Unit 1Acknowledgements…………………………………………….…………….… 1
Foreword………………………………………………………..………………. 3 Background
………………………………………………………..…………… 4 Introduction Purpose of the
Document………………………………………………. 5 Beliefs About Students and
Mathematics Learning……………………. 5 Affective
Domain………………………………………………………. 6 Early
Childhood………………………………………………………… 6 Goal for
Students……………………………………………………….. 7 Conceptual Framework for K–9
Mathematives Mathematical Processes………………………………………………... 8 Nature
of Mathematics…………………………………………………. 13
Strands………………………………………………………………….. 16 Outcomes and Achievement
Indicators………………………………... 17 Summary……………………………………………………………….. 17
Instructional Focus Planning for Instruction………………………………………………… 18
Resources……………………………………………………………….. 18 General and Specific
Outcomes...........................................................................
19 General and Specific Outcomes by Strand
Grades 3 - 5……………….…………………………………………… 20 Outcomes with Achievement
Indicators Unit: Numeration Unit…………………………………….……………. 35 Unit:
Addition and Subtraction………………………………………… 57 Unit: Patterns in
Mathematics………..………………………………..... 75 Unit: Data
Relationships…………………………………………….….. Unit: 2-D Geometry
…………………………………………….……… Unit: Multiplication and Division
Facts………………..………………. Unit: Fractions and
Decimals…………………………………………... Unit:
Measurement…….……………………………………………….. Unit: Multiplying Multi-Digit
Numbers………………..………………. Unit: Dividing Multi-digit
Numbers………………..…………………... Unit: 3-D Geometry …………………………………………………….
Appendix A: Outcomes with Achievement Indicators (Strand)……………. 97
Appendix B: References………………………….………………………... 109
iv
Acknowledgements
The Department of Education would like to thank Western and
Northern Canadian Protocol (WNCP) for Collaboration in Education.
The Common Curriculum Framework for K-9 Mathematics – May 2006 and
The Common Curriculum Framework for Grades 10-12 – January 2008.
Reproduced (and/or adapted) by permission. All rights reserved. We
would also like to thank the provincial Grade 4 Mathematics
curriculum committee, the Alberta Department of Education, the New
Brunswick Department of Education, and the following people for
their contribution:
Trudy Porter, Program Development Specialist - Mathematics,
Division of Program Development, Department of Education Elizabeth
Dubeau, Teacher / Assistant Principal -Woodland Elementary, Dildo/
New Harbour Dina Healey, Teacher – Lakeside Academy, Buchans Gail
Keats, Teacher – Cowan Heights Elementary, St. John’s Gillian
Normore, Teacher – Paradise Elementary, Paradise John Power,
Numeracy Support Teacher – Eastern School District Sharon Power,
Numeracy Support Teacher – Eastern School District Tracy Templeman,
Numeracy Support Teacher – Nova Central School District Patricia
Maxwell – Program Development Specialist - Mathematics, Division of
Program Development, Department of Education
Every effort has been made to acknowledge all sources of
contribution to the development of this document. Any omissions or
errors will be amended in final print.
2 / K-9 Mathematics Curriculum NL
K-9 Mathematics Curriculum NL / 3
Foreword
The Curriculum Focal Points for Prekindergarten through Grade 8
Mathematics released in 2006 by the National Council of Teachers in
Mathematics (NCTM) and the WNCP Common Curriculum Frameworks for
Mathematics K – 9 (WNCP, 2006), assists provinces in developing a
mathematics curriculum framework. Newfoundland and Labrador has
used this curriculum framework to direct the development of this
curriculum guide. This curriculum guide is intended to provide
teachers with the overview of the outcomes framework for
mathematics education. It also includes suggestions to assist
teachers in designing learning experiences and assessment
tasks.
4 / K-9 Mathematics Curriculum NL
K-9 Mathematics Curriculum NL / 5
BACKGROUND
The province of Newfoundland and Labrador commissioned a review of
mathematics curriculum in the summer of 2007. This review resulted
in a number of significant recommendations. In March of 2008 it was
announced that this province accepted all recommendations. The
first and perhaps most significant of the recommendations were as
follows:
That the WNCP Common Curriculum Frameworks for Mathematics K – 9
and Mathematics 10 – 12 (WNCP, 2006 and 2008) be adopted as the
basis for the K – 12 mathematics curriculum in this province.
That implementation commence with Grades K, 1, 4, 7 in
September 2008, followed by in Grades 2, 5, 8 in 2009 and Grades 3,
6, 9 in 2010.
That textbooks and other resources specifically designed to
match the WNCP frameworks be adopted as an integral part of the
proposed program change.
That implementation be accompanied by an introductory
professional development program designed to introduce the
curriculum to all mathematics teachers at the appropriate grade
levels prior to the first year of implementation.
As recommended, implementation at grades K, 1, 4 and 7 begins in
September 2008. All teachers assigned to those grades in the spring
of 2008 received a two-day professional development opportunity
related to the new curriculum and resources. Newly hired teachers
will have the same opportunity in September. All teachers will
receive follow-up professional development in late fall.
6 / K-9 Mathematics Curriculum NL
INTRODUCTION Mathematical understanding is fostered when students
build on their own experiences and prior knowledge.
PURPOSE OF THE DOCUMENT The Mathematics Curriculum Guides for
Newfoundland and Labrador have been derived from The Common
Curriculum Framework for K-9 Mathematics: Western and Northern
Canadian Protocol, May 2006 (the Common Curriculum Framework).
These guides incorporate the conceptual framework for Kindergarten
to Grade 9 Mathematics and the general outcomes, specific outcomes
and achievement indicators established in the common curriculum
framework. They also include suggestions for teaching and learning,
suggested assessment strategies, and an identification of the
associated resource match between the curriculum and authorized as
well as recommended resource materials. BELIEFS ABOUT STUDENTS AND
MATHEMATICS LEARNING
Students are curious, active learners with individual interests,
abilities and needs. They come to classrooms with varying
knowledge, life experiences and backgrounds. A key component in
successfully developing numeracy is making connections to these
backgrounds and experiences. Students learn by attaching meaning to
what they do, and they need to construct their own meaning of
mathematics. This meaning is best developed when learners encounter
mathematical experiences that proceed from the simple to the
complex and from the concrete to the abstract. Through the use of
manipulatives and a variety of pedagogical approaches, teachers can
address the diverse learning styles, cultural backgrounds and
developmental stages of students, and enhance within them the
formation of sound, transferable mathematical understandings. At
all levels, students benefit from working with a variety of
materials, tools and contexts when constructing meaning about new
mathematical ideas. Meaningful student discussions provide
essential links among concrete, pictorial and symbolic
representations of mathematical concepts. The learning environment
should value and respect the diversity of students’ experiences and
ways of thinking, so that students are comfortable taking
intellectual risks, asking questions and posing conjectures.
Students need to explore problem-solving situations in order to
develop personal strategies and become mathematically literate.
They must realize that it is acceptable to solve problems in a
variety of ways and that a variety of solutions may be
acceptable.
K-9 Mathematics Curriculum NL / 7
To experience success, students must be taught to set achievable
goals and assess themselves as they work toward these goals.
Curiosity about mathematics is fostered when children are actively
engaged in their environment.
AFFECTIVE DOMAIN A positive attitude is an important aspect of the
affective domain and has a profound impact on learning.
Environments that create a sense of belonging, encourage risk
taking and provide opportunities for success help develop and
maintain positive attitudes and self-confidence within students.
Students with positive attitudes toward learning mathematics are
likely to be motivated and prepared to learn, participate willingly
in classroom activities, persist in challenging situations and
engage in reflective practices. Teachers, students and parents need
to recognize the relationship between the affective and cognitive
domains, and attempt to nurture those aspects of the affective
domain that contribute to positive attitudes. To experience
success, students must be taught to set achievable goals and assess
themselves as they work toward these goals. Striving toward success
and becoming autonomous and responsible learners are ongoing,
reflective processes that involve revisiting the setting and
assessing of personal goals.
EARLY CHILDHOOD Young children are naturally curious and develop a
variety of mathematical ideas before they enter Kindergarten.
Children make sense of their environment through observations and
interactions at home, in daycares, in preschools and in the
community. Mathematics learning is embedded in everyday activities,
such as playing, reading, beading, baking, storytelling and helping
around the home. Activities can contribute to the development of
number and spatial sense in children. Curiosity about mathematics
is fostered when children are engaged in, and talking about, such
activities as comparing quantities, searching for patterns, sorting
objects, ordering objects, creating designs and building with
blocks. Positive early experiences in mathematics are as critical
to child development as are early literacy experiences.
8 / K-9 Mathematics Curriculum NL
Mathematics education must prepare students to use mathematics
confidently to solve problems. .
