FRACTIONS, DECIMALS, MULTIPLICATION, DIVISION, PLACE VALUE
PROFESSIONAL LEARNING BOOKLET
Greetings fellow mathematicians!
This professional learning booklet is in 4 sections, providing opportunities for group professional learning using a variety of implementation strategies:
Section 1: Professional Learning Video
Section 2: Professional learning of related concepts using a jigsaw model
Section 3: Professional Learning combining related concepts
Section 4: Professional Learning using related concept sequences
Simply provide participants with the section that you wish to implement.
Each section provides:
 complete information on the concept
 complete instructions for the professional learning
 complete recording sheet
Professional Learning using the video: (pages 2 – 3)
 allows each participant to develop and share their understanding of the concept sequences of related concepts using the video
Professional learning of related concepts using a jigsaw model: (pages 4 – 18)




allows each participant to become an ‘expert’ in one concept with an ‘expert’ group
provides opportunity for participants to share their ‘expertise’ with their ‘home’ group of teachers they teach closely with
provides opportunity for home group participants to investigate relationships between their concepts
means that 1 teacher in each home group will always have ‘expert’ understanding of 1 concept, allowing them to bring this deep
understanding to future incidental and planned professional sessions, and planning meetings.
Professional Learning combining related concepts: (pages 19 – 29)

allows each participant to develop and share their understanding of related concepts
Professional Learning using related concept sequences: (pages 30 – 42)
 allows each participant to develop and share their understanding of the concept sequences of related concepts
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1
PROFESSIONAL LEARNING USING THE VIDEO
PROFESSIONAL LEARNING USING THE VIDEO: INSTRUCTIONS
1. Watch the video, pausing it to pausing to engage in substantive conversation to deepen your understanding of the
concept. Identify and explain the levels of understanding and the relationships to other concepts. Record answers to the questions.
2. Identify how you could differentiate learning across the concept and related concepts.
Could you identify the current level of understanding of each child through observation, questioning and work samples?
For example,
 some children investigate the concept, using playing cards to generate numbers they are ready to investigate
 some children investigate one of the related concepts, using playing cards to generate numbers they are ready to investigate
Children sit next to children who are investigating a different level of the same concept, or a different related concept, sharing their
understanding at appropriate moments.
When a child is investigating a concept at a level just above their level of current understanding, they can do so independently. They do not need
to be in the company of an adult!
Differentiating in this way, allows every child to develop independence, resilience, problem solving, reasoning, meta-language, deep understanding
and capacity to explain and engage in mathematical conversations.
Website: http://www.alearningplace.com.au
Email: [email protected]
Twitter: @learn4teach
Youtube: A Learning Place A Teaching Place
Facebook: A Learning Place
2
What levels of understanding of this concept do children go through?
How are these levels of understanding related to levels of understanding in related concepts?
How could you differentiate learning across levels of these related concepts?
Website: http://www.alearningplace.com.au
Email: [email protected]
Twitter: @learn4teach
Youtube: A Learning Place A Teaching Place
Facebook: A Learning Place
3
PROFESSIONAL LEARNING USING THE JIGSAW MODEL
PROFESSIONAL LEARNING USING A JIGSAW MODEL: INSTRUCTIONS
1. Break into groups of 4. This group will be your home group.
2. Each of you select the concept that you’d like to become expert in.
3. Join the group with others interested in the same concept. This group is your expert group.
4. Spend 20 minutes reading through the levels of understanding of your concept, pausing to engage in substantive
conversation to deepen your understanding of your concept with your expert group. Record answers to the questions.
5. Go back to your home group.
6. Spend 30 minutes sharing the levels of understanding of your concept. As your home group members are sharing the levels of
understanding of their concept, identify and explain relationships to the concept that you are an expert in. Record answers to the questions.
7. Identify how you could differentiate learning across the related concepts.
Could you identify the current level of understanding of each child through observation, questioning and work samples?
For example,
 some children investigate the concept, using playing cards to generate numbers they are ready to investigate
 some children investigate one of the related concepts, using playing cards to generate numbers they are ready to investigate
Children sit next to children who are investigating a different level of the same concept, or a different related concept, sharing their
understanding at appropriate moments.
When a child is investigating a concept at a level just above their level of current understanding, they can do so independently. They do not need
to be in the company of an adult!
Differentiating in this way, allows every child to develop independence, resilience, problem solving, reasoning, meta-language, deep understanding
and capacity to explain and engage in mathematical conversations.
Website: http://www.alearningplace.com.au
Email: [email protected]
Twitter: @learn4teach
Youtube: A Learning Place A Teaching Place
Facebook: A Learning Place
4
What levels of understanding of this concept do children go through?
EXPERT GROUP
How could we engage children in differentiated investigation of the concept?
Website: http://www.alearningplace.com.au
Email: [email protected]
Twitter: @learn4teach
Youtube: A Learning Place A Teaching Place
Facebook: A Learning Place
5
Share the levels of understanding of your concept.
HOME GROUP
How do the levels of understanding of your concept relate to the levels of understanding of others’ concepts?
How could you differentiate learning across these related concepts?
Website: http://www.alearningplace.com.au
Email: [email protected]
Twitter: @learn4teach
Youtube: A Learning Place A Teaching Place
Facebook: A Learning Place
6
MULTIPLICATION: LEVELS OF UNDERSTANDING
Grade
Year 1:
Additive
Year 2:
Additive
Level of Understanding
Level Description
Divide into ‘groups of’
and skip count by 2s, 5s,
10s
Children divide by 2, 5 or 10 into groups of 2, 5 or 10 and skip
count to find total.
Divide into ‘groups of’,
‘equal groups’, arrays,
describe using division
and multiplication and
fractions
Children divide by dividing
into ‘groups of’ and into
‘equal groups’, recording a
division number sentence.
