Multiplication and Division 13, Patterns and Algebra 18_Explicit Learning Plan
(Year 3) ACMNA056, ACMNA057, NSW MA2-6NA
Multiplication and division by 5 using properties and relationships.
Multiply single-digit, two-digit numbers by 5 using the distributive property and relationships.
Divide low two-digit, higher two-digit numbers by 5 using properties and relationships, associating dividing into equal groups with fractions.
Resources: Playing cards, pencil, paper
EXPLICIT LEARNING
Focuses
Children think about, talk and listen to a friend about, then have the
children’s
opportunity to share what they already know.
thoughts on the
concept, exposing
current
understanding and
any
misconceptions
Reviews thinking
additively in
Years 1 and 2,
and thinking
multiplicatively
from Year 3 on
►
Today brings an investigation about multiplication and
division.
►
What do you know about multiplication and division?
►
Talk about multiplication and division with a friend.
►
Is anyone ready to share what they are thinking about
multiplication and division?

In Year 1 and 2, you thought of Multiplication additively
as repeatedly adding equal groups and skip or rhythmic
counting.

We found that when we skip count by 5, we are saying
the multiples of 5.

We made an array of equal rows of counters.

And we described it using multiplication.
Record, for example, 5, 10, 15, 20, 25,
Record, for example, multiples of 5
Display an array, for example,
Record, for example, 5 rows of 3 = 15
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1
Reviews arrays
(Multiplication
and Division 5)
Record, for example, 3 rows of 5 = 15
Record, for example, 5 x 3 = 15
Reviews recording
multiplication
using the
multiplication sign
(Multiplication
and Division 5)
Reviews
multiplying by 2
or 4 makes a
number 2 or 4
times larger
(Multiplication
and Division 11, 12
Patterns and
Algebra 17)
Record, for example, 7 x 4 =
5 + 2
Record, for example, 5 x 4 = 20

We found we could describe the array as 5 rows of 3.

And we found we could describe the array as 3 rows of 5.

We found we could record the number sentence using a
multiplication sign.

And we found the number sentence said, 5 times 3.

We found the number sentence said, 5 times 3, because
we didn’t have 3 one time, we didn’t have 3 two times,
we didn’t have 3 three times, we didn’t have 3 four
times, we had 3 five times.

Since Year 3, we have been thinking about multiplication
multiplicatively

When we multiplied by a whole number greater than
one, the number becomes a number of times larger. If
we multiply by 2, the number becomes 2 times larger. If
we multiply by 4, the number becomes 4 times larger.
This is how we now think about multiplication.

We’ve investigated multiplying by 2, by 4 and by 3.

And we found we could use relationships and properties.

We found we could multiply by 2, by 4 and by 3 using the
distributive property.

We could distribute the multiplication over addition. So if
we didn’t know what 7 times 4 equalled, we could
partition 7 into numbers that we could multiply by 4,
multiply them by 4, then add the products.

Do you think the distributive property would work when
we multiply by 5?
2x4=8
Record, for example, 20 + 8 = 28
Record, for example, 7 x 4 = 28
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2
Introduces
multiplying a
single-digit
number by 5
using the
distributive
property (top)
Record, for example, 5 × 7 =
NB: When a child knows what 5 times 7 equals without skip counting, they don't
need to partition. Children partition as much as they need to.
Reviews multiples
(Multiplication
and Division 3)
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
Let's investigate multiplying a single-digit number by 5.

How could we multiply 7 by 5?

Could we skip count by 5?

If we skip count by 5, would we be adding or
multiplying?

When you were in Year 2 you would have skip counted.

Could we learn what 5 times 7 equals by memory?

We could, but that would stop us from learning about
wonderful properties and relationships, and would only
help up to multiply 5 by single-digit numbers.

Could we multiply 5 times 7 without skip counting? Let's
investigate!

If we don't know what 5 times 7 equals, how could we
partition 7 to multiply it by 5?

Could we use other multiples of 5 that we know?

What multiples of 5 do you know?

Do you know what 5 times 5 equals?

What does 5 times 5 equal?

Do you know that 5 times 5 equals 25?

Do you know what 5 times 2 equals?

What does 5 times 2 equal?

Do you know that 5 times 2 equals 10?

Could we use the multiples of 5 that we know to multiply
5 by 7? Let's investigate!
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3
Partition 7, for example, into 5 and 2
Reviews
partitioning
Record, for example,
(Place Value 3, 8,
5 x 7 =
11)
5 + 2
Record, for example, 5 x 5 = 25
Introduces
multiplying the
parts by 5
Record, for example, 5 x 2 = 10
Reviews adding
the products
Record, for example, 25 + 10 = 35
(Multiplication
and Division 11, 12,
Record, for example, 5 x 7 = 35
13)
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
Let’s partition 7 into 5 and 2.

Now that we have partitioned the 7 into 2 and 5, could
we multiply 5 by 5, then multiply 5 by 5?

What does 5 times 5 equal?

Does 5 times 5 equal 25?

What does 5 times 2 equal?

Does 5 times 2 equal 10?

So we didn’t know what 5 times 7 equalled.

So we partitioned 7 into 5 and 2, and multiplied these
parts by 5.

What could we now do to work out what 5 times 7
equals?

Would we add 5 and 2 together to make 7?

So would we add these answers together to make 5
times 7?

Let’s add 25 and 10.

What does 25 plus 10 equal?

Does 25 plus 10 equal 35?

Does 5 times 7 equal 35?

Do you now know what 5 times 7 equals?

Do you now know a multiple of 5?

Do you know 5 times 7 equals 35?

Does that make sense? If we make 7 five times larger,
would it be 35?
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4
Introduces
knowing a
multiple of 5

When we multiply, the answer is called the product.

You’ve started calling the answers ‘products’ now that
you are in Year 3!
Reviews product
(Multiplication
and Division 11, 12
Patterns and
Algebra 17)

What is the product of 5 and 7?

