Download Quebec Operations - Nelson

PQ Mathematics Curriculum (K–6)
Correlation to the Operations Developmental Map
Concept 1
Addition leads to a
total and subtraction
indicates what’s
missing. Addition
and subtraction are
intrinsically related.
Phase 1
Phase 2
Phase 3
Phase 4
Phase 5
Beginner
Concrete
Whole Number Comfort
More Abstract
Flexible
Focus on Counting to Solve Problems
Formal Operations with Numbers to
20; Concrete Operations with
Numbers to 100
Formal Operations with Whole
Numbers; Concrete Operations with
Decimals
Fluency with Whole Number
Operations; Formal Operations with
Decimals
Fluency with Whole Number and
Decimal Operations; Concrete
Operations with Integers and Fractions
Natural numbers
Meaning of operations
Cycle 1: operation, operation
sense: addition (adding,
uniting, comparing), sum,
subtraction (taking away,
complement, comparing),
difference, term, missing
term, number line,
multiplication (repeated
addition, Cartesian product)
and division (repeated
subtraction, sharing, number
of times x goes into y)
Natural numbers
Meaning of operations
Cycle 1: operation, operation
sense: addition (adding,
uniting, comparing), sum,
subtraction (taking away,
complement, comparing),
difference, term, missing
term, number line,
multiplication (repeated
addition, Cartesian product)
and division (repeated
subtraction, sharing, number
of times x goes into y)
Natural numbers
Meaning of operations
Cycle 1: operation, operation
sense: addition (adding,
uniting, comparing), sum,
subtraction (taking away,
complement, comparing),
difference, term, missing
term, number line,
multiplication (repeated
addition, Cartesian product)
and division (repeated
subtraction, sharing, number
of times x goes into y)
Natural numbers
Meaning of operations
Cycle 2: property of
operations: associative law
Natural numbers
Meaning of operations
Cycle 3: meaning of an
equality relation (equation),
meaning of an equivalence
relation
Cycle 1: choice of operation:
addition, subtraction
Cycle 1: choice of operation:
addition, subtraction
Cycle 1: meaning of an
equality relation (equation),
meaning of an equivalence
relation
Cycle 1: meaning of an
equality relation (equation),
meaning of an equivalence
relation
Cycle 1: relationships
between the operations
Cycle 1: relationships
between the operations
Cycle 1: property of
operations: commutative law
Cycle 2: property of
operations: associative law
Operations
Cycle 1: own processes for
mental computation:
addition, subtraction
Operations
Cycle 2: approximating the
result of an operation:
addition, subtraction
Cycle 1: patterns: series of
numbers, family of operations
Cycle 2: own processes for
mental computation:
addition, subtraction
Cycle 3: order of operations
(series of operations involving
natural numbers)
Operations
Cycle 2: own processes for
mental computation:
addition, subtraction
Cycle 2: conventional
processes for written
computation: adding two
four-digit numbers
Cycle 2: conventional
processes for written
computation: subtracting a
four-digit number from a
four-digit number such that
the difference is greater than 0
Cycle 3: series of operations
in accordance with the order
of operations
Cycle 2: patterns: series of
numbers, family of operations
(Cont.)
1 of 8
January 2006
0-17-632376-7
(Cont.)
(Cont.)
Professional Resources and Instruction for Mathematics Educators, © 2005 Nelson, a division of Thomson Canada Limited
(Cont.)
PQ Mathematics Curriculum (K–6)
Correlation to the Operations Developmental Map (Cont.)
Phase 1
Phase 2
Phase 3
Phase 4
Phase 5
Beginner
Concrete
Whole Number Comfort
More Abstract
Flexible
Focus on Counting to Solve Problems
Formal Operations with Numbers to
20; Concrete Operations with
Numbers to 100
Formal Operations with Whole
Numbers; Concrete Operations with
Decimals
Fluency with Whole Number
Operations; Formal Operations with
Decimals
Fluency with Whole Number and
Decimal Operations; Concrete
Operations with Integers and Fractions
Decimals
Meaning of operations
Cycle 2: operation sense:
addition and subtraction
Decimals
Meaning of operations
Cycle 2: operation sense:
addition and subtraction
Decimals
Meaning of operations
Cycle 2: operation sense:
addition and subtraction
Concept 1 (Cont.)
