Mathematics: Ages Six to Nine
Jump to sections:
2.MA.010 History of Mathematics
2.MA.020 Numbers to Ten
2.MA.030 Counting
Study of the Decimal System
2.MA.040 The Decimal System: introduction
2.MA.050 The Decimal System: operations with whole numbers
2.MA.060 Inter-relationships between Operations
Memorisation
2.MA.070 Memorisation: addition facts
2.MA.080 Memorisation: subtraction facts
2.MA.090 Memorisation: multiplication facts
2.MA.100 Memorisation: division facts
2.MA.110 Decanomial: Table of Pythagoras
2.MA.120 Multiples of numbers
Passage to abstraction (operations)
2.MA.130 Passage to Abstraction: addition and subtraction
2.MA.140 Decimal System Hierarchies: extension
2.MA.150 Passage to Abstraction: multiplication
2.MA.160 Passage to Abstraction: division
Fractions
2.MA.170 Introduction to Fractions
2.MA.180 Equivalent Fractions
2.MA.190 Operations with Fractions
Money
2.MA.200 Money
Mathematics Overview
The Children’s House provides children with a strong foundation on which to build the more advanced mathematical understandings they gain in Montessori environments prepared for children over the age of six. In the Children’s House children work with a variety of concrete materials embodying mathematical concepts. When the hand and the mind work together, young children are able to absorb concepts, sometimes without the need for words.
Each Montessori mathematics lesson, in both the Children’s House and in environments prepared for children over six, has two aims, one indirect and one direct. The direct aim is the immediate purpose that is obvious when the concrete materials are presented to the children. The indirect aim, often a more abstract aim, is achieved when children work with the concrete materials independently. The indirect aim prepares children for more advanced concepts they will meet later in the curriculum. In the Montessori view, when children work first with concrete materials and move at their own pace toward abstraction with paper and pencil, or towards mental arithmetic, they internalise concepts at a deeper, more lasting level than if they had memorised them by rote.
Beyond the age of six children continue to learn mathematical principles through the manipulation of concrete materials but they do not want to be tied to concrete materials and strive towards the freedom of working at the abstract level. The concrete materials become stepping-stones only, keys to open the door to abstraction whenever a child is ready. By generalising from their experience with the concrete materials, children over the age of six are able to work out mathematical formulae and definitions for themselves. Montessori teachers do not offer help too early but wait until children have had a chance to work with a problem and come up with a few solutions for themselves. Children also encounter problems to use in their independent work in the form of command cards prepared for all areas of the mathematics curriculum.
In the study of mathematics at this age, children use quantities with precision. They also reason using logic and abstract patterning based on observation and imagination. Children work with measurements, patterns, sequences and mathematical relationships, applying these concepts to practical projects. They learn that the creative potential of mathematics is as great as its potential to order and to provide understanding. In the Montessori view, constructing a ‘mathematical mind’ in this way is the birthright of all children.
The Montessori mathematics curriculum extends children aged from six to twelve further than is normally expected of children at this age and stage. For example, from the age of six children, explore and practise operations with the culture’s system of numeration, the decimal system. Once children have mastered this system, usually by the age of nine, they are ready to examine other number systems, with bases other than ten, from both a mathematical and a historical perspective. Children may even perform operations in number systems with bases other than ten. Extension activities of this type allow children to exercise their mental capacities as well as consolidate and reinforce their existing understanding.
As children progress through the Montessori mathematics curriculum, they learn to make connections between concepts. For example, they learn to apply compound multiplication to fractions and decimals, or progress from the four operations of addition, subtraction, multiplication and division to squaring and cubing, and to solving square roots and cube roots.
History of Mathematics 2.MA.010
Knowledge, Skills and Understanding
Typically, children will:
.01 Explore stories about the history of numbers and mathematics
.02 Research ancient number systems
First Nations Australians
Babylonian
Egyptian
Mayan
Chinese
Hindu-Arabic
Roman
.03 Explore the history of the number system used in our culture: the decimal system
＋ Materials and Activity
Activities include:
- Great lesson: the Story of Numbers
- independent research
- creating charts, models and timelines
- counting and calculating using earlier number systems eg Roman numerals.
Resources include:
- charts, card material and artefacts
- research and reference materials (paper-based, digital, web-based, multimedia).
Numbers to Ten 2.MA.020
Knowledge, Skills and Understanding
Typically, children will:
.01 Consolidate knowledge of numbers to ten
.02 Recognise odd and even numbers
＋ Materials and Activity
Activities include:
- individual and small group lessons and exercises to introduce, consolidate and/or review and assess knowledge of quantities and symbols as needed
- individual tutorials for students needing to review and practise basic knowledge
- memory game of numbers
- odds and evens lesson.