Multiply and divide by
single-digit numbers,
recalling multiples,
using properties
(distributivity) and
relationships between
multiplication, division
and fractions (Year 3
and 4)
Select cards to make 2 numbers
you are ready to multiply.
Multiply the numbers:
1. making equal groups with
counters in rows, recording
the counters and number
sentence
Children divided into arrays, describing what they did in
2 division number sentences, then describe the array in 2
multiplication number sentences.
Years 3-6:
Multiplicative
Differentiated Guided and Independent
Investigation
2. using the distributive
property to partition into
numbers you can multiply
Children multiply
single-digit numbers
using the distributive
property (Year 3, x 2, x
4, x 3, x 10, x 5, Year 4 x
9, x 6, x 8, x 7).
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Email: [email protected]
Twitter: @learn4teach
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7
Highest common factor
(Year 5)
Children find highest common factors of 2
numbers.
Multiply 2 two-digit
numbers
Children multiply 2 two-digit numbers using
the distributive property in the area model.
Multiply decimals (Year
6)
Children multiply decimals
to thousandths by whole
numbers and powers of 10
Division is multiplication
by a fraction (Year 6)
Children explain that division
by a number greater than 1
makes a number a number of
TIMES smaller, or a fraction of
the number, so we are
multiplying by a fraction.
Website: http://www.alearningplace.com.au
Email: [email protected]
Twitter: @learn4teach
Youtube: A Learning Place A Teaching Place
Facebook: A Learning Place
8
DIVISION: LEVELS OF UNDERSTANDING
Grade
K / P / R:
Informal
Level of Understanding
Make and describe
groups with and
without group markers
Level Description
Children make and describe groups with group markers.
Year 2:
Additive
Children divide by 2 by dividing
into groups of 2 and into 2
equal groups, recording a
division number sentence.
Pick up half
Children divide by 2 by dividing into 2 equal groups
and pick up half of their counters.
Children divide by dividing
into ‘groups of’ and into
‘equal groups’, recording a
division number sentence.
Website: http://www.alearningplace.com.au
Email: [email protected]
1. divide in 2 ways – into
‘groups of’ and count groups
and into ‘equal groups’ and
count the number in each
group
2. dividing into equal rows,
recording the counters and
number sentence
Explain and record how
counters can be divided
by 2 by dividing into
both ‘groups of’ and
‘equal groups’
Divide into ‘groups of’,
‘equal groups’, arrays,
describe using division
and multiplication and
fractions
Select cards to make 2 numbers
you are ready to divide.
Divide the numbers:
Children make and describe groups without group markers.
Year 1:
Additive
Differentiated Guided and Independent
Investigation
3. partitioning into preferred
multiples (multiples of the
number that you already
know), recording remainders
as remainders
4. partitioning into preferred
multiples (multiples of the
number that you already
know), dividing remainders to
create fractions
5. divide by multiplying by a
fraction
Twitter: @learn4teach
Youtube: A Learning Place A Teaching Place
Facebook: A Learning Place
9
Children divided into arrays, describing what they did in
2 division number sentences, then describe the array in 2
multiplication number sentences.
Quarter shapes and
groups
Years 3-6:
Multiplicative
Divide by single-digit
numbers, recalling
multiples, using
properties and
relationships between
multiplication, division
and fractions (Year 3
and 4)
Children divided shapes and
groups in quarters, relating
dividing groups into quarters
with division into 4 equal groups.
Children divide by single-digit
numbers using prefered
multiples and relationships to
multiplication and fractions
(Year 3, ÷ 2, ÷ 4, ÷ 3, ÷ 10, ÷ 5.
Year 4 ÷ 9, ÷ 6, ÷ 8, ÷ 7),
recording remainders as
remainders
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Email: [email protected]
Twitter: @learn4teach
Youtube: A Learning Place A Teaching Place
Facebook: A Learning Place
10
Dividing by single-digit
numbers, dividing the
remainder to create a
fraction (Year 5)
Children divide by singledigit numbers, dividing the
remainder to create a
fraction.
Divide decimals (Year 6) Children divide decimals to thousandths by
whole numbers and powers of 10
c
Division is multiplication
by a fraction (Year 6)
Children explain that division by a
number greater than 1 makes a
number a number of TIMES smaller,
or a fraction of the number, so we
are multiplying by a fraction.
Website: http://www.alearningplace.com.au
Email: [email protected]
Twitter: @learn4teach
Youtube: A Learning Place A Teaching Place
Facebook: A Learning Place
11
FRACTIONS AND DECIMALS: LEVELS OF UNDERSTANDING
Grade
K / P / R:
Informal
Year 1:
Additive
Level of Understanding
Halve shapes
Halve shapes and
groups
Year 2:
Additive
Quarter shapes and
groups
Years 3-6:
Multiplicative
Meaning of the
denominator as the
number we divided by
(Year 3)
Level Description
Children divided shapes in halves
Children divided shapes and groups in halves, relating dividing groups
into halves with division into 2 equal groups.
Children divided shapes and
groups in quarters, relating
dividing groups into quarters
with division into 4 equal groups.
Differentiated Guided and Independent
Investigation
Investigate fractions and
decimals:
1. placing a fraction of the
counters onto each fraction
of the shape
2. creating a fraction wall,
identifying multiplicative
relationships between
fractions, the meaning of the
denominator and numerator
3. extending multiplicative place
value of whole numbers to
include decimals by dividing
‘one’ by 10, explaining
decimals as both fractions
and decimals
4. creating a fraction wall,
identifying equivalent
fractions, and the relationship
between the numerator and
denominator
Children explain the
denominator as the number
we divided by.
