mathematics grade3 curriculum guide
TRANSCRIPT
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Mathematics
Grade Three
Interim Edition
Curriculum Guide
September 2010
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TABLE OF CONTENTS
Table of Contents
Acknowledgements....................................iii
Foreword......................................v
Background ...........................................1
Introduction Purpose of the Document........................................2
Beliefs About Students and Mathematics Learning.....................................2
Affective Domain...........................................3
Early Childhood..........................................3
Goal for Students...........................................4
Conceptual Framework for K9 Mathematics...................................4Mathematical Processes................................................5
Nature of Mathematics............................................9
Strands....................................................12
Outcomes and Achievement Indicators.......................................13
Summary.................................................13
Instructional FocusPlanning for Instruction............................................ 14
Resources................................................14
Teaching Sequence..........................................15
Instruction Time per Unit..................................................15
General and Specifc Outcomes.............................................................16
General and Specifc Outcomes by Strand Grades 2 4.......17Patterning....................................................................................31
Numbers to 1000..........................................................83
Data Analysis.....................................................125
Addition and Subtraction........................................................149Geometry..............................................................................203
Multiplication and Division................................................... 237
Fractions............................................................................... 273
Measurement .................................................291
Appendix A: Outcomes with Achievement Indicators (Strand).......325
References................................................................339
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The Department of Education would like to thank the 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, which has been reproduced and/oradapted by permission. All rights reserved.
We would also like to thank the provincial Grade 3 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
Kimberly Pope, Teacher Greenwood Academy, Campbellton
Nicole Kelly, Teacher Smallwood Academy, Gambo
Shannon Best, Teacher Gander Academy, Gander
Lisa Piercey, Teacher Mary Queen of Peace, St. Johns
Valerie Wells, Teacher Bishop Abraham Elementary, St. Johns
Yolanda Anderson, Teacher St. Edwards, Kelligrews
Sherry Mullett, Teacher Lewisporte Academy, Lewisporte
Every effort has been made to acknowledge all sources that contributed to the development of this document.
Any omissions or errors will be amended in future printings.
Acknowledgements
ACKNOWLEDGEMENTS
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GRADE 3 MATHEMATICS CURRICULUM GUIDE - INTERIM v
Foreword
The WNCP Common Curriculum Frameworks for Mathematics
K 9 (WNCP, 2006), formed the basis for the development of this
curriculum guide. While minor adjustments have been made, the
outcomes and achievement indicators established through the WNCP
Common Curriculum Framework are used and elaborated on for
teachers in this document. Newfoundland and Labrador has used
the WNCP 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.
FOREWORD
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GRADE 3 MATHEMATICS CURRICULUM GUIDE - INTERIM 1
INTRODUCTION
The province of Newfoundland and Labrador commissioned an
independent 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 four and perhaps most significiant 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, the implementation schedule for K - 6 mathematics is
as follows:
Implementation Year Grade Level
2008 K, 1 and 4
2009 2, 5
2010 3, 6
BACKGROUND
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INTRODUCTION
Purpose of the
Document
INTRODUCTION
The Mathematics Curriculum Guides for Newfoundland and Labrador
have been derived from The Common Curriculum Framework for K9
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.
The curriculum guide
communicates high
expectationsfor students.
Beliefs AboutStudents and
Mathematics
Learning
Students are curious, active learners with individual interests, abilitiesand 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 inorder 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.
Mathematical
understanding is fostered
when students build on
their own experiences and
prior knowledge.
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INTRODUCTION
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 mathematicsare 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 thesetting and assessing of personal goals.
Affective Domain
Early Childhood Young children are naturally curious and develop a variety ofmathematical 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.
To experience success,students must be taught
to set achievable goals and
assess themselves as theywork toward these goals.
Curiosity about mathematicsis fostered when children
are actively engaged in their
environment.
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INTRODUCTION
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.
Goals For
Students
Mathematics education
must prepare students
to use mathematics
confidently to solve
problems.