GOALS FOR STUDENTS The main goals of mathematics education are to
prepare students to: • use mathematics confidently to solve
problems • communicate and reason mathematically • appreciate and
value mathematics • make connections between mathematics and
its
applications • commit themselves to lifelong learning • become
mathematically literate adults, using
mathematics to contribute to society. Students who have met these
goals will: • gain understanding and appreciation of the
contributions
of mathematics as a science, philosophy and art • exhibit a
positive attitude toward mathematics • engage and persevere in
mathematical tasks and projects • contribute to mathematical
discussions • take risks in performing mathematical tasks • exhibit
curiosity.
K-9 Mathematics Curriculum NL / 9
CONCEPTUAL FRAMEWORK FOR K–9 MATHEMATICS The chart below provides
an overview of how mathematical processes
and the nature of mathematics influence learning outcomes.
• Communicatio
[V]
MATHEMATICAL PROCESSES There are critical components that students
must encounter in a mathematics program in order to achieve the
goals of mathematics education and embrace lifelong learning in
mathematics. Students are expected to: • communicate in order to
learn and express their understanding • connect mathematical ideas
to other concepts in mathematics, to
everyday experiences and to other disciplines • demonstrate fluency
with mental mathematics and estimation • develop and apply new
mathematical knowledge through problem
solving • develop mathematical reasoning • select and use
technologies as tools for learning and for solving
problems • develop visualization skills to assist in processing
information, making
connections and solving problems. The program of studies
incorporates these seven interrelated mathematical processes that
are intended to permeate teaching and learning.
10 / K-9 Mathematics Curriculum NL
Students must be able to communicate mathematical ideas in a
variety of ways and contexts. Through connections, students begin
to view mathematics as useful and relevant.
Communication [C] Students need opportunities to read about,
represent, view, write about, listen to and discuss mathematical
ideas. These opportunities allow students to create links between
their own language and ideas, and the formal language and symbols
of mathematics. Communication is important in clarifying,
reinforcing and modifying ideas, attitudes and beliefs about
mathematics. Students should be encouraged to use a variety of
forms of communication while learning mathematics. Students also
need to communicate their learning using mathematical terminology.
Communication helps students make connections among concrete,
pictorial, symbolic, oral, written and mental representations of
mathematical ideas.
Connections [CN] Contextualization and making connections to the
experiences of learners are powerful processes in developing
mathematical understanding. This can be particularly true for First
Nations, Métis and Inuit learners. When mathematical ideas are
connected to each other or to real-world phenomena, students begin
to view mathematics as useful, relevant and integrated. Learning
mathematics within contexts and making connections relevant to
learners can validate past experiences and increase student
willingness to participate and be actively engaged. The brain is
constantly looking for and making connections. “Because the learner
is constantly searching for connections on many levels, educators
need to orchestrate the experiences from which learners extract
understanding.… Brain research establishes and confirms that
multiple complex and concrete experiences are essential for
meaningful learning and teaching” (Caine and Caine, 1991, p.
5).
K-9 Mathematics Curriculum NL / 11
Mental mathematics and estimation are fundamental components of
number sense.
Mental Mathematics and Estimation [ME] Mental mathematics is a
combination of cognitive strategies that enhance flexible thinking
and number sense. It is calculating mentally without the use of
external memory aids. Mental mathematics enables students to
determine answers without paper and pencil. It improves
computational fluency by developing efficiency, accuracy and
flexibility. “Even more important than performing computational
procedures or using calculators is the greater facility that
students need—more than ever before—with estimation and mental
math” (National Council of Teachers of Mathematics, May 2005).
Students proficient with mental mathematics “become liberated from
calculator dependence, build confidence in doing mathematics,
become more flexible thinkers and are more able to use multiple
approaches to problem solving” (Rubenstein, 2001, p. 442). Mental
mathematics “provides the cornerstone for all estimation processes,
offering a variety of alternative algorithms and nonstandard
techniques for finding answers” (Hope, 1988, p. v). Estimation is
used for determining approximate values or quantities or for
determining the reasonableness of calculated values. It often uses
benchmarks or referents. Students need to know when to estimate,
how to estimate and what strategy to use. Estimation assists
individuals in making mathematical judgements and in developing
useful, efficient strategies for dealing with situations in daily
life.
12 / K-9 Mathematics Curriculum NL
Learning through problem solving should be the focus of mathematics
at all grade levels. Mathematical reasoning helps students think
logically and make sense of mathematics.
Problem Solving [PS] Learning through problem solving should be the
focus of mathematics at all grade levels. When students encounter
new situations and respond to questions of the type How would you
…? or How could you …?, the problem-solving approach is being
modelled. Students develop their own problem-solving strategies by
listening to, discussing and trying different strategies. A
problem-solving activity must ask students to determine a way to
get from what is known to what is sought. If students have already
been given ways to solve the problem, it is not a problem, but
practice. A true problem requires students to use prior learnings
in new ways and contexts. Problem solving requires and builds depth
of conceptual understanding and student engagement. Problem solving
is a powerful teaching tool that fosters multiple, creative and
innovative solutions. Creating an environment where students openly
look for, and engage in, finding a variety of strategies for
solving problems empowers students to explore alternatives and
develops confident, cognitive mathematical risk takers.
Reasoning [R] Mathematical reasoning helps students think logically
and make sense of mathematics. Students need to develop confidence
in their abilities to reason and justify their mathematical
thinking. High-order questions challenge students to think and
develop a sense of wonder about mathematics. Mathematical
experiences in and out of the classroom provide opportunities for
students to develop their ability to reason. Students can explore
and record results, analyze observations, make and test
generalizations from patterns, and reach new conclusions by
building upon what is already known or assumed to be true.
Reasoning skills allow students to use a logical process to analyze
a problem, reach a conclusion and justify or defend that
conclusion.
K-9 Mathematics Curriculum NL / 13
Technology contributes to the learning of a wide range of
mathematical outcomes and enables students to explore and create
patterns, examine relationships, test conjectures and solve
problems. Visualization is fostered through the use of concrete
materials, technology and a variety of visual
representations.
Technology [T] Technology contributes to the learning of a wide
range of mathematical outcomes and enables students to explore and
create patterns, examine relationships, test conjectures and solve
problems. Calculators and computers can be used to: • explore and
demonstrate mathematical relationships and patterns • organize and
display data • extrapolate and interpolate • assist with
calculation procedures as part of solving problems • decrease the
time spent on computations when other mathematical
learning is the focus • reinforce the learning of basic facts •
develop personal procedures for mathematical operations • create
geometric patterns • simulate situations • develop number sense.
Technology contributes to a learning environment in which the
growing curiosity of students can lead to rich mathematical
discoveries at all grade levels.
Visualization [V] Visualization “involves thinking in pictures and
images, and the ability to perceive, transform and recreate
different aspects of the visual-spatial world” (Armstrong, 1993, p.
10). The use of visualization in the study of mathematics provides
students with opportunities to understand mathematical concepts and
make connections among them. Visual images and visual reasoning are
important components of number, spatial and measurement sense.
Number visualization occurs when students create mental
representations of numbers. Being able to create, interpret and
describe a visual representation is part of spatial sense and
spatial reasoning. Spatial visualization and reasoning enable
students to describe the relationships among and between 3-D
objects and 2-D shapes. Measurement visualization goes beyond the
acquisition of specific measurement skills. Measurement sense
includes the ability to determine when to measure, when to estimate
and which estimation strategies to use (Shaw and Cliatt,
1989).
14 / K-9 Mathematics Curriculum NL
• Change • Constancy • Number
Sense • Patterns • Relationships • Spatial Sense • Uncertainty
Change is an integral part of mathematics and the learning of
mathematics. Constancy is described by the terms stability,
conservation, equilibrium, steady state and symmetry.
NATURE OF MATHEMATICS Mathematics is one way of trying to
understand, interpret and describe our world. There are a number of
components that define the nature of mathematics and these are
woven throughout this program of studies. The components are
change, constancy, number sense, patterns, relationships, spatial
sense and uncertainty.
Change It is important for students to understand that mathematics
is dynamic and not static. As a result, recognizing change is a key
component in understanding and developing mathematics. Within
mathematics, students encounter conditions of change and are
required to search for explanations of that change. To make
predictions, students need to describe and quantify their
observations, look for patterns, and describe those quantities that
remain fixed and those that change. For example, the sequence 4, 6,
8, 10, 12, … can be described as: • the number of a specific colour
of beads in each row of a beaded design • skip counting by 2s,
starting from 4 • an arithmetic sequence, with first term 4 and a
common difference of 2 • a linear function with a discrete domain
(Steen, 1990, p. 184).