5. explaining the meaning of the
vinculum
6. creating fractions in their
simplest form
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Email: [email protected]
Twitter: @learn4teach
Youtube: A Learning Place A Teaching Place
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12
Multiplicative
relationships between
fractions (Year 3)
Children explain
multiplicative
relationships
between fractions
using a fraction wall.
Meaning of the
numerator as the
number of parts we are
concerned with (Year 3)
Children explain the numerator in unit and
non-unit fractions as the number of parts
we are concerned with.
Multiplicative place
value of decimals to
tenths, hundredths
(Year 4), thousandths
(Year 5)
Children explain
multiplicative place
value of decimal
numbers by multiplying
by 10 to get the value
of the digit to the left,
and dividing by 10 to
get the value of the
digit to the right.
Numbers with decimals
– additive, standard,
non-standard
Children explain numbers with
decimals using additive, standard
and non-standard place value.
Website: http://www.alearningplace.com.au
Email: [email protected]
Twitter: @learn4teach
Youtube: A Learning Place A Teaching Place
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13
Equivalent fractions
with concrete material
and the relationship
between numerator and
denominator (Year 4)
Children explain equivalent
fractions using concrete
material and the relationship
between numerator and
denominator.
Role of the vinculum as
meaning divided by
(Year 5)
Children explain the vinculum means divided by.
Fractions in simplest
form (Year 6)
Children create fractions in simplest form, by dividing the
numerator and denominator by highest common factor, creating
an equivalent fraction calculation because they are dividing by 1.
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Email: [email protected]
Twitter: @learn4teach
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14
PLACE VALUE: LEVELS OF UNDERSTANDING
Grade
K / P / R:
Informal,
Additive
Year 1:
Formal,
Additive
Level of Understanding
Differentiated Guided and Independent
Investigation
Level Description
Teen numbers are 10
and … , recording
informally
Children use two 10 frames to explain teen
numbers are 10 and …, and that teen means 10.
Friends of 10, recording
informally
Children use a 10 frame to investigate friends
of 10.
Partitioning single-digit
and teen numbers,
recording informally
Children use large connecting
blocks to investigate partitioning
single-digit and teen numbers.
Place value of 10, teen
numbers – additive,
standard, non-standard
Children use connecting centicubes to
explain 10 and teen numbers using
additive, standard and non-standard place
value.
Friends of 10, recording
formally
Children use a 10 frame to
investigate friends of 10.
Partitioning single-digit
and teen numbers,
recording formally
Children use large connecting blocks
to investigate partitioning single-digit
and teen numbers.
Website: http://www.alearningplace.com.au
Email: [email protected]
Twitter: @learn4teach
4 and 6
Select cards to make numbers
you are ready to find friends of 10,
20, any decade, 100.
1. place a single-digit number of
counters on a 10 frame
2. place a teen number of
counters on two 10 frames
3. use understanding of friends
of 10 to work out friends of
any decade, 100
Select cards to make numbers
you are ready to partition.
1. use connecting larger cubes
to partition single-digit
numbers
2. use connecting larger cubes
to partition 10 and teen
numbers
3. partition two-digit numbers
4. partition 100 and three-digit
numbers
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15
Year 2:
Formal,
Additive
Friends of 20, any
decade recording
formally
Children investigate
friends of 20, any
decade.
Partitioning single-digit
and teen numbers,
recording formally
Children partition two-digit numbers,
identifying place value partitions.
Place value of two-digit
numbers – additive,
standard, non-standard
Children explain two-digit numbers
using additive, standard and nonstandard place value.
100, three-digit
numbers – additive,
standard, non-standard
Children explain
three-digit numbers
using additive,
standard and nonstandard place value.
Count by 10s on and off
the decade
Select cards to make numbers
you are ready to record using
additive, standard and nonstandard place value.
1. use connecting centicubes to
record teen numbers using
additive, standard and nonstandard place value
2. use connecting centicubes to
record twenty-something
numbers using additive,
standard and non-standard
place value
3. record two-, three-, four-,
five-digit numbers using
additive, standard and nonstandard place value
4. record numbers with
decimals using additive,
standard and non-standard
place value
Children count
forwards and
backwards by 10s
on and off the
decade, explaining
which digits
change and why.
Website: http://www.alearningplace.com.au
Email: [email protected]
Twitter: @learn4teach
Youtube: A Learning Place A Teaching Place
Facebook: A Learning Place
16
Years 3 - 6:
Formal,
Additive and
Multiplicative
Friends of 100
Children investigate
friends of 100.
Partitioning tens and
two-digit numbers
Children partition tens
numbers and three-digit
numbers, identifying place
value partitions.
1000, four- and fivedigit numbers –
additive, standard, nonstandard
Children explain 1000, four- and five-digit numbers using additive, standard and
non-standard place value.
Count by 10s, 100s and
1000s on and off the
decade, hundred and
thousand
Children count forwards and backwards by 10s, 100s and 1000s on and off the
decade, hundred and thousand, explaining which digits change and why.
Partitioning 1000, fourand five-digit numbers
Children partition 1000, four- and five-digit numbers, identifying place value
partitions.
Multiplicative place
value of whole numbers
(Year 3)
Children explain multiplicative place value of
whole numbers by multiplying by 10 to get the
value of the digit to the left, and dividing by 10
to get the value of the digit to the right.
Multiplicative place
value of decimals to
tenths, hundredths
(Year 4), thousandths
(Year 5)
Children explain
multiplicative place
value of decimal
numbers by multiplying
by 10 to get the value
of the digit to the left,
and dividing by 10 to
get the value of the
digit to the right.
Website: http://www.alearningplace.com.au
Email: [email protected]
Twitter: @learn4teach
Youtube: A Learning Place A Teaching Place
Facebook: A Learning Place
17
Numbers with decimals
– additive, standard,
non-standard
Children explain numbers with
decimals using additive, standard
and non-standard place value.
