Mathematics Scope and Continuum: Years P-7
Broad Developmental Concepts.
Broad Developmental Skills.
Symbology and Ordering
Counting and subitising
Dimension/Strand: Number and Algebra.
Substrand: Number & Place Value
Value of numbers
Sequencing
Place Values & Combinations
Modelling Addition & Subtraction
Foundation
Establish understanding of the language and processes of counting by naming numbers in
sequences, initially to and from 20, moving from any starting point (ACMNA001)
• reading stories from other cultures featuring counting in sequence to assist students to recognise
ways of counting in local languages and across cultures
• identifying the number words in sequence, backwards and forwards, and reasoning with the number
sequences, establishing the language on which subsequent counting experiences can be built
• developing fluency with forwards and backwards counting in meaningful contexts, including stories
and rhymes
• understanding that numbers are said in a particular order and there are patterns in the way we say
them.
Connect number names, numerals and quantities, including zero, initially up to 10 and then beyond
(ACMNA002)
• understanding that each object must be counted only once, that the arrangement of objects does
not affect how many there are, and that the last number counted answers the ‘how many’ question.
• using scenarios to help students recognise that other cultures count in a variety of ways, such as by
placing one pebble in a bag to represent one object (for example to count the number of cattle).
Subitise small collections of objects. (ACMNA003)
• using subitising as the basis for ordering and comparing collections of numbers.
• comparing and ordering items of like and unlike characteristics using the words ‘more’, less’, ‘same
as’ and ‘not the same as’ and giving reasons for these answers.
• understanding and using terms such as ‘first’ and ‘second’ to indicate ordinal position in a sequence.
• using objects which are personally and culturally relevant to students.
Compare, order and make correspondences between collections, initially to 20, and explain
reasoning (ACMNA289)
Represent practical situations to model addition and sharing (ACMNA004)
• using a range of practical strategies for adding and subtracting small groups of numbers, such as
visual displays or concrete materials.
• using Aboriginal and Torres Strait Islander methods of adding and subtracting, including spatial
patterns and reasoning.
Level 1
Develop confidence with number sequences to and from 100 by ones from any starting point. Skip
count by twos, fives and tens starting from zero (ACMNA012)
• using the traditional Korean counting game (sam yew gew) for skip counting.
• developing fluency with forwards and backwards counting in meaningful contexts such as circle
games.
Recognise, model, read, write and order numbers to at least 100. Locate these numbers on a
number line (ACMNA013)
• modelling numbers with a range of material and images identifying numbers that are represented on
a number line and placing numbers on a prepared number line.
Count collections to 100 by partitioning numbers using place value (ACMNA014)
• understanding partitioning of numbers and the importance of grouping in tens
• understanding two-digit numbers as comprised of tens and ones/units.
Represent and solve simple addition and subtraction problems using a range of strategies including
counting on, partitioning and rearranging parts (ACMNA015)
• developing a range of mental strategies for addition and subtraction problems.
Facts & Procedures
Use of the number line to subtract by skipping
•Relationship b/w quantities, e.g. more - less.
back
•Quantity: one-to-one correspondence.
• Cover up model for subtraction.
•Groups of objects in a collection.
1. Subtraction of 1.
• Establish dot patterns.
2. Subtract 2
•Link Word, Number of dots, symbol.
Include Zero
•Subtraction but not sharing equally.
Use of Zero "0" as the symbol for "no-thing" or
e.g. 5-1, 5-3, 10-1, then 2-2 is 0 (nothing)
the start of a count.
•Whole numbers 0 to 10
Much later.
•Position and order of numbers relative to other
•Addition and subtraction of whole number
numbers. e.g. 0 - 9
•Comparison of collections. More - Less
totals to 10, two or more addends
– make to 10 Use of Ten Frames.
• Simple number sequences on a number line.
• Mental strategies:
•Quantity: conservation of whole numbers 0 to
– count on, count back in 1s, 2s, and 3s
10, subitising (seeing groups of 2 or 3 objects
– commutative addition (turn around facts),
without counting)
e.g. when calculating 2 + 7 start with 7 and add 2.
Number line.
Language.
• Position and order of numbers 0 to 10.
– counting (forward to 10, backward in 1s from 5,
• Use of number line to skip count and “Hop and
next number in the chant of the counting
Jump”.
sequence, e.g. 1,2,3,4,?)