CONCEPTUALFRAMEWORKFOR K-9MATHEMATICS
The chart below provides an overview of how mathematical processes
and the nature of mathematics influence learning outcomes.
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PROCESS STANDARDS
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.
This curriculum guide incorporates these seven interrelated
mathematical processes that are intended to permeate teaching and
learning.
MathematicalProcesses
Communication [C] Connections [CN]
Mental Mathematics
and Estimation [ME]
Problem Solving [PS]
Reasoning [R]
Technology [T]
Visualization [V]
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 theformal 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.
Students must be able to
communicate mathematical
ideas in a variety of ways
and contexts.
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Contextualization and making connections to the experiences
of learners are powerful processes in developing mathematical
understanding. 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 relevantto 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 experiencesfrom 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).
Connections [CN]
PROCESS STANDARDS
Through connections,
students begin to viewmathematics as useful and
relevant.
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 needmore thanever beforewith 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 orfor 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.
Mental mathematics and
estimation are fundamental
components of number sense.
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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.
Problem Solving [PS]
Reasoning [R]
PROCESS STANDARDS
Learning through problemsolving should be the focus
of mathematics at all grade
levels.
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 provideopportunities 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.
Mathematical reasoning
helps students thinklogically and make sense of
mathematics.
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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.
Technology [T]
Visualization [V]
PROCESS STANDARDS
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 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).
Visualization is fostered
through the use of concrete
materials, technology
and a variety of visual
representations.
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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 curriculum
guide. The components are change, constancy, number sense, patterns,
relationships, spatial sense and uncertainty.
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).
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.
Nature ofMathematics
Change
Constancy
NATURE OF MATHEMATICS
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 andsymmetry.
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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 developtheir 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.
Number Sense
Patterns
NATURE OF MATHEMATICS
An intuition about numberis the most important
foundation of a numerate
child.
Mathematics is about recognizing, describing and working with
numerical and non-numerical patterns. Patterns exist in all strands of
mathematics.
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.
Mathematics is about
recognizing, describing andworking with numerical
and non-numerical
patterns.
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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 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.
NATURE OF MATHEMATICS
Relationships
Spatial Sense
Uncertainty In mathematics, interpretations of data and the predictions made fromdata 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.
Mathematics is used to
describe and explain
relationships.
Spatial sense offers a way tointerpret and reflect on the
physical environment.
Uncertainty is an inherent
part of making predictions.
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The learning outcomes in the mathematics program are organized
into four strands across the grades K9. Some strands are subdivided
into substrands. There is one general outcome per substrand across the
grades K9.
The strands and substrands, including the general outcome for each,
follow.
Strands
Number
Patterns and Relations
Shape and Space
Statistics and Probability
STRANDS
Number
Patterns and Relations
Shape and Space
Statistics and
Probability
Number
Develop number sense.
Patterns
Use patterns to describe the world and to solve problems.
Variables and Equations
Represent algebraic expressions in multiple ways.
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.
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.
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The mathematics program is stated in terms of general outcomes,
specific outcomes and achievement indicators.
General outcomesare overarching statements about what students areexpected 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.
In the specific outcomes, the word includingindicates that any ensuing
items must be addressed to fully meet the learning outcome. The phrase
such asindicates that the ensuing items are provided for illustrative
purposes or clarification, and are not requirements that must beaddressed to fully meet the learning outcome.
Achievement indicatorsare 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.
The conceptual framework for K9 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.
Outcomes andAchievementIndicators
General Outcomes
Specific Outcomes
Achievement Indicators
Summary
OUTCOMES
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Resources
INSTRUCTIONAL FOCUS
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 andsymbolically.
Students bring a diversity of learning styles and cultural backgrounds
to the classroom. They will be at varying developmental stages.
The resource selected by Newfoundland and Labrador for students and
teachers is Math Makes Sense 3(Pearson). Schools and teachers have
this as their primary resource offered by the Department of Education.
Column four of the curriculum guide references Math Makes Sense 3 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.
