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Elaborations Number - Number concepts

By Allen Kennedy,2014-06-28 21:39
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Elaborations Number - Number concepts ...

The following elaborations are examples only of what students know and can do with what they know and should not be considered prescriptive or exhaustive.

    Foundation Level: Level statement

    Students are developing a notion of counting and an awareness of number and money. Number names are becoming more meaningful. Example learning outcomes:

    Students show an awareness of ‘more’, ‘less’ and ‘same’ in life situations.

    Elaborations To support investigations that emphasise thinking, reasoning and working mathematically Students know: Students may:

    ? participate in familiar songs and games that involve adding more to a collection and taking away one or some ? ‘more’ means adding

    some to a collection ? notice/request ‘more’ in familiar situations (e.g. serving food or drinks, sharing toys or stickers) ? ‘same’ means collections ? notice when one person is given ‘more’ of something than another person match ? indicate that one collection has ‘more’ when two collections have significantly different quantities ? everyday language that ? notice/request ‘same’ in familiar situations (e.g. serving food or drinks, sharing toys or stickers) relates to ‘more’ (adding ? indicate that two collections have the ‘same’ quantity by matching items one to one on) and ‘taking away’

    ? notice that when two collections are significantly different in quantity they are ‘not the same’ (items covered or

    removed from a ? indicate ‘take away’ by covering or removing objects from a collection in familiar situations. collection).

    ? The State of Queensland (Queensland Studies Authority) 2005 U 21

The following elaborations are examples only of what students know and can do with what they know and should not be considered prescriptive or exhaustive.

    Level 1: Level statement

    Students are developing a sense of number by knowing number names and counting in sequence. They recognise, compare, order and represent small whole numbers and use concrete

    materials to explore the concept of parts of a whole. They are developing an awareness of the cost of goods and recognise and represent notes and coins. Students identify and distinguish between situations that require them to add or subtract, to share equally or to create equal groups.

    Core learning outcome: N 1.2

    Students identify and solve addition and subtraction problems involving small whole numbers.

    Elaborations To support investigations that emphasise thinking, reasoning and working mathematically Core content Students know: Students may: Addition

    ? identify ways to solve problems ? totals to 10 ? addition involves joining two or

    more numbers to find a total ? explain the reasons for choosing an operation to solve a problem ? joining model ? subtraction involves taking one ? calculate solutions mentally ? language of joining quantity away from another ? calculate solutions using written methods ? two or more addends ? how to distinguish between ? identify the number that results when numbers are joined together or identify what is left after items are Subtraction situations that require addition covered up or subtraction of whole ? whole numbers to 10 ? explain the addition or subtraction strategy used to solve a problem numbers ? take away model ? describe what is happening to the numbers when adding or subtracting ? addition and subtraction are ? language of take away the inverse of each other ? explain that subtraction undoes addition and vice versa Connections ? check the reasonableness of answers using different computation strategies ? mental computation strategies ? inverse and computation methods for ? check answers using alternative computation methods or using the inverse operation solving addition and ? addition undoes subtraction ? use and explain different combinations of numbers that make the same total subtraction problems ? subtraction undoes addition ? pose problems involving addition and subtraction using everyday language ? how to solve addition and Mental computation strategies ? identify the number patterns or the skip counting sequence (e.g. counting in 2s, 3s) subtraction problems involving ? count on (in 1s, 2s) small whole numbers. ? relate patterns to the counting sequence to 100

    ? count back (in 1s, 2s) ? explore the use of known strategies and computation methods for extensions to larger numbers.

    Computation methods

    ? mental computations

    ? written recordings

    ? words for addition (add)

    ? words for subtraction (cover up,

    take away, left)

    ? calculators, computers

    ? symbols

    ? addition (+)

    ? subtraction () ? The State of Queensland (Queensland Studies Authority) 2005 U 22

The following elaborations are examples only of what students know and can do with what they know and should not be considered prescriptive or exhaustive.