• Recognition of patterns within number line.
•Use of blank number line, number line to
• Use of attributes of the number line, e.g. twos,
beyond 100,
fives, tens…..
Special number chart to 100, starting at 0, ending
Use of the number line to add by skipping.
at 9 and cascading each decade
Combinations.
Introducing
• Joining model for addition.
Introduce
1. Combinations to 5. – make to 5
• Sharing equally, groups of.
2. Combinations to 10. – make to 10
Couple is 2
Few is 3, 4 or more
• breaking up numbers to make them
Several is 5, 6 or more
Many is 9,10 or more
manageable,
Special notes: Avoid bigger or smaller as they
e.g. 7 add 5,
relate to size. Collections, counting relate to
first, break up 5, add 3, make to 10 then add 2
more or less.
7+5 = 7 + (3+2) = (7+3) + 2
Numbers on Number line are lesser to left,
(start of associative law)
greater to right.
Facts & Procedures
Subtraction.
On a number line, ....
• Develop partitioning of groups and focus on
1. Identify lesser or greater numbers.
taking away as an extension of cover up.
OR Identify lower or higher numbers.
•Introduce the – symbol. No use of the = sign yet.
Avoid saying smaller or bigger numbers as this
•Developing Part-Part Whole relationships.
brings confusion later with negative numbers.
Balance Sign (=)
Use number comprehension tests - Van DeWalle. Develop the balance between Part Part and a
1. Patterned sets. How many without counting.
Whole. E.g 3 + 2 is the same as 5 altogether.
2. One and two more, one and two less.
Establish that the = sign is a BALANCE not a
3. Anchors or “benchmarks” of 5 and 10
result.
4. Part-part-whole relationships.
Reinforce the balance aspect with 5 = 3+2
Patterns in numbers:
Relationships between numbers,
•Calculator displays created using constant
e.g. more, less, same as, odd, even.
function. e.g. On calculator 1+3= = = results in
•Identify smaller and larger collections.
calculator constantly adding 3. Similar to skip
•Conservation of number as in "trust the count"
counting from any number.
is 10. Count on from 10, teen numbers.
The next column.
•Position and order of numbers relative to other
Introduce the 2nd place value column as Tens
numbers, to the nearest 5 or 10, ordinal numbers (10s). Use of Icy pole sticks bundled is preferred.
to 10.
Language.
•Establish that the ordinal numbers are read
• counting (forward to 100, forward in 2s to 20,
from the “starting end” and depend on the way
backward in 1s from 10) Include zero in all
the line faces.
counts.
Quantity and Place Value.
• operation to be used, add ,take away, subtract,
•Quantity: grouping using place value.
cover up, part, whole, number names 0 to 10.
– numbers, e.g. more, less, equal to, not equal to
• estimates, about, near.
– subtraction and addition.
Visual:
Establishing Place Value in the “tens” column.
•Pictures, five frame, ten frame, number line
Consolidate: (what is the effect on 2 tens and 3
with decades, number charts with 0 to 110
ones if we add another 1 ten.)
•Subitising (seeing groups of two or three objects
Addition.
and patterns of larger groups without
•Addition of numbers using the joining model
counting), e.g. five domino pattern.
and extending to the + symbol.
•symbolic forms are: equals (=), does not
No use of the = sign yet.
equal (≠)
Level 2
Investigate number sequences, initially those increasing and decreasing by twos, threes, fives and
ten from any starting point, then moving to other sequences.(ACMNA026)
• developing fluency and confidence with numbers and calculations by saying number sequences.
•recognising patterns in number sequences, such as adding 10 always results in the same final digit.
Recognise, model, represent and order numbers to at least 1000 (ACMNA027)
• recognising there are different ways of representing numbers and identifying patterns going beyond
100. • developing fluency with writing numbers in meaningful contexts
Group, partition and rearrange collections up to 1000 in hundreds, tens and ones to facilitate more
efficient counting. (ACMNA028)
• using an abacus to model and represent numbers
• understanding three-digit numbers as comprised of hundreds, tens and ones/units demonstrating
and using models such as linking blocks, sticks in bundles, place-value blocks and Aboriginal bead
strings and explaining reasoning.