Resources include:
- concrete material and corresponding symbols, including number rods (1-10), spindle boxes (0-9), number cards and counters (1-10), stair of colour-coded bead bars (1-10), snake game
- basket with numbers (0-10) written on separate paper squares folded to hide the number.
＋ Links to Australian Curriculum (ACv9)
Counting 2.MA.030
Knowledge, Skills and Understanding
Typically, children will:
.01 Recognise, represent and order numbers 11-20
.02 Recognise, represent and order numbers 1-100
.03 Recognise, represent and order numbers 1-1000
＋ Materials and Activity
Activities include:
- individual and small group lessons and exercises to introduce, consolidate and/or review and assess counting knowledge and skills
- individual tutorials for students needing to review and practise basic skills
- using concrete material to count in a variety of ways within the range of 1-1000.
Resources include:
- teen boards and colour-coded bead bars
- ten boards and colour-coded bead bars
- golden 100-chain (102) and number labels (multiples of 10) colour-coded to indicate hierarchies
- golden 1000-chain (103) and number labels (multiples of 10) colour-coded to indicate hierarchies
- short colour-coded skip counting chains (12 - 92) with number labels for multiples
- long colour-coded skip counting chains (13 - 93) with number labels for multiples.
＋ Links to Australian Curriculum (ACv9)
The Decimal System: introduction 2.MA.040
Knowledge, Skills and Understanding
Typically, children will:
.01 Use concrete materials (bead material) to represent units, tens, hundreds and thousands
.02 Form numerals to 4-digits using manipulable, colour-coded cards representing units, tens, hundreds and thousands
.03 Associate concrete quantities (bead material) with symbols (numerals on cards)
.04 Explore place value using zero as a place holder to build numerals to four digits using manipulable colour-coded number cards
.05 Exchange 10 of each hierarchy for one of the next hierarchy using concrete quantities (bead material)
＋ Materials and Activity
Activities include:
- demonstrations of the hierarchies of the decimal system (units, tens, hundreds, thousands) with golden bead material
- counting the zeroes on the colour-coded number cards to identify hierarchies
- building arrays of beads and cards
- forming numbers using concrete quantities
- forming numbers with number cards (hiding zeroes)
- playing the change game with golden bead material and colour-coded number cards
- expanding numbers using colour-coded number cards
Resources include:
- golden bead material
- colour-coded numerals on cards.
＋ Links to Australian Curriculum (ACv9)
The Decimal System: operations with whole numbers 2.MA.050
Knowledge, Skills and Understanding
Typically, children will:
.01 Consolidate concepts and processes of addition and subtraction
.02 Use vocabulary for talking about the parts of an addition: first addend, second addend, sum
.03 Use vocabulary for talking about the parts of a subtraction: minuend, subtrahend, difference
.04 Use concrete quantities to add and subtract 4-digit numbers, first without and later with exchanging
.05 Notate addition and subtraction problems to four digits both horizontally and vertically
.06 Consolidate concept and process of multiplication as repeated addition of the same number
.07 Use vocabulary for talking about the parts of a multiplication: multiplicand, multiplier, product
.08 Use concrete quantities to multiply 4-digit numbers by a 1-digit (unit) multiplier, first without and later with exchanging
.09 Notate multiplication calculations (multiplicand to four digits and 1-digit multiplier) both horizontally and vertically
.10 Consolidate the concept and process of division as sharing equally (distributive division)
.11 Use vocabulary for talking about the parts of a division: dividend, divisor, quotient
.12 Use concrete material to divide 4-digit numbers by a 1-digit (unit) divisor, first without and later with exchanging
.13 Notate division calculations (dividend to four digits and 1-digit divisor) both horizontally and vertically
.14 Consolidate the concept and process of distributive division with 2- and 3-digit divisors
.15 Explore the concept and process of group division with 1-digit, 2-digit and 3-digit divisors
.16 Notate division calculations with the dividend to four digits and 2-digit and 3-digit divisors horizontally
＋ Materials and Activity
Activities include:
- games and activities with concrete materials to introduce the concepts of addition and subtraction
- using concrete material to add and subtract, with accompanying notation
- calculation using addition and subtraction, progressing from concrete representation accompanied by notation (golden bead material) to more abstract representation (stamp game, dot game), and from calculation without exchanging to calculation with exchanging, including calculation involving zeroes
- completing problems on command cards and/or creating own problems
- activities to build familiarity with graphic representations of addition and subtraction e.g., number lines.
