 7 = 22 to work out 22 – = 15) use the inverse relationship for calculator work to find missing numbers* (eg for 18 + = 25, key in ‘25 – 18 = ’) use the inverse relationship for calculator work to find missing numbers (eg for 45 + = 63, key in ‘63 – 45 =’) use the inverse relationship for calculator work to find missing numbers (eg for 45 + = 63, key in ‘63 – 45 =’) © Department of Education and Training Western Australia, Middle childhood: Mathematics/Number scope and sequence, December 2007 9 Organisation of content into year levels is advisory. Teachers will continue to make professional judgements about when to introduce content based on children’s prior learning and achievement. *National Consistency in Curriculum Outcomes, Statement of Learning – Mathematics Typical sequence of content: Year 4 Year 1 Year 2 work Year 3 out the unknown Year 5 Year 6 Year 7 Understand operations (continued) (eg 40 + 60 = 100 so 40 + 70 is ten more, 110) Represent number stories involving equal groups, sharing, simple rates, arrays, combinations and ratio comparison using materials, drawing a picture or acting out, and using addition, subtraction, multiplication and division number sentences Represent the full range of multiplicative problems using diagrams as needed and multiplication and division number sentences (equations) Represent the full range of multiplicative problems using diagrams as needed and multiplication and division number sentences (equations) identifying unknowns Use multiplication and division equations (and diagrams as needed) to represent the full range of multiplicative situations and situations that involve more than one problem type multiplication can be used for: o repeating equal quantities (eg three and three oranges is six, which is the same as two lots of three oranges) o rates (eg ‘If velvet ribbon costs $1 a metre, how much does four metres of ribbon cost?’) o arrays (eg ‘If there are five rows each with six books, how many books in total?’) o simple combination problems (eg ‘How many different outfits do I have for the holiday if I pack three tops and two skirts?) multiplication can be used for: o repeating equal quantities* (eg eight and eight caravans is six, which is the same as two lots of eight caravans) o rates (eg ‘If ice-cream costs $4 for 4 litres, how much would 20 litres of ice-cream cost?’) o arrays (eg ‘If there are eight rows each with nine cars, how many cars are in the car park in total?’) o simple combination problems (eg ‘How many different pizza topping combinations can I have if I can use three different meats and three vegetables?’) multiplication and division can be used for repeated equal quantities, ratio comparisons, arrays, scales and combinations, and products of measures (eg ‘What is the area of a rectangle 12 cm by 14 cm?’) multiplication and division can be used for repeated equal quantities, ratio comparisons, arrays, scales and combinations, and products of measures (eg ‘If the area of a rectangle is 50 m2, what are the lengths of the sides if the length is twice the width?’) © Department of Education and Training Western Australia, Middle childhood: Mathematics/Number scope and sequence, December 2007 10 Organisation of content into year levels is advisory. Teachers will continue to make professional judgements about when to introduce content based on children’s prior learning and achievement. *National Consistency in Curriculum Outcomes, Statement of Learning – Mathematics Typical sequence of content: Year 4 Year 1 Year 2 division can be used for sharing or Year 3 partitioning a collection, quantity or object (eg Tom arranged his 15 trophies evenly on three shelves, how many were on each shelf?) Year 5 Year 6 Year 7 Understand operations (continued) division can be used for: o arrays (eg ‘Amy has 45 cabbages in five equal rows, how many in a row?’) o ratios (eg ‘Lee has 28 marbles which is four times the amount Pete owns. How many marbles does Pete have?) the relationship between fractions, decimals and division (eg enter into the calculator by keying in ‘3 4 = ’) 3 4 the relationship between fractions, decimals, percentages and division* (eg enter 7 = 0.7 = 70%) 10 for division the result