Multiplication and Division 17, Patterns and Algebra 18_Overview of Learning Plan
(Year 4) ACMNA056, ACMNA057, NSW MA2-6NA
Multiply and divide by 7 using properties and relationships.
THIS IS A SUMMARY OF THE LEARNING PLAN, DESCRIBING THE SEQUENCE OF LEARNING WHICH WILL OCCUR OVER MULTIPLE LESSONS. COMPLETE LEARNING PLAN STARTS ON THE NEXT PAGE.
Multiply
singledigit
numbers
by 7.
Divide
singledigit and
teen
numbers
by 7.
Multiply
and
divide
two- and
threedigit digit
numbers
by 7.
Children:
 multiply single-digit numbers by 7
using properties and relationships,
recalling facts, for example,
 divide two-digit
numbers by 7 using
properties and
relationships, and
multiplication facts,
with and without
remainders, for
example,
Children
 ask one another questions about multiplying and dividing by
7 using properties and relationships, for example:

How could we partition this number to multiply it by 7?

What multiples of 7 do you know that would help?

What new multiple of 7 do you now know?

How could we partition this number to divide it by 7?

What multiples of 7 do you know that would help?

How could we partition using place value to divide by 7?
 multiply teen, twoand three-digit
numbers by 7 using
properties and
relationships
 divide three-digit
numbers by 7
using properties
and relationships
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Multiplication and Division 17, Patterns and Algebra 18_Explicit Learning Plan
(Year 4) ACMNA056, ACMNA057, NSW MA2-6NA
THIS IS THE FULL LEARNING PLAN, WITH DETAILS OF ACTIONS AND QUESTIONS THAT MAY BE USED TO DEVELOP DEEP UNDERSTANDING OVER MULTIPLE LESSONS.
Multiply single-digit, teen, two-digit numbers by 7 using the distributive property and relationships
Divide a two-digit, three-digit numbers by 7 using properties and relationships, associating dividing by 8 with eighthing
Resources: cards, pencil, paper
EXPLICIT LEARNING
What could we do?
Focuses
children’s
Children think about, talk and listen to a friend about, then have the
thoughts on the opportunity to share what they already know.
concept, exposing
current
understanding and
any
misconceptions
Reviews
thinking additively
in Years 1 and 2,
and thinking
multiplicatively
from Year 3 on.
Introduces
multiplying by 7
makes a number
7 times larger.
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What language could we use to explain and ask questions?
►
Today brings an investigation about multiplying by 7.
►
What do you know about multiplying by 7?
►
Talk about multiplying by 7with a friend.
►
Is anyone ready to share what they are thinking about
multiplying by 7?

In Year 1 and 2, you thought of Multiplication additively
as repeatedly adding equal groups and skip or rhythmic
counting

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 7, the number becomes 7 times larger.
This is how we now think about multiplication
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Introduces

Let’s investigate multiplication by a single-digit number.
How could we multiply 6 by 7 using relationships and
properties?

Let’s work out 7 x 7 using the distributive property.

How could we partition 6 to multiply it by 7?

What multiples of 7 do you know? Do you know 7 x 3?

Could we partition 6 into 3 and 3, and then use the
distributive property to multiply? Let’s investigate!
Record, for example, 7 x 3 = 21

What does 7 times 3 equal?
Record, for example, 7 x 3 = 21

What does 7 times 3 equal?
Record, for example, 21 + 21 = 42

What does 21 plus 21 equal?

So what does 7 times 6 equal?

Does 7 times 6 equal 42?

How else could we work out what 7 times 6 equals?
Introduces

Could we work out what 7 times 3 equals and double it?
multiplying by 3
then by 2 to
multiply 7 x 6.

What does 7 times 3 equal?

Does 3 times 7 equal 21?

What does double 21 equal?

Does double 21 equal 42?

So 7 x 6 equals 42. Does that make sense? If we make 6
seven times larger, would it be around 42?
multiplying a
single-digit
number by 7
using the
distributive
property (top)
Reviews
Record, for example, 7 x 6 =
NB: When a child knows what 7 times 6 equals, they don’t need to
partition. Children partition as much as THEY need to
Partition 6 into, for example, 3 and 3
Record, for example,
partitioning
(Place Value 3, 8,
7 x 6 =
11, 13)
3 + 3
Introduces
multiplying the
parts by 7.
Reviews
adding the
products.
Record, for example, 7 x 6 = 42
Record, for example, 7 x 3 = 21
Record, for example, Double 21 = 42 or 2 x 21 = 42
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Introduces
knowing a
multiple of 7

Did you notice that we partitioned 6 into multiples of 7
that we knew. Then we multiplied 7 by those parts of 6.
Then we added the products. Do you think that would
work for all numbers that we want to multiply?