- games and activities with concrete materials to introduce the concept of multiplication as repeated addition of the same number
- using concrete material to multiply with accompanying notation
- multiplication calculation progressing from concrete representation accompanied by notation (golden bead material) to more abstract representation (stamp game), and from calculation without exchanging to calculation with exchanging, including calculation involving zeroes in multiplicand
- activities to build familiarity with geometric representation of multiplication.
- games and activities with concrete materials to introduce the concept of division as sharing equally (from highest hierarchy)
- using concrete material to divide with accompanying notation
- division calculation progressing from concrete representation accompanied by notation (golden bead material) to more abstract representation (stamp game), and from calculation without exchanging to calculation with exchanging, including calculation involving zeroes in dividend
- early games and exploration in calculating division with 2- and 3-digit divisors using concrete material (golden bead material and colour-coded ribbons; stamp game with skittles), including calculation with zeroes in the divisor (stamp game with skittles and counters) and group division
- completing problems on command cards and/or creating own problems (progressing from concrete representation accompanied by notation to increasingly abstract representation) verifying answers.
Resources include:
- golden bead material, colour-coded numerals on cards and notation paper with colour-coded numerals
- stamp game and notation paper with colour-coded lines
- dot game board and/or paper
- colour-coded ribbons
- stamp game, colour-coded skittles and notation paper
- command cards.
＋ Links to Australian Curriculum (ACv9)
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Inter-relationships between operations 2.MA.060
Knowledge, Skills and Understanding
Typically, children will:
.01 Explore inverse relationships of addition and subtraction
.02 Explore inverse relationships of multiplication and division
＋ Materials and Activity
Activities include:
- using concrete materials to explore inverse relationships of addition subtraction
- using concrete materials to explore inverse relationships of multiplication and division
- using inverse relationships to verify answers.
Resources include:
- golden bead material
- stamp game
- bead frames and notation paper.
＋ Links to Australian Curriculum (ACv9)
Memorisation: addition facts 2.MA.070
Knowledge, Skills and Understanding
Typically, children will:
.01 Explore and record all essential addition facts for sums from 2-18 using concrete material
.02 Explore and record all addition facts with zero as an addend for sums from 1-18 using concrete material
.03 Explore and record the doubles of numbers 1-9 using concrete material
.04 Memorise essential addition facts for sums to 18 using finger charts
.05. Use knowledge of addition facts to solve word problems
.06 Find equivalent combinations using concrete material (commutative law)
.07 Prepare for mental calculation of addition combinations (sums to 99) using concrete material:
sums < 10
sums > 10
addends > 10
more than 2 addends
.08 Form and solve addition combinations with brackets using colour-coded bead bars and manipulable brackets, addition and equal signs (associative law)
＋ Materials and Activity
Activities include:
- activities and games with concrete material (addition strip board) to research addition facts for sums to 18
- recording sums in prepared tables
- games with addition combinations on loose cards and finger charts to develop memorisation of addition facts
- games with strip board in which all the ways to make each sum to 18 are researched and recorded
- games with concrete material and charts in which the order of the addends is changed and equivalent (turn around) combinations are substituted (commutative law)
- solving additions (sums to 18) with different unknowns (first addend, second addend or sum)
- games with bead material and manipulable symbols (addition signs, equal signs) to prepare progressively for mental addition of addends to at least 2-digits (sums to 99)
- games with bead material and manipulable symbols (brackets, addition signs, equal signs) to form and solve additions and to expand addends (commutative and associative laws)
- recording and verifying answers in all games and activities
- solving word problems using knowledge of addition facts
- follow-up activities to reinforce memorisation of addition facts and to achieve mastery, including online websites (e.g., Mental Math)
Resources include:
- addition snake game
- addition strip board
- booklets of prepared tables (colour-coded red) and notation paper
- box of loose addition combinations (flash cards) and four addition finger charts progressing in abstraction, including bingo game with sums on tiles
- word problem card material
- box of colour-coded bead bars
- manipulable symbols (brackets, addition signs, equal signs
- online websites that promote memorisation (e.g., Mental Math)
＋ Links to Australian Curriculum (ACv9)
Memorisation: subtraction facts 2.MA.080
Knowledge, Skills and Understanding
Typically, children will:
.01 Explore and record all essential subtraction facts for minuends from 2-18 using concrete materials
.02 Explore and record all subtraction facts with zero as a subtrahend for minuends from 1-18 using concrete materials
.03 Memorise essential subtraction facts for minuends to 18 using finger charts
.04 Use knowledge of subtraction facts to solve word problems
.05 Prepare for mental calculation of subtraction (minuends to 99)
＋ Materials and Activity
Activities include:
- activities and games with subtraction strip board for researching subtraction facts for minuends to 18
- recording differences in prepared tables
- games with loose cards and finger charts to develop memorisation of subtraction facts
- solving subtractions (minuends to 18) with different unknowns (minuend, subtrahend or difference)
- recording and verifying answers in all games and activities
- solving word problems using knowledge of subtraction facts