Is the product of 5 and 7, 35?

Does product mean we multiplied 5 times 7?
5 x 7 =

Did you notice that we partitioned 7 into 5 and 2?
5 + 2

Then we multiplied 5 by those parts of 7.

Then we added the products.

You have just discovered a very important property of
multiplication!

When we partition the number to multiply it, then add
the products, we are using the distributive property.

It’s called the distributive property because we
distributed the multiplication over addition.

You’re going to investigate the distributive property over
the next 4 years!

Did we use addition to make our multiplication easier?

Did we distribute our multiplication over addition?

Do you think that would work for all numbers that we
want to multiply by 5?

How else could we multiply 5 by 7?
Reviews the
distributive
property
(Multiplication
and Division 11, 12
Patterns and
Algebra 17)
Point to where we partitioned 7, for example,
Point to where we multiplied the parts of 7 by 5, for example,
5 x 5 = 25
5 x 2 = 10
Point to where we added the products, for example,
35 + 10 = 35
Record, for example, distributive
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5
Introduces
multiplying by 5
by multiplying by
10 and halving the
product
NB: Ensure
children initially
investigate BOTH
ways of
multiplying by 5
concurrently,
because dividing
by 10 and doubling
is easy for tens
numbers but
becomes
challenging when Record, for example, 10 x 7 = 70
the division by 10
results in a tenth

Do you think there is a relationship between multiplying
by 10 and multiplying by 5?

What is the relationship between 5 and 10?

Is 5 half of 10?

And is 10, double 5?

Let’s start by multiplying 7 by 10

We’ve investigated multiplying by 10 using multiplicative
place value.

We found that we can multiply by 10 by moving the
digits one place to the left because the value of the
column to the left is 10 times greater than the column on
the right.

So when we multiply 7 by 10, the 7 will simply move into
the column on the left.

So 10 times 7 equals 70.

So we have ten 7s, and we only want five 7s.

Do we twice as many 7s as we want?

Do we only want half as many 7s?

If we only want half as many 7s, could we halve ten
sevens?

What is half of 70?

If we can’t half 70, could we partition 70 into 60 and 10,
and halve 60 and halve 10?

Is half of 70, 35?
1
Record, for example, 2 of 70 = 35
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6
Record, for example, 5 x 7 = 35
NB: Ensure children develop a range of strategies for multiplying by 5,
and don’t rely on multiplying by 10 and halving, because dividing by 5
by dividing by 10 and doubling is problematic when dividing by 10
leaves a remainder.
Reviews
differentiating the
investigation for
children as they
demonstrate
understanding

So we multiplied 7 by 5, by multiplying by 10 and then
halving the product.

So 5 x 7 equals 35.

Does that make sense? If we make 7 five times larger,
would it be around 35?

Do we have 2 ways to multiply by 5?

So do we now know a multiple of 5?

What does 5 times 7 equal? Does 5 times 7 equal 35?

If we know that 5 times 7 equals 35, do we also know
that 7 times 5 equals 35?

Today brings an investigation about dividing by 5.
Allow children time now to engage in guided and independent
investigation (at the end of this teaching plan) of multiplying singledigit numbers by 5 using the distributive property, and by
multiplying by 10 and halving, learning multiplication facts through
properties and relationships. When children are able to recall, with
understanding, multiples of 5 up to 10 times 5, they begin to divide
by 5
A child who has not demonstrated understanding of multiplication by
5 by making 5 groups of … will continue to investigate at this level.
(Multiplication and Division 5)
Focuses
children’s
thoughts on the
concept, exposing
current
understanding and
A child could be sitting next to a child who is investigating at a
different level. They will explain their current levels of understanding
to one another as they investigate. This is a research-based way to
accelerate learning for children at all levels.
Children think about, talk and listen to a friend about, then have the
opportunity to share what they already know.
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7
any
misconceptions
Reviews thinking
additively in
Years 1 and 2,
and thinking
multiplicatively
from Year 3 on
(Multiplication
and Division 11, 12
Patterns and
Algebra 17)
Reviews seeing
division in 2
ways, as ‘groups
of …’ and as ‘…
equal groups’
(Multiplication
and Division 7)

What do you know about dividing by 5?

Talk about dividing by 5 with a friend.

Is anyone ready to share what they are thinking about
dividing by 5?

In Year 1 and 2, you thought of Division additively as
repeatedly subtracting equal groups and skip or rhythmic
counting.

We found that when we skip count by 5, we are saying
the multiples of 5.

We found that there are 2 ways we can see dividing by 5.

We found we could divide by 5 by making ‘groups of 5’.

And we found that we counted the number of groups.

So we found that 15 divided into groups of 5 equals 3
groups.

We found we could record the number sentence using a
division sign so that everyone around the world could
read it.
Divide the counters by 3 by
making ‘3 equal groups’, for
example,

We found we could divide by 5 by making ‘5 equal
groups’.

And we found that we counted the number of counters
in each group.
Record, for example, 15 ÷ 5 = 3

So we found that 15 divided into 5 equal groups equals 3
counters in each group.
When children can recall most multiples of 3 up to 10 x 3 with
understanding and without skip counting, they begin to divide by 3
Record, for example, 5, 10, 15, 20, 25, …
Record, for example, multiples of 5
Display 15 counters, for example,
Divide the counters by 5 by
making ‘groups of 5’, for
example,
Record, for example, 15 ÷ 5 = 3
Display 12 counters, for example,
Introduces seeing
‘5 equal groups’
as efficient with
larger numbers
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8
Reviews arrays
(Multiplication
and Division 5)

When we divide a small number by 5, it doesn’t matter
which way we see it,

But when we divide larger numbers by 5, it is more
efficient to see dividing by 5 as making ‘5 equal groups’,
because then we are just fifthing!

We divided counters into an array of equal rows.

And we found that an array makes it easier to see if the
groups are equal.

We found we could describe what we did in a number
sentence; we started with 15 counters, and we divided
them into 5 equal rows, and we had 3 in each row.

