Transcript CAMPPP K to 2 Session 2A Counting and Multiplication

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
1
2
6
1
3
4
5
2
5
6
4
3
6 in this group
OR
6 in this group
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
5
4
7
6
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.”
0
1
2
3
4
5
6
7
8
9
2 + 2 + 2 + 2 + 2 + 2 =12
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
6x2
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)
5 groups of 3
make 15
3 groups of
5 make 15
5 x 3 = 15
3 x 5 = 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
2 x 6 = 12
1 x 12 = 12
and
and
and
4 x 3 +12
6 x 2 = 12
12 x 1 = 12
Late Primary
• Geometric models
– Focus on the commutative property
• axb=bxa
• 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
2 x 6 = 12
1 x 12 = 12
and
and
and
4 x 3 +12
6 x 2 = 12
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.
4
2
3
1
6
12
Late Primary
• Geometric models
– Focus on the relationship of equality among
the different representations.
4
2
3
1
6
12
4 x 3 = 2 x 6 = 1 x 12