may take many forms, depending upon the context of the problem (eg 18 4 can be represented as: o 4 1 units each 2 o each ribbon is 4.5 m long o $18 4 = $4.50 o 18 people with 4 in each car will need 5 cars) © Department of Education and Training Western Australia, Middle childhood: Mathematics/Number scope and sequence, December 2007 11 Organisation of content into year levels is advisory. Teachers will continue to make professional judgements about when to introduce content based on children’s prior learning and achievement. *National Consistency in Curriculum Outcomes, Statement of Learning – Mathematics Typical sequence of content: Year 4 Year 1 Year 2 Match Year 3 the semantics of multiplication and division problem types with the numbers and symbols in number sentences, identifying the unknowns select appropriate operations for number sentences (eg ‘Sam bought 4 lollipops costing $1.20 each. How much change did he get from $5?’ requires multiplication and subtraction) Year 5 Year 6 Year 7 Understand operations (continued) Match the semantics of multiplication and division problem types with the numbers and symbols in number sentences, identifying the unknowns The order of operations The order of operations select appropriate operations for number sentences (eg ‘Jade, Katie and Mark have a combined weight of 100 kg. Mark is 5 kg heavier than Katie who weighs 30 kg. What is Jade’s weight?’ requires addition and subtraction) the acronym BIMDAS represents the order: o Brackets o Indices o Multiplication and o Division (in the order they appear) o Addition and o Subtraction (in the order they appear) (eg use simple examples such as ‘26 – 5 x 4 = 26 – [5 x 4] = 26 – 20 = 6’) the acronym BIMDAS represents the order: o Brackets o Indices o Multiplication and o Division (in the order they appear) o Addition and o Subtraction (in the order they appear)* (eg ‘45 – 36 + 8 x 5.8 = 45 – 36 + [8 x 5.8] = 45 – 36 + 46.4 = 55.4) © Department of Education and Training Western Australia, Middle childhood: Mathematics/Number scope and sequence, December 2007 12 Organisation of content into year levels is advisory. Teachers will continue to make professional judgements about when to introduce content based on children’s prior learning and achievement. *National Consistency in Curriculum Outcomes, Statement of Learning – Mathematics Typical sequence of content: Year 4 Year 1 Year 2 Use the Year 3 size of the groups, the number of groups and the whole amount within number stories to write and link multiplication and division number sentences for small quantities remainders in division can be left or shared into parts of a whole (eg when 13 apples are shared between 4, each person receives 3 whole apples and 1 apple is left) Year 5 Year 6 Year 7 Understand operations (continued) Use the size of the groups, the number of groups and the whole amount within number stories, and in open number sentences, to link and choose between multiplication and division for whole numbers Use the size of the groups, the number of groups and the whole amount within number stories and in open number sentences to link and choose between multiplication and division number sentences for whole numbers, money and familiar measures Use the inverse relationship between multiplication and division for whole and decimal numbers remainders in division can be left or shared into parts of a whole (eg when 13 chalks are shared between four each person receives three whole sticks of chalk and a quarter piece each) remainders in division can be shared into parts of a whole which is a fraction of the divisor remainders in division can be represented in decimal form (eg 29 ÷ 4 = 7.25) (eg 27 4 = 6 3 ) 4 the numbers in a multiplication can refer to the number of groups or the number of objects in each the inverse relationship (eg 3 x 8 = 24, so 24 ÷ 8 = 3 and 24 3 = 8) the inverse relationship* (eg 3 x 8 = 24, so 24 8 = 3 and 24 3 = 8) use the inverse relationship to solve missing number problems (eg 5 x 8 = 40 to work out 40 = 8) use the inverse relationship to solve missing number problems* (eg 7 x 8 = 56 to work out 56 = 8) © Department of Education and Training Western Australia, Middle childhood: Mathematics/Number scope and sequence, December 2007 13 Organisation of content into year levels is advisory. Teachers will continue to make professional judgements about when to introduce content based on children’s prior learning and achievement. *National Consistency in Curriculum Outcomes, Statement of Learning – Mathematics Typical sequence of content: Year 4 Year 1 Year 2 Identify Year 3 different equal groupings Year 5 Year 6 Year 7 Understand operations (continued) within whole numbers and represent as arrays and multiplication and division number sentences multiplication can be performed in any order, without altering the answer (eg use materials to demonstrate that 3 x 4 results in the same quantity as 4 x 3) collections may be arranged into equal sized groups (eg students arrange 24 blocks into arrays: 1 row of 24, 24 rows of 1, 2 rows of 12, 12 rows of 2, 3 rows of 8, 8 rows of 3, 4 rows of 6, 6 rows of 4) identify ‘multiples’, ‘products’ and ‘factors’ (eg 12 is a multiple of 4, 12 is the product of 3 x 4 and 3 and 4 are factors of 12) identify ‘prime’ numbers (eg prime numbers to 20 are 2, 3, 5, 7, 11, 13, 17 and 19) and determine that they have exactly two factors* (eg the factors of 5 are 1 and 5; the factors of 17 are 1 and 17) © Department of Education and Training Western Australia, Middle childhood: Mathematics/Number scope and sequence, December 2007 14 Organisation of content into year levels is advisory. Teachers will continue to make professional judgements about when to introduce content based on children’s prior learning and achievement. *National Consistency in Curriculum Outcomes, Statement of Learning – Mathematics Typical sequence of content: Year 4 Year 1 Year 2 Strategies to mentally add and Year 3 Year 5 Year 6 Calculate subtract to 100, drawing on remembered facts Strategies to mentally add and subtract numbers to 100 and extension of remembered addition facts into the hundreds change a subtraction into an addition (eg 13 – 8 can be thought of as 8 + = 13) change a subtraction into an addition* (eg 56 – 23 can be thought of as 23 + = 56) use standard partitioning (eg 24 + 37 = [20 + 30] + [4 + 7] = 50 + 11 = 61; for 34 – 27 recall 30 – 27 = 3 then add back the 4) Year 7 Strategies to mentally add and subtract two-digit numbers Strategies to mentally add and subtract two-digit numbers use known basic facts in all operations to work out those they do not remember ‘bridge the tens’ (eg 9 + 4 can be thought of as ‘one more to make 10’, so it becomes 10 + 3) ‘bridge the tens and/or hundreds’ (eg 98 + 14 can be thought of as ‘two more to make 100’, so it becomes 100 + 12) ‘bridge the tens and/or hundreds’* (eg 89 + 27 can be thought of as 90 + 26 and 68 – 34 can be thought of as 70 – 36) use standard partitioning (eg 24 + 37 = [20 + 30] + [4 + 7]) and non-standard partitioning* (eg 39 – 17 = 19 – 17 + 20) use standard partitioning (eg 24 + 37 = [20 + 30] + [4 + 7]) and non-standard partitioning (eg 39 – 17 = 19 – 17 + 20) use standard and non-standard partitioning* (eg 36 + 59 = 30 + 50 + 10 + 5 = 95) use standard partitioning for decimals (eg 6.4 + 7.3 = 6 + 7 + 0.4 + 0.3 = 13 + 0.7 = 13.7) extend basic addition and subtraction facts using patterns (eg 7 + 2 = 9 so 70 + 20 = 90) extend basic addition and subtraction facts using patterns (eg 5 + 4 = 9 so 50 + 40 = 90 and 90 – 40 = 50) extend basic addition and subtraction facts using patterns (eg 5 + 7 = 12 so 50 + 70 is 120 and 120 – 50 = 70) © Department of Education and Training Western Australia, Middle childhood: Mathematics/Number scope and sequence, December 2007 15 Organisation of content into year levels is advisory. Teachers will continue to make professional judgements about when to introduce content based on children’s prior learning and achievement. *National Consistency in Curriculum Outcomes, Statement of Learning – Mathematics Typical sequence of content: Year 4 Year 1 Year 2 use Year 3 doubles and near doubles (eg 8 + 14 = 10 + 10 + 2 = 20 plus 2 more so 32) Year 5 Year 6 Year 7 Calculate (continued) use doubles and near doubles (eg 15 + 17 = 15 + 15 + 2 = 30 plus 2 more so 32) rearrange the order to use compatible numbers (eg look for tidy sums when