- follow-up activities to reinforce memorisation of subtraction facts and to achieve mastery, including online websites (e.g., Mental Math)
Resources include:
- subtraction snake game
- subtraction strip board
- booklets of prepared tables (colour-coded green) and notation paper
- box of loose subtraction combinations (flash cards) and three subtraction finger charts progressing in abstraction, including bingo game with differences on tiles
- word problem card material
- online websites (e.g., Mental Math)
＋ Links to Australian Curriculum (ACv9)
Memorisation: Multiplication facts 2.MA.090
Knowledge, Skills and Understanding
Typically, children will:
.01 Explore and record all essential multiplication facts for products from 1-100 using concrete materials
.02 Memorise essential multiplication facts for products to100 using finger charts
.03 Use knowledge of multiplication facts to solve word problems
.04 Find equivalent combinations using concrete material (commutative law)
.05 Memorise the multiples of numbers 1-10 by completing and/or reviewing skip counting of colour-coded bead chains (short chains: 12 - 102; long chains: 13 - 103)
.06 Find squares and cubes of numbers 1-10 using the bead chains, bead squares and bead cubes
.07 Discover that, when the multiplier is zero, the product is always zero for all multiplicands
.08 Form, manipulate and solve multiplication combinations with brackets using concrete material (commutative and distributive laws)
＋ Materials and Activity
Activities include:
- activities and games with multiplication bead board for researching multiplication facts for products to 100
- recording products in prepared tables
- games with loose cards and finger charts to develop memorisation of multiplication facts
- solving multiplications (products to 100) with different unknowns (multiplicand, multiplier or product)
- games with concrete material and charts in which the order of the factors is changed and equivalent (turn around) combinations are substituted (commutative law)
- games in which all ways to make products to 100 are researched and recorded
- solving word problems using knowledge of multiplication facts
- reciting and recording the series of products/multiples used to skip count the colour-coded bead chains, gradually turning over the labels until reciting each series from memory
- folding the bead chains and substituting bead squares and bead cubes
- series of games with colour-coded bead bars including multiplication snake game; building multiplication tables; multiplying by10; small multiplications with bead bars; changing the order of the factors (commutative law); building the squares of numbers 1-10; comparing bead bar layouts of combinations to find ‘common’ multiples
- using bead bars to find and record all ways to make the same product (1-100)
- series of games using colour-coded bead bars and manipulable symbols (brackets, multiplication signs, addition signs, equal signs) to explore distributive law, including multiplying binomial and trinomial expressions by numbers 1-10 [e.g., (3+5)x6; (4+7+2)x3], multiplying a sum by a sum [e.g., (3+5)(4+7)]; analysing squares into binomial and trinomial representations [e.g., 102 represented as (3+7) 2]; using binomials to represent and analyse the progression of squares from 12 to 102, both successive (e.g., 32 🡪 42) and non-successive (e.g., 52 🡪 82)
- drawing representations, recording and verifying answers in all games and activities
- follow-up activities to reinforce memorisation of multiplication facts and to achieve mastery, including online websites (e.g., Mental Math)
Resources include:
- multiplication snake game
- multiplication bead board
- booklets of prepared tables (colour-coded yellow) and notation paper
- box of loose multiplication combinations (flash cards) and four multiplication finger charts progressing in abstraction, including bingo game with products on tiles
- word problem card material
- cabinet of colour-coded bead chains, squares and cubes
- box of colour-coded bead bars
- manipulable symbols (brackets, addition signs, multiplication signs, equal signs)
- online websites (e.g., Mental Math)
＋ Links to Australian Curriculum (ACv9)
Memorisation: Division facts 2.MA.100
Knowledge, Skills and Understanding
Typically, children will:
.01 Explore and record all essential division facts for dividends from 1-100 using concrete materials
.02 Memorise essential division facts for dividends to 100 using finger charts
.03 Use knowledge of division facts to solve word problems
＋ Materials and Activity
Activities include:
- activities and games using division bead board for researching division facts for dividends to 100
- recording quotients in booklets of prepared tables
- games with loose cards and finger charts to develop memorisation of division facts
- solving divisions (dividends to 100) with different unknowns (dividend, divisor or quotient)
- solving word problems using knowledge of division facts
- follow-up activities to reinforce memorisation of division facts and to achieve mastery, including online websites (e.g., Mental Math)
Resources include:
- division bead board
- booklets (colour-coded blue) and notation paper
- box of loose division combinations (flash cards) and division finger charts progressing in abstraction, including bingo game with products on tiles
- word problem card material
- online websites (e.g., Mental Math)
＋ Links to Australian Curriculum (ACv9)
AC9M1N06
Decanomial: Table of Pythagoras 2.MA.110
Knowledge, Skills and Understanding
Typically, children will:
.01 Build the decanomial, or Table of Pythagoras, bead array (concrete representation of all multiplication combinations for numbers 1 – 10)
.02 Use the commutative law of multiplication to re-arrange the Decanomial bead array in a variety of ways
＋ Materials and Activity
Activities include:
- using knowledge of multiplication facts and colour-coded bead bars to build the Decanomial array/layout, first vertically and then horizontally
- re-arranging the Decanomial bead bar array using the commutative law
- commuting the Decanomial bead bar array to bead squares and then to bead cubes (13-103) arranged in a tower (tower of jewels)
- using card material to build the Decanomial array expressed in numerals and algebraic notation.