Constancy Different aspects of constancy are described by the terms
stability, conservation, equilibrium, steady state and symmetry
(AAAS–Benchmarks, 1993, p. 270). Many important properties in
mathematics and science relate to properties that do not change
when outside conditions change. Examples of constancy include the
following: • The ratio of the circumference of a teepee to its
diameter is the same
regardless of the length of the teepee poles. • The sum of the
interior angles of any triangle is 180°. • The theoretical
probability of flipping a coin and getting heads is 0.5. Some
problems in mathematics require students to focus on properties
that remain constant. The recognition of constancy enables students
to solve problems involving constant rates of change, lines with
constant slope, direct variation situations or the angle sums of
polygons.
K-9 Mathematics Curriculum NL / 15
An intuition about number is the most important foundation of a
numerate child. Mathematics is about recognizing, describing and
working with numerical and non-numerical patterns.
Number Sense Number sense, which can be thought of as intuition
about numbers, is the most important foundation of numeracy
(British Columbia Ministry of Education, 2000, p. 146). A true
sense of number goes well beyond the skills of simply counting,
memorizing facts and the situational rote use of algorithms.
Mastery of number facts is expected to be attained by students as
they develop their number sense. This mastery allows for facility
with more complex computations but should not be attained at the
expense of an understanding of number. Number sense develops when
students connect numbers to their own real- life experiences and
when students use benchmarks and referents. This results in
students who are computationally fluent and flexible with numbers
and who have intuition about numbers. The evolving number sense
typically comes as a by-product of learning rather than through
direct instruction. However, number sense can be developed by
providing rich mathematical tasks that allow students to make
connections to their own experiences and their previous
learning.
Patterns Mathematics is about recognizing, describing and working
with numerical and non-numerical patterns. Patterns exist in all
strands of this program of studies. Working with patterns enables
students to make connections within and beyond mathematics. These
skills contribute to students’ interaction with, and understanding
of, their environment. Patterns may be represented in concrete,
visual or symbolic form. Students should develop fluency in moving
from one representation to another. Students must learn to
recognize, extend, create and use mathematical patterns. Patterns
allow students to make predictions and justify their reasoning when
solving routine and nonroutine problems. Learning to work with
patterns in the early grades helps students develop algebraic
thinking, which is foundational for working with more abstract
mathematics in higher grades.
16 / K-9 Mathematics Curriculum NL
Mathematics is used to describe and explain relationships. Spatial
sense offers a way to interpret and reflect on the physical
environment. Uncertainty is an inherent part of making
predictions.
Relationships Mathematics is one way to describe interconnectedness
in a holistic worldview. Mathematics is used to describe and
explain relationships. As part of the study of mathematics,
students look for relationships among numbers, sets, shapes,
objects and concepts. The search for possible relationships
involves collecting and analyzing data and describing relationships
visually, symbolically, orally or in written form.
Spatial Sense Spatial sense involves visualization, mental imagery
and spatial reasoning. These skills are central to the
understanding of mathematics. Spatial sense is developed through a
variety of experiences and interactions within the environment. The
development of spatial sense enables students to solve problems
involving 3-D objects and 2-D shapes and to interpret and reflect
on the physical environment and its 3-D or 2- D representations.
Some problems involve attaching numerals and appropriate units
(measurement) to dimensions of shapes and objects. Spatial sense
allows students to make predictions about the results of changing
these dimensions; e.g., doubling the length of the side of a square
increases the area by a factor of four. Ultimately, spatial sense
enables students to communicate about shapes and objects and to
create their own representations. Uncertainty In mathematics,
interpretations of data and the predictions made from data may lack
certainty. Events and experiments generate statistical data that
can be used to make predictions. It is important to recognize that
these predictions (interpolations and extrapolations) are based
upon patterns that have a degree of uncertainty. The quality of the
interpretation is directly related to the quality of the data. An
awareness of uncertainty allows students to assess the reliability
of data and data interpretation. Chance addresses the
predictability of the occurrence of an outcome. As students develop
their understanding of probability, the language of mathematics
becomes more specific and describes the degree of uncertainty more
accurately.
K-9 Mathematics Curriculum NL / 17
• Number • Patterns
and Probability
STRANDS The learning outcomes in the program of studies are
organized into four strands across the grades K–9. Some strands are
subdivided into substrands. There is one general outcome per
substrand across the grades K–9. The strands and substrands,
including the general outcome for each, follow.
Number • Develop number sense.
Patterns and Relations Patterns • Use patterns to describe the
world and to solve problems.
Variables and Equations • Represent algebraic expressions in
multiple ways.
Shape and Space Measurement • Use direct and indirect measurement
to solve problems.
3-D Objects and 2-D Shapes • Describe the characteristics of 3-D
objects and
2-D shapes, and analyze the relationships among them.
Transformations • Describe and analyze position and motion of
objects and shapes. Statistics and Probability Data Analysis •
Collect, display and analyze data to solve problems.
Chance and Uncertainty • Use experimental or theoretical
probabilities to represent and solve
problems involving uncertainty.
General outcomes Specific outcomes Achievement indicators
OUTCOMES AND ACHIEVEMENT INDICATORS The program of studies is
stated in terms of general outcomes, specific outcomes and
achievement indicators. General outcomes are overarching statements
about what students are expected to learn in each strand/substrand.
The general outcome for each strand/substrand is the same
throughout the grades. Specific outcomes are statements that
identify the specific skills, understanding and knowledge that
students are required to attain by the end of a given grade.
Achievement indicators are samples of how students may demonstrate
their achievement of the goals of a specific outcome. The range of
samples provided is meant to reflect the scope of the specific
outcome. Achievement indicators are context-free. In the specific
outcomes, the word including indicates that any ensuing items must
be addressed to fully meet the learning outcome. The phrase such as
indicates that the ensuing items are provided for illustrative
purposes or clarification, and are not requirements that must be
addressed to fully meet the learning outcome.
SUMMARY The conceptual framework for K–9 mathematics describes the
nature of mathematics, mathematical processes and the mathematical
concepts to be addressed in Kindergarten to Grade 9 mathematics.
The components are not meant to stand alone. Activities that take
place in the mathematics classroom should stem from a
problem-solving approach, be based on mathematical processes and
lead students to an understanding of the nature of mathematics
through specific knowledge, skills and attitudes among and between
strands.
K-9 Mathematics Curriculum NL / 19
Instructional Focus
Planning for Instruction The curriculum is arranged into eleven
units. These units are not intended to be discrete units of
instruction. The integration of outcomes across strands makes
mathematical experiences meaningful. Students should make the
connection between concepts both within and across strands.
Consider the following when planning for instruction: • Integration
of the mathematical processes within each strand is expected. • By
decreasing emphasis on rote calculation, drill and practice, and
the
size of numbers used in paper and pencil calculations, more time is
available for concept development.
• Problem solving, reasoning and connections are vital to
increasing mathematical fluency and must be integrated throughout
the program.
• There is to be a balance among mental mathematics and estimation,
paper and pencil exercises, and the use of technology, including
calculators and computers. Concepts should be introduced using
manipulatives and be developed concretely, pictorially and
symbolically.
• Students bring a diversity of learning styles and cultural
backgrounds to the classroom. They will be at varying developmental
stages.
. Resources
The resource selected by Newfoundland and Labrador for students and
teachers is Math Focus 4 (Nelson). Schools and teachers have this
as their primary resource offered by the Department of Education.
Column four of the curriculum guide references Math Focus 4 for
this reason. Teachers may use any resource or combination of
resources to meet the required specific outcomes listed in column
one of the curriculum guide.
20 / K-9 Mathematics Curriculum NL
GENERAL AND SPECIFIC OUTCOMES
GENERAL AND SPECIFIC OUTCOMES BY STRAND (pages 17 – 30) This
section presents the general and specific outcomes for each strand,
for Grades 3, 4 and 5 GENERAL AND SPECIFIC OUTCOMES WITH
ACHIEVEMENT INDICATORS (pages 95 – 106) This section presents
general and specific outcomes with corresponding achievement
indicators and is organized by unit. The list of indicators
contained in this section is not intended to be exhaustive but
rather to provide teachers with examples of evidence of
understanding that may be used to determine whether or not students
have achieved a given specific outcome. Teachers may use any number
of these indicators or choose to use other indicators as evidence
that the desired learning has been achieved. Achievement indicators
should also help teachers form a clear picture of the intent and
scope of each specific outcome. Refer to Appendix A for the general
and specific outcomes with corresponding achievement indicators
organized by strand.
K-9 Mathematics Curriculum NL 21 /
GENERAL AND SPECIFIC OUTCOMES BY STRAND (Grades 3, 4 and 5)
Number
Grade 3 Grade 4 Grade 5 General Outcome Develop number sense.
General Outcome Develop number sense.