adding 6 + 7 + 5 + 4 + 3 thus add 6 + 4 then 7 + 3 and finally add 5) rearrange the order to use compatible numbers* (eg look for tidy sums when adding 68 + 27 + 12 = 68 + 12 + 27 = 80 + 27 = 107) take some from one number to give to another (eg 9 + 5 is the same as 10 + 4) take some from one number to give to another* (eg 38 + 7 is the same as 40 + 5) change the numbers by adding or subtracting the same amount (eg change ‘32 – 17’ to ‘35 – 20’ or ‘30 – 15’) change the numbers by adding or subtracting the same amount* (eg change ‘32 – 17’ to ‘35 – 20’ or ‘30 – 15’) start with the biggest place (front loading) (eg 26 + 35 is 36, 46, 56 and 5 more; 20 + 30 and 6 + 5, so 50 + 11 is 61) start with the biggest place (front loading)* (eg 26 + 35 is 36, 46, 56 and 5 more; 20 + 30 and 6 + 5, so 50 + 11 is 61) use known addition and subtraction facts to work out those they do know (eg find 7 + 9 by relating it to 8 + 8) use known addition and subtraction facts to work out those they do know (eg find 7 + 9 by relating it to 8 + 8) adding in tens to solve (eg 24 + 32 = 24, 34, 44, 54 add 2 = 56) adding in tens, twenties and thirties, hundreds etc from any starting point (eg 24, 44, 64 …) rearrange the order to use compatible numbers (eg look for tidy sums when adding 26 + 34 + 74 thus adding 26 + 74 to get 100 then add 34 to get 134) rearrange the order to use compatible numbers* (eg look for tidy sums when adding 26 + 34 + 74 thus adding 26 + 74 to get 100 then add 34 to get 134) describe methods used for mentally adding and subtracting in a way that shows they understand why it works describe methods used for calculating mentally in a way that shows they understand why it works © Department of Education and Training Western Australia, Middle childhood: Mathematics/Number scope and sequence, December 2007 16 Organisation of content into year levels is advisory. Teachers will continue to make professional judgements about when to introduce content based on children’s prior learning and achievement. *National Consistency in Curriculum Outcomes, Statement of Learning – Mathematics Typical sequence of content: Year 4 Year 1 Year 2 Strategies to mentally multiply and Year 3 divide single digit numbers and remember some multiplication facts observe patterns in multiplication tables (eg repeating final digit patterns): o 0, 2, 4, 6, 8, 10, 12 for the 2 x table o 0, 5, 0, 5, 0, 5 for the 5 x table o the 9 x table has decreasing digits in the ones column because you add 10 and take 1 make links between the tables (eg the 4 x table is double the 2 x tables plus the 8 x table is double the 4 x table) Year 5 Year 6 Year 7 Calculate (continued) Strategies to mentally multiply and divide single digit numbers and remember most multiplication facts Strategies to mentally multiply and divide larger numbers related to basic facts change a division into a multiplication* (eg 24 6 can be thought of as ‘How many sixes make 24?’) change a division into a multiplication (eg 63 9 can be thought of as ‘How many nines make 63?’) make links between the tables (eg the 6 x table is double the 3 x tables) Strategies to mentally multiply and divide larger numbers rounding a number and adjusting it (eg 6 x 9 is 6 tens take 6 ones) rounding a number and adjusting it (eg 99 x 7 is 100 times 7 take 7) doubling and halving (eg 4 x 6 = [2 x 6] x 2) doubling and halving (eg 35 x 12 is the same as 70 x 6 so 420) take a factor from one to give to another (eg 15 x 6 is 15 x 2 x 3, which is 30 x 3 = 90) use factors of numbers, including prime factors to assist in mental computation* (eg 72 = 6 x 12 = 2 x 3 x 2 x 6 = 2 x 3 x 2 x 2 x 3 = 23 x 32) change the numbers by multiplying or dividing by the same amount (eg 90 ÷ 5 is the same as 180 ÷ 10) change the numbers by multiplying or dividing by the same amount (eg 87 ÷ 5 is the same as 174 ÷ 10) © Department of Education and Training Western Australia, Middle childhood: Mathematics/Number scope and sequence, December 2007 17 Organisation of content into year levels is advisory. Teachers will continue to make professional judgements about when to introduce content based on children’s prior learning and achievement. *National Consistency in Curriculum