Resources include:
- box of coloured bead bars (55 of each bead bar)
- colour-coded bead squares and bead cubes from the bead chain cabinet
- numerical and algebraic decanomial cards and envelopes.
Multiples of Numbers 2.MA.120
Knowledge, Skills and Understanding
Typically, children will:
.01 Explore, analyse and record multiples and factors
.02 Explore, analyse and record prime numbers and composite numbers
＋ Materials and Activity
Activities include:
- lessons and activities as needed to introduce vocabulary (multiple, factor, common multiple, prime factor, prime number)
- using concrete materials (long bead chains, products formed with colour-coded bead bars) and/or memory to identify and record multiples of number 1-10
- identifying multiples for numbers 1-10 on Table of Multiples (numbers 1-100 laid out in a square), to find patterns and locate common multiples
- using tables A and B to record all the possible factors of each multiple
- using table C to identify prime numbers.
Resources include:
- cabinet of colour-coded bead chains, squares and cubes from the bead chain cabinet
- box of colour-coded bead bars
- Table of Multiples charts (to 100)
- student records of multiples and factors
- tables A, B, C.
Passage to Abstraction: addition and subtraction 2.MA.130
Knowledge, Skills and Understanding
Typically, children will:
.01 Combine knowledge of addition and subtraction (concept and process) with memorised essential facts to achieve passage to abstraction (calculation without the support of concrete material)
＋ Materials and Activity
Activities include:
- introduction to small bead frame with abacus and golden bead material
- counting through the hierarchies (units, tens, hundreds, thousands) as represented on the small bead frame
- forming and reading numbers to 9999 on the small bead frame
- notating numbers to 9999 on small bead frame paper
- forming and solving 4-digit additions and subtractions on the small bead frame (from calculation without exchanging to calculation with exchanging, including calculation involving zeroes) and recording the calculations on small bead frame notation paper
- completing the passage to abstraction (calculating additions and subtractions without the support of concrete material; notating addition and subtraction calculations without small bead frame notation paper as a guide)
- extend calculations to addition and subtraction with 7-digit numbers using large bead frame and large bead frame paper aligned to interest and need.
Resources include:
- abacus and golden bead material for introduction
- small bead frame and small bead frame notation paper
- large bead frame and large bead frame notation paper
- command cards; word problems.
＋ Links to Australian Curriculum (ACv9)
Decimal System Hierarchies: extension 2.MA.140
Knowledge, Skills and Understanding
Typically, children will:
.01 Manipulate, name and notate numbers to 6-digits
.02 Recognise number ‘families’:
‘simple family’ of units, tens, hundreds
‘family of thousands’: thousands, tens of thousands, hundreds of thousands’
‘family of millions’
.03 Mark groupings of digits with comma or separation
＋ Materials and Activity
Activities include:
- introduction to wooden hierarchical material (observing the colour-coding and relative size; link with golden bead material)
- naming the concrete quantities (units, tens, hundreds, thousands, ten thousands, hundred thousands, millions)
- sorting and labelling quantities by ‘family’ (hierarchy) using geometric shape, size and colour-coding as a guide
- reading aloud and writing the symbols (words and numerals) for the quantities
- matching symbols to quantities
- expanding numerals into constituent hierarchies
- making charts and cards.
Resources include:
- quantities to 1,000 represented in golden bead material (bead cube, square, bar, unit and colour-coded numbers on cards)
- quantities to 1,000,000 represented in wooden hierarchical material and labels (words and numerals).