Explore the connection between addition and subtraction. (ACMNA029)
• becoming fluent with partitioning numbers to understand the connection between addition and
subtraction. • using counting on to identify the missing element in an additive problem
Solve simple addition and subtraction problems using a range of efficient mental and written
strategies. (ACMNA030)
• becoming fluent with a range of mental strategies for addition and subtraction problems, such as
commutativity for addition, building to 10, doubles, 10 facts and adding 10
• modelling and representing simple additive situations using materials such as 10 frames, 20 frames
and empty number lines.
Recognise and represent multiplication as repeated addition, groups and arrays. (ACMNA031)
• representing array problems with available materials and explaining reasoning
• visualising a group of objects as a unit and using this to calculate the number of objects in several
identical groups.
Recognise and represent division as grouping into equal sets and solve simple problems using these
representations. (ACMNA032)
• dividing the class or a collection of objects into equal-sized groups
• identifying the difference between dividing a set of objects into three equal groups and dividing the
same set of objects into groups of three.
Facts & Procedures
Number characteristics.
Introduce sharing. e.g. Share equally between,
• Focus on differences between numbers.
share equally among.
• Consolidate odd & even numbers.
Develop equal parts of whole objects
•Use of a number chart as a more efficient
and equal parts of collections.
number line. Use charts from 0 to 110 and
Half, Quarter: part of a whole, half of half.
beyond.
Introduce repeated sharing as division.
• List things that are equal to 10.
Language:
Addition.
– counting (forward in 2s, 5s, 10s
• Basic addition facts to 10 and subtraction facts
to 100, strategies for operations
as the inverse. Addition and subtraction totals as
• counting from different numbers,
a balance. Left side balances the right side.
• extensions to larger numbers,
Subtraction.
• backward in 1s from any number)
• Use subtraction as 7-5 not "what do we add to
• counting (forward to 1000, forward in 2s, 5s,
5 to get 7".
10, to 100 or 1000, backward in 1s from 30)
Number Facts
Include zero in all counts.
• Frequent use of number facts,
• number names to 1000, fraction names,
e.g. in triangle fact form.
add, subtract, left, multiply and divide, groups of,
5+2 = 7, 7-2 = 5, 7-5 = 2
rows of, jumps of, share between, share, odd,
Also 7=5+2, 5=7-2, 2=7-5
even.
• Establish two addends to 18. Develop fluency
Visual:
with number facts to 18 in preparation for two
Number charts 0-110 and beyond.
column addition.
Number charts showing patterns of odds.
Place value system.
10 frames.
• Consolidate Place value system: – tens – ones.
Open ended tasks should begin for students at
-Use of Icy pole sticks bundled is preferred.
this level.
-lead to 2 column addition using “Regrouping,
Special Notes: Generally, the word ALTOGETHER
th
renaming” Booker et al. 4 Edition.
at the start of a sentence indicates subtraction,
Two addend addition to 99, two or more
end of a sentence indicates addition.
addends, missing addends.
Groups and Arrays.
• Identify repeated addition, (e.g. Constant
function on calculator).
Repeated addition leads to multiplication.
Allow for column and row arrays.
Make connections to charts.
Mathematics Scope and Continuum: Years P-7
Broad Developmental Concepts. Inter-relations between operations. Forward Operations
Broad Developmental Skills.
Mental and written computations. Recall (Algorithms)
Level 3
Investigate the conditions required for a number to be odd or even and identify odd and even numbers. (ACMNA051)
•identifying even numbers using skip counting by twos or by grouping even collections of objects in twos
• explaining why all numbers that end in the digits 0, 2, 4, 6 and 8 are even and that
numbers ending in 1, 3, 5, 7 and 9 are odd
Recognise, model, represent and order numbers to at least 10 000. (ACMNA052)
• placing four-digit numbers on a number line using an appropriate scale
• reproducing numbers in words using their numerical representations and vice versa.
Apply place value to partition, rearrange and regroup numbers to at least 10 000 to assist calculations and solve problems. (ACMNA053)
• recognising that 10 000 equals 10 thousands, 100 hundreds, 1000 tens and 10 000 ones
• justifying choices about partitioning and regrouping numbers in terms of their
usefulness for particular calculations
Recognise and explain the connection between addition and subtraction. (ACMNA054)
• demonstrating the connection between addition and subtraction using partitioning or by writing equivalent number sentences.
