Friday, 18 March 2016

WEEK 3 - MULTIPLICATION

THE BIG IDEAS
The first big idea this week surrounded the concept of multiplication (repeated addition) with an emphasis on the importance of understanding and successfully completing addition before beginning multiplication. This is important because the concept of addition is only extended for multiplication. This means that before you begin to teach students multiplication you must first assess their understanding of addition to ensure they have a solid foundation to build upon. 

 Secondly, unlike addition and subtraction their is only one 'type' of multiplication. However there are four different ways to display or model multiplication. These include:

  1. The set model - This is the most common multiplicative structure where you are dealing with a certain number of groups, all the same size. Both the number and the size of the groups are known but the total is unknown. An example of a question using the set model is: "There are three acorns on each of the two plates. How many altogether?"
  2. The array model - Area and arrays are typical examples of multiplicative structure. The area of any rectangle (in square units) can be found  either by covering the rectangle with unit squares and counting the all individually or by multiplying the width by the length. Similarly, in a rectangular array - an arrangement of discrete, countable objects - the total number of objects can be found by multiplying the number of rows by the number of objects in each row.
  3. Measurement model - The measurement model is used when dealing with measurement. An example of a question using this model is “I bought 4 hair ribbons each 2 metres long. How many metres of ribbon did I buy?” 
  4. Combination model - Combination models involve two factors representing the sizes of two different  sets and the product indicates how many different pairs of things that can be formed. An example of this is "I have 3 different coloured shirts and 2 different coloured pairs of trousers. How many different outfits can I make?" (Helping children learn mathematics, 2012)

Video: Multiplication as repeated addition
Video: Set model
Video: Array model
Video: Combinations model
THE MATHEMATICAL PROPERTIES
Null factor
When a number is multiplied by 0, the product will be 0
e.g. 3x0=0 and 0x3=0

Identity property
Any number multiplies by 1 remains the same
e.g.  3x1=3 and 1x3=3 

Commutative property
Also known as 'Turn arounds', this states that two numbers multiplied together have the same total no matter the order of the addends
e.g  3x4=12 and 4x3=12

Associative property
Used when multiplying three or more numbers together, it states that the problem can be solved out of sequence and still arrive at the correct total
e.g.  3x(4x5) is equal to (3x4)x5

Distributive property
Allows us to break up larger multiplication problems into smaller multiplication problems
e.g. 4x7 is equal to (4x5)+(4x2)


THE CONCEPT, SKILL, STRATEGIES AND SUPPORTING RESOURCES

Concept 
In multiplication, the concept is repeated addition. This concept can be taught using a set up as simple as pens in cups, gems in trays or acorns on plates - using a system such as this allows children to see the relationship between addition and multiplication as 3 x 3 is equal to 3 + 3 + 3. 




   

Skill
The ability to successfully complete repeated addition. This skill can be practiced using resources such as the ones suggested above and multiplication stories. Multiplication stories are stories that use multiplication throughout the plot of the book. Teachers can use this resource to allow children to practice the skill of multiplication in a way that seems new and exciting. 














Strategies 
For multiplication, there are four main strategies:

  1. Use counting (for 5x and 10x) - Using our hands, rods or MAB blocks to add groups of  5 or 10
  2. Think real world (for 0x and 1x) - Showing real world examples of '0 groups of' and '1 group of' to demonstrate
  3. Use doubles (for 2x, 4x and 8x) - similar to the addition strategy of doubles, for 2x the number is doubled, for 4x it is double doubled and 8x is double double doubled. 
  4. Build-up (for 3x and 6x) and build-down (for 9x) - For 3x, we build up from 2x (e.g. 3x3= 2x3+3), for 6x we build up from 5x (e.g. 6x3= 5x3+3) and for 9x we build-down for 10x (e.g. 9x3= 10x3-3). 
A resource to assist with the development of these strategies (particularly use doubles and build-up/build-down) with children is to use folding addition cards. These cards allow the teacher to demonstrate each strategy to the children.
THE LANGUAGE MODEL
The language model is used to display the relationship between the visual, verbal and symbolic elements of mathematics and forms 'stages' of learning. 

Student language - During student language, all language used should be familiar to the child and accompanied with familiar objects. This may vary depending on the 'story'. For example, 'There were three birds in three different trees. How many birds were there altogether?' in this story we may use an addition mat, toy birds, puppets, etc.
Materials language - At this stage, there is still no introduction of mathematical terms. The language is very similar to student language however the visuals used have become more abstract. For example, 'There are three counters in each of these three trays. How many counters altogether?'

Mathematics language - Moving away from 'stories', this stage introduces mathematical terms. For example, 'What does three multiplied by three equal?'