And we found the number sentence said, 15 divided by 5
equals 3.

We found that we can think about dividing by 5 as
fifthing.

When we fifth we divided the group into 5 equal groups.

We found that we could look at the array in a different
way see that we have also divided 15 into 3 equal rows
of 5.

And the number sentence says 15 divided by 3 equals 5.
Display 15 counters, for example,
Divide the counters into 5 equal rows, array,
for example,
Reviews recording
number sentences
using a division
sign
(Multiplication
and Division 5)
Reviews seeing
arrays in 2 ways
(Multiplication
and Division 7)
Reviews seeing
multiplication
multiplicatively
(Multiplication
and Division 11, 12
Patterns and
Algebra 17)
Record, for example, 15 ÷ 5 = 3
Record, for example, fifth
Record, for example, a fifth of 15 = 3
Display the array, for example,
Record, for example, 15 ÷ 3 = 5
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9
Introduces
dividing by 5
makes a number
5 times smaller, a
fifth times as big
Introduces
dividing a low
two-digit number
that is a multiple
of 5, by 5 using
the properties
and relationships
(top)
Introduces linking
division by 5 to
multiplication by 5
as inverse
operations
Introduces
dividing by 5 is
fifthing
When we divide, we are actually using the distributive property,
although technically the distributive property only applies to
multiplication. In Year 6, children recognise that they have been using
the distributive property for division when they investigate that
division is really multiplication by a fraction, for example, division by 5
is multiplication by a fifth.
Record, for example, 35 ÷ 5 =
NB: When a child knows what 35 divided by 5 equals without skip
counting, they don't need to partition. Children partition as much as
THEY need to
Record, for example, 3 x ___ = 15
Display, for example,
35 ÷ 5 =
1
5
of 35 =
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
We’ve investigated seeing multiplication by 5 additively
and multiplicatively.

We found that when we skip count by 5, we are seeing
multiplication additively because we are repeatedly
adding 5.

We found that when we see multiplication by 5
multiplicatively, we are making a number 5 times larger.

Today we are going to investigate seeing division by 5
multiplicatively.

We’re going to investigate seeing division by 5 as making
a number 5 times smaller, a fifth times as much.

Let's investigate how we can divide a low two-digit
number by 5.

How could we divide 35 by 3?

We could get 35 counters and divide them into 5 equal
groups or rows.

We could use what we know about multiplying by 5.

That’s because multiplication and division are inverse –
they undo one another.

35 divided by is asking us what number multiplied by 5
equals 35.

Do you what number multiplied by 5 equals 35?

Let’s investigate how we could work this out without skip
counting.
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10
Reviews
preferred
multiples
Reviews
partitioning
(Place Value 3, 8,
11, 13)
Record, for example,
35 ÷ 5 =
20 + 15
Introduces
dividing the parts
by 5
1
5
of 35 =
20 + 15
NB: Children suggest partitions that are their preferred multiples. The
multiples above are suggestions only. Demonstrating using children's
preferred multiples develops deep understanding of division and the
relationship to multiplication and fractions
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
When we divide by 5, what fraction do we get?

Will we get a number that is a fifth as big?

Will we get a fifth of the number?

How could we record this in a number sentence?

Let's record our number sentence as both a division and
as a fraction

How could we divide 35 by 5?

How could we find a fifth of 35?

Could we partition 35 into multiples of 5 that we know?

Could we partition 35 into our preferred multiples of 5?

Is 10 a multiple of 5?

Because 10 is a multiple of 5, will all tens numbers be
multiples of 5?

So 10 is a multiple of 5. Is 20 a multiple of 5?

Could we partition 35 into 20?

If we partition 35 into 20, what will the other part be?

Will the other part be 15?

Is 15 a multiple of 5?
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11
Record, for example,
20 ÷ 5 =
1
5
of 20 =
Record, for example,
20 ÷ 5 = 4
1
5
of 20 = 4
Record, for example,
15 ÷ 5 =
1
5
of 15 =

Now that we have our preferred multiples of 5, let’s
divide our parts by 5 and find a fifth.

Let's record our first division number sentence - 20
divided by 5 equals.

Let's record our number sentence using fractions – a fifth
of 20 equals.

Do both of these number sentences say the same thing?

When we divide by 5, are we finding a fifth?

What does 20 divided by 5 equal?

Does 20 divided by 5 equal 4?

What is a fifth of 20?

Is a fifth of 20, 4?

Let's record our second division number sentence - 15
divided by 5 equals.

Let's record our number sentence using fractions – a fifth
of 15 equals.

Do both of these number sentences say the same thing?

When we divide by 5, are we finding a fifth?

What does 15 divided by 5 equal?

Does 15 divided by 5 equal 3?

What is a fifth of 15?

Is a fifth of 15, 3?
Record, for example,
15 ÷ 5 = 3
1
5
of 15 = 3
Reviews adding
the quotients
(Multiplication
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12
and Division 11, 12
Patterns and
Algebra 17)
Introduces
knowing a
multiple of 3

So we didn’t know what 35 divided by 5 equalled.

So we partitioned 35 into 20 and 15, and divided these
parts by 5.

What could we now do to work out what 35 divided by 5
equals?

Would we add 20 and 15 together to make 35?

So would we add these answers together to make 35
divided by 5?

Let’s add 4 and 3.

What does 4 plus 3 equal?

Does 4 plus 3 equal 7?
of 35 = 7

Does 35 divided by 5 equal 7?
20 + 15

Do you now know what 35 divided by 5 equals?

Do you now know a multiple of 5?

Do you know 35 divided by 5 equals 7?

Does that make sense?

If we make 35 five times smaller, would it be 7?

If we make 35 a fifth as big, would it be 7?

When we divide, the answer is called the quotient.

You’ve started calling the answers ‘quotients’ now that
you are in Year 3!

What is the quotient of 35 and 5?