INSTRUCTIONALFOCUS
Planning for Instruction
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Instruction Time Per Unit The suggested number of weeks of instruction per unit is listed in theguide at the beginning of each unit. The number of suggested weeks
includes time for completing assessment activities, reviewing andevaluating.
INSTRUCTIONAL FOCUS
Teaching Sequence The curriculum guide for Grade 3 is organized by units from Unit 1 toUnit 8. The purpose of this timeline is to assist in planning. The use of
this timeline is not mandatory; however, it is mandtory that all outcomes
are taught during the school year so a long term plan is advised. There
are a number of combinations of sequences that would be appropriate for
teaching this course. The arrow showing estimated focus does not mean
the outcomes are never addressed again. The teaching of the outcomes is
ongoing and may be revisited as necessary.
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GENERAL AND SPECIFIC OUTCOMES
GENERAL AND SPECIFIC OUTCOMES BY STRAND
(pages 1730)
This section presents the general and specific outcomes for each strand,
for Grade 2, 3 and 4.
Refer toAppendix Afor the general and specific outcomes with
corresponding achievement indicators organized by strand for Grade 3.
GENERAL AND SPECIFIC OUTCOMES WITH ACHIEVEMENT
INDICATORS(beginning at page 31)
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 to be used
to determine whether or not students have achieved a given specific
outcome. Teachers should use these indicators but other indicators
may be added 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.
GENERAL
AND SPECIFICOUTCOMES
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GENERAL AND SPECIFIC OUTCOMES BY
STRAND
(Grades 2, 3 and 4)
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GENERAL AND SPECIFIC OUTCOMES BY STRAND
[C] Communication [PS] Problem Solving[CN] Connections [R] Reasoning[ME] Mental Mathematics [T] Technology and Estimation [V] Visualization
Grade 2 Grade 3 Grade 4
General OutcomeDevelop number sense.
General OutcomeDevelop number sense.
General OutcomeDevelop number sense.
Specific Outcomes Specific Outcomes Specific Outcomes
1. Say the number sequence from0 to 100 by: 2s, 5s and 10s, forward andbackward, using starting pointsthat are multiples of 2, 5 and 10respectively 10s, using starting points from1 to 9 2s, starting from 1.[C, CN, ME, R]
2. Demonstrate if a number (up to100) is even or odd.[C, CN, PS, R]
3. Describe order or relativeposition, using ordinal numbers(up to tenth).[C, CN, R]
4. Represent and describenumbers to 100, concretely,pictorially and symbolically.
[C, CN, V]
1. Say the number sequence 0 to1000 forward and backward by:
5s, 10s or 100s, using anystarting point 3s, using starting points thatare multiples of 3 4s, using starting points thatare multiples of 4 25s, using starting points thatare multiples of 25.
[C, CN, ME]
2. Represent and describenumbers to 1000, concretely,pictorially and symbolically.[C, CN, V]
3. Compare and order numbers to1000.[C, CN, R, V]
4. Estimate quantities less than1000, using referents.
[ME, PS, R, V]
1. Represent and describe wholenumbers to 10 000, pictoriallyand symbolically.[C, CN, V]
2. Compare and order numbers to10 000.[C, CN, V]
3. Demonstrate an understandingof addition of numbers with
answers to 10 000 and theircorresponding subtractions(limited to 3- and 4-digitnumerals) by:
using personal strategies foradding and subtracting estimating sums anddifferences solving problems involvingaddition and subtraction.
[C, CN, ME, PS, R]
Number
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GRADE 3 MATHEMATICS CURRICULUM GUIDE - INTERIM 19
GENERAL AND SPECIFIC OUTCOMES BY STRAND
[C] Communication [PS] Problem Solving[CN] Connections [R] Reasoning[ME] Mental Mathematics [T] Technology and Estimation [V] Visualization
Grade 2 Grade 3 Grade 4
General OutcomeDevelop number sense.
General OutcomeDevelop number sense.
General OutcomeDevelop number sense.