Symbolic language - This is the only stage where symbols (including symbolic numbers) are used. An example of a question from this stage would be '3 x 3 =' 





THE LANGUAGE MODEL FOR MULTIPLICATION
"The same sequence of experiences used for developing understanding of addition and subtraction - moving from concrete to pictorial to symbolic - should also be followed for multiplication and division. 


 THE MISCONCEPTION
  1. The student may have overspecialized his knowledge of multiplication and restricted it to “fact tests” or one particular problem format. If I was to come across this issue I would take the student back to the childrens' or materials language stage and give examples for all methods of multiplication. To avoid students overspecializing, I will ensure I am providing equal amounts of questions using each format.
  2. The student knows how to multiply but does not know when to multiply (other than because he was told to do so, or because the computation was written as a multiplication problem). This student may not understand the relationship between addition and multiplication and may benefit from spending more time and the childrens' or materials language stage with scaffolding and discussion. 

THE ACARA LINK
Addition is first introduced in year two. 
Strand: Number and Algebra
Substrand: Number and Place Value
Code: ACMNA031
Content descriptions: Recognise and represent multiplication as repeated addition, groups and arrays
Elaborations: 
  • representing array problems with available materials and explaining reasoning
  • visualising a group of objects as a unit and using this to calculate the number of objects in several identical groups
Scootle resource ideas:

  1. The array game is an online, interactive game that introduces children to the array model.
  2. The share and group biscuit factory is an online, interactive game that explores the relationship between multiplication and division at the childrens' language stage.
  3. The making arrays game is an online, interactive game that allows children to create arrays using the set number given.
Link: Australian curriculum - Year two
Link: The array game
Link: Share and group the biscuit factory
Link: Making arrays game

 THE RESOURCES AND IDEAS

  1. Pipe cleaner and beads - used for the set model. Each colour has a certain amount of beads (purple has two, green has three, etc). An example of a question where this resource can be used is finding four groups of three. A child would look for the pipe cleaners with three beads (green), pick up four pipe cleaners and count how many there are altogether. 
  2. Chocolate chip cookies - another resource used for the set model, felt circles or the "cookie" represent the group and felt "chocolate chips" represent the number in each group. So, four groups of three would be displayed as four cookies with three chips on each cookie. 
  3. Dice and a grid - used to introduce the array model, one die represents the rows and the other represents the columns which is then coloured in on the grid. The opposite can the be drawn to demonstrate the commutative nature of multiplication (for example, four rows of three and three rows of four).











THE TEXTBOOK SUMMERY
  • There are four distinct types of multiplicative structure:

  1. The set model - This is the most common multiplicative structure where you are dealing with a certain number of groups, all the same size. Both the number and the size of the groups are known but the total is unknown. An example of a question using the set model is: "There are three acorns on each of the two plates. How many altogether?".
  2. The array model - Area and arrays are typical examples of multiplicative structure. The area of any rectangle (in square units) can be found  either by covering the rectangle with unit squares and counting the all individually or by multiplying the width by the length. Similarly, in a rectangular array - an arrangement of discrete, countable objects - the total number of objects can be found by multiplying the number of rows by the number of objects in each row.
  3. Measurement model - The measurement model is used when dealing with measurement. An example of a question using this model is “I bought 4 hair ribbons each 2 metres long. How many metres of ribbon did I buy?” 
  4. Combination model - Combination models involve two factors representing the sizes of two different  sets and the product indicates how many different pairs of things that can be formed. An example of this is "I have 3 different coloured shirts and 2 different coloured pairs of trousers. How many different outfits can I make?" 
  • Before children tackle written strategies for multiplication, they must first have a firm grasp of place value, expanded notation and the distributive property. 
THE TEXTBOOK SUMMERY
ACU,. (2016). Learning Environment Online. Leo.acu.edu.au. Retrieved 3 March 2016, from http://leo.acu.edu.au/course/view.php?id=18458

Australia, E. (2016). Home - Scootle. Scootle.edu.au. Retrieved 3 April 2016, from https://www.scootle.edu.au/ec/p/home

Australian government,. (2016). Home - The Australian Curriculum v8.1.Australiancurriculum.edu.au. Retrieved 3 April 2016, from http://www.australiancurriculum.edu.au/

Reys, Lindquist, Lambdin, Smith, Rogers, & Falle, et al. (2012). Helping children learn mathematics. Milton, QLD: John Wiley & Sons.

YouTube. (2016). Youtube.com. Retrieved 3 April 2016, from https://www.youtube.com/
Posted by at

No comments:

Post a Comment

Subscribe to: Post Comments (Atom)