＋ Links to Australian Curriculum (ACv9)
Passage to Abstraction: Multiplication 2.MA.150
Knowledge, Skills and Understanding
Typically, children will:
.01 Combine knowledge of multiplication (concept and process) with memorised essential facts to achieve passage to abstraction (calculation without the support of concrete material) for multiplications with:
1-digit multiplier
products to 4-digits
.02 Combine knowledge of multiplication (concept and process) with memorised essential facts to achieve passage to abstraction (calculation without the support of concrete material) for multiplications with:
2-digit and 3-digit multipliers
products to 6-digits
.03 Expand 4-digit numbers to constituent hierarchies
.04 Calculate mentally simple multiplications, including exchanging
.05 Combine knowledge of multiplication (concept and process) with memorised essential facts to achieve passage to abstraction (calculation without the support of concrete material) for multiplications with:
multipliers to 4-digits
multiplicands to 9-digits
＋ Materials and Activity
Activities include:
- introduction to large bead frame with wooden hierarchical material
- counting through the hierarchies (units, tens, hundreds, thousands, ten thousands, hundred thousands, millions) as represented on the large bead frame
- forming and reading numbers to 6-digits on the large bead frame
- notating numbers to 6-digits on large bead frame notation paper
- forming and solving multiplications (2-digit and 3-digit multipliers; 6-digit products) on the large bead frame and recording the calculations on large bead frame notation paper using expanded hierarchies and partial products
- forming and solving multiplications (2-digit and 3-digit multipliers; 6-digit products) on the flat (golden) bead frame and recording the calculations, first without and later with partial products
- introduction to multiplication checkerboard: colours and values
- using colour-coded squares to build the checkerboard
- exploring hierarchies by multiplying and dividing x10
- forming quantities with colour-coded bead bards on the checkerboard, and reading and writing the numbers
- representing small multiplications with bead bars
- forming and solving multiplications (9-digit multiplicands; multi-digit multipliers) on the checkerboard and recording the calculations, first without and later with partial products
- using the checkerboard to make charts
- introduction to the bank game: material, roles and rules
- using the bank game to perform short and long multiplications and develop skills in mental multiplication
- completing the passage to abstraction (calculating short and long multiplications without the support of concrete material; notating short and long multiplication calculations without guide paper).
Resources include:
- large bead frame and large bead frame notation paper
- flat (golden) bead frame
- command cards; word problems.
- multiplication checkerboard, colour-coded number tiles and quantities represented in colour-coded bead bars
- bank game card material.
＋ Links to Australian Curriculum (ACv9)
Passage to Abstraction: Division 2.MA.160
Knowledge, Skills and Understanding
Typically, children will:
.01 Combine knowledge of division (concept and process) with memorised essential facts to achieve passage to abstraction (calculation without the support of concrete material)
dividends to 4-digits; 1-digit divisor
dividends to 6-digits; multi-digit divisors
zeroes in dividend and divisor
.02 Perform division calculations using both distributive and group division
.03 Calculate mentally simple divisions, with exchanging
＋ Materials and Activity
Activities include:
- introduction to racks and tubes (test tube material); observing the colour-coding, verifying 10 beads/tube
- forming and solving short divisions (1-digit multiplier) using racks and tubes, and recording calculation, including divisions with zero in the quotient and the divisor
- progressing step-by-step from horizontal notation with single and multi-digit divisors, including recording remainders, to vertical notation with multi-digit divisors recording intermediate remainders (‘the next amount to be shared’) and ‘what has been used’
- exploring processes for both distributive and group division
- forming and solving long divisions (multi-digit divisors) using racks and tubes, and recording calculation (both distributive and group division)
- completing the passage to abstraction (calculating and notating short and long divisions without the support of concrete material).
Resources include:
- racks and tubes
- stamp game
- command cards; word problems.
Introduction to Fractions 2.MA.170
Knowledge, Skills and Understanding
Typically, children will:
.01 Understand the concepts ‘whole’ and ‘fraction’
.02 Name and notate fractions, identify common uses and relate to shapes, objects and events
.03 Use vocabulary for talking about the parts of a fraction (numerator, denominator, fraction line)
.04 Manipulate and name fractions from halves to tenths
＋ Materials and Activity
Activities include:
- stories, demonstrations, games and activities with concrete materials to build knowledge about the possibilities for dividing a unit into smaller quantities, concluding with emphasis on fractions (dividing the unit into equal parts)
- forming fractions with concrete material
- learning the names of the parts of a fraction, and their etymology
- saying the names of fractions and labelling the parts of fractions
- name, label and notate all possible fractions formed with concrete material from halves to tenths (words and manipulable symbols)
- making fraction charts and booklets
- activities to build familiarity with a range of graphic representations of fractions e.g., shaded squares in a grid or shaded sectors of a pie graph
- activities linking knowledge about fractions with the solution of real life problems.