Website: http://www.alearningplace.com.au
Email: [email protected]
Twitter: @learn4teach
Youtube: A Learning Place A Teaching Place
Facebook: A Learning Place
18
PROFESSIONAL LEARNING COMBINING RELATED CONCEPTS
PROFESSIONAL LEARNING COMBINED CONCEPTS: INSTRUCTIONS
1. Spend 30 minutes reading through the levels of understanding of the related concepts, pausing to engage in substantive
conversation to deepen your understanding of your concept with your group. Identify and explain relationships between
concepts. Record answers to the questions.
2. Spend 30 minutes identifying how you could differentiate learning across the related concepts.
Could you identify the current level of understanding of each child through observation, questioning and work samples?
For example,
 some children investigate the concept, using playing cards to generate numbers they are ready to investigate
 some children investigate one of the related concepts, using playing cards to generate numbers they are ready to investigate
Children sit next to children who are investigating a different level of the same concept, or a different related concept, sharing their
understanding at appropriate moments.
When a child is investigating a concept at a level just above their level of current understanding, they can do so independently. They do not need
to be in the company of an adult!
Differentiating in this way, allows every child to develop independence, resilience, problem solving, reasoning, meta-language, deep understanding
and capacity to explain and engage in mathematical conversations.
Website: http://www.alearningplace.com.au
Email: [email protected]
Twitter: @learn4teach
Youtube: A Learning Place A Teaching Place
Facebook: A Learning Place
19
COMBINED CONCEPTS PROFESSIONAL LEARNING
Identify and discuss the levels of understanding of each concept with others in your group.
How do the levels of understanding of each concept relate to the levels of understanding of the related concepts?
How could you differentiate learning across levels of these related concepts?
Website: http://www.alearningplace.com.au
Email: [email protected]
Twitter: @learn4teach
Youtube: A Learning Place A Teaching Place
Facebook: A Learning Place
20
MULTIPLICATION, DIVISION, FRACTIONS, DECIMALS, PLACE VALUE
LEVELS OF UNDERSTANDING
Grade
K / P / R:
Informal
Level of Understanding
Make and describe
groups with and
without group markers
Differentiated Guided and Independent
Investigation
Level Description
Children make and describe groups with group markers.
Multiply the numbers:
Children make and describe groups without group markers.
Halve shapes
Children divided shapes in halves
Teen numbers are 10
and … , recording
informally
Children use two 10 frames to explain teen
numbers are 10 and …, and that teen means 10.
Friends of 10, recording
informally
Partitioning single-digit
and teen numbers,
recording informally
Select cards to make 2 numbers
you are ready to multiply.
Children use a 10 frame to investigate friends
of 10.
3. making equal groups with
counters in rows, recording
the counters and number
sentence
4. using the distributive
property to partition into
numbers you can multiply
4 and 6
Children use large connecting blocks
to investigate partitioning singledigit and teen numbers.
Website: http://www.alearningplace.com.au
Email: [email protected]
Twitter: @learn4teach
Youtube: A Learning Place A Teaching Place
Facebook: A Learning Place
21
Year 1:
Additive
Explain and record how
counters can be divided
by 2 by dividing into
both ‘groups of’ and
‘equal groups’
Children divide by 2 by dividing
into groups of 2 and into 2
equal groups, recording a
division number sentence.
Pick up half
Children divide by 2 by dividing into 2 equal groups
and pick up half of their counters.
Halve shapes and
groups
Children divided shapes and groups in halves, relating dividing groups
into halves with division into 2 equal groups.
Divide into ‘groups of’
and skip count by 2s, 5s,
10s
Select cards to make 2 numbers
you are ready to divide.
Divide the numbers:
Children divide by 2, 5 or 10 into groups of 2, 5 or 10 and skip
count to find total.
Place value of 10, teen
numbers – additive,
standard, non-standard
Children use connecting centicubes to
explain 10 and teen numbers using
additive, standard and non-standard place
value.
Friends of 10, recording
formally
Children use a 10 frame to
investigate friends of 10.
Website: http://www.alearningplace.com.au
Email: [email protected]
Twitter: @learn4teach
6. divide in 2 ways – into
‘groups of’ and count groups
and into ‘equal groups’ and
count the number in each
group
7. dividing into equal rows,
recording the counters and
number sentence
8. partitioning into preferred
multiples (multiples of the
number that you already
know), recording remainders
as remainders
9. partitioning into preferred
multiples (multiples of the
number that you already
know), dividing remainders to
create fractions
10.divide by multiplying by a
fraction
Youtube: A Learning Place A Teaching Place
Facebook: A Learning Place
22
Year 2:
Additive
Partitioning single-digit
and teen numbers,
recording formally
Children use large connecting blocks
to investigate partitioning single-digit
and teen numbers.
Friends of 20, any
decade recording
formally
Children investigate
friends of 20, any
decade.
Partitioning single-digit
and teen numbers,
recording formally
Children partition two-digit numbers,
identifying place value partitions.
Place value of two-digit
numbers – additive,
standard, non-standard
Children explain two-digit numbers
using additive, standard and nonstandard place value.
Divide into ‘groups of’,
‘equal groups’, arrays,
describe using division
and multiplication and
fractions
Children divide by dividing
into ‘groups of’ and into
‘equal groups’, recording a
division number sentence.
Select cards to make numbers
you are ready to find friends of 10,
20, any decade, 100.
4. place a single-digit number of
counters on a 10 frame
5. place a teen number of
counters on two 10 frames
6. use understanding of friends
of 10 to work out friends of
any decade, 100
Select cards to make numbers
you are ready to partition.
5. use connecting larger cubes
to partition single-digit
numbers
6. use connecting larger cubes
to partition 10 and teen
numbers
7. partition two-digit numbers
8. partition 100 and three-digit
numbers
Children divided into arrays, describing what they did in
2 division number sentences, then describe the array in 2
multiplication number sentences.
