Partitive/Sharing Division
Definition

You know the number of groups and you are trying to find the amount inside each group. Number of shares are known.

Partitive/Sharing Division
Example

20 biscuits altogether. They are equally shared between 5 dogs, how many biscuits does each dog get?
20/5 = 4 biscuits each.

Quotative/Measuring Division
Definition

You know the amount in each group, you are trying to find out how many groups there are. Size of the share is known.

Quotative/Measuring Division
Example

20 biscuits altogether. Each dog gets an equal amount of 4 biscuits each. How many dogs get a share of the biscuits?
20/4 = 5 dogs.

Multiplication/Division Properties

Commutative
Associative
Distributive
Inverse/Reversibility

Commutative Definition

Change the order of factors.

Associative Definition

Doubling, halving, thirding and trebling.
Division: Factorising
Multiplication: Proportional adjustment

Distributive Definition

Splitting factors and using tidy numbers.
Division: Chunking
Multiplication: Place value partitioning
Both: Rounding and compensating

Inverse Definition

Doing and undoing.
Reversing.

Counting All (CA) Features

Counting using units of one.
Multiplication: forming sets and counting objects by one
Division: Sharing objects one by one or forming sets and counting by one.

Advanced Counting (AC)  Early Additive (EA)
Stage 4  Stage 5

Composite counting.
Multiplication: skip counting or repeated addition.
Division: sharing or subtraction by skip counting.

Advanced Additive (AA)
Stage 6

Known facts and deriving.
Multiplication: knowing facts and using commutative, associative and distributive properties to find unknown facts.
Division: sharing, repeated subtraction through reverse multiplication (inverse).

Problem Types

Equal groups
Rate
Comparison
PartWhole
Cartesian Product
Rectangular area

Equal Groups

Sharing and forming of equal groups.

Rate

Use of measurable units such as time.

Comparison

Comparing a quantity to find another.

PartWhole

Uses more than one method to find whole answer e.g ratio andmultiplication

Cartesian Product

Combinations made from two different quantities

Rectangular Area

Two factors combined to make a product. Changing the order of factors doesn't change the product.

Factorisation

Division proportional adjustment:
136/8 = 136/2/2/2 = 68/2 = 34/2 = 17

Number Knowledge

Division:
130/5 = 130/10 x2 = 13x2 = 26

Chunking

Division place value partitioning:
136/8 = 80/8 + 56/8 = 10+7 = 17

Rounding and Compensating
Division

Division:
136/8 = 160/8 24/8 = 203 = 17

Place Value Partioning

Multiplication:
38x5 = 30x5 +8x5 = 150+40 = 190

Proportional Adjustment

Multiplication:
38x5 = 19x10 = 190

Rounding and Compensating

Multiplication:
38x5 = 40x5  2x5 = 20010 = 190

Division is

The inverse of multiplication and repeated subtraction.

Multiplication is

The inverse of division and repeated addition.

Place value partitioning extra examples

3x37= 3x30 + 3x7
5x42= 5x40 + 5x2
4x29= 4x20 + 4x9

Rounding and Compensating extra examples

6x24= 6x25  6x1
3x29= 3x30  3x1
7x19= 7x20  7x1

Proportional Adjustment extra examples

8x12= 4x24 = 2x48

YoungLoveridge (2005) key points

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YoungLoveridge (2005) Implications

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Neill (2008) key points

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Neill (2008) Implications

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