Is the quotient of 35 and 5, 7?
Record, for example,
4+3=7
Record, for example,
1
35 ÷ 5 = 7
5
20 + 15
1
20 ÷ 5 = 4
5
1
15 ÷ 5 = 3
Reviews quotient
(Multiplication
and Division 11, 12
Patterns and
Algebra 17)
5
of 20 = 4
of 15 = 3
4+3=7
Point to where you partitioned 35 into preferred multiples, for
example,
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13
35 ÷ 5 =
20 + 17
1
5
of 35 =
20 + 17

Does quotient mean we divided 35 by 5?

Did you notice that when we didn’t know what 35
divided by 5 equalled, we partitioned 35 into our
preferred multiples of 5?

Then we divided our preferred multiples by 5.

Then we added the quotients.

Do you think that would work for all numbers that we
want to divide?

If you know that 35 divided by 5 equals 7, do you also
know that 5 times 7 equals 35?

If we know what 7 times 5 equals, can we work out what
35 divided by 7 equals?

Are multiplication and division inverse?

If we multiply by 5, then divide by 5 do we get back to
where we started from?

If we multiply 7 by 5, do we get 35?

Then if we divide 35 by 5, do we get back to 7?
Point to where we divided the parts of 35 by 5, for example,
20 ÷ 5 = 4
15 ÷ 5 = 3
Reviews using the
inverse
relationship
between
multiplication and
division
(Multiplication
and Division 11, 12,
13 Patterns and
Algebra 17)
5
1
5
of 20 = 4
of 15 = 3
Point to where we added the quotients, for example,
4+3=7
Record, for example,
Introduces
dividing a nonmultiple low twodigit number by 5
using properties
and relationships
1
35 ÷ 5 = 7
1
5
of 35 = 7
5 x 7 = 35
Record, for example,
5 x 7 = 35 and
35 ÷ 5 = 7
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14
Reviews dividing
by 5 is fifthing
Reviews
preferred
multiples
Record, for example, 37 ÷ 5 =
NB: When a child knows what 37 divided by 5 equals without skip
counting, they don't need to partition. Children partition as much as
THEY need to
Reviews
partitioning
(Place Value 3, 8,
11, 13)
Display, for example,
37 ÷ 5 =
1
5
of 37 =
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
We’ve investigated dividing a low two-digit number by 5.

And we found that dividing by 5 and finding a fifth is the
same thing.

We found we could use the distributive property.

Today we’re going to investigate dividing another low
two-digit number by 5.

How could we divide 37 by 3?

We could get 37 counters and divide them into 5 equal
groups or rows.

Let’s investigate how we could work this out without skip
counting.

When we divide by 5, what fraction do we get?

Will we get a number that is a fifth as big?

Will we get a fifth of the number?

How could we record this in a number sentence?

Let's record our number sentence as both a division and
as a fraction

How could we divide 37 by 5?

How could we find a fifth of 37?

Could we partition 37 into multiples of 5 that we know?

Could we partition 37 into our preferred multiples of 5?

Is 10 a multiple of 5?
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
Because 10 is a multiple of 5, will all tens numbers be
multiples of 5?

So 10 is a multiple of 5. Is 20 a multiple of 5?

Could we partition 37 into 20?

If we partition 35 into 20, what will the other part be?

Will the other part be 17?

Is 17 a multiple of 5?

Because 17 is not a multiple of 5, how could we partition
17 into multiples of 5 that we know?

Is 15 a multiple of 5?

Could we partition 17 into 15 and 2?

Is 2 less than 5?

Because 2 is less than 5, we’ll leave it remaining.

Now that we have our preferred multiples of 5, let’s
divide our parts by 5 and find a fifth.

Let's record our first division number sentence - 20
divided by 5 equals.
Record, for example,
37 ÷ 5 =
1
5
20 + 17
of 37 =
20 + 17
Record, for example,
37 ÷ 5 =
Reviews dividing
the parts by 5
20 + 17
15 + 2
1
5
of 37 =
20 + 17
15 + 2
NB: Children suggest partitions that are their preferred multiples. The
multiples above are suggestions only. Demonstrating using children's
preferred multiples develops deep understanding of division and the
relationship to multiplication and fractions
Record, for example,
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20 ÷ 5 =
1
5
of 20 =
Record, for example,
20 ÷ 5 = 4
1
5
of 20 = 4
Record, for example,
15 ÷ 5 =
Reviews adding
the quotients
1
5
15 ÷ 5 = 3
5
Let's record our number sentence using fractions – a fifth
of 20 equals.

Do both of these number sentences say the same thing?

When we divide by 5, are we finding a fifth?

What does 20 divided by 5 equal?

Does 20 divided by 5 equal 4?

What is a fifth of 20?

Is a fifth of 20, 4?

Let's record our second division number sentence - 15
divided by 5 equals.

Let's record our number sentence using fractions – a fifth
of 15 equals.

Do both of these number sentences say the same thing?

When we divide by 5, are we finding a fifth?

What does 15 divided by 5 equal?

Does 15 divided by 5 equal 3?

What is a fifth of 15?

Is a fifth of 15, 3?

So we didn’t know what 35 divided by 5 equalled.

So we partitioned 35 into 20 and 15, and divided these
parts by 5.
of 15 =
Record, for example,
1

of 15 = 3
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Record, for example,
Reviews quotient
4+3=7
Record, for example,
1
37 ÷ 5 =
5
20 + 17
of 37 =
20 + 17
15 + 2
15 + 2
1
20 ÷ 5 = 4
5
1
15 ÷ 5 = 3
5
15 + 2

Would we add 20 and 15 together to make 35?

So would we add these answers together to make 35
divided by 5?

Let’s add 4 and 3.

What does 4 plus 3 equal?

Does 4 plus 3 equal 7?

Does 37 divided by 5 equal 7 with 2 remaining?

Does that make sense?

If we make 37 five times smaller, would it be 7 with 2
remaining?

If we make 37 a fifth as big, would it be 7 with 2
remaining?

