Module 6 - Multiplicative Thinking
Typical Progression
Common Difficulties
Manipulatives, Games and Technology
Pedagogy/Strategies for Teaching and Assessment
Mathematical Representation
Initially introduced informally without emphasis on formal language or recording
Experience multiplication as repeated addition and start thinking multiplicatively, such as 'making arrays and combinations'
Combinations - ie how many different outfits can be made from 2 pants, 3 skirts
Rate problems ie. $5 per student, 38 students
Recipes - how would you modify a recipe for 4 people to feed 8
Properties
Commutative a x b = b x a
Associative (a x b) x c = a x (b x c)
Distributive a x (b+c) = (a x b) (a x c)
Multiplication property of 1 (n x 1 = n)
Multiplication property of zero (n x 0 = 0)
Array Presentation
Useful concrete or pictorial representation
3 x 5 and 5 x 3, rotating arrays can get understanding of commutativity
Arrays
The Arrays Game - roll dice and mark out arrays on grid paper
Multiplication Strategies
Counting in Multiples
Relate to known facts
Using Patterns
Multiplication
Break a tower
Number Windows
Cover Ups
Step 1 - Understanding, Step 2 Networking, Step 3 Recalling (See Addition/Subtraction notes)
Tables - Commutative properties
MAB Blocks - Trading
Alternative Methods of Multiplication
Venetian Grid Method
Grid multiplication
Division
Arises naturally out of real world situations
Sharing (how many toys..) and Grouping (how many teams...) aspects
Division
"The Doorbell Rang" Pat hutchins
Division strategies
Relate to multiplication
Repeated subtraction
Doubling and Halving
Compatible numbers strategy - estimation ie 3388 / 76 -> 3200 / 80
Function Machine
Order of Operations BIMDAS
Repeated Aggregation Struction
Multiplication = so many sets of
Scaling Structure
Increase quantity by a certain amount/scale factor
Expanded notation 100 / 20 / 7
Factoring
Commutative Property
Partitioning
Distributive property
Doubling & Halving
Logical Reasoning
eg. 15x120 -> 10x120 plus 5x120, but 5x120 is half of 10x120, so 15x120 =1,200 + 600 = 1,800
14 x 55 = (2x7) x (5x11)
Multiplication Algorithm uses DISTRIBUTIVE PROPERTY - 84x6 = 4x6 + 80x6
Difficulty learning standard written algorithms
In process of learning algorithm, children forget to make sense of numbers they are dealing with
Many errors due to faulty application of written algorithm
Should be encouraged to use estimation to check the answers they obtained are reasonable
Difficulty with multiplying with zeroes