Multiplication and Division 10, Patterns and Algebra 18_Guided and Independent Investigations
(Year 3) ACMNA056, ACMNA057, ACMNA051, NSW MA2-6NA, MA2-8NA
Multiply and divide by 2 using properties and relationships, explaining odd and even numbers.
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: cards, pencil, paper
What could we do?
Children:
1. select cards to make numbers to
multiply by 2 as guided by the
teacher
2. record their number sentence
3. multiply by 2 using the distributive
property and / or by doubling
What language could we use to ask
questions and explain?
 How could we multiply this
number by 2?
 How could we use the
distributive property?
 How could we double?
 How could we divide this
number by 2?
1. select cards to make numbers to
 How could we halve this
divide by 2 / halve as guided by the
number?
teacher
 How could we partition the
2. record their number sentence
number?
3. divide the number by 2 by
 How could we divide the parts?
partitioning, dividing the parts by 2  How could we add the
/ halving the parts, adding the
quotients?
quotients
What language could we use to
explain?
What could we do?
Children:
1. sit in pairs
2. select cards to make numbers to multiply by
2 that are neither too easy nor too
challenging
3. record their number sentence
4. multiply by 2 using the distributive property
and / or by doubling
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
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1. display cards to make numbers to divide by
2 / halve that are neither too easy nor too
challenging
2. record their number sentence
3. divide the number by 2 by partitioning,
dividing the parts by 2 / halving the parts,
adding the quotients
 The number I selected is ...
 I multiplied … by 2 by ...
 I partitioned the number into …
 I multiplied the parts …
 I added the products …
 I doubled by …
 The number I selected is ...
 I divided … by 2 by ...
 I halved this number by …
 I partitioned the number into …
 I divided the parts …
 I added the quotients …
What is multiplication?
What is division?
How could we multiply by 2?
How could we divide by 2?
What is the distributive property?
What fraction do we get when we divide by 2?
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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,
Multiplication and
Division by 2 /
Halving
 Children sit with a friend. They individually select:
►
one card to make a single-digit number they feel ready to multiply by 2
►
an Ace and another card to make a teen number they feel ready to divide by 2
►
an Ace and another card to make a teen number they feel ready to multiply by 2
►
2 cards to make a two-digit number they feel ready to divide by 2
►
2 cards to make a two-digit number they feel ready to multiply by 2
►
3 cards to make a three-digit number they feel ready to divide by 2
►
3 cards to make a three-digit number they feel ready to multiply by 2
They independently multiply / divide their number by 2. They explain to their friend how they used the distributive property to
multiply the number by 2. They explain to their friend how they partitioned the number into their preferred multiples of 2, to
divide it by 2. They explain why they do or do not have a remainder
Alternating between multiplication and division will deep understanding of both and the inverse relationship.
Inverse and
commutative
number
sentences.
 Children flip cards to make single-digit numbers to multiply by 2. They record the multiplication number sentence. They use the
commutative property to record another multiplication sentence. They then use the inverse relationship between multiplication
and division to record 2 division number sentences. For example, they flip 6 and record 6 x 2 = 12 and 2 x 6 = 12. They then
record 12 ÷ 2 = 6 and 12 ÷ 6 = 2.
Calculator
multiplication and
division by 2 /
halving.
 Children use a calculator to multiply a number by 2, then check using the distributive property. They explain to their friend how
they multiplied using the distributive property.
 Children use a calculator to divide a number by 2, then check using partitioning. They explain to their friend how they divided.
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What number did
I multiply by 2?
Array multiples
of 2 (tables!).
 In pairs, each child has 2 cards. Each child places their cards face down and records each number’s product
when multiplied by 2 on paper next to the pair, for example,
Children take turns to guess each of their friend’s numbers, saying, for example, ‘I think the number is 9
because 2 times 9 equals 18’ or ‘I think the number is 9 because 18 divided by 2 equals 9’. If correct, the child
takes the card. The first child to have both of their friend’s cards, wins. Variation 1: Children select 3, 4 or 5
cards. Variation 2: Children play in small groups and take it in turns to guess any of the other players’ numbers.
 Children have a multiplication by single-digit numbers array and use it to recall
multiples of 2 through properties and relationships. For example, 2 x 7 could be
represented using 2 sheets of card as, for example,
Children develop recall of single-digit multiplication by 2 with understanding of, for example, the commutative property, the
relationship to multiplying by 4, 6 and 8 and the relationship to multiplying by 10.
NB: This array replaces the traditional multiplication table chart where products are in the place of the dots. Any multiplication
table charts with products listed actually ensure children don’t learn to recall single-digit multiplication because it is easier to
look at the product on the chart than to use properties and relationships to work out a single-digit multiplication.
Recall multiples
of 2 (tables!).