    At each level, investigations should occur in a range of contexts. For example, students could investigate:

    ? number of items to be ordered for the school tuckshop

    ? number of squares that could be combined to make a patchwork quilt ? amount of ingredients added or removed from labelled containers ? ingredients that need to be added for a recipe

    ? musical games and rhymes involving numbers

    ? a set number of cards dealt in two or three rounds

    ? combinations of tokens or materials to be allocated for activities ? how to create work groups of a specified size

    ? payment for goods using whole dollars

    ? gardening activities, such as planting borders or edible gardens ? number patterns on a computer or calculator.

? The State of Queensland (Queensland Studies Authority) 2005 U 23

The following elaborations are examples only of what students know and can do with what they know and should not be considered prescriptive or exhaustive.

    Level 2: Level statement

    Students demonstrate their developing number sense by comparing, ordering and representing whole numbers to 999 and understanding that the value of a digit in a number determines its place value. They understand that a whole can be made up of equal parts and use concrete materials to represent halves and quarters. When using money to purchase goods, they tender

    different combinations of notes and coins.

    Students are beginning to recall or work out some addition, subtraction and multiplication number facts. They use a range of computation methods, including mental, written and calculator, to

    solve problems.

    Core learning outcome: N 2.2

    Students identify and solve addition and subtraction problems involving whole numbers, selecting from a range of computation methods, strategies and known number facts.

    Elaborations To support investigations that emphasise thinking, reasoning and working mathematically Core content Students know: Students may: Addition

    ? totals to 999 ? how to distinguish between ? identify whether problems encountered are solved by addition or subtraction and explain reasons for

    the decision situations that require addition ? two or more addends or subtraction of whole ? identify number patterns created when working out addition facts to 9 + 9 ? recall or work out addition facts numbers to 9 + 9 ? identify and explain how different strategies such as doubles and their inverses, near doubles and make ? addition and subtraction are to 10 could be used when calculating a solution Subtraction the inverse of each other ? solve addition and subtraction problems using identified strategies and computation methods ? whole numbers to 999 ? numbers can be added ? explain different ways of solving problems, including the turnaround strategy together in any order ? models and language ? represent a problem visually using a five or ten frame, number line, hundred board or student-generated ? take away ? addition and subtraction facts model ? missing addend ? mental computation strategies ? identify extensions of mental computation strategies and computation methods, such as make to 100 or and computation methods for ? comparison (difference) 1000, and explain the similarities and differences solving addition and ? recall or work out subtraction facts ? check the reasonableness of answers and justify reasoning subtraction problems Connections ? check answers using alternative computation methods or using the inverse operation ? how to solve addition and ? missing addend ? use and explain different combinations of numbers to make the same total subtraction problems involving

    whole numbers. ? inverse (backtracking) ? pose problems involving addition and subtraction using everyday language

    ? related addition and subtraction ? explore the use of known strategies and computation methods for extensions to larger numbers. facts

    Mental computation strategies

    ? to work out basic facts

    ? count on

    ? count back

    ? doubles

    ? near doubles

    ? make to 10

    ? turnarounds (commutativity)

    ? The State of Queensland (Queensland Studies Authority) 2005 U 24

The following elaborations are examples only of what students know and can do with what they know and should not be considered prescriptive or exhaustive.

    ? generalisations about addition and

    subtraction

    ? extension of strategies to larger

    numbers

    ? student-generated

    Computation methods

    ? mental computations

    ? written recordings

    ? student-generated

    ? traditional methods

    ? calculators, computers At each level, investigations should occur in a range of contexts. For example, students could investigate:

    ? number of students in the school or year group ordering burgers for a burger and free drink day

    ? people or objects required for a team

    ? groups of workers or players required for a game or activity

    ? the number of people or objects needed to leave a bus or lift for safety reasons

    ? symmetrical aspects of the natural or built environments (doubles or near doubles)

    ? combinations of art prints or objects in a ten frame

    ? lunch packages of 10 items

    ? children absent from the class group at particular times during the day

    ? regular savings for a particular purchase.