Website: http://www.alearningplace.com.au
Email: [email protected]
Twitter: @learn4teach
Youtube: A Learning Place A Teaching Place
Facebook: A Learning Place
23
Quarter shapes and
groups
Children divided shapes and
groups in quarters, relating
dividing groups into quarters
with division into 4 equal groups.
100, three-digit
numbers – additive,
standard, non-standard
Children explain
three-digit numbers
using additive,
standard and nonstandard place value.
Count by 10s on and off
the decade
Children count
forwards and
backwards by 10s
on and off the
decade, explaining
which digits
change and why.
Friends of 100
Children investigate
friends of 100.
Partitioning tens and
two-digit numbers
Children partition tens
numbers and three-digit
numbers, identifying place
value partitions.
Website: http://www.alearningplace.com.au
Email: [email protected]
Twitter: @learn4teach
Select cards to make numbers
you are ready to record using
additive, standard and nonstandard place value.
5. use connecting centicubes to
record teen numbers using
additive, standard and nonstandard place value
6. use connecting centicubes to
record twenty-something
numbers using additive,
standard and non-standard
place value
7. record two-, three-, four-,
five-digit numbers using
additive, standard and nonstandard place value
8. record numbers with
decimals using additive,
standard and non-standard
place value
Youtube: A Learning Place A Teaching Place
Facebook: A Learning Place
24
Years 3-6:
Multiplicative
Multiply and divide by
single-digit numbers,
recalling multiples,
using properties
(distributivity) and
relationships between
multiplication, division
and fractions (Year 3
and 4)
Children multiply single-digit numbers using the distributive property
(Year 3, x 2, x 4, x 3, x 10, x 5, Year 4 x 9, x 6, x 8, x 7).
Investigate fractions and
decimals:
7. placing a fraction of the
counters onto each fraction
of the shape
8. creating a fraction wall,
identifying multiplicative
relationships between
fractions, the meaning of the
denominator and numerator
9. extending multiplicative place
value of whole numbers to
include decimals by dividing
‘one’ by 10, explaining
decimals as both fractions
and decimals
Children divide by single-digit
numbers using prefered
multiples and relationships to
multiplication and fractions
(Year 3, ÷ 2, ÷ 4, ÷ 3, ÷ 10, ÷ 5.
Year 4 ÷ 9, ÷ 6, ÷ 8, ÷ 7),
recording remainders as
remainders
10.creating a fraction wall,
identifying equivalent
fractions, and the relationship
between the numerator and
denominator
11.explaining the meaning of the
vinculum
12.creating fractions in their
simplest form
Website: http://www.alearningplace.com.au
Email: [email protected]
Twitter: @learn4teach
Youtube: A Learning Place A Teaching Place
Facebook: A Learning Place
25
Meaning of the
denominator as the
number we divided by
(Year 3)
Children explain the
denominator as the number
we divided by.
1000, four- (Year 3) and
five-digit (Year 4)
numbers – additive,
standard, non-standard
Children explain 1000, four- and five-digit numbers using additive, standard and
non-standard place value.
Count by 10s, 100s and
1000s on and off the
decade, hundred and
thousand
Children count forwards and backwards by 10s, 100s and 1000s on and off the
decade, hundred and thousand, explaining which digits change and why.
Partitioning 1000, four(Year 3) and five-digit
(Year 4) numbers
Children partition 1000, four- and five-digit numbers, identifying place value
partitions.
Multiplicative place
value of whole numbers
(Year 3)
Children explain multiplicative place value of
whole numbers by multiplying by 10 to get the
value of the digit to the left, and dividing by 10
to get the value of the digit to the right.
Multiplicative
relationships between
fractions (Year 3)
Children explain
multiplicative
relationships
between fractions
using a fraction wall.
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Meaning of the
numerator as the
number of parts we are
concerned with (Year 3)
Children explain the numerator in unit and
non-unit fractions as the number of parts
we are concerned with.
Multiplicative place
value of decimals to
tenths, hundredths
(Year 4), thousandths
(Year 5)
Children explain
multiplicative place
value of decimal
numbers by multiplying
by 10 to get the value
of the digit to the left,
and dividing by 10 to
get the value of the
digit to the right.
Numbers with decimals
– additive, standard,
non-standard (Year 4, 5,
6)
Children explain numbers with
decimals using additive, standard
and non-standard place value.
Equivalent fractions
with concrete material
and the relationship
between numerator and
denominator (Year 4)
Children explain equivalent
fractions using concrete
material and the relationship
between numerator and
denominator.
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Highest common factor
(Year 5)
Children find highest common factors of 2
numbers.
Dividing by single-digit
numbers, dividing the
remainder to create a
fraction (Year 5)
Children divide by singledigit numbers, dividing the
remainder to create a
fraction.
Role of the vinculum as
meaning divided by
(Year 5)
Children explain the vinculum means divided by.
Multiply 2 two-digit
numbers
Children multiply 2 two-digit numbers using
the distributive property in the area model.
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Multiply and divide
decimals (Year 6)
Children multiply and
divide decimals to
thousandths by whole
numbers and powers of 10
Division is multiplication
by a fraction (Year 6)
Children explain that division
by a number greater than 1
makes a number a number of
TIMES smaller, or a fraction of
the number, so we are
multiplying by a fraction.
Fractions in simplest
form (Year 6
Children create fractions in simplest form, by dividing the
numerator and denominator by highest common factor, creating
an equivalent fraction calculation because they are dividing by 1.
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PROFESSIONAL LEARNING USING RELATED CONCEPT SEQUENCES
PROFESSIONAL LEARNING USING RELATED CONCEPT SEQUENCES: INSTRUCTIONS
1. Spend 30 minutes reading through the levels of understanding of the concept, pausing to engage in substantive
conversation to deepen your understanding of the concept with your group. Identify and explain the levels of understanding and
the relationships to other concepts. Record answers to the questions.