Recall addition facts for single digit Numbers and related subtraction facts to develop increasingly efficient mental strategies for computation
(ACMNA055)
• recognising that certain single-digit number combinations always result in the same answer for addition and subtraction, and using this knowledge for
addition and subtraction of larger numbers
• combining knowledge of addition and subtraction facts and partitioning to aid
computation (for example 57 + 19 = 57 + 20 – 1)
Recall multiplication facts of two, three, five and ten and related division facts. (ACMNA056)
• establishing multiplication facts using number sequences.
Represent and solve problems involving multiplication using efficient mental and written strategies and appropriate digital technologies. (ACMNA057)
• writing simple word problems in numerical form and vice versa
• using a calculator to check the solution and reasonableness of the answer.
Facts & Procedures
Place Value.
Multiplication.
• Whole numbers to tens of thousands.
• Repeated addition and therefore establishing the early need for multiplication.
•Use of the HTO and extend to the Thousands house. Not
• Introduce Multiplication facts , 1s, 2, 5s, 10s
separate columns.
• Move from skip counting to multiplication facts.
•Use of multiplication charts.
•Fact triangles for related division facts using single-digit divisors as
the inverse of multiplication facts,
e.g. Use of Multiplication fact triangles.
Limit to 1, 2, 4, 5, 10
Some schools chant tables, in the sequence form 1x _, 2x_, 5x_
• Use of extended notation e.g. 123 is 100 + 20 + 3.
Multiplication – Area links
This allows for easier multiplication of 2 and 3 digit numbers alter Begin making direct links between Multiplication and Area as represented on the
on at Level 4
multiplication chart.
•Relationship between numbers, e.g. greater than, less than,
Requires special chart with the multiplicands separate from the body of the chart
equivalent to
Arrays. (Multiplication and Area)
• Order numbers to 1000, Introduce less than and greater than
•Arrays, rows of, equal groups of… Equal groups (quotition)
symbols. Use relationship that the symbol always points to the
•Sharing parts equally (partition). Use loops on chart method.
lesser number e.g. 5 > 3, 6 < 7. Avoid use of eating as symbology.
Language
Number Facts.
multiply and divide, groups of, rows of, jumps of, share between, share, odd, even.
•Consolidate number facts to 18.
Visual:
•Extensions of basic addition facts and subtraction facts as the
Number charts showing patterns of odds.
inverse.
10 frames.
• Extend addition algorithm to the thousands column.
Number charts 0-110 and beyond.
• Addition and subtraction totals to 999, two or more addends,
Specialised Area charts for Multiplication and division.
missing addends.
Multiplication charts with square numbers highlighted.
Rules for addition and subtraction:
Symbols for the operations, multiplication and division. Use of the symbols > < = ≠
Mental strategies:
2 odds, 2 evens, odd and even.
Recognising patterns within patterns. E.g. 4 are subset of 2s
• related facts, e.g. calculate 14 take 8 by recalling 8+6 equals 14
• build up, build down to preferred reference point, e.g. to the nearest decade or
Early use of the associative laws for addition (Friendly numbers).
e.g. in a set of numbers, add the complementary number first.
100
Complementary pairs to make 10, 20, 50s 100s to aid addition.
• extensions of count on and count back strategies from single-digit facts to 2- and
3- digit numbers
Language.
• doubles (x 2), double doubles (x 4)
• counting (forward to 1000, forward in 2s, 5s, 10, to 100 or
• skip counting (x 2, x 5, x 10).
1000, backward in 1s from 30) Include zero in all counts.
• Use of composite numbers.
• number names to 1000, add, subtract, left over
Dimension/Strand: Number & Algebra. Substrand: Number & Place Value
Inter-relations between operations. Inverse Operations
Mental and written computations. Recall (Algorithms)
Level 4
Investigate and use the properties of odd and even numbers. (ACMNA071)
• using the four operations with pairs of odd or even numbers or one odd and one even number, then using the relationships established to
check the accuracy of calculations.
Recognise, represent and order numbers to at least tens of thousands. (ACMNA072)
• reproducing five-digit numbers in words using their numerical representations, and vice versa.
Apply place value to partition, rearrange and regroup numbers to at least tens of thousands to assist calculations and solve problems.