General Outcome Develop number sense.
Specific Outcomes
Specific Outcomes
Specific Outcomes
1. Say the number sequence 0 to 1000 forward and backward by: • 5s,
10s or 100s, using any
starting point • 3s, using starting points
that are multiples of 3 • 4s, using starting points
that are multiples of 4 • 25s, using starting points
that are multiples of 25. [C, CN, ME]
2. Represent and describe numbers to 1000, concretely, pictorially
and symbolically. [C, CN, V]
3. Compare and order
4. Estimate quantities less than
1000, using referents. [ME, PS, R, V]
1. Represent and describe whole numbers to 10 000, pictorially and
symbolically. [C, CN, V]
2. Compare and order numbers
to 10 000. [C, CN, V]
3. Demonstrate an
understanding of addition of numbers with answers to
10 000 and their corresponding subtractions (limited to 3- and
4-digit numerals) by: • using personal strategies
for adding and subtracting • estimating sums and
differences • solving problems
[C, CN, ME, PS, R]
1. Represent and describe whole numbers to 1 000 000. [C, CN, V,
T]
2. Use estimation strategies,
in problem-solving contexts. [C, CN, ME, PS, R, V]
3. Apply mental mathematics
strategies and number properties, such as: • skip counting from
a
known fact • using doubling or
halving • using patterns in the
9s facts • using repeated doubling
or halving to determine, with fluency,
answers for basic multiplication facts to 81 and related division
facts. [C, CN, ME, R, V]
[C] Communication [PS] Problem Solving [CN] Connections [R]
Reasoning [ME] Mental Mathematics [T] Technology and Estimation [V]
Visualization
22 / K-9 Mathematics Curriculum NL
Number (continued)
Grade 3 Grade 4 Grade 5 General Outcome Develop number sense.
General Outcome Develop number sense.
General Outcome Develop number sense.
Specific Outcomes
Specific Outcomes
Specific Outcomes
5. Illustrate, concretely and pictorially, the meaning of place
value for numerals to 1000. [C, CN, R, V]
6. Describe and apply mental
mathematics strategies for adding two 2-digit numerals, such as: •
adding from left to right • taking one addend to the
nearest multiple of ten and then compensating
• using doubles. [C, CN, ME, PS, R, V]
7. Describe and apply mental
mathematics strategies for subtracting two 2-digit numerals, such
as: • taking the subtrahend to
the nearest multiple of ten and then compensating
• thinking of addition • using doubles. [C, CN, ME, PS, R, V]
4. Explain and apply the properties of 0 and 1 for multiplication
and the property of 1 for division. [C, CN, R]
5. Describe and apply mental
mathematics strategies, such as: • skip counting from a
known fact • using doubling or halving • using doubling or
halving
and adding or subtracting one more group
• using patterns in the 9s facts
• using repeated doubling to determine basic multiplication facts
to 9 × 9 and related division facts. [C, CN, ME, R]
4. Apply mental mathematics strategies for multiplication, such as:
1. annexing then adding
zero 2. halving and doubling 3. using the distributive
property. [C, CN, ME, R, V]
5. Demonstrate, with and
6. Demonstrate, with and
without concrete materials, an understanding of division (3-digit
by 1-digit), and interpret remainders to solve problems. [C, CN,
ME, PS, R, V]
[C] Communication [PS] Problem Solving [CN] Connections [R]
Reasoning [ME] Mental Mathematics [T] Technology and Estimation [V]
Visualization
K-9 Mathematics Curriculum NL / 23
Number (continued)
Grade 3 Grade 4 Grade 5 General Outcome Develop number sense.
General Outcome Develop number sense.
General Outcome Develop number sense.
Specific Outcomes
Specific Outcomes
Specific Outcomes
8. Apply estimation strategies to predict sums and differences of
two 2- digit numerals in a problem-solving context. [C, ME, PS,
R]
9. Demonstrate an
understanding of addition and subtraction of numbers with answers
to 1000 (limited to 1-, 2- and 3- digit numerals), concretely,
pictorially and symbolically, by:
• using personal strategies for adding and subtracting with and
without the support of manipulatives
• creating and solving problems in context that involve addition
and subtraction of numbers.
[C, CN, ME, PS, R, V]
6. Demonstrate an understanding of multiplication (2- or 3-digit by
1-digit) to solve problems by: • using personal strategies
for multiplication with and without concrete materials
• using arrays to represent multiplication
• connecting concrete representations to symbolic
representations
• estimating products • applying the distributive
property. [C, CN, ME, PS, R, V]
7. Demonstrate an
understanding of division (1-digit divisor and up to 2-digit
dividend) to solve problems by: • using personal strategies
for dividing with and without concrete materials
• estimating quotients • relating division to
multiplication. [C, CN, ME, PS, R, V]
7. Demonstrate an understanding of fractions by using concrete,
pictorial and symbolic representations to: • create sets of
equivalent
fractions • compare fractions with
like and unlike denominators.
decimals (tenths, hundredths, thousandths), concretely, pictorially
and symbolically. [C, CN, R, V]
9. Relate decimals to
fractions and fractions to decimals (to thousandths). [CN, R,
V]
10. Compare and order
decimals (to thousandths) by using: • benchmarks • place value •
equivalent decimals. [C, CN, R, V]
[C] Communication [PS] Problem Solving [CN] Connections [R]
Reasoning [ME] Mental Mathematics [T] Technology and Estimation [V]
Visualization
24 / K-9 Mathematics Curriculum NL
Number (continued)
Grade 3 Grade 4 Grade 5 General Outcome Develop number sense.
General Outcome Develop number sense.
General Outcome Develop number sense.
Specific Outcomes
Specific Outcomes
Specific Outcomes
10. Apply mental mathematics strategies and number properties, such
as: • using doubles • making 10 • using the commutative
property • using the property of zero • thinking addition for
subtraction for basic addition facts and related subtraction facts
to 18. [C, CN, ME, PS, R, V]
8. Demonstrate an understanding of fractions less than or equal to
one by using concrete, pictorial and symbolic representations to: •
name and record fractions
for the parts of a whole or a set
• compare and order fractions
• model and explain that for different wholes, two identical
fractions may not represent the same quantity
• provide examples of where fractions are used.
[C, CN, PS, R, V] 9. Represent and describe
decimals (tenths and hundredths), concretely, pictorially and
symbolically. [C, CN, R, V]
11. Demonstrate an understanding of addition and subtraction of
decimals (limited to thousandths). [C, CN, PS, R, V]
[C] Communication [PS] Problem Solving [CN] Connections [R]
Reasoning [ME] Mental Mathematics [T] Technology and Estimation [V]
Visualization
K-9 Mathematics Curriculum NL / 25
Number (continued)
Grade 3 Grade 4 Grade 5 General Outcome Develop number sense.
General Outcome Develop number sense.
Specific Outcomes
Specific Outcomes
11. Demonstrate an understanding of multiplication to 5 × 5 by: •
representing and
explaining multiplication using equal grouping and arrays
• creating and solving problems in context that involve
multiplication
• modelling multiplication using concrete and visual
representations, and recording the process symbolically
• relating multiplication to repeated addition
• relating multiplication to division.
[C, CN, PS, R]
10. Relate decimals to fractions and fractions to decimals (to
hundredths). [C, CN, R, V]
11. Demonstrate an
understanding of addition and subtraction of decimals (limited to
hundredths) by: • using personal strategies
to determine sums and differences
• estimating sums and differences
• using mental mathematics strategies
to solve problems. [C, ME, PS, R, V]
[C] Communication [PS] Problem Solving [CN] Connections [R]
Reasoning [ME] Mental Mathematics [T] Technology and Estimation [V]
Visualization
26 / K-9 Mathematics Curriculum NL
Number (continued)
Grade 3 Grade 4 Grade 5 General Outcome Develop number sense.
Specific Outcomes
12. Demonstrate an understanding of division (limited to division
related to multiplication facts up to 5 × 5) by: • representing
and
explaining division using equal sharing and equal grouping
• creating and solving problems in context that involve equal
sharing and equal grouping
• modelling equal sharing and equal grouping using concrete and
visual representations, and recording the process
symbolically
• relating division to repeated subtraction
• relating division to multiplication.
• comparing fractions of the same whole that have like
denominators.
[C, CN, ME, R, V]
[C] Communication [PS] Problem Solving [CN] Connections [R]
Reasoning [ME] Mental Mathematics [T] Technology and Estimation [V]
Visualization
K-9 Mathematics Curriculum NL / 27
Patterns and Relations (Patterns)
Grade 3 Grade 4 Grade 5 General Outcome Use patterns to describe
the world and to solve problems.
General Outcome Use patterns to describe the world and to solve
problems.
General Outcome Use patterns to describe the world and to solve
problems.