    ? The State of Queensland (Queensland Studies Authority) 2005 U 25

The following elaborations are examples only of what students know and can do with what they know and should not be considered prescriptive or exhaustive.

    Level 3: Level statement

    Students compare, order and represent whole numbers to 9 999, common and decimal fractions and recognise the value of each digit. They tender appropriate amounts of money for cash

    transactions and identify other methods of paying for goods and services.

    Students recall or work out all addition, subtraction and multiplication number facts and some division facts. They use a range of computation methods, including mental, written and

    calculator, to solve problems that involve whole numbers and decimal fractions in context.

    Core learning outcome: N 3.2

    Students identify and solve addition and subtraction problems involving whole numbers and decimal fractions in context, selecting from a range of computation methods, strategies and known number facts.

    Elaborations To support investigations that emphasise thinking, reasoning and working mathematically Core content Students know: Students may: Numeration

    ? identify whether problems encountered require addition and/or subtraction ? totals to 9 999 ? how to distinguish between

    situations that require addition ? explain whether exact or approximate answers are required ? decimals to 2 places in context with or subtraction of whole the same number of places ? decide on the most efficient method of performing the computation required to solve a problem (e.g. numbers and decimal fractions mental, written or calculator) ? recall addition facts to 9 + 9 ? how to use the inverse ? select from a range of computation methods and strategies to solve problems and give reasons for the Subtraction relationship between addition selection and subtraction to solve ? whole numbers to 9 999 ? make numbers manageable where necessary and explain the process e.g. make to nearest thousand, problems ? mental computations with money nearest one ? addition and subtraction facts (change) ? select from known number facts when calculating ? mental computation strategies ? recall subtraction facts ? explain the relationship between addition and subtraction facts and check computations using backtracking and computation methods for Connections (inverse relationship) solving addition and ? inverse (backtracking) subtraction problems involving ? pose addition or subtraction problems involving whole numbers and decimal fractions in context using

    whole numbers and decimal everyday language. ? related addition and subtraction fractions in context facts ? how to apply and interpret Mental computation strategies decimal fractions in context ? for larger numbers and decimal ? how to solve addition and fractions in context subtraction problems involving ? making numbers manageable whole numbers and decimal ? count on and back fractions in context. ? doubles

    ? changing operations

    ? turnarounds (commutativity)

    ? generalisations about addition and

    subtraction

    ? student-generated

    ? The State of Queensland (Queensland Studies Authority) 2005 U 26

The following elaborations are examples only of what students know and can do with what they know and should not be considered prescriptive or exhaustive.

    Computation methods

    ? mental computations

    ? exact

    ? approximate

    ? written recordings

    ? student-generated

    ? traditional methods

    ? calculators, computers At each level, investigations should occur in a range of contexts. For example, students could investigate:

    ? resources for class craft activities

    ? set and prop construction for drama performances

    ? purchasing goods directly or ordering from a catalogue within a set budget

    ? catering costs for a school or class activity

    ? measurement problems involving lengths of sides and boundaries

    ? shopping situations involving change

    ? travel distances using road maps.

    ? The State of Queensland (Queensland Studies Authority) 2005 U 27

The following elaborations are examples only of what students know and can do with what they know and should not be considered prescriptive or exhaustive.

    Level 4: Level statement

    Students compare and order whole numbers and common and decimal fractions. They identify fractions expressed in different ways and make connections between common fractions, decimal fractions and percentages. They identify a range of factors such as advertising, discounts and methods of payment that may influence financial decisions. Students recall all addition, subtraction, multiplication and division number facts. They use a range of computation methods to solve problems that involve whole numbers, common and

    decimal fractions, percentages and rates.

    Core learning outcome: N 4.2

    Students identify and solve addition and subtraction problems involving whole numbers and common and decimal fractions, selecting from a range of computation methods, strategies and known number facts.