(ACMNA073)
• recognising and demonstrating that the place-value pattern is built on the operations of multiplication or division of tens.
Investigate number sequences involving multiples of 3, 4, 6, 7, 8, and 9. (ACMNA074)
• recognising that number sequences can be extended indefinitely, and determining any patterns in the sequences.
Recall multiplication facts up to 10 × 10 and related division facts. (ACMNA075)
• using known multiplication facts to calculate related division facts.
Develop efficient mental and written strategies and use appropriate digital technologies for multiplication and for division where there
is no remainder. (ACMNA076)
•using known facts and strategies, such as commutativity, doubling and halving for
multiplication, and connecting division to multiplication when there is no remainder.
Facts & Procedures
Relationships
•between multiplication facts (2s, 4s, and 8s, 3s, 6s, and 9s)
•Inverse operations and the sides of rectangles.
•Introduce Multiples and factors as parts of multiplication and
division. (sides of rectangles)
Language
– inverse (backtracking)
– doubles (x 2), double doubles (x 4), double double doubles (x 8)
– build up (x 7 facts), build down (x 9 facts)
• Establish the value of the place e.g. 641 has 6 hundreds or 64 tens,
– halving.
or 641 ones.
– counting – estimates
– explanations of reasoning
• Position and order of numbers relative to other numbers and to
– calculation strategies and reasonableness of solutions
zero, to the nearest 5 or 10, and their extensions to 2, 3 and 4 digit
– mathematical language: number names to beyond 10,000.
numbers.
Visual:
Multiplication.
Number charts showing the connection between sides of
• Move from skip counting to multiplication facts.
rectangles and factors.
•Use of multiplication charts.
Number charts 0-110 and beyond.
• Review Multiplication facts 1s, 2s, 5s, 10s
Number charts showing patterns of odds.
• Consolidate Multiplication facts, 2s, 4s, 8s, 11s
10 frames.
• Introduce Multiplication facts, 3s, 6s, 12s.
Multiplication charts with square numbers highlighted.
• Develop strategies for Multiplication facts 9s, 7s.
Symbols for the operations, multiplication and division.
• Fact triangles for related division facts using single-digit divisors as
Use of the symbols > < = ≠
the inverse of multiplication facts, e.g.
Mental strategies:
Multiplying by 10
– inverse operations
Multiplication by whole numbers increases the resulting number
– manageable numbers
shifting the digits further to the left in the place value columns.
– extensions of basic number facts, e.g. 6 + 3 = 9 extension of
Therefore, multiplying by 10 increases the number shifting the same
basic fact 600 + 300 = 900
digits to the left. It is not a simple “adding a zero”.
– doubles (x 2), double doubles (x 4), double double doubles (x 8)
• Multiplying and dividing by 10 and 100. Movement of digits left or
– multiplying and dividing by 10 and 100.
right with the zero playing an important part.
Use of associative and commutative laws for quicker mental
Multiplication and Area.
addition and subtraction.
Make direct links between Multiplication and Area as represented on
E.g. (Associative) 1 + 2 + 98 +99 = 1 + 99 + 2 + 98
the special multiplication chart.
E.g. (commutative) 8 x 3 = 3 x 8
•Arrays, rows of, equal groups of… Equal groups (quotition)
•Sharing parts equally (partition)
Place Value
• Extend place value
• Whole numbers beyond 10,000
• Consolidation of composite numbers.
Mathematics Scope and Continuum: Years P-7
Broad Developmental Concepts.
Broad Developmental Skills.
Multiplicative Thinking
Factors and Multiples
Dimension/Strand: Number and Algebra.
Substrand: Number & Place Value
Multiplicative Thinking
Efficient Processes
Composition of numbers
Efficient Processes
Level 5
Identify and describe factors and multiples of whole numbers and use them to solve problems. (ACMNA098)
• exploring factors and multiples using number sequences
• using simple divisibility tests
Use estimation and rounding to check the reasonableness of answers to calculations. (ACMNA099)
• recognising the usefulness of estimation to check calculations
•applying mental strategies to estimate the result of calculations, such as estimating the cost of a supermarket trolley load.
Solve problems involving multiplication of large numbers by one or two digit
numbers using efficient mental, written strategies and appropriate digital technologies (ACMNA100)
•exploring techniques for multiplication such as the area model, the Italian lattice
method or the partitioning of numbers
•applying the distributive law and using arrays to model multiplication and explain
calculation strategies.
