Kindergarten to Grade 2 / Session #4

Developing Early Multiplication Concepts Welcome to… …the exciting world of Counting and Developing Early Multiplication Concepts

Everything we do in Kindergarten to Grade 2 is the foundation of later mathematical understanding.

Big Ideas in NS&N that pertain to PR Numbers tell how many or how much. Classifying numbers or numerical relationships provides information about the characteristics of the numbers or the relationship. There are many equivalent representations for a number or numerical relationship. Each representation may emphasize something different about that number or relationship. Our number system of ones, tens, and hundreds helps us know whether we have some, many and very many. The operations of addition, subtraction, multiplication and division hold the same fundamental meaning no matter the domain to which they are applied.

Counting Counting involves both reciting a series of numbers and representing a quantity by a symbol First experiences with counting are not initially attached to an understanding of the quantity or value of the numerals Counting is a powerful early tool and is intricately connected to the other four ‘Big Ideas’

Let’s find out more about the Counting Principles…

Stable Order Principle Principles of Counting Stable Order Principle 1,2,3,4,5,6… not 1,2,3,4,6,8,9,10

Order Irrelevance Principle Principles of Counting Order Irrelevance Principle 6 1 1 2 3 4 5 2 It doesn’t matter where we start the count, as long as we do not add or take away, the count will be the same. 5 6 4 3 OR 6 in this group 6 in this group

Conservation Principle Principles of Counting Even 6 month old babies have a sense or “moreness”; if you offer them a whole cookie or a half a cookie, they will reach for the whole one. This seems to be tied to the amount of space something takes up. In this principle, young children will assume that, since the dots are spread out, and therefore take up more space, there must be more of them. They need to disregard the space and focus on the quantity. Conservation Principle

Abstraction Principle Principles of Counting Abstraction Principle Following what was just said about young children’s sense of “moreness”, children will first assume that there are more elephants because they are bigger. They need instead to focus on the “five-ness” – the abstract concept of “howmanyness”.

The Abstraction Principle Can Also Look Like… Principles of Counting The Abstraction Principle Can Also Look Like…

One-to-One Correspondence Principles of Counting One-to-One Correspondence Each item gets one, and only one count.

Cardinality Principle Principles of Counting Cardinality Principle 1 3 5 7 2 4 6 8 The last number counted names the count and tells us how many. Early counters who have not mastered this principle will count the items, but when asked how many, will recount – perhaps many times! 8 hearts

Movement is Magnitude Principle Principles of Counting Movement is Magnitude Principle 4 2 5 1 3 To go up one number, you must have one more thing. This seems fairly obvious for counting by ones, but often we overlook frequently repeating opportunities to actually count 2 things when we count by twos, to count groups of 5 things when we count by fives, to count groups of ten things when we count by tens. We move too quickly into assuming students understand this.

Principles of Counting Unitizing Hundreds Tens Ones Our place value system takes ten units and bundles them together and treats the bundles as groups that can themselves be counted. In this instance, we have unitized the group of ten. 50 has 5 groups of 10. However, this also works when we multiply. For example, multiplying by 6 means we can count groups of 6. If we have 6 sixes, we might see 6 bundles of 6 – we have unitized six. This understanding is at the heart of multiplicative thinking.

From Five to Ten! Children build on their concept of 5 to develop a concept of 10. They consolidate their concept of quantities of 10 in relation to the teens and decades. They can use this foundation to understand that the digit 1 in 10 represents a bundle of ten.

Quantity Quantity represents the “howmanyness” of a number and is a crucial concept in developing number sense. Having a conceptual understanding of the quantity of five and then of ten are important prerequisites to understanding place value, the operations and fractions. An early understanding of quantity helps with concepts around estimating and reasoning with number, particularly proportional reasoning.