2. Spend 30 minutes identifying how you could differentiate learning across the concept and related concepts.
Could you identify the current level of understanding of each child through observation, questioning and work samples?
For example,
 some children investigate the concept, using playing cards to generate numbers they are ready to investigate
 some children investigate one of the related concepts, using playing cards to generate numbers they are ready to investigate
Children sit next to children who are investigating a different level of the same concept, or a different related concept, sharing their
understanding at appropriate moments.
When a child is investigating a concept at a level just above their level of current understanding, they can do so independently. They do not need
to be in the company of an adult!
Differentiating in this way, allows every child to develop independence, resilience, problem solving, reasoning, meta-language, deep understanding
and capacity to explain and engage in mathematical conversations.
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CONCEPT SEQUENCE PROFESSIONAL LEARNING
Identify and discuss the levels of understanding of the concept with others in your group.
How do the levels of understanding of the concept relate to the levels of understanding of the related concepts?
How could you differentiate learning across levels of these related concepts?
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YEAR 2
YEAR 1
MULTIPLICATION AND DIVISION CONCEPT SEQUENCE
1
Divide by making 'groups of' and 'equal groups'.
2
Divide by 2 by making 2 equal groups, determine how many in each group, and describe part left
over, describe halves.
Essential related concept is
Fractions and Decimals 2.
Divide by 2 by making groups of 2, determine the number of groups, and describe part left over.
3
Rhythmic / skip count forwards and backwards by 2s, 5s and 10s naming multiples on a number line and on a hundred chart.
4
Divide into groups of 2, 5 and 10, and find total using skip and rhythmic counting.
5
Divide into equal rows (array) with no remainder, then describe using multiplication.
Find total using skip counting and by number of rows and number in each row.
Record multiplication and division as repeated addition and subtraction on a number line.
Explain even numbers as counters paired in 2 equal rows.
Essential related concept is
Measurement and
Geometry 23.
6
Multiplication of coins and notes to make equivalent values, 20 cents = 2 x 10 cents, 5 x 20 cents
= $1, 4 x $5 = $20.
Essential related concept is
Money and Financial
Mathematics 8.
7
Divide by making ‘groups of …’ and count groups, and making ‘… equal groups’ and count
counters in each group.
Divide into 'groups of …' and '… equal groups' and describe part left over.
Essential related concept is
Fractions and Decimals 2, 4.
8
Divide into 'groups of 4' and '4 equal groups' and describe part left over.
Divide by 4 by grouping into 4 equal groups, determine how many in each group, describe part
left over, quarters.
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YEAR 3
YEAR 4
9
Multiplication and division by 10 using multiplicative place value.
10
Multiplication and division by 2 using properties and relationships.
11
Multiplication and division by 4 using properties and relationships.
12
Multiplication and division by 3 using properties and relationships.
13
Multiplication and division by 5 using properties and relationships.
14
Multiplication and division by 9 using properties and relationships.
15
Multiplication and division by 6 using properties and relationships.
16
Multiplication and division by 8 using properties and relationships.
17
Multiplication and division by 7 using properties and relationships.
18
Describe patterns formed by skip and rhythmic counting forwards and backwards by 3, 4, 6, 7,
8, 9, 10 identifying the terms as multiples of 3, 4, 6, 7, 8, 9, 10 identifying the rule and terms
through multiplication
Describe patterns formed by skip and rhythmic counting forwards and backwards by 3, 4, 6, 7,
8, 9, 10 from any point on number line identifying the rule.
Essential related concept is
Patterns and Algebra 23.
19
Solve word problems using number sentences involving multiplication and division.
Essential related concept is
Patterns and Algebra 25.
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Twitter: @learn4teach
Essential related concepts are
Place Value 6, 7, 8, 11, 12, 13,
14, 15, 16, 17,
Patterns and Algebra 15, 18,
Fractions and Decimals 7,
Measurement and Geometry
42.
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20
Highest common factor.
21
Explain equivalent division calculations result if both numbers are divided by the same factor.
YEAR 6
YEAR 5
Create and solve equivalent number sentences involving multiplication and division.
Essential related concept is
Patterns and Algebra 26.
Essential related concept is
Fractions and Decimals 19.
22
Dividing by single-digit numbers, dividing the remainder to create a fraction.
23
Divisibility Tests.
24
Multiplication of 2 two-digit numbers using properties and relationships, including the distributive property in the area model.
25
Exposure to formal algorithms for Multiplication and Division (NSW).
26
Multiply and divide decimals to thousandths by whole numbers and powers of 10.
Record remainders as fractions and decimals when dividing by 10.
27
Division is multiplication by a fraction.
28
Prime and composite numbers.
Explain composite numbers as the product of prime factors.
29
30
Use prime factors to simplify calculations.
Identify and explain square numbers.
Essential related concept is
Patterns and Algebra 29.
Missing and equivalent number sentences using order of operations and grouping
symbols.
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Essential related concept is
Fractions and Decimals 26,
Place Value 29
Measurement and Geometry 59, 64.
Essential related concept is
Fractions and Decimals 28.
Essential related concept is
Patterns and Algebra 32.
Twitter: @learn4teach
Essential related concepts are
Addition and Subtraction 36,
Patterns and Algebra 31.
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KINDERGARTEN
2
Halve shapes, lengths and groups.
3
Quarter shapes and lengths by quartering and by halving a half.
4
Quarter groups by quartering (dividing into 4 equal groups) and by halving a half.
5
Eighth shapes and lengths by eighthing, by halving a quarter and by quartering a half.
6
Eighth groups by eighthing (dividing into 8 equal groups), by halving a quarter and by quartering a half.