Specific Outcomes
Specific Outcomes
Specific Outcomes
1. Demonstrate an understanding of increasing patterns by: •
describing • extending • comparing • creating numerical (numbers to
1000) and non-numerical patterns using manipulatives, diagrams,
sounds and actions. [C, CN, PS, R, V]
2. Demonstrate an understanding of decreasing patterns by: •
describing • extending • comparing • creating numerical (numbers to
1000) and non-numerical patterns using manipulatives, diagrams,
sounds and actions. [C, CN, PS, R, V]
3. Sort objects or numbers, using one or more than one attribute.
[C, CN, R, V]
1. Identify and describe patterns found in tables and charts,
including multiplication chart. [C, CN, PS, V]
2. Translate among different representations of a pattern, such as
a table, a chart or concrete materials. [C, CN, V]
3. Represent, describe and
extend patterns and relationships, using charts and tables, to
solve problems. [C, CN, PS, R, V]
4. Identify and explain mathematical relationships, using charts
and diagrams, to solve problems. [CN, PS, R, V]
1. Determine the pattern rule to make predictions about subsequent
elements. [C, CN, PS, R, V]
[C] Communication [PS] Problem Solving [CN] Connections [R]
Reasoning [ME] Mental Mathematics [T] Technology and Estimation [V]
Visualization
28 / K-9 Mathematics Curriculum NL
Patterns and Relations (Variables and Equations)
Grade 3 Grade 4 Grade 5 General Outcome Represent algebraic
expressions in multiple ways.
General Outcome Represent algebraic expressions in multiple
ways.
General Outcome Represent algebraic expressions in multiple
ways.
Specific Outcomes
Specific Outcomes
Specific Outcomes
4. Solve one-step addition and subtraction equations involving a
symbol to represent an unknown number. [C, CN, PS, R, V]
5. Express a given problem as an equation in which a symbol is used
to represent an unknown number. [CN, PS, R]
6. Solve one-step equations
involving a symbol to represent an unknown number. [C, CN, PS, R,
V]
2. Express a given problem as an equation in which a letter
variable is used to represent an unknown number (limited to whole
numbers). [C, CN, PS, R]
3. Solve problems involving
single-variable, one-step equations with whole number coefficients
and whole number solutions. [C, CN, PS, R]
[C] Communication [PS] Problem Solving [CN] Connections [R]
Reasoning [ME] Mental Mathematics [T] Technology and Estimation [V]
Visualization
K-9 Mathematics Curriculum NL / 29
Shape and Space (Measurement)
Grade 3 Grade 4 Grade 5 General Outcome Use direct and indirect
measurement to solve problems.
General Outcome Use direct and indirect measurement to solve
problems.
General Outcome Use direct and indirect measurement to solve
problems.
Specific Outcomes
Specific Outcomes
Specific Outcomes
1. Relate the passage of time to common activities, using
nonstandard and standard units (minutes, hours, days, weeks,
months, years). [CN, ME, R]
2. Relate the number of
seconds to a minute, the number of minutes to an hour and the
number of days to a month in a problem-solving context. [C, CN, PS,
R, V]
3. Demonstrate an
understanding of measuring length (cm, m) by: • selecting and
justifying
referents for the units cm and m
• modelling and describing the relationship between the units cm
and m
• estimating length, using referents
[C, CN, ME, PS, R, V]
1. Read and record time, using digital and analog clocks, including
24-hour clocks. [C, CN, V]
2. Read and record calendar
dates in a variety of formats. [C, V]
3. Demonstrate an
understanding of area of regular and irregular 2-D shapes by: •
recognizing that area is
measured in square units • selecting and justifying
referents for the units cm2 or m2
• estimating area, using referents for cm2 or m2
• determining and recording area (cm2 or m2)
• constructing different rectangles for a given area (cm2 or m2) in
order to demonstrate that many different rectangles may have the
same area.
[C, CN, ME, PS, R, V]
1. Identify 90º angles. [ME, V]
2. Design and construct different rectangles, given either
perimeter or area, or both (whole numbers), and make
generalizations. [C, CN, PS, R, V]
3. Demonstrate an understanding of measuring length (mm) by: •
selecting and justifying
referents for the unit mm
• modelling and describing the relationship between mm and cm
units, and between mm and m units. [C, CN, ME, PS, R, V]
[C] Communication [PS] Problem Solving [CN] Connections [R]
Reasoning [ME] Mental Mathematics [T] Technology and Estimation [V]
Visualization
30 / K-9 Mathematics Curriculum NL
Shape and Space (Measurement) (continued)
Grade 3 Grade 4 Grade 5 General Outcome Use direct and indirect
measurement to solve problems.
General Outcome Use direct and indirect measurement to solve
problems.
Specific Outcomes
Specific Outcomes
4. Demonstrate an understanding of measuring mass (g, kg) by:
• selecting and justifying referents for the units g and kg
• modelling and describing the relationship between the units g and
kg
• estimating mass, using referents
• measuring and recording mass. [C, CN, ME, PS, R, V]
5. Demonstrate an
• estimating perimeter, using referents for cm or m
• measuring and recording perimeter (cm, m)
• constructing different shapes for a given perimeter (cm, m) to
demonstrate that many shapes are possible for a perimeter.
[C, ME, PS, R, V]
4. Demonstrate an understanding of volume by: • selecting and
justifying
referents for cm3 or m3
units • stimating volume,
• measuring and recording volume (cm3 or m3)
• constructing right rectangular prisms for a given volume.
[C, CN, ME, PS, R, V] 5. Demonstrate an
understanding of capacity by: • describing the
relationship between mL and L
• selecting and justifying referents for mL or L units
• stimating capacity, using referents for mL or L
• measuring and recording capacity (mL or L).
[C, CN, ME, PS, R, V]
[C] Communication [PS] Problem Solving [CN] Connections [R]
Reasoning [ME] Mental Mathematics [T] Technology and Estimation [V]
Visualization
K-9 Mathematics Curriculum NL / 31
Shape and Space (3-D Objects and 2-D Shapes)
Grade 3 Grade 4 Grade 5 General Outcome Describe the
characteristics of 3-D objects and 2-D shapes, and analyze the
relationships among them.
General Outcome Describe the characteristics of 3-D objects and 2-D
shapes, and analyze the relationships among them.
General Outcome Describe the characteristics of 3-D objects and 2-D
shapes, and analyze the relationships among them.
Specific Outcomes
Specific Outcomes Specific Outcomes
6. Describe 3-D objects according to the shape of the faces and the
number of edges and vertices. [C, CN, PS, R, V]
7. Sort regular and irregular
polygons, including: • triangles • quadrilaterals • pentagons •
hexagons • octagons according to the number of sides. [C, CN, R,
V]
4. Describe and construct right rectangular and right triangular
prisms. [C, CN, R, V]
6. Describe and provide examples of edges and faces of 3-D objects,
and sides of 2-D shapes that are: • parallel • intersecting •
perpendicular • vertical • horizontal.
[C, CN, R, T, V] 7. Identify and sort
quadrilaterals, including: • rectangles • squares • trapezoids •
parallelograms • rhombuses according to their attributes. [C, R,
V]
[C] Communication [PS] Problem Solving [CN] Connections [R]
Reasoning [ME] Mental Mathematics [T] Technology and Estimation [V]
Visualization
32 / K-9 Mathematics Curriculum NL
Shape and Space (Transformations)
Describe and analyze position and motion of objects and
shapes.
General Outcome Describe and analyze position and motion of objects
and shapes.
Specific Outcomes
Specific Outcomes
• identifying symmetrical 2-D shapes
• creating symmetrical 2-D shapes
• drawing one or more lines of symmetry in a 2-D shape.
[C, CN, V]
[CN, R, V]
8. Identify and describe a single transformation, including a
translation, rotation and reflection of 2-D shapes. [C, T, V]
9. Perform, concretely, a
single transformation (translation, rotation or reflection) of a
2-D shape, and draw the image. [C, CN, T, V]
[C] Communication [PS] Problem Solving [CN] Connections [R]
Reasoning [ME] Mental Mathematics [T] Technology and Estimation [V]
Visualization
K-9 Mathematics Curriculum NL / 33
Statistics and Probability (Data Analysis)
Grade 3 Grade 4 Grade 5 General Outcome Collect, display and
analyze data to solve problems.
General Outcome Collect, display and analyze data to solve
problems.
General Outcome Collect, display and analyze data to solve
problems.