    Elaborations To support investigations that emphasise thinking, reasoning and working mathematically Core content Students know: Students may: Addition and subtraction

    ? identify whether problems encountered require addition and/or subtraction ? whole numbers ? how to distinguish between

    situations involving any whole ? explain how to add and/or subtract common fractions ? common fractions (same numbers and common and denominators) ? explain how to add and/or subtract decimal fractions involving different numbers of decimal places such decimal fractions that require as hundredths and thousandths ? decimal fractions including different addition or subtraction numbers of decimal places ? explain whether exact or approximate answers are required ? how to use the inverse Connections ? decide on the most efficient method of performing the computation required to solve a problem relationship between addition (e.g. mental, written or calculator) and subtraction to solve ? inverse (backtracking) problems ? select from a range of computation methods and strategies to solve problems and give reasons for the Mental computation strategies selection ? addition and subtraction facts ? for whole numbers and decimal ? make numbers manageable where necessary and explain the process e.g. make to nearest thousand or ? mental computation strategies fractions nearest one and computation methods for ? making numbers manageable ? select from known number facts when calculating solving addition and ? count on and back subtraction problems involving ? use the relationship (inverse relationship) between addition and subtraction to check computations ? doubling whole numbers and common ? pose addition and subtraction problems involving whole numbers and common and decimal fractions. and decimal fractions ? changing operations ? how to solve addition and ? for common fractions subtraction problems involving ? generalisations about addition and whole numbers and common subtraction and decimal fractions. Computation methods

    ? mental computations

    ? exact

    ? approximate

    ? written recordings

    ? student-generated

    ? traditional methods

    ? calculators, computers

    ? The State of Queensland (Queensland Studies Authority) 2005 U 28

The following elaborations are examples only of what students know and can do with what they know and should not be considered prescriptive or exhaustive.

    At each level, investigations should occur in a range of contexts. For example, students could investigate: ? global population and production patterns

    ? total rainfalls for specific time periods

    ? tallying amounts of money or measurements

    ? catering costs for a function

    ? measurements of track and field events to determine places and margins

    ? distances to and between planets.

? The State of Queensland (Queensland Studies Authority) 2005 U 29

The following elaborations are examples only of what students know and can do with what they know and should not be considered prescriptive or exhaustive.

    Level 5: Level statement

    Students compare and order positive and negative integers and explain and record index notation. They interpret and use conventions for expressing rates and ratios. They identify methods of saving and investigate the factors affecting debit and credit transactions. They understand that the purchase of goods and services may attract fees or charges. Students use a range of computation methods to solve problems that involve positive rational numbers, rates, ratios and direct proportions.

    Core learning outcome: N 5.2

    Students identify and solve addition and subtraction problems involving positive rational numbers using a range of computation methods and strategies.

    Elaborations To support investigations that emphasise thinking, reasoning and working mathematically Core content Students know: Students may: Addition and subtraction

    ? explain whether an exact or approximate answer is required ? how to identify situations ? positive rational numbers involving positive rational ? whole numbers ? select and justify computation methods and strategies used to solve problems numbers that require addition ? decimal fractions ? explain how to solve problems involving rational numbers including index notation or subtraction or both ? common fractions ? pose and solve real-life problems involving addition and subtraction of positive rational numbers ? mental computation strategies ? related denominators ? decide if common or related denominators are required for common fractions and explain reasoning and computation methods for Connections solving addition and ? check reasonableness of solutions.

    subtraction problems involving ? inverse (backtracking) positive rational numbers Mental computation strategies ? common or related ? relevant to whole numbers, denominators are required for common fractions and decimal addition and subtraction of fractions common fractions ? generalisations about addition and ? how to add and subtract subtraction common fractions Computation methods ? how to solve addition and

    subtraction problems involving ? mental computations positive rational numbers. ? exact

    ? approximate

    ? written recordings

    ? student-generated

    ? traditional methods

    ? calculators, computers At each level, investigations should occur in a range of contexts. For example, students could investigate:

    ? financial transactions involving profit or personal bank balances

    ? budgeting for a range of personal needs

    ? mobile phone plans

    ? quantities required for catering for large numbers at school events

    ? distances covered during legs of car rallies.

    ? The State of Queensland (Queensland Studies Authority) 2005 U 30

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