7
Role of the denominators 2, 4, 3, 5, 8, 10, as the number we have divided by.
8
Multiplicative relationships between fractions while building a fraction wall.
9
Non-unit fractions and the role of numerator as the number of parts we are concerned with.
10
Locate fractions on a number line and identify that 𝟐, 𝟒, 𝟑, 𝟓, 𝟖, 𝟏𝟎 = 1 including on a number line.
YEAR 3
YEAR 2
1
YEAR 1
FRACTIONS AND DECIMALS CONCEPT SEQUENCE
Essential related concept is
Measurement and Geometry
(Area) 10.
Halve shapes and lengths, explaining it is half as big / long.
Halve different shapes, explaining why halves are different sizes / areas.
Essential related concepts are
Patterns and Algebra 9,
Measurement and Geometry 27.
Essential related concept is
Measurement and Geometry 27.
Essential related concepts are
Multiplication and Division 9, 10,
11, 12, 13, 14, 15, 16, 17, Patterns
and Algebra 15.
𝟐 𝟒 𝟑 𝟓 𝟖 𝟏𝟎
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11
Explain multiplicative place value of decimals to tenths by dividing 1 by 10 to get tenths.
Explain multiplicative place value of decimals to tenths by multiplying tenths by 10 to get 1.
Explain standard and non-standard place value of decimals to tenths.
Express tenths as both fraction and decimal.
Essential related concepts are
Place Value 18, 20,
Measurement and Geomery
39, 45, 47.
12
Multiplicative place value of decimals to hundredths by dividing a tenth by 10.
Essential related concepts are
Place Value 21,
Measurement and Geomery
39, 45, 47.
YEAR 4
Standard and non-standard place value of decimals to hundredths, expressing hundredths as both
fraction and decimal.
13
Equivalent fractions with concrete material and the relationship between numerator and denominator.
14
Equivalent fractions on a number line.
15
Number patterns involving fractions, (halves, quarters and thirds), that increase through addition
and decrease through subtraction, including on number line.
Essential related concepts are
Addition and Subtraction 26,
Patterns and Algebra 22.
16
Recognise that amounts of money are written with two decimal places.
Recognise cents as a fraction of a dollar.
Identify other countries’ currencies as decimal.
Essential related concepts are
Money and Financial
Mathematics 11,
Place Value 22.
17
Round a number with one or two decimal places to the nearest whole number.
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Multiplicative place value of whole numbers and decimals to hundredths by multiplying and
dividing by 10, 100 and 1000.
Standard and non-standard place value of decimals to hundredths, expressing hundredths as both
fraction and decimal.
Essential related concepts are
Place Value 24,
Measurement and Geometry
51, 57.
19
Role of the vinculum as meaning divided by.
20
Division with remainders divided to create fractions.
Essential related concept is
Multiplication and Division
28.
21
Multiplicative place value of whole numbers and decimals to thousandths by multiplying and
dividing by 10, 100 and 1000.
Express decimals to thousandths as both fractions and decimals.
Standard and non-standard place value of decimals to thousandths, expressing thousandths as
both fraction and decimal.
Essential related concepts are
Place Value 25,
Measurement and Geometry
51, 57.
22
Order fractions and decimals on a number line.
Essential related concept is
Place Value 26.
23
Add and subtract fractions and mixed numerals with the same denominator.
Essential related concept is
Addition and Subtraction 29.
24
Number patterns involving fractions, decimals and whole numbers, that increase through addition
and decrease through subtraction, including on number line.
Essential related concepts are
Patterns and Algebra 27,
Place Value 27.
YEAR 5
18
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25
Multiplicative place value of whole numbers and decimals of any size by multiplying and
dividing by 10, 100 and 1000.
Express decimals as both fractions and decimals.
Essential related concepts are
Place Value 28,
Measurement and Geometry 59, 64.
26
Multiply and divide decimals to thousandths by whole numbers and powers of 10.
Essential related concepts are
Multiplication and Division 26,
Place Value 29,
Measurement and Geometry 59, 64.
Essential related concepts are
Multiplication and Division 28
YEAR 6
Record remainders as fractions and decimals when dividing by 10.
27
Division is multiplication by a fraction.
28
Fractions in simplest form, dividing numerator and denominator by highest common factor, equivalent fraction calculation.
29
Percentages as hundredths, related to fractions and decimals including on a number line.
30
Calculate percentage discounts of 10%, 25% and 50% on sale items.
Essential related concept is Money
and Financial Mathematics 13.
31
Describe probabilities using fractions, decimals and percentages, including on a number
line.
32
Add and subtract whole numbers and decimals of any size using place value.
33
Estimate and add and subtract fractions and mixed numerals with related denominators.
34
Number patterns involving fractions, decimals and whole numbers, including geometric
number patterns, that increase through addition and decrease through subtraction,
including on number line, in a table, describing the rule.
Essential related concept is Statistics
and Probability 18, Place Value 30.
Essential related concept is
Addition and Subtraction 30.
Essential related concept is
Addition and Subtraction 31.
Essential related concepts are
Patterns and Algebra 28,
Place Value 33,
Addition and Subtract 32.
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YEAR 1
KINDERGARTEN
PLACE VALUE CONCEPT SEQUENCE
1
Partition single-digit numbers, and 10, into 2 or more equal or unequal parts, informal recording.
2
Make friends of 10 using 10 frames, informal recording.
3
Partition teen numbers into 2 or more equal and unequal parts, informal recording.
4
Make teen numbers as '10 and ...' using two 10 frames, informal recording.
5
Make friends of 20 using two 10 frames, informal recording.
6
Make friends of 10, through addition and commutativity, and through subtraction.
7
Explain place value of teen numbers as 10 and ..., recording on a place value chart.