 To continue their focus on recalling multiples of 2 with understanding, children flip cards one at a time and name the product
when the single-digit number is multiplied by 2. If the child knows the product when multiplied by 2 efficiently (quite quickly,
and without much conscious thought), they place the card in the ‘know’ pile. If the child does not know the product when
multiplied by 2 efficiently, they place the card in the ‘learn’ pile. Once they have 2 cards with different numbers in the ‘learn’
pile, the child stops flipping cards. The child look at the numbers that they cannot yet multiply by 2, and consider some
strategies to assist them to efficiently multiply the number by 2. If they cannot think of any strategies themselves, they ask their
friend and classmates in either a structured sharing session, or while investigating multiplying and dividing by 2. Their friend
and classmates share their successful strategies for efficiently recalling multiples of 2 up to 2 x 10.
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Email: [email protected]
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A word about learning tables (multiplication of single-digit numbers):
We may be tempted to have children attempt to ‘learn’ tables by rote or by memory. Memory is not reliable, does not a large
storage area and is regularly cleared out to allow room for more memories.
Many children are unsuccessful at learning tables (multiplication of single-digit numbers) by rote. Children who are successful at
learning tables (multiplication of single-digit numbers), learn up to 10 times (or sometimes 12 times) the single-digit number with
no understanding of

how to multiply their single-digit number by two-digit numbers and higher,

important properties of multiplication including the distributive property, the commutative property and the associative
property,

the relationships between numbers including multiplying by 2 is doubling, dividing by 2 is halving
Children who investigate tables (multiplication of single-digit numbers) through properties and relationships develop understanding
and capacity to apply these properties and relationships to multiply and divide numbers of any size by single-digit numbers.

It is important that children develop their capacity to efficiently recall tables (multiplication of single-digit numbers) with
understanding of properties and relationships, a capacity that continues to develop over the 2 years of Year 3 and Year 4.
Children then use and further develop this capacity over the 2 years of Year 5 and Year 6.
Website: http://www.alearningplace.com.au
Email: [email protected]
Twitter: @learn4teach
YouTube: A Learning Place A Teaching Place
Facebook: A Learning Place
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Investigating Multiplication and Division by 2 Using Properties and Relationships
MULTIPLICATION AND DIVISION 10, PATTERNS AND ALGEBRA 18 Multiply and divide by 2 using properties and relationships, explaining
odd and even numbers.
Select a card to make a single-digit number to multiply by 2 that is neither too easy
nor too challenging.
Record your multiplication number sentence.
Multiply by 2 using the distributive property and / or by doubling.
http://www.alearningplace.com.au
Investigating Multiplication and Division by 2 Using Properties and Relationships
MULTIPLICATION AND DIVISION 10, PATTERNS AND ALGEBRA 18 Multiply and divide by 2 using properties and relationships, explaining
odd and even numbers.
Select an Ace and another card to make a teen number to divide by 2 that is neither
too easy nor too challenging.
Record your division number sentence.
Divide by 2 by partitioning, dividing the parts by 2 / halving the parts, adding the
quotients.
Explain why you do or do not have a remainder.
http://www.alearningplace.com.au
Investigating Multiplication and Division by 2 Using Properties and Relationships
MULTIPLICATION AND DIVISION 10, PATTERNS AND ALGEBRA 18 Multiply and divide by 2 using properties and relationships, explaining
odd and even numbers.
Select an Ace and another card to make a teen number to multiply by 2 that is
neither too easy nor too challenging.
Record your multiplication number sentence.
Multiply by 2 using the distributive property and / or by doubling.
http://www.alearningplace.com.au
Investigating Multiplication and Division by 2 Using Properties and Relationships
MULTIPLICATION AND DIVISION 10, PATTERNS AND ALGEBRA 18 Multiply and divide by 2 using properties and relationships, explaining
odd and even numbers.
Select 2 cards to make a two-digit number to divide by 2 that is neither too easy nor
too challenging.
Record your division number sentence.
Divide by 2 by partitioning, dividing the parts by 2 / halving the parts, adding the
quotients.
Explain why you do or do not have a remainder.
http://www.alearningplace.com.au
Investigating Multiplication and Division by 2 Using Properties and Relationships
MULTIPLICATION AND DIVISION 10, PATTERNS AND ALGEBRA 18 Multiply and divide by 2 using properties and relationships, explaining
odd and even numbers.
Select 2 cards to make a two-digit number to multiply by 2 that is neither too easy
nor too challenging.
Record your multiplication number sentence.
Multiply by 2 using the distributive property and / or by doubling.
http://www.alearningplace.com.au
Investigating Multiplication and Division by 2 Using Properties and Relationships
MULTIPLICATION AND DIVISION 10, PATTERNS AND ALGEBRA 18 Multiply and divide by 2 using properties and relationships, explaining
odd and even numbers.
Select 3 cards to make a three-digit number to divide by 2 that is neither too easy
nor too challenging.
Record your division number sentence.
Divide by 2 by partitioning, dividing the parts by 2 / halving the parts, adding the
quotients.
Explain why you do or do not have a remainder.
http://www.alearningplace.com.au
Investigating Multiplication and Division by 2 Using Properties and Relationships
MULTIPLICATION AND DIVISION 10, PATTERNS AND ALGEBRA 18 Multiply and divide by 2 using properties and relationships, explaining
odd and even numbers.
