MULTIPLICATION AND DIVISION 21, PATTERNS AND ALGEBRA 26_INVESTIGATION AND REFLECTION
(Year 5) ACMNA098, NSW MA3-4NA
Equivalent simpler division calculations result if both numbers are divided by a common factor, create equivalent multiplication.
GUIDED AND INDEPENDENT INVESTIGATIONS and REFLECTION
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.
Children investigate and explain independently over many lessons at just beyond their current level of understanding, informing
both themselves and the teacher of their current level of understanding. It is during independent investigation that deep understanding and
metalanguage develops.
As they investigate, allow children to experience confusion (problematic knowledge) and to make mistakes to develop
resilience and deep understanding, If children knew what it was they were doing, it wouldn’t be called learning!
GUIDE children through the INVESTIGATION process until they are ready to investigate INDEPENDENTly.
Children DISCUSS then RECORD their response to the REFLECTION question.
Teaching Segment and Video 1:
Equivalent division.
These investigations and reflections are directly linked to Explicit Teaching, and also appear on the Explicit Teaching Plan. Instructions for students
appear on this PDF, on the corresponding Video and on the Explicit Teaching PowerPoint.

In pairs, children use cards to create a division number sentence. They 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. Reflection: How can we simplify division by creating equivalent divisions by
dividing by a common factor?
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Teaching Segment and Video 2:
Equivalent multiplication.
More investigations.

In pairs, children use cards to create a multiplication number sentence. They create an equivalent
multiplication number sentence by dividing one number and multiplying the other number by the same
number. They check that the number sentences are equivalent by performing the 2 multiplications. Reflection:
How can we create equivalent multiplications by dividing and multiplying by a factor?
These investigations are not directly linked to Explicit Teaching. Instructions for students appear here and on the Explicit Teaching PowerPoint.
Halve and double dimensions.

In pairs, children construct rectangles using square centimetres. They work out the rectangle’s area. They halve
the length of one dimension while doubling the length of the other dimension. They work out the new
rectangle’s area. They discuss why the area remained the same. Reflection: How can we create shapes with
equivalent areas by dividing one dimension and multiplying the other dimensions by a factor?
Hectares.

Children create areas of hectares in shapes other than squares using common factors. For example, a square
hectare is 100 m by 100m. If we halve 1 dimension and double the other dimension, 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)
Reflection: How can we create shapes with equivalent areas by dividing one dimension and multiplying the
other dimensions by a factor?
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Email: [email protected]
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PROBLEM SOLVING directly linked to explicit teaching, investigations and reflections
Problems allow children to investigate concepts in new and varied situations. Any problem worth solving takes time and effort
– that’s why they’re called problems!
Problems are designed to develop and use higher order thinking. Allowing children to grapple with problems, providing minimal
support by asking strategic questions, is key. Differentiating problems allows children to solve simpler problems, before solving
more complex problems on a concept.
Problems may not always be solved the first time they are presented – or at all. The focus of problem solving is the development
of problem solving understanding and capacity – not mastery! Returning to a problem after further learning, develops both
resilience and increased confidence as children take the necessary time and input the necessary effort.
After solving problems, children also create their own problems.
Create 3 levels of a problem. GUIDE children through the first level using the problem solving steps. Allow children to investigate the second level
with friends, with minimal guidance. Allow children to investigate the third level INDEPENDENTly. Children create their own problem.
Teaching Segment and Video 1:
Equivalent division
Teaching Segment and Video 2:
Equivalent multiplication
These problems are directly linked to Explicit Teaching, are embedded in the Explicit Teaching Plan, and appear on the Explicit Teaching PowerPoint.
These, and more problems, appear as blackline masters on the Problem Solving PDF and are differentiated on the Problem Solving PowerPoint.

256 ÷ 8 has the same value as: 256 ÷ 4 64 ÷ 8 64 ÷ 4 64 ÷ 2 (64 ÷ 2)

23 x 8 has the same value as:

Three of these calculations give the same value. Which one gives a different value?
244 × 2 122 × 4 61 x 8 366 x 1 (366 x 1)

Complete the missing number sentence: 62 x 16 =
23 x 4
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23 x 16
46 x 4
46 x 16 (46 x 4)
x 8 (124)
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Investigating Equivalent Number Sentences Using Common Factors
Multiplication and Division 21, Patterns and Algebra 26 Equivalent simpler division calculations result if both numbers are divided by a
common factor, create equivalent multiplication.
Use cards to create a division number sentence.
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.
Reflection: How can we simplify division by creating equivalent divisions by dividing
by a common factor?
Problem Solving
256 ÷ 8 has the same value as:
256 ÷ 4
64 ÷ 8
64 ÷ 4
64 ÷ 2
Hint: Change the number sentence, and allow children to solve again!
http://www.alearningplace.com.au
Investigating Equivalent Number Sentences Using Common Factors
Multiplication and Division 21, Patterns and Algebra 26 Equivalent simpler division calculations result if both numbers are divided by a
common factor, create equivalent multiplication.
Use cards to create a multiplication number sentence.
Create an equivalent multiplication number sentence by dividing one number and
multiplying the other number by the same number.
Check that the number sentences are equivalent by performing the 2
multiplications.
Reflection: How can we create equivalent multiplications by dividing and
multiplying by a factor?
Problem Solving
23 x 8 has the same value as:
a. 23 x 4
b. 23 x 16
c. 46 x 4
d. 46 x 16
Hint: Change the number sentences, and allow children to solve again!
Problem Solving
Three of these calculations give the same value.
Which one gives a different value?
a. 244 × 2
b. 122 × 4
c. 61 x 8
d. 366 x 1
Hint: Change the number sentences, and allow children to solve again!
Problem Solving
Complete the missing number sentence:
62 x 16 = x 8
Hint: Change the number sentence, and allow children to solve again!
http://www.alearningplace.com.au
Investigating Equivalent Number Sentences Using Common Factors
Multiplication and Division 21, Patterns and Algebra 26 Equivalent simpler division calculations result if both numbers are divided by a
common factor, create equivalent multiplication.
Sit with a friend.
Construct a rectangle using square centimetres.
Work out the rectangle’s area.
Halve the length of one dimension while doubling the length of the other
dimension.
Work out the new rectangle’s area.
Discuss why the area remained the same.
Reflection: How can we create shapes with equivalent areas by dividing one
dimension and multiplying the other dimensions by a factor?
http://www.alearningplace.com.au
Investigating Equivalent Number Sentences Using Common Factors
Multiplication and Division 21, Patterns and Algebra 26 Equivalent simpler division calculations result if both numbers are divided by a
common factor, create equivalent multiplication.
Sit with a friend.
Create areas of hectares in shapes other than squares by halving the length of one
dimension while doubling the length of the other dimension.
For example, a square hectare is 100 m by 100m.
If we halve 1 dimension and double the other dimension, we get 50 m by 200 m.
Does this shape still have the area of a hectare? Why?
Reflection: How can we create shapes with equivalent areas by dividing one
dimension and multiplying the other dimensions by a factor?
http://www.alearningplace.com.au