Download MULTIPLICATION and DIVISION

MULTIPLICATION and DIVISION
Partitioning, Place Value, Compensation, Estimation, Doubling, Halving, Inverse
Operations, Equality, Commutativity, Associativity, Distributivity, Multiples, Factors
Concept Development
Before children can begin to multiply or divide, they develop their understanding of groups.
Initially children may be able to make groups, but when asked to pick up a group, they may pick up
just one item. Understanding that we start with a group, break into smaller groups, and that a
group may be one item, involves understanding groups flexibly
Children then develop their understanding of equality - that 2 things are the same size and can
be represented by the same number
Children develop their understanding of division in two ways. They can divide by 2 by making
groups of 2. They can divide by 2 by making 2 equal groups. When we divide by 2 by making
groups of 2, we count the groups. When we divide by 2 by making 2 equal groups, we count the
items in the group. Seeing division in 2 ways is vital to further development of the division
concept!
When children first begin to multiply and divide, they do it by counting items by ones. They will
be able to make 3 groups of 2, but when they count the items, they will count by ones paying no
attention to the group structure eg 1 2 3 4 5 6
As they develop their understanding of groups, children will begin to count the items using skip or
rhythmic counting. They will be able to make 3 groups of 2, and when they count the items, they
will skip count eg 2 4 6 or rhythmic count 1 2 3 4 5 6. Rhythmic is just as advanced as skip
counting - and it demonstrates that the child understands they are repeatedly adding 2
As they further develop their understanding of groups, children will begin to multiply and divide
without having to see the items, but they will need group markers. In the absence of group
markers, they will raise a finger to represent each group. Children will be able to imagine groups of
2, and will skip count eg 2 4 6 or rhythmic count 1 2 3 4 5 6. Using arrays is a great way to
develop children's capacity to visualise the groups
Children will begin to use non-counting strategies such as knowing that 2 twos are 4, and adding
another 2 to make 3 twos
By the end of Year 2, children have been investigating multiplication and division additively. They
have been seeing multiplication and division as adding and subtracting equal groups
From Year 3 onwards, children need to change the way they think of multiplication and division.
Children need to think of multiplication and division multiplicatively - and they need to know it!
They need to begin to see multiplication by 2 as making a number 2 times greater - twice as big double. If they continue to think of multiplication as repeated addition, they will never be able to
multiply large numbers and division will never make sense!
They need to begin to see division by 2 as making a number 2 times smaller - 2 times less - half as
big. This follows on from seeing division in two ways and links division to fractions
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Now to the old bug bear - tables!!!!!
We prefer to call tables 'multiplication by single-digit numbers. Thinking of tables as multiplication
by single-digit numbers, allows children to develop their understanding of related concepts,
relationships and properties! Learning tables by rote deprives them of this understanding!
Let's investigate multiplication by 2 to see how we use these concepts, relationships and
properties:
If children do not know what 8 times 2 equals, they can partition the 8 into
numbers that they can multiply by 2. They could partition 8 into 5 and 3, 4
and 4 or 2 and 6
What does 2 times 5 equal?
What does 2 times 3 equal?
Because we partitioned 8 into 5 plus 3, we now need to add the products.
What is 10 plus 6?
Are you thinking it is easier just to learn what 2 times 8 equals by rote? Are you thinking you don't
have to think nearly so much if you just learn your tables? That's the point! We only think when
we have a problem, and the reason we learn Mathematics is to think!
If children just learnt what 2 times 8 equals by rote, they would be deprived of an opportunity to
discover and use an important property of multiplication - the distributive property. The
distributive property is a natural property of multiplication and it means we can partition a
number in numbers that add to make it, multiply each part, then add the products - and it work
with numbers of any size! And children will naturally use it! Children will learn what 2 times 8
equals - with deep understanding of the concepts, relationships and properties they used to learn
it!
If children learn tables by rote, they only know up to multiplication by 10 (or possibly 12).
Investigating multiplication using concepts, relationships and properties means that children can
use exactly the same concepts, relationships and properties to multiply numbers of any size!
Let's investigate how we can use the same concepts, relationships and properties to multiply a
larger number by 2:
If children do not know what 28 times 2 equals, they can partition the 28
into numbers that they can multiply by 2. They could partition 28 using
place value, into 20 and 8
What does 2 times 20 equal?
What does 2 times 8 equal?
Because we partitioned 28 into 20 plus 8, we now need to add the
products. What is 40 plus 16?
How does learning to multiply by single-digit numbers (tables) using concepts, relationships and
properties, help children to understand the inverse relationship between multiplication and
division and the relationship between division and fractions?
Let's investigate division by 3 to see how we use these concepts, relationships and properties:
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If children do not know what 15 divided by 3 equals, they can partition
the 15 into numbers that they can divide by 3. Numbers that they know
are divisible by 10 are called 'preferred multiples'. 'Preferred' because
the child prefers them because they know them. 'Multiples' because
when we multiply by 3 we get a multiple of 3. Children may know that 6
is a multiple of 3, they may also know 9 is a multiple of 3. Multiples will
differ between children - children use multiples that they prefer
What does 9 divided by 3 equal? What is a third of 9?
What does 6 divided by 3 equal? What is a third of 6?
Because we partitioned 15 into 9 plus 6, we now need to add the
quotients. What is 3 plus 2?
Did you notice we used division by 3 as making '3 equal groups', rather than as making 'groups of
3'? Did you notice we related division by 3 as finding a third?
Children can now divide numbers of any size using preferred multiples. They develop their
understanding that partitioning using place value is not always efficient because not every 10s or
100s number is divisible by 3
Children move onto dividing numbers by 3 that are not divisible by 3. Initially they will record a
remainder
As children develop their understanding of the relationship between division and fractions, they
investigate why the distributive property appears to work with division! They investigate what
they are doing when they find a fraction of a number. If we were multiplying 30 by 3, we would
have 3 times as much as 30. When we find a third of 30, will we have a third times as much as 30?
Are we multiplying by a third? If we multiply 30 by a third, we would have a third times as much as
30. So when we find a third of 30, we are multiplying 30 by a third
This is why it is very important we don't tell children rules like 'multiplication makes a number
larger'. This is clearly incorrect! Multiplying by a number larger than 1 makes a number larger.
Multiplying by a fractions makes a number smaller. Multiplying by 1 keeps a number the same
size. Multiplying by 0 makes 0! And multiplying by a fraction is the same as dividing by a whole
number. So many beautiful concepts and relationships!
Children are exposed to the algorithms for multiplication and division in Year 5, but continue to
multiply and divide using concepts, relationships and properties all of their lives!
The Scope and Sequences, Teaching Programs, Teaching Plans, Teaching Videos and Learning
Videos develop the concepts of multiplication and division, and the concepts, relationships and
properties of Partitioning, Place Value, Compensation, Estimation, Doubling, Halving, Inverse
Operations, Equality, Commutativity, Associativity, Distributivity, Multiples, Factors from
Kindergarten to Year 6
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Email: [email protected] Twitter: @carol_learning Facebook: A Learning Place
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