Multiplication and Division 21, Patterns and Algebra 26_Guided and Independent Investigations
(Year 5) ACMNA098, NSW MA3-4NA
Equivalent division calculations result if both numbers are divided by a common factor.
GUIDED INVESTIGATION
Children learn
how to investigate the concept by following teacher’s instructions until they
are ready to investigate the concept independently.
INDEPENDENT INVESTIGATION
Children investigate and explain independently over many lessons at their
current level of understanding informing both themselves and the teacher of their current level
of understanding
Resources: playing cards, pencils, paper
What could we do?
Children:
1. sit in pairs
2. select cards to make 2 numbers to divide
as guided by the teacher
3. create an equivalent division number
sentence by dividing both numbers by a
common factor
4. select cards to make 2 numbers to multiply
as guided by the teacher
5. create an equivalent multiplication number
sentence by dividing one number and
multiplying the other number by the same
factor
REFLECTION Before, during and after lessons,
children discuss then record responses to reflection
questions to inform themselves and the teacher of their
current level of understanding
What language could we use to
ask questions and explain?
 How could we create an
equivalent division number
sentence by dividing both
numbers by a common
factor?
 How could we create an
equivalent multiplication
number sentence by
dividing one number and
multiplying the other
number by the same
factor?
What language could we use
to explain?
What could we do?
Children:
1. sit in pairs
2. select cards to make 2 numbers that are
neither too easy nor too challenging to divide
3. create an equivalent division number sentence
by dividing both numbers by a common factor
4. select cards to make 2 numbers that are
neither too easy nor too challenging to multiply
5. create an equivalent multiplication number
sentence by dividing one number and
multiplying the other number by the same
factor
 I created an equivalent
division number sentence
by dividing both numbers
by a common factor by …
 I created an equivalent
multiplication number
sentence by dividing one
number and multiplying
the other number by the
same factor by …
What is a factor?
How can we create an equivalent division number sentence by dividing both numbers by a common
factor?
How can we create an equivalent multiplication number sentence by dividing one number and
multiplying the other number by the same factor?
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Email: [email protected]
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1
CONGRUENT INVESTIGATIONS
If you can’t explain it
simply, you don’t
understand it well enough.
These investigations allow children to investigate and explain the concept in new and varied situations, providing formative assessment
data for both the child and the teacher.
‘Doing’ mathematics is not enough and is not a good indicator of understanding.
Anyone who has
Investigation takes time as children develop both the capacity and meta-language to explain mathematical concepts
never made a
at their current level of understanding.
mistake has never
tried anything new.
As they investigate, allow children to experience confusion (problematic knowledge) and to make mistakes to develop
resilience and deep understanding,

In pairs, each child creates a division number sentence, for example, 34 ÷ 4 = . They hand their division number sentence to a
friend to create an equivalent division number sentence by dividing both numbers by a common factor. They check that the
number sentences are equivalent by performing the 2 divisions.

In pairs, each child creates a multiplication number sentence, for example, 34 x 4 = . They hand their multiplication number
sentence to a friend to create an equivalent multiplication number sentence by dividing one number and multiplying the
other number by the same factor. They check that the number sentences are equivalent by performing the 2 multiplications.

In pairs, children construct quadrilaterals using square centimetres. They work out the quadrilateral’s area. They halve the
length of one dimension while doubling the length of the other dimension. They work out the new quadrilateral’s area. They
discuss why the area remained the same.