Outcomes, Statement of Learning – Mathematics Typical sequence of content: Year 4 Year 5 Year 6 Year 1 Year 2 Year 3 Year 7 Calculate (continued) place value partitions (eg 6 x 25 is 6 x 20 add 6 x 5, so 120 + 30) place value partitions (eg 6 x 25 is 6 x 20 add 6 x 5, so 120 + 30) use known multiplication facts to work out unknown multiplication facts (eg 5 x 8 = 40, so 6 x 8 is 40 + 8 = 48) use known multiplication facts to work out unknown multiplication facts (eg 5 x 8 = 40, so 6 x 8 is 40 + 8 = 48) use known multiplication facts to work out unknown multiplication facts (eg 5 x 8 = 40, so 6 x 8 is 40 + 8 = 48) multiply and divide numbers by 10 and 100 and display changes using models such as place value charts* (eg multiply 1.5 repeatedly by 10 and colour in the 1 000s chart) extend basic multiplication facts to multiply and divide numbers by multiples of 10 and 100 (eg 3 x 4 = 12 so 1 200 ÷ 3 = 400) extend basic multiplication facts to multiply and divide numbers by multiples of 10 and 100* (eg 27 x 1 000 = 27 x 10 x 10 x 10 = 27 x 103) Use counting by unit fractions to mentally add and subtract orally presented fractions with like denominators Use counting by unit fractions to mentally add and subtract orally presented fractions with like denominators Strategies to mentally add and subtract readily visualised fractions and find a unit fraction of a whole number Strategies to mentally add and subtract readily visualised fractions and find a unit fraction of a whole number count forwards or backwards (eg for count forwards or backwards* use known number facts add and subtract fractions with like 1 4 + 3 4 find 1) = count on 2 4 , 3 4 , 4 4 to (eg for ‘I took home 5 8 7 8 of a cake and was eaten, how much was left? (eg for 3 8 + 7 8 10 eighths = = [3 + 7] eighths = 12 8 or 11 4 ) denominators (eg solve 2 1 + 1 1 by 3 3 visualising) count back 7 , 6 , 5 , 4 , 3 and 2 ) 8 8 8 8 8 8 © Department of Education and Training Western Australia, Middle childhood: Mathematics/Number scope and sequence, December 2007 18 Organisation of content into year levels is advisory. Teachers will continue to make professional judgements about when to introduce content based on children’s prior learning and achievement. *National Consistency in Curriculum Outcomes, Statement of Learning – Mathematics Typical sequence of content: Year 4 Year 1 Year 2 Year 3 Year 5 Year 6 Year 7 Calculate (continued) use number lines, arrays and fraction walls to perform mental calculations with common fractions* (eg use a fraction wall to answer 1 2 + 1 4 = ) find a unit fraction of a whole number by dividing the whole number by the amount of groups (eg find 1 3 1 3 of 18, divide 18 by 3, find a unit fraction of a whole number by dividing the whole number by the amount of groups (eg to find a quarter of 24, divide 24 by 4, therefore 1 equals 6) 4 of 18 = 6) Add and subtract whole numbers into the 100s and money using their own and pencil methods Add and subtract whole numbers into the 100s, money and simple measures using their own and pencil methods and informal written methods for readily visualised fractions Add and subtract whole numbers, money and measures using their own methods or conventional algorithms and informal written methods for readily visualised fractions Add and subtract whole numbers, money and measures using their own methods or conventional algorithms and informal written methods for readily visualised fractions informal paper-and-pencil methods (eg diagrams, jottings) when problems go beyond mental calculation capabilities (eg 128 + 379 = ) informal and formal paper-and-pencil strategies* (eg diagrams, jottings, algorithms) when problems go beyond mental calculation capabilities (eg 6.83 m – 1.25 m = ) use own methods or conventional algorithms to add and subtract larger whole numbers and decimals, where numbers go beyond mental capabilities use own methods or conventional algorithms to add and subtract larger whole numbers and decimals, where numbers go beyond mental capabilities* everyday language to explain strategies used to solve a problem or number sentence problem or number sentence everyday language to explain strategies used to solve a problem or number