Addition leads to a
total and subtraction
indicates what’s
missing. Addition
and subtraction are
intrinsically related.
Operations
Cycles 2 & 3: approximating
the result of an operation
Fractions
Meaning of operations
Cycle 3: operation sense
(using objects and diagrams):
addition
Operations
Cycle 3: adding fractions
using objects and diagrams,
when the denominator of one
fraction is a multiple of the
denominator of the other
fraction
Symbols
Cycle 1: , , 2 of 8
January 2006
0-17-632376-7
Cultural References
Operations
Cycle 1: interdisciplinary or
social context
Cultural References
Operations
Cycle 2: interdisciplinary or
social context
Cultural References
Operations
Cycle 2: interdisciplinary or
social context
Symbols
Cycle 1: , , Symbols
Cycle 2: , , Symbols
Cycle 3: , , Professional Resources and Instruction for Mathematics Educators, © 2005 Nelson, a division of Thomson Canada Limited
PQ Mathematics Curriculum (K–6)
Correlation to the Operations Developmental Map (Cont.)
Concept 2
Multiplication and
division are extensions
of addition and
subtraction.
Multiplication and
division are intrinsically
related.
Phase 1
Phase 2
Phase 3
Phase 4
Phase 5
Beginner
Concrete
Whole Number Comfort
More Abstract
Flexible
Focus on Counting to Solve Problems
Formal Operations with Numbers to
20; Concrete Operations with
Numbers to 100
Formal Operations with Whole
Numbers; Concrete Operations with
Decimals
Fluency with Whole Number
Operations; Formal Operations with
Decimals
Fluency with Whole Number and
Decimal Operations; Concrete
Operations with Integers and Fractions
Natural numbers
Meaning of operations
Cycle 1: operation, operation
sense: addition (adding,
uniting, comparing), sum,
subtraction (taking away,
complement, comparing),
difference, term, missing
term, number line,
multiplication (repeated
addition, Cartesian product)
and division (repeated
subtraction, sharing, number
of times x goes into y)
Natural numbers
Meaning of operations
Cycle 1: operation, operation
sense: addition (adding,
uniting, comparing), sum,
subtraction (taking away,
complement, comparing),
difference, term, missing
term, number line,
multiplication (repeated
addition, Cartesian product)
and division (repeated
subtraction, sharing, number
of times x goes into y)
Natural numbers
Meaning of operations
Cycle 2: operation sense:
multiplication (e.g., repeated
addition, Cartesian product),
product, factor, multiples of a
natural number, division
(repeated subtraction, sharing,
number of times x goes into
y), quotient, remainder,
dividend, divisor, set of
divisors of a natural number,
properties of divisibility
Natural numbers
Meaning of operations
Cycle 2: operation sense:
multiplication (e.g., repeated
addition, Cartesian product),
product, factor, multiples of a
natural number, division
(repeated subtraction, sharing,
number of times x goes into
y), quotient, remainder,
dividend, divisor, set of
divisors of a natural number,
properties of divisibility
Natural numbers
Meaning of operations
Cycle 3: operation sense:
multiplication (e.g., repeated
addition, Cartesian product),
product, factor, multiples of a
natural number, division
(repeated subtraction, sharing,
number of times x goes into
y), quotient, remainder,
dividend, divisor, set of
divisors of a natural number,
properties of divisibility
Cycle 2: choice of operation:
multiplication, division
Cycle 2: choice of operation:
multiplication, division
Cycle 3: choice of operation:
multiplication, division
Cycle 2: meaning of an
equality relation (equation),
meaning of an equivalence
relation
Cycle 2: meaning of an
equality relation (equation),
meaning of an equivalence
relation
Cycle 3: meaning of an
equality relation (equation),
meaning of an equivalence
relation
Cycle 2: property of
operations: associative law
Cycle 2: relationships
between the operations
Operations
Cycle 2: own processes for
mental computation:
multiplication, division
Cycle 3: property of
operations: distributive law
Cycle 1: meaning of an
equality relation (equation),
meaning of an equivalence
relation
Cycle 1: relationships
between the operations
Cycle 2: own processes for
written computation:
multiplying a three-digit
number by a one-digit
number
Cycle 2: patterns: series of
numbers, family of operations
(Cont.)