Specific Outcomes Specific Outcomes Specific Outcomes
5. Compare and order numbersup to 100.[C, CN, ME, R, V]
6. Estimate quantities to 100,using referents.[C, ME, PS, R]
7. Illustrate, concretely andpictorially, the meaning of placevalue for numerals to 100.
[C, CN, R, V]
8. Demonstrate and explainthe effect of adding zero to,or subtracting zero from, anynumber.[C, R]
5. Illustrate, concretely andpictorially, the meaning of placevalue for numerals to 1000.[C, CN, R, V]
6. Describe and apply mentalmathematics strategies for addingtwo 2-digit numerals, such as:
adding from left to right taking one addend to thenearest multiple of ten and then
compensating using doubles.[C, CN, ME, PS, R, V]
7. Describe and apply mentalmathematics strategies forsubtracting two 2-digit numerals,such as:
taking the subtrahend to thenearest multiple of ten andthen compensating thinking of addition
using doubles.[C, CN, ME, PS, R, V]
4. Explain and apply theproperties of 0 and 1 formultiplication and the property of1 for division.[C, CN, R]
5. Describe and apply mentalmathematics strategies, such as:
skip counting from a knownfact using doubling or halving
using doubling or halvingand adding or subtracting onemore group using patterns in the 9s factsusing repeated doubling
to determine basic multiplicationfacts to 9 9 and related divisionfacts.[C, CN, ME, R]
Number
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GRADE 3 MATHEMATICS CURRICULUM GUIDE - INTERIM20
GENERAL AND SPECIFIC OUTCOMES BY STRAND
[C] Communication [PS] Problem Solving[CN] Connections [R] Reasoning[ME] Mental Mathematics [T] Technology and Estimation [V] Visualization
Grade 2 Grade 3 Grade 4
General OutcomeDevelop number sense.
General OutcomeDevelop number sense.
General OutcomeDevelop number sense.
Specific Outcomes Specific Outcomes Specific Outcomes
9. Demonstrate an understandingof addition (limited to 1- and2-digit numerals) with answersto 100 and the correspondingsubtraction by:
using personal strategies foradding and subtracting withand without the support ofmanipulatives creating and solving problemsthat involve addition and
subtraction using the commutativeproperty of addition (the orderin which numbers are addeddoes not affect the sum) using the associative propertyof addition (grouping a set ofnumbers in different ways doesnot affect the sum) explaining that the order inwhich numbers are subtractedmay affect the difference.
[C, CN, ME, PS, R, V]
8. Apply estimation strategiesto predict sums and differencesof two 2-digit numerals in aproblem-solving context.[C, ME, PS, R]
9. Demonstrate an understandingof addition and subtraction ofnumbers with answers to 1000(limited to 1-, 2- and 3-digitnumerals), concretely, pictorially
and symbolically, by: using personal strategies foradding and subtracting withand without the support ofmanipulatives creating and solving problemsin context that involve additionand subtraction of numbers.
[C, CN, ME, PS, R, V]
6. Demonstrate an understandingof multiplication (2- or 3-digit by1-digit) to solve problems by:
using personal strategies formultiplication with and withoutconcrete materials using arrays to representmultiplication connecting concreterepresentations to symbolicrepresentations
estimating products applying the distributiveproperty.
[C, CN, ME, PS, R, V]
7. Demonstrate an understandingof division (1-digit divisor andup to 2-digit dividend) to solveproblems by:
using personal strategies fordividing with and withoutconcrete materials
estimating quotients relating division tomultiplication.
[C, CN, ME, PS, R, V]
Number
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GRADE 3 MATHEMATICS CURRICULUM GUIDE - INTERIM 21
GENERAL AND SPECIFIC OUTCOMES BY STRAND
[C] Communication [PS] Problem Solving[CN] Connections [R] Reasoning[ME] Mental Mathematics [T] Technology and Estimation [V] Visualization
Grade 2 Grade 3 Grade 4
General OutcomeDevelop number sense.
General OutcomeDevelop number sense.