When we divide, the answer is called the quotient.

You’ve started calling the answers ‘quotients’ now that
you are in Year 3!

What is the quotient of 37 and 5?

Is the quotient of 37 and 5, 7 with 2 remaining?

Does quotient mean we divided 37 by 5?

Did you notice that when we didn’t know what 37
divided by 5 equalled, we partitioned 37 into our
preferred multiples of 5?
of 15 = 3
Point to where you partitioned 37 into preferred multiples, for
example,
1
37 ÷ 5 =
of 37 =
5
20 + 17
What could we now do to work out what 35 divided by 5
equals?
of 20 = 4
4+3=7
Reviews
differentiating the
investigation for
children as they

20 + 17
15 + 2
Point to where we divided the parts of 37 by 5, for example,
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demonstrate
understanding
20 ÷ 5 = 4
15 ÷ 5 = 3
1
5
of 20 = 4
1
of 15 = 3
5
Point to where we added the quotients, for example,

Then we divided our preferred multiples by 5.

Then we added the quotients.

Do you think that would work for all numbers that we
want to divide?
4+3=7
Allow children time now to engage in guided and independent
investigation (at the end of this teaching plan) of dividing low twodigit numbers by 5 using properties and relationships, relating
dividing multiples to multiplication facts using the inverse
relationship between multiplication and division, and recording
remainders when dividing non-multiples.
A child who has not demonstrated understanding of multiples of 5 up
to 10 times 5, will continue to investigate this before they begin to
divide by 5.
A child who has not demonstrated understanding of division by 5 by
making ‘groups of 5’ and by making ‘5 equal groups’ will continue to
investigate at this level. (Multiplication and Division 7)
A child who has not demonstrated understanding of multiplication by
5 by making 5 groups of … will continue to investigate at this level.
(Multiplication and Division 5)
A child could be sitting next to a child who is investigating at a
different level. They will explain their current levels of understanding
to one another as they investigate. This is a research-based way to
accelerate learning for children at all levels.
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Focuses
children’s
thoughts on the
concept, exposing
current
understanding and
any
misconceptions
Reviews seeing
multiplication
multiplicatively
Children think about, talk and listen to a friend about, then have the
opportunity to share what they already know.
Children move onto multiplying teen numbers by 5 when they have
learnt multiples of 5 up to 10 times 5, with understanding.
Today brings an investigation about multiplying by 5.
►
What do you know about multiplying by 5?
►
Talk about multiplying by 5 with a friend.
►
Is anyone ready to share what they are thinking about
multiplying by 5?

We’ve investigated multiplying a single-digit number by
5.

We found that multiplying by 5 makes a number 5 times
bigger.

We began to call the answers to a multiplication,
products.

We investigated multiplying single-digit numbers by 5.

We found that if we didn’t know a product when we
multiplied by 5, we could partition the number then
multiply the parts and add the products.

We found that we then knew a multiple of 5.
Record, for example, 3 times bigger
Record, for example, product
Reviews
Point to where we partitioned the 7, for example,
partitioning (Place
Value 3, 8, 11, 13)
Point to where we multiplied the parts, for example,
Reviews
multiplying the
parts by 5
Reviews adding
the products
►
5 x 7 =
5 + 2
5 x 5 = 25
5 x 2 = 10
Point to where added the products, for example,
Record, for example, 5 x 7 = 35
25 + 10 = 35
Record, for example, 5 × 17 =
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Introduces
multiplying a teen
number by 5
using the
distributive
property (top)
Partition 17, for example,
Reviews
partitioning (Place
5 x 17 =
Value 3, 8, 11, 13)
10 + 7
Reviews
multiplying the
parts by 3
Record, for example, 5 x 10 = 50
Reviews adding
the products
Record, for example, 5 x 7 = 35
Record, for example, 50 + 35 = 85
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
Today we’re going to investigate multiplying a teen
number by 5.

How could we multiply 17 by 5?

How could we partition 17 to multiply it by 5?

Could we partition 17 using place value?

Could we partition 17 into 10 and 7, the multiply 3 by 10
and by 7, then add the products?

How could we multiply 5 by 10?

We’ve investigated by 10 and we found that the digits
move one place to the left.

We found that this is because the column on the left is
10 times the value of the column on the right.

So does 5 times 10 equal 50?

So we’ve multiplied 5 by 10.

What is 5 times 7?

We've been multiplying single-digit numbers by 5 for a
while now so we know what 5 times 7 equals.

We know that 5 times 7 equals 35.

Because we add 10 and 7 to make 17, we’ll now add the
products.

What does 50 plus 35 equal?

Does 50 plus 35 equal 85?
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Record, for example,
Reviews the
distributive
property
5 x 17 = 85
Point to where we partitioned the 17, for example,
5 x 17 =

What does 5 times 17 equal?

Does 5 times 17 equals 85?

Does that make sense?

If we make 17 five times larger, would it be around 85?

Did you notice that we partitioned 17?

Then we multiplied those parts of 17 by 5.

Then we added the products.

Do you think that works for all numbers that we want to
multiply?

Did we just used a very important property of
multiplication?

When we partition the number to multiply it, then add
the products, are we using the distributive property?

Did we use addition to make our multiplication easier?