3 of 8
January 2006
0-17-632376-7
(Cont.)
Cycle 3: order of operations
(series of operations involving
natural numbers)
Operations
Cycle 2: approximating the
result of an operation:
addition, subtraction,
multiplication, division
Cycle 2: own processes for
mental computation:
multiplication, division
(Cont.)
Professional Resources and Instruction for Mathematics Educators, © 2005 Nelson, a division of Thomson Canada Limited
(Cont.)
PQ Mathematics Curriculum (K–6)
Correlation to the Operations Developmental Map (Cont.)
Phase 1
Phase 2
Phase 3
Phase 4
Phase 5
Beginner
Concrete
Whole Number Comfort
More Abstract
Flexible
Focus on Counting to Solve Problems
Formal Operations with Numbers to
20; Concrete Operations with
Numbers to 100
Formal Operations with Whole
Numbers; Concrete Operations with
Decimals
Fluency with Whole Number
Operations; Formal Operations with
Decimals
Fluency with Whole Number and
Decimal Operations; Concrete
Operations with Integers and Fractions
Concept 2 (Cont.)
Cycle 2: patterns: series of
numbers, family of operations
Multiplication and
division are extensions
of addition and
subtraction.
Multiplication and
division are intrinsically
related.
Cycle 3: series of operations
in accordance with the order
of operations
Decimals
Meaning of operations
Cycle 3: operation sense:
multiplication and division
Decimals
Meaning of operations
Cycle 3: operation sense:
multiplication and division
Operations
Cycle 3: approximating the
result of an operation:
addition, subtraction,
multiplication, division
Symbols
Cycle 2: , ÷, 4 of 8
January 2006
0-17-632376-7
Cultural References
Operations
Cycle 1: interdisciplinary or
social context
Cultural References
Operations
Cycle 2: interdisciplinary or
social context
Cultural References
Operations
Cycle 2: interdisciplinary or
social context
Symbols
Cycle 2: , ÷, Symbols
Cycle 2: , ÷, Symbols
Cycle 3: , ÷, Professional Resources and Instruction for Mathematics Educators, © 2005 Nelson, a division of Thomson Canada Limited
PQ Mathematics Curriculum (K–6)
Correlation to the Operations Developmental Map (Cont.)
Concept 3
There are many
algorithms for
performing a given
operation.