Select 3 cards to make a three-digit number to multiply by 2 that is neither too easy
nor too challenging.
Record your multiplication number sentence.
Multiply by 2 using the distributive property and / or by doubling.
http://www.alearningplace.com.au
Investigating Multiplication and Division by 2 Using Properties and Relationships
MULTIPLICATION AND DIVISION 10, PATTERNS AND ALGEBRA 18 Multiply and divide by 2 using properties and relationships, explaining
odd and even numbers.
Investigate multiplying and dividing by 2 using cards to make numbers you are ready
to multiply or divide:
 Select a card to make a single-digit number to multiply by 2 that is neither too
easy nor too challenging. OR
 Select an Ace and another card to make a teen number to divide by 2 that is
neither too easy nor too challenging. OR
 Select an Ace and another card to make a teen number to multiply by 2 that is
neither too easy nor too challenging. OR
 Select 2 cards to make a two-digit number to divide by 2 that is neither too
easy nor too challenging. OR
 Select 2 cards to make a two-digit number to multiply by 2 that is neither too
easy nor too challenging. OR
 Select 3 cards to make a three-digit number to divide by 2 that is neither too
easy nor too challenging. OR
 Select 3 cards to make a three-digit number to multiply by 2 that is neither
too easy nor too challenging.
Record your number sentence.
Multiply by 2 using the distributive property and / or by doubling OR divide by 2 by
partitioning into preferred multiples of 2.
http://www.alearningplace.com.au
Investigating Multiplication and Division by 2 Using Properties and Relationships
MULTIPLICATION AND DIVISION 10, PATTERNS AND ALGEBRA 18 Multiply and divide by 2 using properties and relationships, explaining
odd and even numbers.
Select a card to make single-digit numbers to multiply by 2.
Record the multiplication number sentence.
Use the commutative property to record another multiplication sentence.
Use the inverse relationship between multiplication and division to record 2 division
number sentences.
For example, flip 6 and record 6 x 2 = 12 and 2 x 6 = 12. Then record 12 ÷ 2 = 6 and
12 ÷ 6 = 2.
http://www.alearningplace.com.au
Investigating Multiplication and Division by 2 Using Properties and Relationships
MULTIPLICATION AND DIVISION 10, PATTERNS AND ALGEBRA 18 Multiply and divide by 2 using properties and relationships, explaining
odd and even numbers.
Sit with a friend.
Each of you has 2 cards.
Place your cards face down and record each number’s product when
multiplied by 2 on paper next to the pair, for example,
Take turns to guess your friend’s numbers, saying, for example, ‘I think the number
is 9 because 2 times 9 equals 18’ or ‘I think the number is 9 because 18 divided by 2
equals 9’.
If correct, take the card.
The first child to have both of your friend’s cards, wins.
Variation 1: Select 3, 4 or 5 cards.
Variation 2: Play in small groups and take it in turns to guess any of the other
players’ numbers. The child with the most cards at the end of the game wins.
http://www.alearningplace.com.au
Investigating Multiplication and Division by 2 Using Properties and Relationships
MULTIPLICATION AND DIVISION 10, PATTERNS AND ALGEBRA 18 Multiply and divide by 2 using properties and relationships, explaining
odd and even numbers.
Sit with a friend.
Lay out 12 cards in a 3 by 4 array.
Take turns select 1 card to make a single-digit number, or 2 cards to make a twodigit number, to multiply by 2.
Multiply the number by 2, explain how you used the distributive property.
Check on a calculator.
If correct, keep the cards.
http://www.alearningplace.com.au
Investigating Multiplication and Division by 2 Using Properties and Relationships
MULTIPLICATION AND DIVISION 10, PATTERNS AND ALGEBRA 15, 18 Multiply and divide by 2 using properties and relationships,
explaining odd and even numbers.
Flip cards one at a time and name the product when the single-digit number is
multiplied by 2.
If you know the product when multiplied by 2 efficiently (quite quickly, and without
much conscious thought), place the card in the ‘know’ pile.
If you do not know the product when multiplied by 2 efficiently, place the card in
the ‘learn’ pile.
Once you have 2 cards with different numbers in the ‘learn’ pile, stop flipping cards.
Look at the numbers that you cannot yet multiply by 2, and consider some strategies
to assist them to efficiently multiply the number by 2.
If you cannot think of any strategies yourself, ask your friend and classmates to
share their successful strategies for efficiently recalling multiples of 2 (tables!) up to
2 x 10.
http://www.alearningplace.com.au
Investigating Multiplication and Division by 2 Using Properties and Relationships
MULTIPLICATION AND DIVISION 10, PATTERNS AND ALGEBRA 15, 18 Multiply and divide by 2 using properties and relationships,
explaining odd and even numbers.
Have a multiplication by single-digit numbers array and use it to recall multiples of 2
(tables!) through properties and relationships.
For example, 2 x 7 could be represented using 2 sheets of card as, for example,
http://www.alearningplace.com.au