You have just used a very important property of
multiplication!

When we partition the number to multiply it, then add
the products, we are using the distributive property

Did we use addition to make our multiplication easier?
Did we distribute our multiplication over addition?

Do you think that’s why it’s called the distributive
property?

So do we now know a multiple of 7?

What does 7 times 6 equal?
Record, for example, 7 x 6 = 42

Does 7 times 6 equal 42?
Record, for example, 6 x 7 = 42

If we know that 7 times 6 equals 42, do we also know
that 6 times 7 equals 42?
Allow children time now to engage in guided and independent investigation of
multiplying single-digit numbers by 7 using the distributive property and the
commutative property. When children are able to recall, with understanding,
multiples of 7 up to 10 times 7, they begin to divide by 7.
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Children think about, talk and listen to a friend about, then have the
Focuses
opportunity to share what they already know.
children’s
thoughts on the
concept, exposing
current
understanding and
any
misconceptions
►
Today brings an investigation about dividing by 7.
►
What do you know about dividing by 7?
►
Talk about dividing by 7with a friend.
►
Is anyone ready to share what they are thinking about
dividing by 7?

In Year 1 and 2, you thought of division additively as
repeatedly subtracting equal groups and skip or rhythmic
counting

We also investigated seeing division as either making
‘groups of ...’ or ‘... equal groups’ I could divide by 7 by
making ‘groups of 7’ or by making ‘7 equal groups’

To divide larger numbers, seeing division as making ‘4
equal groups’ is more efficient, because it means that we
are just quartering! Let’s investigate!

When we divide by a number greater than one, the
number becomes a number of times smaller. If we divide
by 4, the number becomes 4 times smaller. If we divide
1
1
by 4, the number becomes 4 times as much, 4 times as
big. If we divide by 7, the number becomes 7 times
1
smaller. If we divide by 7, the number becomes 7 times
Reviews
thinking additively
in Years 1 and 2,
and thinking
Record, for example,
multiplicatively
from Year 3 on.
35 ÷ 7 =
Reviews
Record, for example,
seeing
1
multiplication
35 ÷ 7 =
of 35 =
7
multiplicatively,
dividing by 7
NB: When a child knows what 35 divided by 7 equals without skip counting, they
makes a number don’t need to partition. Children partition as much as THEY need to.
7 times smaller, a Record, for example,
seventh times as
1
35 ÷ 7 =
of 35 =
big.
7
21 + 14
21 + 14
1
as much, 7 times as big.

This is seeing division, multiplicatively, and this is how we
now think about division
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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Introduces
dividing a twodigit number by 7
using properties
and relationships.
(top)
Record, for example,
1
35 ÷ 7 =
7
21 + 14
21 ÷ 7 =
of 35 =
𝟏
𝟕
of 21 =
partitioning using
standard place
value is not useful
when dividing by
7 because not
every tens
number is
divisible by 7.
Record, for example,
1
35 ÷ 7 =
partitioning using
non-standard
place value and
preferred
multiples of 7 to
divide by 7.
7
21 + 14
21 ÷ 7 = 3
Let’s investigate how we can divide a two-digit number
by 7

How could we divide 35 by 7?

When we divide by 7, what fraction do we get? Will we
get a number that is a seventh as big? Will we get a
seventh 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 7? How could we find a
seventh of 35?

Could we partition 35 into multiples of 7 that we know
from our investigation of multiplying by 7? Could we
partition 35 into our preferred multiples of 7?

Is 30 a multiple of 7? Will partitioning 35 using place
value help us to divide by 7? Why not? Is every tens
number a multiple of 7?

What number do you know, without skip counting, is a
multiple of 7?

Do you know 21 is a multiple of 7? If we partition 35 into
21, what will the other part be? Will the other part be
14? Is 14 a multiple of 7?