Phase 1
Phase 2
Phase 3
Phase 4
Phase 5
Beginner
Concrete
Whole Number Comfort
More Abstract
Flexible
Focus on Counting to Solve Problems
Formal Operations with Numbers to
20; Concrete Operations with
Numbers to 100
Formal Operations with Whole
Numbers; Concrete Operations with
Decimals
Fluency with Whole Number
Operations; Formal Operations with
Decimals
Fluency with Whole Number and
Decimal Operations; Concrete
Operations with Integers and Fractions
Natural numbers
Operations
Cycle 1: own processes for
mental computation:
addition, subtraction
Natural numbers
Meaning of operations
Cycle 1: operation, operation
sense: addition (adding,
uniting, comparing), sum,
subtraction (taking away,
complement, comparing),
difference, term, missing
term, number line,
multiplication (repeated
addition, Cartesian product)
and division (repeated
subtraction, sharing, number
of times x goes into y)
Natural numbers
Operations
Cycle 2: own processes for
mental computation:
addition, subtraction,
multiplication, division
Natural numbers
Operations
Cycle 3: own processes for
mental computation:
addition, subtraction,
multiplication, division
Cycle 2: own processes for
written computation:
multiplying a three-digit
number by a one-digit
number
Cycle 3: conventional
processes for written
computation: multiplying a
three-digit number by a twodigit number
Cycle 2: own processes for
written computation: dividing
a three-digit number by a onedigit number
Cycle 3: conventional
processes for written
computation: dividing a fourdigit number by a two-digit
number, expressing the
remainder as a decimal that
does not go beyond the
second decimal place
Cycle 1: own processes for
written computation:
addition, subtraction
Cycle 2: operation sense:
multiplication (e.g., repeated
addition, Cartesian product),
product, factor, multiples of a
natural number, division
(repeated subtraction, sharing,
number of times x goes into
y), quotient, remainder,
dividend, divisor, set of
divisors of a natural number,
properties of divisibility
Cycle 2: conventional
processes for written
computation: adding two
four-digit numbers
Cycle 3: patterns: series of
numbers, family of operations
Operations
Cycle 1: own processes for
written computation:
addition, subtraction
Cycle 2: own processes for
mental computation:
addition, subtraction,
multiplication, division
(Cont.)
5 of 8
January 2006
0-17-632376-7
(Cont.)
(Cont.)
Professional Resources and Instruction for Mathematics Educators, © 2005 Nelson, a division of Thomson Canada Limited
(Cont.)
PQ Mathematics Curriculum (K–6)
Correlation to the Operations Developmental Map (Cont.)
Phase 1
Phase 2
Phase 3
Phase 4
Phase 5
Beginner
Concrete
Whole Number Comfort
More Abstract
Flexible
Focus on Counting to Solve Problems
Formal Operations with Numbers to
20; Concrete Operations with
Numbers to 100
Formal Operations with Whole
Numbers; Concrete Operations with
Decimals
Fluency with Whole Number
Operations; Formal Operations with
Decimals
Fluency with Whole Number and
Decimal Operations; Concrete
Operations with Integers and Fractions
Decimals
Meaning of operations
Cycle 2: operation sense:
addition and subtraction
Decimals
Meaning of operations
Cycle 2: operation sense:
addition and subtraction
Concept 3 (Cont.)
Cycle 2: own processes for
written computation:
multiplying a three-digit
number by a one-digit
number
There are many
algorithms for
performing a given
operation.
Cycle 2: own processes for
written computation: dividing
a three-digit number by a onedigit number
Cycle 2: mental computation:
addition and subtraction
Cycle 3: operation sense:
multiplication and division
Cycle 3: written computation:
multiplication whose product
does not go beyond the
second decimal
Operations
Cycle 3: mental computation:
multiplication and division of
decimals by 10, 100, 1000
Cultural References
Operations
Cycle 1: own or conventional
computation processes:
development, limitations,
advantages and disadvantages
6 of 8
January 2006
0-17-632376-7
Cultural References
Operations
Cycle 2: own or conventional
computation processes:
development, limitations,
advantages and disadvantages
Cultural References
Operations
Cycle 2: own or conventional
computation processes:
development, limitations,
advantages and disadvantages
Cycle 1: technology:
development, limitations,
advantages and disadvantages
Cycle 2: technology:
development, limitations,
advantages and disadvantages
Cultural References
Operations
Cycle 3: technology:
development, limitations,
advantages and disadvantages
Professional Resources and Instruction for Mathematics Educators, © 2005 Nelson, a division of Thomson Canada Limited
PQ Mathematics Curriculum (K–6)
Correlation to the Operations Developmental Map (Cont.)
Skill 1
Memorizes facts.