sentence problem or number sentence* © Department of Education and Training Western Australia, Middle childhood: Mathematics/Number scope and sequence, December 2007 19 Organisation of content into year levels is advisory. Teachers will continue to make professional judgements about when to introduce content based on children’s prior learning and achievement. *National Consistency in Curriculum Outcomes, Statement of Learning – Mathematics Typical sequence of content: Year 4 Year 1 Year 2 use Year 3 standard partitioning to add and subtract two-digit whole numbers without regrouping (eg 42 + 37 can be written as 40 + 30 = 70 and 2 + 7 = 9, so 79, and recorded with words, numbers and/or signs) Year 5 Year 6 Year 7 Calculate (continued) use standard partitioning, into the hundreds, to add and subtract twodigit whole numbers with and without regrouping (eg 54 – 28 can be written as (40 – 20) + (14 – 8) = 20 + 6 = 26) partition larger numbers and decimals to add and subtract with and without regrouping (eg 6.5 – 2.8 can be written as (5 – 2) and [1.5 – 0.8] = 3 + 0.7 = 3.7) partition larger numbers and decimals to add and subtract with and without regrouping (eg 6.5 – 2.8 can be written as [5 – 2] and [1.5 – 0.8] = 3 + 0.7 = 3.7) add and subtract fractions with unlike denominators by writing the fractions in different ways add and subtract fractions with unlike denominators by writing the (eg 1 = 2 so 1 2 4 2 2 1 3 + = ) 4 4 4 + 1 4 can be fractions in different ways* (eg 1 = 2 4 so 1 2 + 1 4 can be 2 4 + 1 4 = 2 3 ) 4 Informal written strategies to multiply and divide, interpreting remainders appropriately Informal written strategies to multiply and divide, interpreting remainders appropriately Informal written strategies to multiply and divide whole numbers, money and measures (multipliers and divisors to at least 10) Use own methods or conventional algorithms to multiply and divide whole numbers, money and measures factorise numbers and represent them in different ways (eg use 12 blocks to find the arrays 12 x 1, 2 x 6, 3 x 4 and 4 x 3) factorise numbers and represent them in different ways* (eg 12 = 2 x 6 and 3 x 4, 12 2 = 6 and 12 4 = 3) factorise products to represent them in different ways (eg 64 = 2 x 32, 8 x 8, 4 x 16, 2 x 2 x 16) rewrite larger whole numbers as a product of factors, including prime factors (eg factor trees) express the ‘left over’ after a division as a remainder (eg 23 ÷ 7 = 3 r 2) express the remainder after division as a remainder or a fraction express the remainder after division as a remainder or a fraction of the express the remainder after division as a remainder or a fraction of the (eg 23 ÷ 4 = 5 r 3 or 5 3 ) 4 divisor (eg 53 ÷ 4 = 13 1 or 13.25) 4 divisor (eg 73 ÷ 8 = 9 r 1 or 9 1 or 8 9.125) © Department of Education and Training Western Australia, Middle childhood: Mathematics/Number scope and sequence, December 2007 20 Organisation of content into year levels is advisory. Teachers will continue to make professional judgements about when to introduce content based on children’s prior learning and achievement. *National Consistency in Curriculum Outcomes, Statement of Learning – Mathematics Typical sequence of content: Year 4 Year 1 Year 2 Year 3 Year 5 Year 6 Year 7 use place value to multiply and divide a two-digit whole number by a single-digit whole number (eg 24 x 6 equals (20 x 6) + (4 x 6) = 120 + 24 = 144) use place value to multiply and divide a two-digit whole number by a single-digit whole number (eg 145 x 4 equals (100 x 4) + (40 x 4) + (5 x 4) = 400 + 160 + 20 = 580) Calculate (continued) interpret remainders in the context of the question (eg 23 children are to be driven in cars to the concert. If only 4 children can fit in each car, how many cars are needed? 