Resources include:
- metal fraction insets (to tenths)
- box of manipulable cut-out fractions
- manipulable labels (words and symbols)
- geometry charts
- command cards.
＋ Links to Australian Curriculum (ACv9)
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Equivalent Fractions 2.MA.180
Knowledge, Skills and Understanding
Typically, children will:
.01 Identify equivalent fractions
.02 Reduce fractions to lowest terms
.03 Explore variation in notation for equivalent fractions
＋ Materials and Activity
Activities include:
- using concrete material to find equivalent fractions within the range halves to tenths
- notating equivalent fractions
- reducing fractions to lowest terms using concrete quantities and manipulable symbols
- creating charts and booklets
- creating own problems (progressing from concrete representation accompanied by notation to increasingly abstract representation).
Resources include:
- fraction insets and labels
- fraction research sheets
- geometry charts
- command cards; word problems.
＋ Links to Australian Curriculum (ACv9)
Operations with Fractions 2.MA.190
Knowledge, Skills and Understanding
Typically, children will:
.01 Add and subtract fractions with the same denominators
.02 Multiply and divide fractions by a whole number
＋ Materials and Activity
Activities include:
- using concrete materials to add and subtract fractions with like denominators, accompanied by notation
- using concrete materials to multiply and divide fractions with whole numbers, accompanied by notation
- completing problems on command cards and/or creating own problems (progressing from concrete representation accompanied by notation to increasingly abstract representation
- activities to build familiarity with a range of graphic representations of operations with fractions e.g., shaded squares in a grid or shaded sectors of a pie graph
- activities linking knowledge about operations with fractions and the solution of real life problems.
Resources include:
- fraction insets and labels
- geometry charts
- command cards; word problems.
＋ Links to Australian Curriculum (ACv9)
Money 2.MA.200
Knowledge, Skills and Understanding
Typically, children will:
.01 Research the use of money over time and in different places
＋ Materials and Activity
Activities include
- research projects e.g., history of money; bartering, currency other than coins and notes e.g., shells, gold, money symbols and abbreviations ($ c); animals and people on Australian coins and notes; money used in the past or in different countries
- going out activities e.g., visiting mint, museums
- making charts, timelines, models and artwork, rubbing coins.
Resources include:
- coin and note collections
- fundamental needs of humans timeline
- research and reference materials (paper-based, digital, web-based, multimedia).
Typically, children will:
.02 Learn units of money (dollars and cents)
.03 Learn values of coins and notes
.04 Apply knowledge of the decimal system to use of money
.05 Performs operations with coins:
add
subtract (make change)
multiply same amounts
divide (share amounts equally)
.06 Performs operations with notes and coins:
.06 Performs operations with notes and coins:
add
subtract (make change)
multiply same amounts
divide (share amounts equally)
＋ Materials and Activity
Activities include
- research projects e.g., history of money; bartering, currency other than coins and notes e.g., shells, gold, money symbols and abbreviations ($ c); animals and people on Australian coins and notes; money used in the past or in different countries
- going out activities e.g., visiting mint, museums
- making charts, timelines, models and artwork, rubbing coins.
- stories and games to introduce and explore money and its use
- individual and group games and exercises with play money and real money in classroom to explore values, notation, exchanging across hierarchies and operations
- making charts and booklets
- practising mental calculation of money problems
- going out e.g., to shops to buy items
Resources include:
- coin and note collections
- Fundamental Needs of Humans charts
- research and reference materials (paper-based, digital, web-based, multimedia).
- real and play money
- command cards; word problems
- commercial resources
＋ Links to Australian Curriculum (ACv9)
Data and Probability 2.MA.210
Knowledge, Skills and Understanding
Typically, children will:
.01 Display objects and images according to previously established criteria
.02 Interpret display to make generalisations
.03 Deduce patterns in data or numbers and continue or extend pattern
.04 Use a tally system when counting to collect data
.05 Collect and collate data and represent visually with pictures and symbols
.06 Read and interpret data represented in simple graphs and charts
picture graphs
tally graphs
column graphs
pie charts
flow charts
.07 Identify practical activities and everyday events involving chance
.08 Describe possible outcomes and events as ‘likely’ or ‘unlikely’ and identify some events as ‘certain’ or ‘impossible’
.09 Conduct repeated chance experiments
.10 Estimate the quantity of objects in collections and make estimates when solving problems to determine the reasonableness of calculations
＋ Materials and Activity
Activities include:
- Sorting objects (e.g., by colour or shape), counting numbers in each category and building a display so data can be used to answer questions (e.g., what is the most common eye-colour in our class? What is the most common colour of the cars travelling down our street? What is the least common colour?)