Phase 1
Phase 2
Phase 3
Phase 4
Phase 5
Beginner
Concrete
Whole Number Comfort
More Abstract
Flexible
Focus on Counting to Solve Problems
Formal Operations with Numbers to
20; Concrete Operations with
Numbers to 100
Formal Operations with Whole
Numbers; Concrete Operations with
Decimals
Fluency with Whole Number
Operations; Formal Operations with
Decimals
Fluency with Whole Number and
Decimal Operations; Concrete
Operations with Integers and Fractions
Natural numbers
Operations
Cycle 1: operations to be
memorized: additions (0 0
to 10 10) related to the
corresponding subtractions
Natural numbers
Operations
Cycle 1: operations to be
memorized: additions (0 0
to 10 10) related to the
corresponding subtractions
Natural numbers
Operations
Cycle 2: operations to be
memorized: multiplications
(0 0 to 10 10) related to
the corresponding divisions
Natural numbers
Operations
Cycle 2: approximating the
result of an operation:
addition, subtraction,
multiplication, division
Natural numbers
Operations
Cycle 3: approximating the
result of an operation:
addition, subtraction,
multiplication, division
Cycle 2: operations to be
memorized: multiplications
(0 0 to 10 10) related to
the corresponding divisions
Skill 2
Uses standard mental
math and estimation
procedures.
Natural numbers
Operations
Cycle 1: approximating the
result of an operation:
addition, subtraction
Natural numbers
Operations
Cycle 1: approximating the
result of an operation:
addition, subtraction
Cycle 1: own processes for
mental computation:
addition, subtraction
Cycle 2: own processes for
mental computation:
addition, subtraction,
multiplication, division
Cycle 2: own processes for
mental computation:
addition, subtraction,
multiplication, division
Decimals
Operations
Cycle 2: approximating the
result of an operation
Decimals
Operations
Cycle 3: approximating the
result of an operation
Cycle 3: mental computation:
addition, subtraction,
multiplication, division
Skill 3
Computes using pencil
and paper with whole
numbers and decimals,
without the aid of a
calculator.
Natural numbers
Operations
Cycle 1: own processes for
mental computation: addition
Natural numbers
Operations
Cycle 1: own processes for
written computation:
addition, subtraction
(Cont.)
7 of 8
January 2006
0-17-632376-7
Natural numbers
Operations
Cycle 2: own processes for
written computation:
multiplying a three-digit
number by a one-digit
number
(Cont.)
Professional Resources and Instruction for Mathematics Educators, © 2005 Nelson, a division of Thomson Canada Limited
(Cont.)
PQ Mathematics Curriculum (K–6)
Correlation to the Operations Developmental Map (Cont.)
Skill 3 (Cont.)
Computes using pencil
and paper with whole
numbers and decimals,
without the aid of a
calculator.
Phase 1
Phase 2
Phase 3
Phase 4
Phase 5
Beginner
Concrete
Whole Number Comfort
More Abstract
Flexible
Focus on Counting to Solve Problems
Formal Operations with Numbers to
20; Concrete Operations with
Numbers to 100
Formal Operations with Whole
Numbers; Concrete Operations with
Decimals
Fluency with Whole Number
Operations; Formal Operations with
Decimals
Fluency with Whole Number and
Decimal Operations; Concrete
Operations with Integers and Fractions
Cycle 2: own processes for
written computation:
multiplying a three-digit
number by a one-digit
number
Cycle 2: own processes for
written computation: dividing
a three-digit number by a onedigit number
Cycle 2: own processes for
written computation: dividing
a three-digit number by a onedigit number
8 of 8
January 2006
0-17-632376-7
Cycle 3: conventional
processes for written
computation: multiplying a
three-digit number by a twodigit number
Decimals
Operations
Cycle 2: written computation:
addition, subtraction; the
result must not go beyond the
second decimal
Decimals
Operations
Cycle 2: written computation:
addition, subtraction; the
result must not go beyond the
second decimal
Cycle 3: written computation:
multiplication whose product
does not go beyond the
second decimal
Cycle 3: written computation:
division by a natural number
less than 11
Professional Resources and Instruction for Mathematics Educators, © 2005 Nelson, a division of Thomson Canada Limited