General OutcomeDevelop number sense.
Specific Outcomes Specific Outcomes Specific Outcomes
10. Apply mental mathematicsstrategies, such as:
counting on and countingback making 10 using doubles using addition to subtract
for basic addition facts and relatedsubtraction factsto 18.[C, CN, ME, PS, R, V]
10. Apply mental mathematicsstrategies and number properties,such as:
1. using doubles2. making 103. using addition to subtract4. using the commutativeproperty5. using the property of zero
for basic addition facts and relatedsubtraction facts to 18.
[C, CN, ME, PS, R, V]
8. Demonstrate an understandingof fractions less than or equal toone by using concrete, pictorialand symbolic representations to:
name and record fractions forthe parts of a whole or a set compare and order fractions model and explain that fordifferent wholes, two identicalfractions may not represent thesame quantity
provide examples of wherefractions are used.[C, CN, PS, R, V]
9. Represent and describedecimals (tenths and hundredths),concretely, pictorially andsymbolically.[C, CN, R, V]
Number
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GRADE 3 MATHEMATICS CURRICULUM GUIDE - INTERIM22
GENERAL AND SPECIFIC OUTCOMES BY STRAND
[C] Communication [PS] Problem Solving[CN] Connections [R] Reasoning[ME] Mental Mathematics [T] Technology and Estimation [V] Visualization
Grade 2 Grade 3 Grade 4
General OutcomeDevelop number sense.
General OutcomeDevelop number sense.
General OutcomeDevelop number sense.
Specific Outcomes Specific Outcomes Specific Outcomes
11. Demonstrate anunderstanding of multiplication to5 5 by:
representing and explainingmultiplication using equalgrouping and arrays creating and solving problemsin context that involvemultiplication modelling multiplicationusing concrete and visual
representations, and recordingthe process symbolically relating multiplication torepeated addition relating multiplication todivision.
[C, CN, PS, R]
10. Relate decimals to fractionsand fractions to decimals (tohundredths).[C, CN, R, V]
11. Demonstrate anunderstanding of addition andsubtraction of decimals (limited tohundredths) by:
using compatible numbers estimating sums and
differences using mental mathematicsstrategies
to solve problems.[C, ME, PS, R, V]
Number
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GENERAL AND SPECIFIC OUTCOMES BY STRAND
[C] Communication [PS] Problem Solving[CN] Connections [R] Reasoning[ME] Mental Mathematics [T] Technology and Estimation [V] Visualization
Grade 2 Grade 3 Grade 4
General OutcomeDevelop number sense.
General OutcomeDevelop number sense.
General OutcomeDevelop number sense.
Specific Outcomes Specific Outcomes Specific Outcomes
12. Demonstrate anunderstanding of division(limited to division related tomultiplication facts up to 5 5)by:
representing and explainingdivision using equal sharingand equal grouping creating and solving problemsin context that involve equalsharing and equal grouping modelling equal sharing andequal grouping using concreteand visual representations,and recording the processsymbolically relating division to repeatedsubtraction relating division tomultiplication.
[C, CN, PS, R]13. Demonstrate anunderstanding of fractions by:
explaining that a fractionrepresents a part of a whole describing situations in whichfractions are used comparing fractions ofthe same whole with likedenominators.
[C, CN, ME, R, V]
Number
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GRADE 3 MATHEMATICS CURRICULUM GUIDE - INTERIM24
GENERAL AND SPECIFIC OUTCOMES BY STRAND
[C] Communication [PS] Problem Solving[CN] Connections [R] Reasoning[ME] Mental Mathematics [T] Technology and Estimation [V] Visualization
Grade 2 Grade 3 Grade 4
General OutcomeUse patterns to describe the
world and to solve problems.
General OutcomeUse patterns to describe the
world and to solve problems.
General OutcomeUse patterns to describe the
world and to solve problems.