Relationship Between Counting and Quantity Children don’t intuitively make the connection between counting and their beginning understanding of quantity. With rich experiences using manipulatives, they gradually learn that the last number in a sequence identifies the quantity in the set that is being counted (cardinality) This is an important beginning step in linking counting and quantity.

Quantity and Mathematical Reasoning Children need continued experience with all types of manipulatives to understand that each quantity also holds within it many smaller quantities. Developing a robust sense of quantity helps children with mathematical reasoning.

We Need to Revisit Often! Quantity is not a simple concept that children either have or do not have. Children need experience in repeating similar types of estimation (and checking) activities to build up their conceptual understanding of the amount of something. Resist the temptation to move too quickly into just using numbers!

Constructing Understanding of Multiplication A continuum of conceptual understanding

Beginning in Early Primary Skip counting Must be connected the actual count of objects Must be connected to many different models Students see the Movement-is-Magnitude principle of counting MUST BE BOTH FORWARD AND BACKWARD

Beginning in Early Primary Skip counting should be seen as a method for counting more quickly and strongly connected to the counting of real objects or people

Beginning in Early Primary and continuing throughout Primary Skip counting models Rote skip counting focusing on the aural aspects of the rhythm of the chant Counting real things Counting money Number line Hundred chart / carpet Rekenrek Five frames and ten frames Base Ten materials (late Primary)

Beginning in Early Primary Skip counting Teachers begin to show the representation of adding the same quantity over and over again. Teachers connect the model and the symbolic representation of repeated addition to the number of times the quantity is added. This should be shown as “I added 2 6 times to get 12.” AND “I added 6 groups of 2 together to get 12 all together.” 2 + 2 + 2 + 2 + 2 + 2 =12

Beginning in Early Primary Skip counting This should be shown as “I added 2 6 times to get 12.” AND “I added 6 groups of 2 together to get 12 all together.” Let's say it together! 0 1 2 3 4 5 6 7 8 9 10 11 12 2 + 2 + 2 + 2 + 2 + 2 =12

Beginning in Early Primary Skip counting This should be shown as “I added 2 6 times to get 12.” AND “I added 6 groups of 2 together to get 12 all together.” Let's say it together! 2 + 2 + 2 + 2 + 2 + 2 =12

Mid - Primary 6 groups of 2 6 x 2 Skip counting All work is initially modeled by the teacher, who shows these connections many times, INFORMALLY at first, then explicitly. Once the students have seen many examples of repeated addition, and the symbolic representation with the addition symbol, the connection can be made to standard notation: 6 groups of 2 6 x 2

Late Primary Skip counting 3 groups of 5 make 15. (5, 20, 15) 3 x 5 = 15

Late Primary Geometric models Using the array model (lining up objects into rows and columns) Begin to work with square tiles to make arrays with different quantities

Late Primary Geometric models Using the array model (lining up objects into rows and columns) 3 groups of 5 make 15 3 x 5 = 15 5 groups of 3 make 15 5 x 3 = 15

Late Primary Geometric models Begin to work with square tiles to make arrays with different quantities How many different rectangles can you make for 12 tiles? What equations are represented by each arrangement? 3 x 4 = 12 and 4 x 3 +12 2 x 6 = 12 and 6 x 2 = 12 1 x 12 = 12 and 12 x 1 = 12

Late Primary Geometric models Focus on the commutative property a x b = b x a Does turning the rectangle ¼ turn yield a different rectangle? Is 3 x 4 always the same as 4 x 3? 3 x 4 = 12 and 4 x 3 +12 1 x 12 = 12 and 12 x 1 = 12 2 x 6 = 12 and 6 x 2 = 12

Late Primary Geometric models Focus on the relationship of the row and column lengths as the array is rearranged: as the number of columns doubles, the number of rows is halved. 4 2 1 3 6 12

Late Primary Geometric models 4 x 3 = 2 x 6 = 1 x 12 Focus on the relationship of equality among the different representations. 4 2 1 3 6 12 4 x 3 = 2 x 6 = 1 x 12