23 ÷ 4 = 5 r 3, so 6 cars are needed) multiply and divide larger whole numbers and decimals by a single or double digit number, where numbers go beyond mental capabilities* how to find the fraction of a number using multiples (eg find saying ‘6 x 5 = 30 so be 6’) 1 5 1 5 of 30 by of 30 must how to find the fraction of a number using multiples* (eg find 3 of 40 by saying 5 ‘8 x 5 = 40 so 1 of 40 must be 8 and 5 3 5 of 40 is 24’ and ‘the number of one-eighth slices in three pizzas is 24’) © Department of Education and Training Western Australia, Middle childhood: Mathematics/Number scope and sequence, December 2007 21 Organisation of content into year levels is advisory. Teachers will continue to make professional judgements about when to introduce content based on children’s prior learning and achievement. *National Consistency in Curriculum Outcomes, Statement of Learning – Mathematics Typical sequence of content: Year 4 Year 1 Year 2 Year 3 Year 5 Year 6 Year 7 Calculate (continued) compare simple fractions from a model or diagram (eg 1 2 > 1 3 > 1 4 ) equivalence between commonly used fractions and decimal fractions (eg 1 = 0.5, 1 = 0.2, 3 = 0.75) 2 5 4 compare simple fractions and percentages based on multiples of 10% and 25% from a model or diagram* (eg 25% of 2 litres equals 50% of 1 litre) equivalence between common fractions, decimal fractions and percentages* (eg 1 = 0.5 = 50%, 2 1 5 = 0.2 = 20%, 3 4 = 0.75 = 75%) use a number line or materials to solve practical problems involving negative numbers* (eg show that an overnight change in temperature from 5˚C to –7˚C is a drop of 12 °C) use rates to calculate cost given price and quantity (eg 9 packets @ $1.25) use rates to calculate: o cost given price and quantity (eg 34 packets @ $1.25) o speed given distance and time* (eg 120 km covered in 2 hours is 60 km/h) simplify ratios (eg 4:6 = 2:3, simplify ratios* (eg 6:8 = 3:4, 1 2 :2 = 1:4) 1 2 :4 = 1:8) © Department of Education and Training Western Australia, Middle childhood: Mathematics/Number scope and sequence, December 2007 22 Organisation of content into year levels is advisory. Teachers will continue to make professional judgements about when to introduce content based on children’s prior learning and achievement. *National Consistency in Curriculum Outcomes, Statement of Learning – Mathematics Typical sequence of content: Year 4 Year 1 Year 2 Year 3 Year 5 Year 6 Year 7 Calculate (continued) use ratio to compare quantities of the same type (eg in a Year 7 class of 14 girls and 8 boys, the girl/boy ratio is 14:8 or 7:4) use ratio to compare quantities of the same type* (eg in a Year 7 class of 14 girls and 8 boys, the girl/boy ratio is 14:8 or 7:4) calculate proportions of a given ratio* (eg change a recipe for four people to cater for two) Use a calculator for all four operations Use a calculator for all four operations Use calculator functions for all four operations and memory, interpreting decimal quotients appropriately Use calculator functions for all four operations and memory, interpreting decimal quotients appropriately, converting between fractions, decimals and percentages the efficiency of using the x key instead of repeated addition, and the ÷ key instead of repeated subtraction on a calculator use a calculator for whole and decimal numbers, where numbers go beyond mental capabilities use a calculator for whole and decimal numbers and fractions where numbers go beyond mental capabilities use a calculator for whole and decimal numbers and fractions, where numbers go beyond mental capabilities* multiplication can be entered on a calculator in a different order and still have the same result (eg 4 x 5 = 5 x 4) calculators are the sensible choice for laborious or repetitive computations calculators are the sensible choice for laborious or repetitive computations use calculators to carry out more complex or repetitive computations with attention to order of operations* (eg include GST and fixed packaging and postage costs for a list of items from a catalogue to calculate total cost for several orders) © Department of Education and Training Western Australia, Middle childhood: Mathematics/Number scope and sequence, December 2007 23 Organisation of content into year levels is advisory. Teachers will continue to make professional judgements about when to introduce content based on children’s prior learning and achievement. *National Consistency in Curriculum Outcomes, Statement of Learning – Mathematics Typical sequence of content: Year 4 Year 1 Year 2 Year 3 Year 5 Year 6 Year 7 Calculate (continued) use calculator functions to find the solution to a sequence of