Specific Outcomes Specific Outcomes Specific Outcomes1. Demonstrate an understandingof repeating patterns (three to fiveelements) by:
describing extending comparing creating
patterns using manipulatives,diagrams, sounds and actions.[C, CN, PS, R, V]
2. Demonstrate an understandingof increasing patterns by:
describing reproducing extending creating
patterns using manipulatives,diagrams, sounds and actions(numbers to 100).[C, CN, PS, R, V]
1. Demonstrate an understandingof increasing patterns by:
describing extending comparing creating
patterns using manipulatives,diagrams, sounds and actions(numbers to 1000).[C, CN, PS, R, V]
2. Demonstrate an understandingof decreasing patterns by:
describing extending comparing creating
patterns using manipulatives,diagrams, sounds and actions(numbers to 1000).[C, CN, PS, R, V]
1. Identify and describe patternsfound in tables and charts,including a multiplication chart.[C, CN, PS, V]
2. Translate among differentrepresentations of a pattern, suchas a table, a chart or concretematerials.[C, CN, V]
3. Represent, describe and extendpatterns and relationships,using charts and tables, to solveproblems.[C, CN, PS, R, V]
4. Identify and explainmathematical relationships, usingcharts and diagrams, to solveproblems.[CN, PS, R, V
Patterns and Relations(Patterns)
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GRADE 3 MATHEMATICS CURRICULUM GUIDE - INTERIM 25
GENERAL AND SPECIFIC OUTCOMES BY STRAND
[C] Communication [PS] Problem Solving[CN] Connections [R] Reasoning[ME] Mental Mathematics [T] Technology and Estimation [V] Visualization
Grade 2 Grade 3 Grade 4
General OutcomeUse patterns to describe the
world and to solve problems.
General OutcomeUse patterns to describe the
world and to solve problems.
General OutcomeRepresent algebraic expressions inmultiple ways.
Specific Outcomes Specific Outcomes Specific Outcomes3. Demonstrate and explainthe meaning of equality andinequality by using manipulativesand diagrams(0 100)[C, CN, R, V]
4. Record equalities andinequalities symbolically, usingthe equal symbol or the not equal
symbol.[C, CN, R, V]
3. Solve one-step addition andsubtraction equations involvingsymbols representing an unknownnumber.[C, CN, PS, R, V]
5. Express a given problem as anequation in which a symbol isused to represent an unknownnumber.[CN, PS, R]
6. Solve one-step equationsinvolving a symbol to represent anunknown number.[C, CN, PS, R, V]
Patterns and Relations(Variables and Equations)
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GRADE 3 MATHEMATICS CURRICULUM GUIDE - INTERIM26
GENERAL AND SPECIFIC OUTCOMES BY STRAND
[C] Communication [PS] Problem Solving[CN] Connections [R] Reasoning[ME] Mental Mathematics [T] Technology and Estimation [V] Visualization
Grade 2 Grade 3 Grade 4
General OutcomeUse direct or indirectmeasurement to solve problems.
General OutcomeUse direct or indirectmeasurement to solve problems.
General OutcomeUse direct or indirectmeasurement to solve problems.
Specific Outcomes Specific Outcomes Specific Outcomes1. Relate the number of days to aweek and the number of monthsto a year in a problem-solvingcontext.[C, CN, PS, R]
2. Relate the size of a unit ofmeasure to the number of units(limited to nonstandard units)used to measure length and mass .
[C, CN, ME, R, V]
3. Compare and order objects bylength, height, distance aroundand mass, using nonstandardunits, and make statements ofcomparison.[C, CN, ME, R, V]
4. Measure length to the nearestnonstandard unit by: using multiple copies of a unit
using a single copy of a unit(iteration process).[C, ME, R, V]
1. Relate the passage of timeto common activities, usingnonstandard and standard units(minutes, hours, days, weeks,months, years).[CN, ME, R]
2. Relate the number of seconds toa minute, the number of minutesto an hour and the number of
days to a month, in a problemsolving context.[C, CN, PS, R, V]
3. Demonstrate an understandingof measuring length (cm, m) by:
selecting and justifyingreferents for the units cm andm modelling and describing therelationship between the unitscm and m
estimating length, usingreferents measuring and recordinglength, width and height.