Solve problems involving division by a one digit number, including those that result in a remainder. (ACMNA101)
•using the fact that equivalent division calculations result if both numbers are divided by the same factor
•interpreting and representing the remainder in division calculations sensibly for the context.
Use efficient mental and written strategies and apply appropriate digital technologies to solve problems. (ACMNA291)
• using calculators to check the reasonableness of answers.
Level 6
Identify and describe properties of prime, composite, square and triangular numbers. (ACMNA122)
•understanding that some numbers have special properties and that these properties can be used to solve
problems
•representing composite numbers as a product of their prime factors and using this
form to simplify calculations by cancelling common primes
•understanding that if a number is divisible by a composite number then it is also
divisible by the prime factors of that number (for example 216 is divisible by 8 because the number
represented by the last three digits is divisible by 8, and hence 216 is also divisible by 2 and 4)
Select and apply efficient mental and written strategies and appropriate digital technologies to solve
problems involving all four operations with whole numbers. (ACMNA123)
•applying strategies already developed for solving problems involving small numbers to those involving large
numbers
•applying a range of strategies to solve realistic problems and commenting on the
efficiency of different strategies.
Investigate everyday situations that use integers. Locate and represent these numbers on a number line.
(ACMNA124)
• understanding that whole numbers can be positive and negative and continue
indefinitely in both directions
•investigating everyday situations that use positive and negative integers, such as temperatures, to
understand how the positive numbers (whole numbers, fractions, decimals and percentages) can be extended
to include negative numbers
• using number lines to position and order positive and negative integers around zero solving everyday
additive problems involving positive and negative integers without developing formal rules for the operations
(for example using a number line and counting to find the resulting outside temperature if it is 5°C at 7pm and
drops by 8°C overnight)
Level 7
Investigate index notation and represent whole numbers as products
of powers of prime numbers. (ACMNA149)
•defining and comparing prime and composite numbers and explaining
the difference between them
•applying knowledge of factors to strategies for expressing whole
numbers as products of powers of prime factors, such as repeated
division by prime factors or creating factor trees
•solving problems involving lowest common multiples and greatest
common divisors (highest common factors) for pairs of whole numbers
by comparing their prime factorisation
Investigate and use square roots of perfect square numbers.
(ACMNA150)
• investigating square numbers such as 25 and 36 and developing
square-root notation
• investigating between which two whole numbers a square root lies.
Apply the associative, commutative and distributive laws to aid mental
and written computation. (ACMNA151)
• understanding that arithmetic laws are powerful ways of describing
and simplifying calculations.
Compare, order, add and subtract integers (ACMNA280)
Facts & Procedures
Order of operations
Addition AND Subtraction.
• Addition and subtraction of whole numbers to 10,000 • Order of operations as the Hierarchical order.
and beyond using column addition algorithm.
Number Characteristics. (Factors)
• Prime numbers (up to at least 20) have only two
distinct factors. Explore the “Sieve of Eratosthenes.”
• Use of Factor Trees to find factors in prime Form,
Expanded form.
• Use of the Prime Factors to find the Lowest Common
Avoid use of Letters standing for Order as this presupposes
Factor of larger numbers.
Ordinal Order rather than Hierarchical order.
• Composite numbers have more than two factors.
A thorough understanding of Hierarchical order of Operations is
• Position and order of numbers relative to other
required
before teaching. The staircase hierarchy above is
numbers and to zero, to the nearest 5 or 10, and their
suggested.
extensions to 2-, 3- and 4- digit numbers
Divisibility and Factors.
• Comparison of number using place value.
Divisibility allows for far better understanding of factors and leads
• Estimation strategies for Operations– familiar
to very efficient mental computation.
reference points, 5, 10, tens, hundreds, thousands
• Focus on Divisibility in the following order –
Rounding is used for estimation purposes.
th
Rules for 5, 10, 2, 4, 8, 3, 6, 9) where there is no remainder.
Booker. G. et al, 4 Ed, Teaching Primary
Note
the patterns within the rules for divisibility
Mathematics, P. 200
Language.
• Use of the rounding strategy.
add, subtract, left, multiply and divide, groups of, rows of, jumps
of, share between, share, odd, even, remainder, divisible, factor,
multiple.