Specific Outcomes
Specific Outcomes
Specific Outcomes
1. Collect first-hand data and organize it using: • tally marks •
line plots • charts • lists to answer questions. [C, CN, PS,
V]
2. Construct, label and
interpret bar graphs to solve problems. [C, PS, R, V]
1. Demonstrate an understanding of many-to-one correspondence. [C,
R, T, V]
2. Construct and interpret
pictographs and bar graphs involving many-to-one correspondence to
draw conclusions. [C, PS, R, V]
1. Differentiate between first-hand and second-hand data. [C, R, T,
V]
2. Construct and interpret double bar graphs to draw conclusions.
[C, PS, R, T, V]
[C] Communication [PS] Problem Solving [CN] Connections [R]
Reasoning [ME] Mental Mathematics [T] Technology and Estimation [V]
Visualization
34 / K-9 Mathematics Curriculum NL
Statistics and Probability (Chance and Uncertainty)
Grade 3 Grade 4 Grade 5
General Outcome Use experimental or theoretical probabilities to
represent and solve problems involving uncertainty.
Specific Outcomes
3. Describe the likelihood of a single outcome occurring, using
words such as: • impossible • possible • certain. [C, CN, PS,
R]
4. Compare the likelihood of
two possible outcomes occurring, using words such as: • less likely
• equally likely • more likely. [C, CN, PS, R]
[C] Communication [PS] Problem Solving [CN] Connections [R]
Reasoning [ME] Mental Mathematics [T] Technology and Estimation [V]
Visualization
UNIT: NUMERATION
Es tim
at ed
C om
pl et
io n
UNIT: NUMERATION
UNIT: NUMERATION
Unit Overview
Background .
Students have already had significant place value experience up to
end of Grade 3. While there may be many students who have not
mastered the topic completely, most should arrive at grade 4 with a
foundation to build upon. The intent in Grade 4 is to extend upon
this foundation and develop place value concepts for 4-digit
numbers to 10 000. It is important for students to gain an
understanding of the relative size (magnitude) of numbers through
real life contexts that are personally meaningful. Use numbers from
student’s experiences, such as capacity for local arenas, or
population of the school/community. Students can use these personal
referents to think of other large numbers. Students can also use
benchmarks that they may find helpful such as multiples of 100 and
1000, as well as 250, 500, 750, 2500, 5000, and 7500. The focus of
instruction should be ensuring students develop flexible thinking
with respect to larger whole numbers.
Process Standards Key
[C] Communication [PS] Problem Solving [CN] Connections [R]
Reasoning [ME] Mental Mathematics [T] Technology and Estimation [V]
Visualization
UNIT: NUMERATION
Strand: Number
Specific Outcome It is expected that students will: N1 Represent
and describe whole numbers to 10 000, concretely, pictorially and
symbolically [C, CN, V]
Suggestions for Teaching and Learning While the unit on
multiplication and division facts does not begin until around
Christmas, it is suggested that the facts with products up to 45 be
incorporated early as part of the 5-10 minutes of morning routine.
Students should recognize the value represented by each digit in a
number, as well as what the number means as a whole. Include
situations in which students use money, place value charts, base
ten materials and number lines. Money
How many $100 bills are there in $8347? Place value charts
Thousands
H T O H T O
Base ten materials
What does 999 look like? What would 1 000 look like? Have students
construct it with the base ten blocks. If you
were to make a base ten block that could represent 10 000, what
would it look like? Why?
If you had ten flats, what is the total value? How would you write
a given base ten collection, as a
numeral? If you could create a new base ten block that would
represent
10 000, what would it look like and why? (One suggestion might be
to model 10 000 as a long train or tower using 10 of the thousand
cubes. It will be a 10 thousand rod. Students should recognize that
this also models 10 000 unit cubes.)
Mathematical language: Words have special meanings in mathematics.
By using a multiplicity of mathematical language, we help children
to develop a rich language base conducive to communicating
understanding. This will lead to higher order thinking as students
move through the grades. For example, when using Base Ten
materials, there is a variety of vocabulary that can be used to
describe these materials. The thousands block (“block” is a generic
term for any of the Base Ten pieces) can be referred to as the
large cube, the hundreds block can be referred to as a flat, the
tens block can be referred to as a rod or a long and the ones block
can be referred to as a unit. Avoid using “thousands cube”,
“hundreds flat”, “tens rod” or “ones” as student will need to be
flexible in their thinking of models. Later, when the Base Ten
materials are used to represent decimals, the individual blocks
will take on different leanings. At that point, for example, the
rod may represent one.
UNIT: NUMERATION
2157
2000+100+50+7
157 more than 2000
General Outcome: Develop Number Sense
Suggested Assessment Strategies Student-Teacher Dialogue N1 Ask
questions about the reasonableness of numbers such as, “Would it be
reasonable for an elementary school to have 9600 students?” or
“Would it be reasonable for an elevator to hold 20 people?” “Would
someone be able to drive 26 hundred kilometres in a day?” “Would it
be reasonable to pay $5 000 for a boat/book/computer?” Investigate
and discuss possible answers. Have students create their own
“reasonable” questions about a variety of topics. Portfolio N1
Exploding the Number: Have the students write any 4-digit number in
the center of a large sheet of paper. Ask the students to represent
the number in as many ways as possible. This should be repeated any
time throughout the unit as children build on their knowledge of
number.
Resources/Notes Authorized Resources MathFocus 4 Chapter 2:
Numeration Getting Started: Modelling Numbers TR pp. 9 - 12 SB pp.
34-35
Additional Resources: Teaching Student-Centered Mathematics, Grades
3-5, Van de Walle and Lovin, p. 45 -49 Making Math Meaningful to
Canadian Students, K-8, Small, p.137 – 148
UNIT: NUMERATION
40 GRADE 4 MATHEMATICS CURRICULUM GUIDE - DRAFT
0 5500 Place 2500 on the number line in relative position
Place 5125 on the number line in relative position
Place 7500 on the number line in relative position
5500 6500
Strand: Number
Specific Outcome It is expected that students will: N1 Represent
and describe whole numbers to 10 000, concretely, pictorially and
symbolically (Cont’d) [C, CN, V] N1.1 Read a given four-digit
numeral without using the word and; e.g., 5321 is five thousand
three hundred twenty-one, NOT five thousand three hundred AND
twenty-one. N1.2 Write a given numeral, using proper spacing
without commas; e.g.4567 or 4 567, 10 000. N1.4 Represent a given
numeral using a place value chart or other models.
Suggestions for Teaching and Learning Number-line Given a number
line with two reference points ask the students to place a given
whole number. Include number lines with different starting and
ending points.
N1.1 The word ‘and’ is reserved for reading decimal numbers. For
example, 3.2 will read “three and two tenths”. Set an example by
not using “and” when reading numbers. This is simply a convention.
Ask students to watch for times when you will use the word “and” in
a number, inappropriately. Do this intentionally on occasion to
observe who notices. Rewards may be given.
N1.2 Students will see four digit numbers written with or without a
space between the thousands and hundreds digits. However, the
conventional use of spaces helps children to read larger
multi-digit numbers without the use of commas.
1.4 When representing numerals, students should model numbers
containing zeros: For example: 1003 means: 1 thousand, 3 ones
UNIT: NUMERATION
General Outcome: Develop Number Sense
Suggested Assessment Strategies (Cont’d) Student-Teacher Dialogue
Tell the student that a snowmobile costs $9130. Ask: If one were to
pay for it in $100 bills, how many would be needed? Extend by
asking how many $10 dollar bills would be needed. Performance Have
students, as a class, create a “ten thousands” chart. Provide each
small group of students with hundred grids (or other pictorial
representations such as arrays of dots) and have them create a
model to represent 1000. Combine these models to create a class
representation of 10 000. Performance Have the students find large
numbers from newspapers and magazines. Ask them to share and
discuss the numbers within their group. Have students read, write,
and model these numbers in different ways. Student-Teacher Dialogue
Teachers could ask the student to imagine flats placed on top of
each other to form a tower. Ask: How many flats would be required
to construct a tower representing 10 000? How high would this be?
Paper and Pencil Teachers model numbers using base ten materials.
For example, show 4 rods, 2 large cubes, 3 units. (Note: It is not
always necessary for the blocks to be presented in the typical
order). Have the students record the number that is represented.
Student-Teacher Dialogue Ask a student to use base ten materials to
model 2046 in three different ways. Have him/her explain the
models. Student-Teacher Dialogue Ask: How are 1003 and 103
different? Similar?
Resources/Notes MathFocus 4 Lesson 1: Modelling Thousands N1 (1.1/
1.2) TR pp.13 - 15 SB pp.36-37
UNIT: NUMERATION
Strand: Number
Specific Outcome It is expected that students will: N1 Represent
and describe whole numbers to 10 000, concretely, pictorially and
symbolically (Cont’d) [C, CN, V] Achievement Indicators: N1.2 Write
a given numeral, using proper spacing without commas; e.g.4567 or 4
567, 10 000. N1.4 Represent a given numeral using a place value
chart or other models. N1.7 Explain the meaning of each digit in a
given 4-digit numeral, including numerals with all digits the same;
e.g., for the numeral 2 222, the first digit represents two
thousands, the second digit two hundreds, the third digit two tens
and the fourth digit two ones.