Now that we have our preferred multiples of 7, we can
start dividing by 7 using these multiples

Our first part of 35 is 21. Let’s record 21 divided by 7 and
a seventh of 21
21 + 14
Introduces
Introduces

of 35 =
21 + 14
1
7
of 21 = 3
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Record, for example,
1
35 ÷ 7 =
7
21 + 14
21 ÷ 7 = 3
14 ÷ 7 = 2
of 35 =

Do both of these number sentences say the same thing?

Which way makes more sense to you? That is the way
that you should think of it - in the way that makes sense
to you

What is 21 divided by 7? We can think of 21 divided by 7
in 3 ways

We can think of 21 divided by 7 as asking ‘how many 7s
in 21?’

We can think of 21 divided by 7 as asking ‘ 7 times what
equals 21?’

We can think of 21 divided by 7 as ‘what number is a
seventh as big as 21?’ or ‘what is a seventh of 21?’

Which way makes more sense to you? That is the way
that you should think of it - in the way that makes sense
to you

What is 21 divided by 7?

What is a seventh of 21?

So we’ve divided our first part by 7. Now we need to
divide our second part by 7.

Our second part is 14

Let’s record 14 divided by 7 and a seventh of 14

Do both of these number sentences say the same thing?
21 + 14
1
7
𝟏
𝟕
of 21 = 3
of 14 = 2
Reviews
dividing by 7 is
seventhing.
(Multiplication
and Division 9)
Reviews
dividing the parts
by 7 / finding a
seventh.
Record, for example, 3 + 2 = 5
Record, for example,
35 ÷ 7 = 5
21 + 14
1
7
of 35 = 5
21 + 14
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7
1
21 ÷ 7 = 3
7
1
14 ÷ 7 = 2
7
of 21 = 3
Record, for example,
Reviews
21 + 14
adding the
quotients.
21 ÷ 7 = 3
1

What is 14 divided by 7? We can think of 14 divided by 7
in 3 ways

We can think of 14 divided by 7 as asking ‘how many 7s
in 14?’

We can think of 14 divided by 7 as asking ‘ 7 times what
equals 14?’

We can think of 14 divided by 7 as ‘what number is a
seventh as big as 14?’ or ‘what is a seventh of 14?’

Which way makes more sense to you? That is the way
that you should think of it - in the way that makes sense
to you

Now that we’ve divided both parts of our number by 7,
we can add the quotients

A quotient is just the mathematical name for the answer
when we divide

Our first quotient is 3 and our second quotient is 2

What does 3 plus 2 equal?

So what does 35 divided by 7 equal?

Does 35 divided by 7 equal 5?
of 35 = 5
7
21 + 14
1
7
1
14 ÷ 7 = 2
Which way makes more sense to you? That is the way
that you should think of it - in the way that makes sense
to you
of 14 = 2
Record, for example, 3 + 2 = 5
35 ÷ 7 = 5

7
of 21 = 3
of 14 = 2
3+2=5
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Website: http://www.alearningplace.com.au
Email: [email protected]
Twitter: @learn4teach

So 35 divided by 7 equals 5. Does that make sense? If we
make 35 seven times smaller, would it be around 5? Is 5
a seventh as big as 35? Is 5 a seventh of 35? Are there 5
sevens in 35?

If we know that 35 divided by 7 equals 5, do we also
know that 5 times 7 equals 35 and 7 times 5 equals 35?
Do we know another multiple of 7?

Did you notice that we partitioned 35 into preferred
multiples of 7? Then we divided our preferred multiples
by 7, then we added the quotients. Do you think that
would work for all numbers that we want to divide?
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Introduces
dividing a nonmultiple two-digit
number by 7
using properties
and relationships.
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
Introduces
partitioning using
standard place
value is not useful
when dividing by
7 because not
every tens
number is
divisible by 7.
Introduces
partitioning using
non-standard
place value and
preferred
multiples of 7 to
divide by 7

Let’s investigate how we can divide a two-digit number
by 7

How could we divide 37 by 7?

When we divide by 7, what fraction do we get? Will we
get a number that is a seventh as big? Will we get a
seventh 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 7? How could we find a
seventh of 37?

Could we partition 37 into multiples of 7 that we know
from our investigation of multiplying by 7? Could we
partition 37 into our preferred multiples of 7?