Visual:
Rounding and Order of Operations staircase charts.
e.g. Area model to represent say 4x3=12 where the factors are
Multiplication.
evident in the dimensions.
• Multiplication of 2 digit by 2 digit.
e.g. set notation {2, 4, 6, 8, ...} set of even numbers.
• Multiplication and division by whole numbers up to 9 e.g. Linear model of an extended number line in both directions.
as any multiplication or division is placed in a column
Mental strategies:
containing no more than one digit.
• Estimate: Round first then add or multiply
• Multiplication and division by 10 is a special case.
• Rounding: Operation first then round.
Division.
• Consolidation of multiplication re-call facts.
• Develop algorithm for division by one digit divisor.
Less reliance on skip counting
• Explore division where there is a remainder.
More reliance on recalling tables
Facts & Procedures
Number Characteristics.
Language.
• Whole numbers, square numbers, triangular
add, subtract, left or remainder, multiply and
numbers.
divide, groups of, rows of, jumps of, share
• Extended use of Factor Trees to find factors in prime
between, share, odd, even, divisible, factor,
Form, Expanded form.
multiple, division, multiplicative thinking, greater,
• Use of the Prime Factors to find the Lowest Common lesser.
Visual:
Factor of larger numbers.
Rounding and Order of Operations charts.
• Use of Pascal's Triangle to identify patterns and
pathways, how numbers can relate.
Area model to represent say 4x3=12 where the
• Whole number and tens links made to Key
factors are evident in the dimensions.
percentages: 10%, 20%, 25%, 30%, 40%, 50%, 100%
Set notation {2, 4, 6, 8, ...} set of even numbers.
• Rates express multiplicative relationships between
Linear model of an extended number line in both
unlike quantities.
directions. Number lines from -20 to +20
Extended Number Line.
Mental strategies:
•Position and order of numbers relative to other
• Rules of divisibility
numbers, and to zero, to the nearest 5 or 10 and their
• Inverse (backtracking)
extensions to 2, 3 and 4 digit numbers.
• Factors of numbers,
• Numbers to left are LESSER, (not smaller),
e.g. 27 x 3 = 9 x 3 x 3 = 9 x 9 = 81
• Numbers to the right are GREATER (not bigger).
• Estimate: Round first then add or multiply
Multiplication.
• Rounding: Operation first, then round.
As an area concept.
• Multiplication of 3 digit by 3 digit.
• Consolidation of multiplication re-call facts.
Division.
Automaticity in Re-call.
• Focus on division, not as reverse multiplication but
Less reliance on skip counting
dividing groups into smaller elements, more groups,
More reliance on recalling tables.
fewer per group. Use special chart for loops.
Students must develop multiplicative thinking skills
• Many examples of the meaning and result of
at this level.
division.
This includes process of division as much as it does
Integers.
of multiplication.
• Position on number line as a distance from zero.
Become aware that multiplication does not always
• Developing the concept of negative numbers in
result in greater numbers. E.g. 250 x 0.1.
Become aware that division does not always result
context.
in lesser numbers. E.g. 10 ÷ 0.5.
The area model demonstrates this effectively.
Facts & Procedures
This segment is for only part of the number strand.
Notation.
• Index notation for square numbers, e.g. 6x6,
• Repeated multiplication of simple numbers. E.g. 2x2x2 = 2 power 3
• Powers of 10.
Characteristics of Numbers.
• Factor trees and the powers of prime numbers.
• Use of factor trees to find other factors and sets of primes.
• Extending the number line in both directions
• Positive and negative numbers (integers)
Lesser or Greater numbers, not bigger or smaller.
• Use of number laws to speed computation.
e.g. Commutative, associative, distributive, (leads to factors)
Language.
Use of prefixes – words, abbreviations,
e.g. 20K/20,000, $1.5m/$1.5 million, $3b/$3 billion and their use with
electronic devices.
• calculations for the operations with and without electronic devices
• estimates.
Visual:
Use of squares to determine square sides (square roots)
Number lines from -20 to +20
Mental strategies:
• Rules of divisibility
• inverse (backtracking)
• factors of numbers,
e.g. 27 x 3 = 9 x 3 x 3 = 9 x 9 = 81
• Estimate: Round first then add or multiply
• Rounding: Operation first then round.