Suggestions for Teaching and Learning N1.2 Students will see four
digit numbers written with or without a space between the thousands
and hundreds digits. However, the conventional use of spaces helps
children to read larger multi-digit numbers without the use of
commas. While a four digit number can be written with or without
the space, five digit numbers should include appropriate spacing.
N1.4 One model of representation may not meet the needs of all
students in the class. Acceptable models include place value
charts, base ten materials and money. Lesson 2 addresses this,
primarily using base ten materials and place value charts. The use
of other models appears in subsequent lessons. N1.7 Students will
describe numbers in several ways. Typically, the number 8 347 is
read as eight thousand, three hundred forty-seven but might also be
expressed as:
8 thousands, 34 tens, 7 ones; 83 hundreds, 4 tens, 7 ones; or 8
thousands, 3 hundreds, 47 ones; etc.
UNIT: NUMERATION
General Outcome: Develop Number Sense
Suggested Assessment Strategies Paper and Pencil Teachers model
numbers using base ten materials. For example, show 4 rods, 2 large
cubes, 3 units. (Note: It is not always necessary for the blocks to
be presented in the typical order). Have the students record the
number that is represented. Student-Teacher Dialogue Pose a problem
such as “Patrick chose 6 base ten blocks. The value of these blocks
is more than 4000 and less than 4804. Which blocks might Patrick
have chosen?” Have the student model and explain. Extension: Ask if
a student can find all possible numbers which fit the criteria. Ask
the student to justify his/her answer. Presentation Tell the
students that a number has 4 digits. The digit in the thousands
place is greater than the digit in the tens place. Ask: What number
might this be? Have students share their responses. Ask: What is
the greatest number this could be? What is the least number this
could be? Paper and Pencil Ask students to write a number that has
at least 20 tens.
Resources/Notes
.
Strand: Number
Specific Outcome It is expected that students will: N1 Represent
and describe whole numbers to 10 000, concretely, pictorially and
symbolically (Cont’d) [C, CN, V] Achievement Indicators: N1.5
Express a given numeral in expanded notation; e.g., 321 = 300 + 20
+ 1. N1.6 Write the numeral represented by a given expanded
notation. N1.7 Explain the meaning of each digit in a given 4-digit
numeral, including numerals with all digits the same; e.g., for the
numeral 2 222, the first digit represents two thousands, the second
digit two hundreds, the third digit two tens and the fourth digit
two ones
Suggestions for Teaching and Learning Indicators N1.5/ N1.6/ N1.7
are addressed together in lesson 3. It is important not to limit
teaching examples to: 1 635 = ___ thousands ___hundreds ____tens
____ones in which the student is likely to put the numbers 1, 6, 3,
5, in the blanks in order, since it is the only information
available even if he/she has no idea what this question means. To
better assess student understanding of place value, provide numbers
such as 1 635. Ask, “Which digit is in the hundreds place?” Student
might answer “6”. Ask, “What does the 3 represent?” Student might
answer “30” or “3 tens”. This type of questioning tells the teacher
more about the student understanding of expanded notation. Students
should be given the opportunity to work with numbers involving
zeros. (e.g. 4062 - It is important to note that that 4062 does not
have a digit in the hundreds place however it still has 40
hundreds.)
UNIT: NUMERATION
General Outcome: Develop Number Sense
Suggested Assessment Strategies Pencil-Paper Have students use a
reference book to find the populations of 2 towns, where
populations are 10 000 or less. Ask students to represent these
populations in expanded notation. Journal Say the standard name for
a number with four digits (e.g. “four thousand forty six”). Using
base ten blocks, students model that number on their desks. Record
what they have made, in a journal. Next students enter the number
on a calculator (standard form) and record. Finally, students write
the number in expanded form. Performance Find Your Partner (card
game) – Prepare 2 sets of cards to suit size of group. For example:
Set A Set B 2023 2000 + 20 + 3 2332 300 + 2 + 30 + 2000 223 200 +
20 + 3 2230 2000 + 200 + 30 2032 2 + 30 + 2000 3202 three thousand
two hundred two In Set A, be sure to include some cards with
numbers having “0” as a digit or cards that have the same digit
repeated (e.g. 2117). In Set B, cards could include base ten
pictures or number lines. Distribute cards, 1 per student, and have
students circulate around the room to find the partner with the
card that corresponds to their own. As students compare cards,
encourage them to discuss why their cards match or do not match.
Once students have found their partner, they will read their
numbers to the teacher for confirmation. Performance/Journal Have
students work in pairs. Provide each pair with 4 number cubes. Roll
the cubes and line them up to form a 4 digit numeral. Record the
numeral in standard form and in expanded notation. Using the same
four numerals, rearrange the cubes to find all possible 4 digit
numerals. Record each one in math journal. Extension: Place these
numbers on a number line.
Resources/Notes MathFocus 4 Lesson 3: Expanded Form N1 (1.1/ 1.2/
1.4/ 1.5/ 1.6/ 1.7)) TG pp. 20 - 23 SB pp. 42 – 44 More practice
may be needed. Extra practice in the black line masters can be
found on page 11.
UNIT: NUMERATION
Strand: Number
Specific Outcome It is expected that students will: N1 Represent
and describe whole numbers to 10 000, concretely, pictorially and
symbolically (Cont’d) [C, CN, V] Achievement Indicators: N1.4
Represent a given numeral using a place value chart or other models
N1.6 Write the numeral represented by a given expanded
notation.
Suggestions for Teaching and Learning Extending students’
conceptual understanding of numbers beyond 1000 is sometimes
difficult to do because physical models for thousands are not
commonly available. Encouraging students to extend the patterns in
the place value system and to create familiar real-world referents
helps students develop a fuller sense of these larger numbers. (Van
de Walle & Lovin, Teaching Student Centered Mathematics 3 -5,
2006) Provide interesting tasks using numbers like 10 000 and these
will become lasting reference points or benchmarks and will provide
meaning to large numbers encountered in everyday life. Big number
tasks need not take up a lot of time but can be done as group or
school-wide projects. Assuming there about 500 pages in your math
book, how many math books would it take to make 10 000 pages?
Introduction of the words “ten thousand” should be done when the
students have demonstrated that they understand such a number does
exist. What does 10 000 look like? Arrange the class into 10 groups
and supply each group with hundred grids. Assign each group the
task of building a rod to represent 1 000 and tape them end to end.
In a large area have students come together to create a 10 000
model using the thousand strips. Ask: How long would it take to
count to 10 000? (You may time how long it takes for the students
to count 100. Then multiply by 100 using a calculator) Ask: Do you
think our school sends home 10 000 newsletters in a year?
UNIT: NUMERATION
General Outcome: Develop Number Sense
Suggested Assessment Strategies Presentation As a class project,
collect some type of object of reaching 10 000. For example,
pennies, buttons, bread tags, soup labels, etc. Performance Create
a large amount such as 10 000 by asking students to draw 100, 200
or 500 dots on a sheet a paper. Ask students to compile their
sheets to form a visual of 10 000 (e.g. book, bulletin board etc.).
10 000 paper chain links can be constructed and hung (with
benchmarks indicated) along the hallway. Let the school be aware of
the ultimate goal. Performance Use a sheet of 1cm dot paper. Cut
the show it shows a 10cm x 20cm array of dots. Ask:
• how many dots are on one sheet? • how many sheets are needed to
show 1 000 dots? • how many are needed to show 5 000 dots? 10 000
dots?
Place the sheets in an array of 50 sheets so that students what an
array of 10 000 dots would look like. Journal Students research to
find examples of situations where the number 10 000 is used. Make
these into posters to display or describe in words and drawings.
Performance Show one package of unopened copier paper to the class.
Discuss how many sheets are in the package and how many packages
would be needed for 10 000 sheets of paper.