Did we distribute our multiplication over addition?
10 + 7
Point to where we multiplied the parts, for example,
5 x 10 = 50
5 x 7 = 35
Point to where added the products, for example,
50 + 35 = 85
Record, for example, 5 x 17 = 85
Reviews
differentiating the
investigation for
children as they
demonstrate
understanding
Allow children time now to engage in guided and independent
investigation (at the end of this teaching plan) of multiplying teen
numbers by 3 using the distributive property, using multiplication
facts learnt through properties and relationships.
A child who has not demonstrated understanding of multiplication of
single-digit numbers by 3 using the distributive property, learning
multiplication facts through properties and relationships, will
continue to investigate at this level.
A child could be sitting next to a child who is investigating at a
different level. They will explain their current levels of understanding
to one another as they investigate. This is a research-based way to
accelerate learning for children at all levels.
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Focuses
children’s
thoughts on the
concept, exposing
current
understanding and
any
misconceptions
Children think about, talk and listen to a friend about, then have the
opportunity to share what they already know.
When children can recall some multiples of 5 up to 10 x 5 with
understanding and without skip counting, they begin to divide larger
two-digit numbers by 5.
Reviews seeing
division in 5
ways, as ‘groups
Record, for example, ‘groups of 5’ and ‘5 equal groups’
of …’ and as ‘…
equal groups’
(Multiplication
and Division 7)
Reviews dividing
low two-digit
numbers by 5
using the
properties and
relationships
Record, for example, fifth
Record, for example, 5 times smaller

Today brings an investigation about dividing by 5.

What do you know about dividing by 5?

Talk about dividing by 5 with a friend.

Is anyone ready to share what they are thinking about
dividing by 5?

We’ve investigated dividing by 5.

And we found that there are 2 ways we can see dividing
by 5.

We found we could divide by 5 by making ‘groups of 5’.

And we found that we could divide by 5 by making ‘5
equal groups’.

We found that seeing dividing by as making ‘5 equal
groups’ is more efficient when dividing larger numbers
because we just fifthing.

We’ve investigated dividing a teen low two-digitnumber
by 5.

And we found that dividing by 5 and fifthing are the
same thing.

We found that dividing by 5 makes a number 5 times
smaller.

We found that dividing by 5 makes a number fifth as big.

We began to call the answer to a division, a quotient.
Record, for example, fifth as big
Record, for example, quotient
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Record, for example, 35 ÷ 3 =
Point to where we partitioned the 35, for example,
35 ÷ 5 =

We’ve investigated dividing low two-digit numbers by 5.

We found that if we didn’t know a quotient when we
divided by 5, we could partition the number then divide
the parts and add the quotients.

And we found that because we knew that 35 divided by 5
equals 7, we also knew that 5 times 7 equals 35.

We found that multiplication and division are inverse.

Because if we multiply by 5, then divide by 5 we get back
to where we started.

So when we multiply 7 by 5, we get 35.

Then if we divide 35 by 5, we get back to 7.

We’ve investigated dividing a non-multiple low two-digit
number by 5

And we found that we had 1, 2, 3 or 4 remaining.
20 + 15
Point to where we divided the parts, for example,
20 ÷ 5 = 4
15 ÷ 5 = 3
4+3=7
Reviews linking Point to where added the products, for example,
division by 3 to
multiplication by 3
1
Record, for example, 35 ÷ 5 = 7
of 35 = 7
as inverse
5
operations
5 x 7 = 35
Record, for example,
5 x 7 = 35 and
Reviews dividing
non-multiple
numbers by 5 and
getting 1, 2, 3 or Record, for example, 37 ÷ 5 = 7 r2
4 remaining
35 ÷ 5 = 7
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Introduces
dividing a higher
two-digit number
by 5 using
properties and
relationships
(top)
When we divide, we are actually using the distributive property,
although technically the distributive property only applies to
multiplication. In Year 6, children recognise that they have been using
the distributive property for division when they investigate that
division is really multiplication by a fraction, for example, division by 5
is multiplication by a fifth.

Today we’re going to investigate dividing a higher twodigit number by 5.

How could we divide a higher two-digit number by 5?

How could we divide 85 by 5?

If we don't know what 85 divided by 5 equals, could we
partition 85 into multiples of 5 that we prefer?
Record 85 ÷ 5 =

Prefer just means that we like them. We like them
because we know them.

Could we partition 85 into our preferred multiples of 5?

What multiples of 5 do you know without skip or
rhythmic counting?

What number do you know is a multiple of 5?

Is 10 a multiple of 5?

Because 10 is a multiple of 5, are all tens numbers
multiples of 5?

Is 50 a multiple of 5?

Could we partition 85 using non-standard place value
into 50 and 35?

Is 35 a multiple of 5?

Now that we have our preferred multiples of 5, could we
start dividing by 5 using these multiples?

When we divide by 5, what fraction do we get?

Will we get a number that is fifth as big?
Record, for example,
85 ÷ 5 =
50 + 35
Reviews dividing
by 5 is fifthing
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
Will we get fifth of the number?

How could we record this in a number sentence?

Let's record our number sentence as both a division and
as a fraction.

Let's record our first division number sentence - 50
divided by 5 equals.

Let's record our number sentence using fractions - fifth
of 50 equals.

Do both of these number sentences say the same thing?

When we divide by 5, are we finding a fifth?

What does 50 divided by 5 equal?
of 50 = 10

What is a fifth of 50?
of 35 = 7

What does 35 divided by 5 equal?

What is a fifth of 35?

So we didn’t know what 85 divided by 5 equalled.

So we partitioned 85 into 50 and 35, and divided these
parts by 5.

What could we now do to work out what 85 divided by 5
equals?

Would we add 50 and 35 together to make 85?

So would we add these answers together to make 45
divided by 3?
Reviews dividing
the parts by 5
Record, for example,
50 ÷ 5 =
Record, for example,
50 ÷ 5 =
Record, for example,
50 ÷ 5 = 10
35 ÷ 5 = 7
1
5
1
5
1
5
of 50 =
Reviews adding
the quotients
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Record, for example, 10 + 7 = 17
Record, for example
Reviews quotient
1
85 ÷ 5 = 17
5
of 85 = 17
Record, for example, quotient
Point to where we partitioned 45, for example,
85 ÷ 5 =
50 + 35
Point to where we divided the parts of 45 by 3, for example,
50 ÷ 5 = 10
35 ÷ 5 = 7
1
5
1
5
of 35 = 7
10 + 7 = 17
85 ÷ 5 = 17
Let’s add 10 and 7.

What does 10 plus 7 equal?

Does 10 plus 7 equal 17?