[C, CN, ME, PS, R, V]
1. Read and record time, usingdigital and analog clocks,including 24-hour clocks.[C, CN, V]
2. Read and record calendar datesin a variety of formats.[C, V]
3. Demonstrate an understanding
of area of regular and irregular 2-D shapes by: recognizing that area ismeasured in square units selecting and justifyingreferents for the units cm2or m2
estimating area, using referentsfor cm2or m2
determining and recordingarea (cm2or m2) constructing differentrectangles for a given area (cm2
or m2) in order to demonstratethat many different rectanglesmay have the same area.
[C, CN, ME, PS, R, V]
Shape and Space(Measurement)
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GRADE 3 MATHEMATICS CURRICULUM GUIDE - INTERIM 27
GENERAL AND SPECIFIC OUTCOMES BY STRAND
[C] Communication [PS] Problem Solving[CN] Connections [R] Reasoning[ME] Mental Mathematics [T] Technology and Estimation [V] Visualization
Grade 2 Grade 3 Grade 4
General OutcomeUse direct or indirectmeasurement to solve problems.
General OutcomeUse direct or indirectmeasurement to solve problems.
General OutcomeUse direct or indirectmeasurement to solve problems.
Specific Outcomes Specific Outcomes Specific Outcomes5. Demonstrate that changingthe orientation of an object doesnot alter the measurements of itsattributes.[C, R, V]
4. Demonstrate an understandingof measuring mass (g, kg) by:
selecting and justifyingreferents for the units g and kg modelling and describing therelationship between the units gand kg estimating mass, usingreferents measuring and recording mass
[C, CN, ME, PS, R, V]
5. Demonstrate an understandingof perimeter of regular andirregular shapes by:
estimating perimeter, usingreferents for cm or m measuring and recordingperimeter (cm, m) constructing differentshapes for a given perimeter(cm, m) to demonstrate that
many shapes are possible for aperimeter.
[C, ME, PS, R, V]
Shape and Space(Measurement)
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GRADE 3 MATHEMATICS CURRICULUM GUIDE - INTERIM28
GENERAL AND SPECIFIC OUTCOMES BY STRAND
[C] Communication [PS] Problem Solving[CN] Connections [R] Reasoning[ME] Mental Mathematics [T] Technology and Estimation [V] Visualization
Grade 2 Grade 3 Grade 4
General OutcomeDescribe the characteristics of3-D objects and 2-D shapes, and
analyze the relationships amongthem.
General OutcomeDescribe the characteristics of3-D objects and 2-D shapes, and
analyze the relationships amongthem.
General OutcomeDescribe the characteristics of3-D objects and 2-D shapes, and
analyze the relationships amongthem.
Specific Outcomes Specific Outcomes Specific Outcomes
6. Sort 2-D shapes and 3-Dobjects, using two attributes, andexplain the sorting rule.[C, CN, R, V]
7. Describe, compare andconstruct 3-D objects, including:
cubes
spheres cones cylinders pyramids.
[C, CN, R, V]
8. Describe, compare andconstruct 2-D shapes, including:
triangles squares rectangles circles.
[C, CN, R, V]
9. Identify 2-D shapes as parts of3-D objects in the environment.[C, CN, R, V]
6. Describe 3-D objects accordingto the shape of the faces and thenumber of edges and vertices.[C, CN, PS, R, V]
7. Sort regular and irregularpolygons, including:
triangles
quadrilaterals pentagons hexagons octagons
according to the number of sides.[C, CN, R, V]
4. Describe and construct rightrectangular and right triangularprisms.[C, CN, R, V]
Shape and Space(3-D Objects and 2-D Shapes)
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GENERAL AND SPECIFIC OUTCOMES BY STRAND
[C] Communication [PS] Problem Solving[CN] Connections [R] Reasoning[ME] Mental Mathematics [T] Technology and Estimation [V] Visualization
Grade 2 Grade 3 Grade 4
General OutcomeDescribe and analyze positionand motion of objects and shapes.