8
Partition single-digit numbers and teen numbers in place value and non-place value.
9
Make friends of 20, through addition and commutativity, and through subtraction.
10
Identify friends of any decade, through addition and commutativity, and through subtraction.
11
Essential related concepts are
Addition and Subtraction 6
and 7,
Multiplication and Division 9,
10, 11, 12, 13, 14, 15, 16, 17.
Essential related concept is
Addition and Subtraction 8.
Essential related concept is
Addition and Subtraction 9.
Read and order two-digit numbers using place value including on a place value chart, on a number Essential related concepts are
Addition and Subtraction 13,
line and on a hundred chart.
14, 15, 16, 17,
Explain standard place value of two-digit numbers grouping in tens and ones, formal recording Multiplication and Division 9,
including a place value chart.
10, 11, 12, 13, 14, 15, 16, 17.
Explain non-standard place value of two-digit numbers grouping flexibly in tens and ones.
Partition two-digit numbers using standard place value and non-place value.
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YEAR 2
Count forwards and backwards by 10s on and off the decade from two-digit numbers.
13
Partition tens numbers.
14
Identify friends of any 100, through addition and subtraction.
Essential related concepts are
Addition and Subtraction 13,
14, 15, 16, 17,
Patterns and Algebra 11, 12,
Multiplication and Division 9,
10, 11, 12, 13, 14, 15, 16, 17,
Measurement and Geometry
21.
15
Explain standard and non-standard place value of three-digit numbers grouping flexibly in
hundreds, tens and ones.
Read and order three-digit numbers on a number line and hundred charts.
Partition three-digit numbers using standard and non-standard place value and non-place value.
16
Count forwards and backwards by 100s, 10s and 1s on and off the hundred and decade from
three-digit numbers.
Essential related concepts are
Addition and Subtraction 13,
14, 15, 16, 17, 21,
Multiplication and Division 9,
10, 11, 12, 13, 14, 15, 16, 17.
17
Count forwards and backwards by 100s and 1000s on and off the decade, hundred and thousand
from four-digit numbers, including over 1000s.
Explain standard and non-standard place value of four-digit numbers.
Partition four-digit numbers using standard and non-standard place value and non-place value.
Essential related concepts are
Addition and Subtraction 14,
21, 23, 24,
Patterns and Algebra 16,
Multiplication and Division 9,
10, 11, 12, 13, 14, 15, 16, 17,
21, 24.
18
Explain multiplicative place value of whole numbers to ten-thousands by multiplying by 10.
Essential related concepts are
Fractions and Decimals 11,
Measurement and Geometry
30, 35, 36, 39.
YEAR 3
12
Explain multiplicative place value of whole numbers to ten-thousands by dividing by 10.
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YEAR 4
19
Explain standard and non-standard place value of five-digit numbers.
Partition five-digit numbers using standard and non-standard place value and non-place value.
Read, order five-digit numbers using place value.
Count forwards and backwards by 1000s, 100s, 10s and 1s on and off the decade, hundred and
thousand from five-digit numbers.
Essential related concepts are
Addition and Subtraction 21,
23, 24,
Patterns and Algebra 20,
Multiplication and Division 9,
10, 11, 12, 13, 14, 15, 16, 17,
21, 24.
20
Explain multiplicative place value of decimals to tenths by dividing 1 by 10 to get tenths.
Explain multiplicative place value of decimals to tenths by multiplying tenths by 10 to get 1.
Explain standard and non-standard place value of decimals to tenths.
Essential related concepts are
Fractions and Decimals 11,
Measurement and Geometry
39, 45, 47.
Express tenths as both fraction and decimal.
21
Explain multiplicative place value of decimals to hundredths by dividing a tenth by 10 to get
hundredths.
Explain multiplicative place value of decimals to hundredths by multiplying hundredths by 10 to
get tenths.
Explain standard and non-standard place value of decimals to hundredths.
Expressing hundredths as both fraction and decimal.
Essential related concepts are
Fractions and Decimals 12,
Measurement and Geometry
39, 45, 47.
22
Recognise amounts of money are written with two decimal places.
Essential related concepts are
Money and Financial
Mathematics 11,
Fractions and Decimals 16.
Identify other countries’ currencies as decimal.
23
Round a number with one or two decimal places to the nearest whole number.
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YEAR 5
24
Multiplicative place value thousandths by multiplying and dividing by 10, 100 and 1000.
25
Express decimals to thousandths and beyond using standard and non-standard place value
as both fractions and decimals.
26
Order decimals on a number line.
27
Explain patterns that increase or decrease by adding or subtracting decimals on a number
line.
28
Explain multiplicative place value of whole numbers and decimals of any size by
multiplying and dividing by 10, 100 and 1000.
29
Multiply and divide decimals to thousandths by whole numbers and powers of 10.
YEAR 6
Record remainders as fractions and decimals when dividing by 10.
30
Describe probabilities using fractions, decimals and percentages, including on a number
line.
31
Explain number patterns that increase and decrease by adding and subtracting decimals,
generalising and describing the rules.
32
Add and subtract decimals using place value.
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Essential related concept is
Fractions and Decimals 18, 21,
Measurement and Geometry 51, 57.
Essential related concept is
Fractions and Decimals 22.
Essential related concepts are
Addition and Subtraction 30,
Fractions and Decimals 24,
Patterns and Algebra 27.
Essential related concept is
Fractions and Decimals 25,
Measurement and Geometry 59, 64.
Essential related concept is
Multiplication and Division 26,
Fractions and Decimals 26,
Measurement and Geometry 59, 64.
Essential related concept is
Fractions and Decimals 32.
Essential related concepts are
Addition and Subtraction 32,
Patterns and Algebra 28,
Fractions and Decimals 34.
Essential related concepts are
Addition and Subtraction 30,
Fractions and Decimals 33.
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