Does 85 divided by 5 equal 17?

Does that make sense?

If we make 85 five times smaller, would it be 17?

If we make 85 a fifth as big, would it be 17?

When we divide, the answer is called the quotient.

You’ve started calling the answers ‘quotients’ since that
you are in Year 3!

What is the quotient of 85 and 5?

Is the quotient of 85 and 5, 17?

Does quotient mean we divided 85 by 5?

Did you notice that when we didn’t know what 85
divided by 5 equalled, we partitioned 85 into our
preferred multiples of 5?

Then we divided our preferred multiples by 5.

Then we added the quotients.

Do you think that would work for all numbers that we
want to divide by 5?

If you know that 85 divided by 5 equals 17, do you also
know that 5 times 17 equals 85?

If we know what 5 times 17 equals, can we work out
what 85 divided by 3 equals?
of 50 = 10
Point to where we added the quotients, for example,
Reviews using the Record, for example,
inverse
relationship
between

1
of 85 = 17
5
5 x 17 = 85
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multiplication and
division
Record, for example,
5 x 17 = 85 and

Are multiplication and division inverse?

If we multiply by 5, then divide by 5 do we get back to
where we started from?

If we multiply 17 by 5, do we get 85?

Then if we divide 85 by 5, do we get back to 17?

How could we divide a non-multiple higher two-digit
number by 5?

How could we divide 87 by 5?

If we don't know what 87 divided by 5 equals, could we
partition 87 into multiples of 5 that we prefer?

Prefer just means that we like them. We like them
because we know them.

Could we partition 87 into our preferred multiples of 5?

We’ve investigated dividing two-digit numbers by 5.

We found that because 10 is t a multiple of 5, every tens
number is a multiple of 5!

We found that 50 is a multiple of 5.

Could we partition 87 using non-standard place value
into 50 and 37?

Is 37 a multiple of 5?

37 is not a multiple of 5.

We’ll need to partition 37 into multiples of 5.
85 ÷ 5 = 17
Introduces
dividing a nonmultiple higher
two-digit number
by 3
Record, for example, 87 ÷ 5 =
Record, for example,
87 ÷ 5 =
50 + 37
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Record, for example,
87 ÷ 5 =
50 + 37
Reviews dividing
by 5 is fifthing
35 + 2

What is a multiple of 5 that is close to 37?

Is 35 a multiple of 5?

Could we partition 37 into 35 and 2?

Because 2 is less than 3, we can’t partition it into a
multiple of 3.

So the 2 will be left over.

Now that we have our preferred multiples of 3, could we
start dividing by 5 using these multiples?

When we divide by 5, what fraction do we get?

Will we get a number that is a fifth as big?

Will we get a fifth of the number?

How could we record this in a number sentence?

Let's record our number sentence as both a division and
as a fraction.

Let's record our first division number sentence - 50
divided by 5 equals.

Let's record our number sentence using fractions – a fifth
of 50 equals.

Do both of these number sentences say the same thing?

When we divide by 5, are we finding a fifth?

What does 50 divided by 5 equal?

What is fifth of 50?
Children partition as much as they need to.
Reviews dividing
the parts by 5
Record, for example,
50 ÷ 5 =
Record, for example,
50 ÷ 5 =
1
5
of 50 =
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Record, for example,
50 ÷ 5 = 10
35 ÷ 5 = 7
1
of 50 = 10
5
1
5
Record, for example, 10 + 7 = 17
Record, for example
What does 35 divided by 5 equal?

What is fifth of 35?

So we didn’t know what 87 divided by 3 equalled.

So we partitioned 87 into 50 and 35 with 2 remaining,
and divided 50 and 35 by 3.

What could we now do to work out what 87 divided by 5
equals?

Would we add 50 and 35 and 2 together to make 87?

So would we add these answers together to make 87
divided by 5?

Let’s add 10 and 7.

What does 10 plus 7 equal?

Does 10 plus 7 equal 17?

Does 47 divided by 5 equal 7, with 2 remaining?

Does that make sense?

If we make 87 five times smaller, would it be 17 with 2
remaining?

If we make 87 a fifth as big, would it be 17 with 2
remaining?

When we divide, the answer is called the quotient.

We’ve started calling the answers ‘quotients’ now that
you are in Year 3!

What is the quotient of 87 and 5?
of 35 = 7
Reviews adding
the quotients
Reviews quotient

87 ÷ 5 = 17 r2
1
5
of 87 = 17 r2
Record, for example, quotient
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Point to where we partitioned 87, for example,
87 ÷ 5 =
50 + 37

Is the quotient of 87 and 5, 17 with 2 remaining?

Does quotient mean we divided 87 by 5?

Did you notice that when we didn’t know what 87
divided by 5 equalled, we partitioned 87 into our
preferred multiples of 5?

Then we divided our preferred multiples by 5.

Then we added the quotients.

And recorded the 2 that we had remaining.

Do you think that would work for all numbers that we
want to divide?
35 + 2
Point to where we divided the parts of 87 by 5, for example,
Reviews
1
50 ÷ 5 = 10
of 50 = 10
5
differentiating the
1
investigation for
35 ÷ 3 = 7
of 35 = 7
5
children as they
Point to where we added the quotients, for example,
demonstrate
understanding
10 + 7 = 17
87 ÷ 5 = 17 r2
Allow children time now to engage in guided and independent
investigation (at the end of this teaching plan) of dividing higher
two-digit numbers by 5 using properties and relationships, relating
dividing to multiplication facts using the inverse relationship
between multiplication and division, and recording remainders
when dividing non-multiples.
A child who has not demonstrated understanding of dividing lower
two-digit numbers by 5 using properties and relationships, will
continue to investigate this before they begin to divide higher twodigit numbers by 5.
A child who has not demonstrated understanding of division by 5 by
making ‘groups of 5’ and by making ‘5 equal groups’ will continue to
investigate at this level. (Multiplication and Division 7)
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A child could be sitting next to a child who is investigating at a
different level. They will explain their current levels of understanding
to one another as they investigate. This is a research-based way to
accelerate learning for children at all levels.
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Twitter: @learn4teach
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Focuses
children’s
thoughts on the
concept, exposing
current
understanding and
any
misconceptions
Reviews seeing
multiplication
multiplicatively
Children think about, talk and listen to a friend about, then have the
opportunity to share what they already know.
Children move onto multiplying two-digit numbers by 5 when they
have learnt multiples of 5 up to 10 times 5, with understanding.
Today brings an investigation about multiplying by 5.
►
What do you know about multiplying by 5?
►
Talk about multiplying by 5 with a friend.
►
Is anyone ready to share what they are thinking about
multiplying by 5?