- Completing or extending graphical and/or number patterns
- Using tallying and simple tables to record data (e.g., number of times particular numbers are thrown on a die).
- Designing questions that can be used as the basis for a survey or data collection activity (e.g., favourite colour or pet)
- Designing simple tables and matrices to record data for a particular purpose
- Using images, icons or symbols to represent data categories
- Creating a scale based on size to represent data accurately
- Displaying data in simple graphs, using simple drawing equipment and/or simple software
- Interpreting data represented in simple graphs
- Applying knowledge of data representation and interpretation to a range of everyday contexts across the curriculum (e.g., recording weather patterns)
- Interpret graphs, and other ways of displaying data, and research their use in a range of paper-based and electronic resources.
- Classifying a list of everyday events or sorting a set of event cards according to how likely they are to happen using the language of chance
- Predicting what could happen next in practical activities that involve an element of chance and using appropriate terminology
- Making predictions and testing what would happen in chance experiments (eg. Pulling names out of a hat, rolling a die, flipping a coin)
Resources include:
- Objects and events in the environment
- Tally sheets
- Graph paper and drawing equipment (paper-based and electronic)
- Paper-based, digital, web-based, and/or multimedia resources.
- Command cards; word problems.
- Objects and events in the environment
- Paper-based, digital, web-based, and/or multimedia resources
- Command cards; word problems
- Chance materials, e.g., coins, dice, spinners, playing cards
＋ Links to Australian Curriculum (ACv9)
AC9M1ST01
AC9M2ST02
AC9MFST01
AC9MFA01
AC9M1A02
AC9M2A01
AC9M3N07
AC9M1ST01
AC9MFST01
AC9M1ST02
AC9M2ST01
AC9M3ST01
AC9M3ST03
AC9TDIFK02
AC9TDI2K02
AC9M3ST02
AC9M3P01
AC9M3P02
AC9M3N05
Digital Systems 2.MA.220
Knowledge, Skills and Understanding
Typically, children will:
.01 Explore the logic and purpose of digital systems (software and hardware)
.02 Explore the digital instructional language of coding
.03 Investigate simple problems for known users that can be solved with digital systems
.04 Follow and describe algorithms involving a sequence of steps, branching (decisions) and iteration (repetition)
.05 Use the basic features of common digital tools to create, collaborate and communicate content demonstrating agreed behaviours, guided by trusted adults
＋ Materials and Activity
Activities include:
- Using simple flowchart format to systematise binary problem-solving solution to everyday problem
- Writing precise directions for an everyday task (e.g., making a pbj ham sandwich, washing hands) which are then acted out literally
- Exploring non-digital (unplugged) coding resources, activities and games
- Using lego wedo mindstorms to program lego robots or scratch to program a tello drone or similar device with simple commands
- Disassemble and reassemble non-working computer tower system to identify parts (e.g., cpu, usb and ethernet ports, memory, keyboard, microphone and other input devices, speakers and other output devices, etc.)
- Learning chess and recording all moves on paper
- Solving wooden logic and sudoku puzzles
- Using simple coding program such as scratch to write simple programs that solve problems on the basis of user choices
- Discuss and implement a series of steps for the class to follow to find certain information online
- Using an online search engine safely and appropriately to find and explore certain information not readily available in books
- Grouping, naming and itemising objects using a logical hierarchy e.g., digitally categorising items on a shopping list using folders and files
- Access their school account with a recorded username and password if appropriate
- Discuss that some websites and apps store their personal data online
- Explore transmitting different types of data between digital systems
- Recognise different types of data and explore how the same data can be represented differently depending on the purpose
- Identify what personal data is stored and shared in their online accounts and discuss any associated risks
Resources include:
- Problem solving materials
- Procedural texts and instructions
- Manipulative building materials (e.g. Lego)
- Programmable devices including drones, robots, etc.
- Computers and other digital devices
- Strategy games (e.g. Chess)
- Logic puzzles and materials
- Coding apps and programs (e.g. Scratch)
＋ Links to Australian Curriculum (ACv9)
AC9TDIFK01
AC9TDI2K01
AC9TDI4K01
AC9TDI2P02
AC9TDI4P02
AC9TDI4P04
AC9TDIFP01
AC9TDI2P01
AC9TDI2P03
AC9TDI4P04
AC9TDI2P02
AC9TDI4P02
AC9TDI2P04
AC9TDI2P05
AC9TDI2P06
AC9TDI2P07
AC9TDI4K02
AC9TDI4P06
AC9TDI4P07
AC9TDI4P08
AC9TDI4P09