calculations (eg enter (2.75 x 35) + (0.54 x 27)) division must be entered on a calculator in the correct order use the context to interpret calculator answers (eg 5 children share $37, the display shows 7.4, interpret as $7.40) use the context to interpret calculator answers (eg 5 children share $37, the display shows 7.4, interpret as $7.40) Use number lines and rounding to check calculations Use appropriate strategies to check calculations Use appropriate strategies to check calculations Use appropriate strategies to check calculations check a calculation by doing it in a different way (eg check the result of an addition by adding the numbers in a different order) check a calculation by doing it in a different way* (eg check the result of addition by subtracting an addend from the total) check a calculation by doing it in a different way (eg check a product by repeatedly subtracting one of the original factors) check a calculation by doing it in a different way (eg check a product by dividing it by one of the original factors) estimate and approximate the answers (eg 16 + 19 cannot be 25 because 15 + 15 is 30, so it has to be more than 30) estimate and approximate the answers* (eg 3 x 46 must be between 120 and 150 because 3 x 40 = 120 and 3 x 50 is 150) estimate and approximate the answers (eg 78 ÷19 ≈ 4 because 78 is about 80 and 19 is about 20) estimate and approximate the answers (eg 3 468 – 876 ≈ 2 400 because 3 468 is about 3 500 and 876 is about 900) round numbers (up or down depending upon the context) to the nearest 10 to approximate calculations (eg 47 + 8 will be about 50 + 10 so the answer should be just under 60) round numbers (up or down depending upon the context) to the nearest 5, 10 or 100 to approximate calculations* (eg 374 + 75 ≈ 500 when rounded to hundreds. The answer is less than 500 as both numbers were rounded up) when rounding whole numbers to the nearest 5, 10, 100 or 1 000, the same number may need to be rounded up or down (eg 4 381 to the nearest 10 goes down to 4 380, but to the nearest hundred rounds up to 4 400) round decimal numbers (up or down depending upon the context) to the nearest whole number or to the nearest one or two decimal places when approximating calculations* (eg 29 643 ÷ 87 will be approximately 30 000 ÷ 100 or 27 000 ÷ 90, that is around 300) © Department of Education and Training Western Australia, Middle childhood: Mathematics/Number scope and sequence, December 2007 24 Organisation of content into year levels is advisory. Teachers will continue to make professional judgements about when to introduce content based on children’s prior learning and achievement. *National Consistency in Curriculum Outcomes, Statement of Learning – Mathematics Typical sequence of content: Year 4 Year 1 Year 2 use Year 3 the properties of operations to check calculations (eg ‘I have taken a whole number from a larger whole number so the answer must be smaller than what I started with’) check the accuracy of results against the context of the problem (eg the winner of the race should have the shortest time for the race) Year 5 Year 6 Year 7 use the properties of operations to check calculations* (eg ‘If I multiplied by a number larger than one and got a smaller answer, it cannot be right’) use the properties of operations to check calculations (eg ‘If I multiplied by a number larger than one and got a smaller answer, it cannot be right’) use the properties of operations to check calculations (eg 0.2 x 0.3 must be less than 0.2 because it is multiplied by a number less than one) check the accuracy of results against the context of the problem* (eg Jack takes 20 seconds to run 100 m but he will probably take longer than 80 seconds [4 x 20] to run 400 m) check the accuracy of results against the context of the problem (eg ‘If I check the accuracy of results against the context of the problem* (eg the answer cannot be right if the average height worked out is greater than that of anybody in the group) Calculate (continued) am looking for 4 of 90, then the 5 answer has to be less than 90’) © Department of Education and Training Western Australia, Middle childhood: Mathematics/Number scope and sequence, December 2007 25