Children create areas of hectares in shapes other than squares using common factors. For example, a hectare is 100 m by 100
m. If we divide the first 100 m by 2 and multiply the second 100 m by 2, we get 50 m by 200 m which is still a hectare because
it is still 10 000 square metres. (Links to Measurement and Geometry 52)

In pairs, children select cards to create a division number sentence. They each solve the division number sentence by dividing
both numbers by a common factor. They compare their solution strategies.
Website: http://www.alearningplace.com.au
Email: [email protected]
Twitter: @learn4teach
YouTube: A Learning Place A Teaching Place
Facebook: A Learning Place
2
Investigating Equivalent Number Sentences Using Common Factors
Multiplication and Division 21, Patterns and Algebra 26 Explain that equivalent division calculations result if both numbers are divided by a
common factor, create and solve equivalent number sentences involving multiplication and division
Select cards to make 2 numbers that are neither too easy nor too challenging to
divide.
Divide the numbers by dividing both numbers by a common factor.
http://www.alearningplace.com.au
Investigating Equivalent Number Sentences Using Common Factors
Multiplication and Division 21, Patterns and Algebra 26 Explain that equivalent division calculations result if both numbers are divided by a
common factor, create and solve equivalent number sentences involving multiplication and division
Select cards to make 2 numbers that are neither too easy nor too challenging to
divide.
Create an equivalent division number sentence by dividing both numbers by a
common factor.
http://www.alearningplace.com.au
Investigating Equivalent Number Sentences Using Common Factors
Multiplication and Division 21, Patterns and Algebra 26 Explain that equivalent division calculations result if both numbers are divided by a
common factor, create and solve equivalent number sentences involving multiplication and division
Select cards to make 2 numbers that are neither too easy nor too challenging to
multiply.
Create an equivalent multiplication number sentence by dividing one number and
multiplying the other number by the same factor.
http://www.alearningplace.com.au
Investigating Equivalent Number Sentences Using Common Factors
Multiplication and Division 21, Patterns and Algebra 26 Explain that equivalent division calculations result if both numbers are divided by a
common factor, create and solve equivalent number sentences involving multiplication and division
Sit with a friend.
Each of you create a division number sentence.
Hand your division number sentence to a friend to create an equivalent division
number sentence by dividing both numbers by a common factor.
Check that the number sentences are equivalent by performing the 2 divisions.
http://www.alearningplace.com.au
Investigating Equivalent Number Sentences Using Common Factors
Multiplication and Division 21, Patterns and Algebra 26 Explain that equivalent division calculations result if both numbers are divided by a
common factor, create and solve equivalent number sentences involving multiplication and division
Sit with a friend.
Each of you create a multiplication number sentence.
Hand your multiplication number sentence to a friend to create an equivalent
multiplication number sentence by dividing one number and multiplying the other
number by the same factor.
Check that the number sentences are equivalent by performing the 2
multiplications.
http://www.alearningplace.com.au
Investigating Equivalent Number Sentences Using Common Factors
Multiplication and Division 21, Patterns and Algebra 26 Explain that equivalent division calculations result if both numbers are divided by a
common factor, create and solve equivalent number sentences involving multiplication and division
Select cards to create a division number sentence.
Solve the division number sentence by dividing both numbers by a common factor.
Compare your solution strategy with a friend.
http://www.alearningplace.com.au
Investigating Equivalent Number Sentences Using Common Factors
Multiplication and Division 21, Patterns and Algebra 26 Explain that equivalent division calculations result if both numbers are divided by a
common factor, create and solve equivalent number sentences involving multiplication and division
Sit with a friend.
Construct a quadrilateral using square centimetres.
Work out the quadrilateral’s area.
Halve the length of one dimension while doubling the length of the other
dimension.
Work out the new quadrilateral’s area.
Discuss why the area remained the same.
http://www.alearningplace.com.au
Investigating Equivalent Number Sentences Using Common Factors
Multiplication and Division 21, Patterns and Algebra 26 Explain that equivalent division calculations result if both numbers are divided by a
common factor, create and solve equivalent number sentences involving multiplication and division
Sit with a friend.
Create areas of hectares in shapes other than squares using common factors.
For example, a hectare is 100 m by 100 m.
If we divide the first 100 m by 2 and multiply the second 100 m by 2, we get 50 m
by 200 m which is still a hectare because it is still 10 000 square metres.
http://www.alearningplace.com.au