Is 30 a multiple of 7? Will partitioning 37 using place
value help us to divide by 7? Why not? Is every tens
number a multiple of 7?

What number do you know, without skip counting, is a
multiple of 7?

Do you know 21 is a multiple of 7? If we partition 37 into
21, what will the other part be? Will the other part be
14? Is 16 a multiple of 7?

Let’s partition 16 into multiples of 7.

Now that we have our preferred multiples of 7, we can
start dividing by 7 using these multiples
Record, for example,
37 ÷ 7 =
21 + 16
14 + 2
1
7
of 37 =
21 + 16
14 + 2
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Record, for example,
21 ÷ 7 =
𝟏
𝟕

Our first part of 35 is 21. Let’s record 21 divided by 7 and
a seventh of 21

Do both of these number sentences say the same thing?

Which way makes more sense to you? That is the way
that you should think of it - in the way that makes sense
to you

What is 21 divided by 7? We can think of 21 divided by 7
in 3 ways

We can think of 21 divided by 7 as asking ‘how many 7s
in 21?’

We can think of 21 divided by 7 as asking ‘ 7 times what
equals 21?’

We can think of 21 divided by 7 as ‘what number is a
seventh as big as 21?’ or ‘what is a seventh of 21?’

Which way makes more sense to you? That is the way
that you should think of it - in the way that makes sense
to you

What is 21 divided by 7?

What is a seventh of 21?

So we’ve divided our first part by 7. Now we need to
divide our second part by 7.

Our second part is 14

Let’s record 14 divided by 7 and a seventh of 14

Do both of these number sentences say the same thing?
of 21 =
Reviews
dividing by 7 is
seventhing.
(Multiplication
and Division 9)
Record, for example,
1
37 ÷ 7 =
7
21 + 16
of 37 =
21 + 16
14 + 2
Reviews dividing
the parts by 7
21 ÷ 7 = 3
14 + 2
1
7
of 21 = 3
Record, for example,
14 ÷ 7 = 2
𝟏
𝟕
of 14 = 2
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
Which way makes more sense to you? That is the way
that you should think of it - in the way that makes sense
to you

What is 14 divided by 7? We can think of 14 divided by 7
in 3 ways

We can think of 14 divided by 7 as asking ‘how many 7s
in 14?’

We can think of 14 divided by 7 as asking ‘ 7 times what
equals 14?’

We can think of 14 divided by 7 as ‘what number is a
seventh as big as 14?’ or ‘what is a seventh of 14?’

Which way makes more sense to you? That is the way
that you should think of it - in the way that makes sense
to you

Now that we’ve divided both parts of our number by 7,
we can add the quotients

A quotient is just the mathematical name for the answer
when we divide

Our first quotient is 3 and our second quotient is 2

What does 3 plus 2 equal?
of 37 = 5 r2

So what does 37 divided by 7 equal?
21 + 16

Does 37 divided by 7 equal 5 with 2 remaining?
Reviews
adding the
quotients.
Record, for example, 3 + 2 = 5
Record, for example,
37 ÷ 7 = 5 r2
21 + 16
14 + 2
1
7
14 + 2
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
So 37 divided by 7 equals 5. Does that make sense? If we
make 37 seven times smaller, would it be around 5? Is 5
about a seventh as big as 37? Is 5 about a seventh of 37?
Are there about 5 sevens in 37?

Did you notice that we partitioned 37 into preferred
multiples of 7? Then we divided our preferred multiples
by 7, then we added the quotients. Do you think that
would work for all numbers that we want to divide?
Allow children time now to engage in guided and independent investigation of
dividing two-digit numbers by 7 using properties and relationships, and using the
inverse relationship between multiplication and division, and recording remainders
when dividing non-multiples.
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Introduces
multiplying teen,
two-digit and
three-digit
numbers by 7. top
When children have demonstrated deep understanding of, and capacity to use their deep understanding to recall multiplication of single-digit numbers by 7,
use the same questioning to allow them to investigate multiplication of teen numbers, two-digit numbers and three-digit number by 7:
Introduces
dividing threedigit numbers by
7. top
When children have demonstrated deep understanding of division by 7, including capacity to use multiplication by 7 up to 10 x 7 to divide by 7, use the
same questioning to allow them to investigate using multiplication by 7 to divide two-digit numbers and three-digit numbers:
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