Kindergarten to Grade 2 / Session #4

Welcome to…

…the exciting world of

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…

Principles of Counting

Stable Order Principle

1,2,3,4,5,6…

not

1,2,3,4,6,8,9,10

Principles of Counting Order Irrelevance Principle

6

5 2

3 4 OR

6 in this group 6 in this group

1

6 5

2

3 4

1

Principles of Counting

Conservation Principle

Principles of Counting

Abstraction Principle

Principles of Counting

The Abstraction Principle Can Also

Look Like…

Principles of Counting

One-to-One Correspondence

Principles of Counting

Cardinality Principle

1 3

2 4 6

5 7

8

Principles of Counting

Movement is Magnitude Principle

1 2 3 4 5

Principles of Counting

Unitizing

Hundreds Tens Ones

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.”

2 + 2 + 2 + 2 + 2 + 2 =12

0 1 2 3 4 5 6 7 8 9 10 11 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.”

2 + 2 + 2 + 2 + 2 + 2 =12

Mid - Primary

• 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

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 relationship of the row and

column lengths as the array is rearranged: as

the number of columns doubles, the number

of rows is halved.

3

4

6

2

12

1

Late Primary

• Geometric models

– Focus on the relationship of equality among

the different representations.

3

4

6

2

12

1

4 x 3 = 2 x 6 = 1 x 12