We’ve investigated multiplying a single-digit number by
5.

We found that multiplying by 5 makes a number 5 times
bigger.

We began to call the answer to a multiplication,
products.

We investigated multiplying single-digit numbers by 5.

We found that if we didn’t know a product when we
multiplied by 5, we could partition the number then
multiply the parts and add the products.

We found that we then knew a multiple of 5.
Record, for example, 5 times bigger
Record, for example, product
Reviews
Point to where we partitioned the 7, for example,
partitioning (Place
Value 5, 8, 11, 15)
Point to where we multiplied the parts, for example,
Reviews
multiplying the
parts by 5
Reviews adding
the products
►
5 x 7 =
5 + 2
5 x 5 = 25
5 x 2 = 10
Point to where added the products, for example,
Record, for example, 5 x 7 = 35
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25 + 10 = 35
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Introduces
multiplying a twodigit number by 5
using the
distributive
property (top)

Today we’re going to investigate multiplying a two-digit
number by 5.

How could we multiply 57 by 5?

How could we partition 57 to multiply it by 5?

Could we partition 57 using place value?

Could we partition 57 into 50 and 7, the multiply 5 by 50
and by 7, then add the products?

How could we multiply 5 by 50?
Record, for example, 5 x 50 =

Is 50, 10 times 5?
Record, for example, 50 = 10 x 5

Is it easy to multiply by 10 because the digit moves one
place to the left because of multiplicative place value?

Is 5 times 50, 5 times 10 times 5?

Do we have to multiply in the order that the numbers are
recorded?

Could we multiply 5 times 5 first, then multiply by 10?

What does 5 times 5 equal?

Does 5 times 5 equal 25?
Record, for example, 5 x 50 = 5 x 10 x 5 = 25 x 10

What does 25 times 10 equal?
Record, for example, 5 x 50 = 5 x 10 x 5 = 25 x 10 = 250

Does 25 times 10 equal 250?

You have just used a very important property of
multiplication.
Record, for example, 5 × 57 =
Reviews
partitioning (Place Partition 57, for example,
Value 5, 8, 11, 15)
5 x 57 =
Reviews
multiplying the
parts by 5
Reviews
multiplying by 10
using
multiplicative
place value
(Multiplication
and Division 10)
50 + 7
Record, for example, 5 x 50 = 5 x 10 x 5
Point to the 5 and the 5, for example, 5 x 50 = 5 x 10 x 5
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Reviews the
associative
property for
multiplication
(Multiplication
and Division 11, 12
Patterns and
Algebra 17)
Record, for example, associative
Record, for example, 5 x 7 = 35
Reviews adding
the products
Record, for example, 250 + 35 = 285
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
When we multiply, we can multiply the numbers in any
order.

This property is called the associative property.

Associative just means they have something the same,
something in common.

Your friend is your associate!

It’s called the associative property because we are
multiplying number that are associated with each other
first.

The associative property makes multiplication easier!

We have already investigated adding numbers in any
order.

And we found that the associative property made
addition easier too!

You will be investigating the associative property over
the next 4 years!

So we’ve multiplied 5 by 50.

What is 5 times 7?

We've been multiplying single-digit numbers by 5 for a
while now so we know what 5 times 7 equals.

We know that 5 times 7 equals 35.

Let’s add the products.

What does 250 plus 35 equal?

Does 250 plus 35 equal 285?
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Reviews product
Record, for example,

What does 5 times 57 equal?

Does 5 times 57 equals 285?

Does that make sense?

If we make 57 three times larger, would it be around
285?
50 + 7

Did you notice that we partitioned 57?
5 x 50 = 250

Then we multiplied those parts of 57 by 5.
5 x 7 = 35

Then we added the products.
250 + 35 = 285

Do you think that works for all numbers that we want to
multiply?

Did we just used a very important property of
multiplication?

When we partition the number to multiply it, then add
the products, are we using the distributive property?

Did we use addition to make our multiplication easier?

Did we distribute our multiplication over addition?
5 x 57 = 285
50 + 7
Point to where we partitioned the 57, for example,
Reviews the
distributive
property
5 x 57 =
Point to where we multiplied the parts, for example,
Point to where added the products, for example,
Record, for example, 5 x 57 = 285
Reviews
differentiating the
investigation for
children as they
demonstrate
understanding
Allow children time now to engage in guided and independent
investigation (at the end of this teaching plan) of multiplying twodigit numbers by 5 using the distributive property, using
multiplication facts learnt through properties and relationships.
A child who has not demonstrated understanding of multiplication of
single-digit and teen numbers by 5 using the distributive property,
learning multiplication facts through properties and relationships, will
continue to investigate at this level.
Website: http://www.alearningplace.com.au
Email: [email protected]
Twitter: @learn4teach
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A child who has not demonstrated understanding of multiplication by
5 by making 5 groups of … will continue to investigate at this level.
(Multiplication and Division 5)
A child could be sitting next to a child who is investigating at a
different level. They will explain their current levels of understanding
to one another as they investigate. This is a research-based way to
accelerate learning for children at all levels.
Website: http://www.alearningplace.com.au
Email: [email protected]
Twitter: @learn4teach
YouTube: A Learning Place A Teaching Place
Facebook: A Learning Place
37