Resources/Notes MathFocus 4 Lesson 4: Describing 10 000 N1 (1.2/
1.4) TG pp. 24 - 26 SB p. 45 Mid-Chapter Review N1 (1.4/ 1.5/ 1.6)
TG. pp 27-29 SB p46 -47 In the interest of time, these may be done
together in one period. Page 28 in TG suggests . . . “Students
should be able to complete Questions 1 to 7 in class. If not,
assign the rest for homework.” Teachers should use own discretion
in assigning class practice and homework. It is not necessary to
assign all questions nor is it beneficial for all students to
complete unfinished practice at home. Teaching Student-Centered
Mathematics, Grades 3-5, Van de Walle and Lovin, p. 50 - 51
UNIT: NUMERATION
Strand: Number
Specific Outcome It is expected that students will: N1 Represent
and describe whole numbers to 10 000, concretely, pictorially and
symbolically (Cont’d) [C, CN, V] Achievement Indicators: N1.1 Read
a given four-digit numeral without using the word and; e.g., 5321
is five thousand three hundred twenty-one, NOT five thousand three
hundred AND twenty-one. N1.2 Write a given numeral, using proper
spacing without commas; e.g.4567 or 4 567, 10 000. N1.3 Write a
given numeral
0–10 000 in words
Suggestions for Teaching and Learning Lesson 5 in the text deals
mainly with cheque writing. Remember that cheque writing is not an
outcome. It is simply the context used. In fact, we do not write
cheques today as often as in the past because of wide use of debit
cards. However, cheque writing is a good example of a practical use
of writing numbers in words. More practice will be needed and can
be found on page 13 of the black line masters. In addition, when
writing numbers from standard form to words and vice versa, it is
important to ensure that one or more zeros occur in some of the
numbers at a various place value position. Students should be able
to represent numbers, which they see or hear, in words.
Examples:
• Say: One thousand nine hundred twenty two people attend a
hockey game. Write this in words. • Write “2 900” on chart. Have
students record in words.
(Acceptable answers include: twenty nine hundred or two thousand
nine hundred).
UNIT: NUMERATION
General Outcome: Develop Number Sense
Suggested Assessment Strategies Performance Have students spin a
spinner 3 or 4 times and write the corresponding number in words.
Performance Have students cut a 4 digit number from a newspaper or
magazine and paste it in their journal. Write the number in words.
Performance Write the current year in words. They can also write
other years such as their birth year, their expected high school
graduation year, the year they will turn forty. Performance What’s
My Number? –Provide materials for the students to create a poster
with a door or flap in the middle. Ask each student to think of a
secret number and write it behind the flap using numerals and/or
words. Next instruct students to write clues around the outside of
the door that will assist classmates in identifying the secret
number. Display posters with a letter assigned to each. Provide
students with recording sheets so that they can visit each poster
and guess the secret numbers in the display. The intent in this
task is that students will record their guesses by using words.
Presentation Have each student choose any 4 digit number and create
a silly rhyme. These rhymes can be combined in a class booklet.
Remind students to use words instead of numerals. For example: “One
thousand four hundred eight Too many peas to put on my plate!” “
Twenty four hundred seventy one Happy days spent in the sun.”
Resources/Notes MathFocus 4 Lesson 5: Writing Number Words N1 (1.1/
1.2/ 1.3) TG. pp 30 -32 SB pp 48 -49
[C] Jordan’s Secret Number Greater than 999 Less than 2 000 Has 4
digits Has repeated digits in the Multiple of 10 hundreds and
thousands place Tens place is 3 less than 7
UNIT: NUMERATION
Strand: Number
Specific Outcome It is expected that students will: N1 Represent
and describe whole numbers to 10 000, concretely, pictorially and
symbolically (Cont’d) [C, CN, V] N2 Compare and order whole numbers
to 10 000. [C, CN, V] Achievement Indicators: N2.2 Create and order
three different 4-digit numerals N2.3 Identify the missing numbers
in an ordered sequence or on a number line (vertical or
horizontal). N2.4 Identify incorrectly placed numbers in an ordered
sequence or on a number line (vertical or horizontal).
Suggestions for Teaching and Learning N2 Comparing and ordering is
fundamental to understanding numbers. Students should investigate
meaningful contexts to compare and order two or more numbers, both
with and without models. Students must realize that when comparing
two numbers with the same number of digits, the digit in the
greatest place value needs to be addressed first. For example, when
asked to explain why one number is greater or less than another,
they might say that 2542 < 3653 because 2542 is less than 3
thousands while 3653 is more than 3 thousands. When comparing 6456
and 6546, students will begin comparing the thousands and then move
to the right until they notice a difference in place value. N2.2
Assign pairs of students the task of making challenging number
cards for their classmates to put in order. Provide the students
with opportunities to use number lines with various starting and
ending points (0 to10 000). Children will encounter instances of
having to read vertical number lines as well as horizontal.
Examples include thermometers, measuring cups, distance above/below
sea level, growth charts, etc.
0 1000 2000 3000 4000 5000 6000
0 75
UNIT: NUMERATION
General Outcome: Develop Number Sense
Suggested Assessment Strategies Paper and Pencil Ask the students
to each write a number that would fall about half way between 9598
and 10 000. Performance Provide students with cards that have
4-digit numbers written on them For example: Ask students to stand
in a line in ascending order. Ask a few students to explain why
they positioned themselves in that particular spot. Numbers can
vary according to student level and small group variations are
possible. This task may be repeated using descending order. Ask
students to space themselves with respect to number size (in
relative position). Performance/Portfolio N2 Provide the following
riddle: I am thinking of a number. It is between 8000 and 10 000.
All the digits are even and the sum of the digits is 16. What are
some possibilities? Have students place their numbers in relative
position on an open number line. Challenge pairs of students to
write similar riddles for one another and to record answers.
Performance Ask two students to hold the ends of a skipping rope
representing a number line. Attach 4-digit number cards to the line
(using clothespins or fold-over cards). Place several cards out of
order. Ask them to identify incorrectly placed numbers and to
justify their reasoning. Variation: Repeat having some blank spaces
for student to identify the missing 4 digit number in the sequence.
For example:
Resources/Notes MathFocus 4 Lesson 6: Locating Numbers on a Number
Line N1 (1.1) N2 (2.3/ 2.4) TG. pp 33-36 SB pp 50-52 Lesson 6
address two outcomes. Work also with number lines that include
fewer markings so that students will draw upon their estimation
skills over a broader range of numbers. This can be done using a
string, a start point card, an end point card and various numbers
that students are asked to place in the number line. This can be
done by individuals within the context of a whole group activity.
Curious Math N1 (1.7) TR pp.37 -38 SB p. 53 (may be omitted)
3000 3300 3003 13003303 3033
1367 1467 1567? 1667
Strand: Number
Specific Outcome It is expected that students will: N1 Represent
and describe whole numbers to 10 000, concretely, pictorially and
symbolically (Cont’d) [C, CN, V] N2 Compare and order whole numbers
to 10 000 (Cont’d) [C, CN, V] Achievement Indicators: N2.2 Create
and order three different 4-digit numerals N2.1 Order a given set
of numbers in ascending or descending order, and explain the order
by making references to place value
Suggestions for Teaching and Learning N2.2 Students must recognize
that when comparing the size of a number, the 4 in 4289 has a
greater value than the 9 and they should be able to provide an
explanation. Tape numbers on students’ backs and have students
identify their number by asking classmates questions. E.g. Am I
greater than 1000? Am I less than 750? Am I an even or odd number?
Am I a multiple of…? Once students have identified their number,
have them correctly place it on a class number line. Discuss
solutions posed by students. The focus of lesson 8 is on
communication in relation to ordering numbers. Recognize that
mathematical communication takes on various forms including oral,
written, symbolic, graphical, pictorial and physical. (Small p.62)
It is important to encourage students to communicate the process
used to compare and order numbers. Instruction should enable
students to: • Organize and consolidate their mathematical thinking
through
communication. • Communicate their mathematical thinking coherently
and clearly
to peers, teachers and others. • Analyze and evaluate the
mathematical thinking and strategies of
others. • Use the language of mathematics to express mathematical
ideas
precisely. (NCTM , 2000)
General Outcome: Develop Number Sense
Suggested Assessment Strategies Performance Provide a stack of 4
sets of shuffled cards numbered 0 - 9. Ask the students to select 4
cards and arrange them to make the greatest possible number. Have
students record and read the number. Then rearrange the cards to
make the least possible number. Performance Have students use a
reference book to find the populations of 2 communities, where
populations are 10 000 or less. Ask them to find another population
that is greater than that of one of the communities, but less than
that of the other. Paper and Pencil Tell the students that you are
thinking of a 4-digit number that has 4 hundreds, a greater number
of tens, and an even greater number of ones. Ask them to give three
possibilities. Paper and Pencil N2.2 Ask the student to record two
numbers: the first has 3 in the thousands place, but is less than
the second which has 3 in the hundreds place. Paper and Pencil N2
Ask the students to find 3 ways to fill in the blanks to make the
following statement true: __245 > 7__84 Student-Teacher Dialogue
Tell the student that Bethany’s number had 9 hundreds, but Fran’s
had only 6 hundreds. Fran’s number was greater. Explain how this
was this possible? Student-Teacher Dialogue: Ask: Which number
below must be greater? Explain why. 4 _ _ 2 9 _ 3 Paper and Pencil
Given a set of whole numbers,