General OutcomeDescribe and analyze positionand motion of objects and shapes.
General OutcomeDescribe and analyze positionand motion of objects and shapes.
Specific Outcomes Specific Outcomes Specific Outcomes5. Demonstrate an understandingof line symmetry by:
identifying symmetrical 2 Dshapes creating symmetrical2-D shapes drawing one or more lines ofsymmetry in a 2-D shape.
[C, CN, V]
6. Demonstrate an understandingof congruency, concretely andpictorially.[CN, R, V]
Shape and Space(Transformations)
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GENERAL AND SPECIFIC OUTCOMES BY STRAND
[C] Communication [PS] Problem Solving[CN] Connections [R] Reasoning[ME] Mental Mathematics [T] Technology and Estimation [V] Visualization
Grade 2 Grade 3 Grade 4
General OutcomeCollect, display and analyze datato solve problems.
General OutcomeCollect, display and analyze datato solve problems.
General OutcomeCollect, display and analyze datato solve problems.
Specific Outcomes Specific Outcomes Specific Outcomes
1. Gather and record dataabout self and others to answerquestions.[C, CN, PS, V]
2. Construct and interpretconcrete graphs and pictographsto solve problems.[C, CN, PS, R, V]
1. Collect first-hand data andorganize it using:
tally marks line plots charts lists
to answer questions.[C, CN, PS, V]
2. Construct, label and interpretbar graphs to solve problems.[C, PS, R, V]
1. Demonstrate an understandingof many-to-one correspondence.[C, R, T, V]
2. Construct and interpretpictographs and bar graphsinvolving many-to-onecorrespondence to draw
conclusions.[C, PS, R, V]
Statistics and Probability(Data Analysis)
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31
Patterning
Suggested Time: 3 Weeks
1
2
This is the first explicit focus on Patterning in Grade 3 but, as with other
outcomes, it is ongoing throughout the year.
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PATTERNING
Unit Overview
Focus and Context
Math Connects
In Grade 3, students continue working with increasing patterns. They
build on what they have learned in Grade 2 by communicating theirunderstanding of increasing patterns and by representing increasing
patterns in a variety of ways: concretely, pictorially and symbolically.
Students verbalize and communicate rules to help them understand the
predictability of a pattern. A large focus in Grade 3 is the introduction
and development of decreasing patterns. Students use their knowledge
of increasing patterns to make connections to the concept of decreasing
patterns, since similar understandings are developed. These patterning
concepts are the basis for further algebraic thinking and will be
extended in later grades.
It is important that students see the connection between increasingand decreasing patterns. Many opportunities should be provided forthem to connect both types of patterns. Since increasing and decreasingpatterns introduce students to a higher level of algebraic thinking,students will also make connections to the patterns embedded in otherstrands of mathematics.
Historically, much of the mathematics used today was developedto model real-world situations, with the goal of making predictionsabout those situations. As patterns are identified, they can be expressednumerically, graphically, or symbolically and used to predict howthe pattern will continue. It is important that students identify thatpatterns exist all around us. Viewing and discussing patterns in a realworld context creates authentic experiences for patterning concepts tobe applied and developed. Identifying patterns in the yearly/monthlycalendar, house numbers and money can be used to solve real lifeproblems.
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PATTERNING
33
STRAND OUTCOME PROCESSSTANDARDS
Patterns andRelations(Patterns)
3PR1 Demonstratean understanding ofincreasing patterns by:
describing
extending
comparing
creating
patterns usingmanipulatives,diagrams, sounds andactions (numbers to1000).
[C, CN, PS, R, V]
Patterns andRelations(Patterns)
3PR2 Demonstratean understanding ofdecreasing patterns by:
describing
extending
comparing
creating
patterns usingmanipulatives,diagrams, sounds and
actions (numbers to1000).
[C, CN, PS, R, V]
Process Standards
Key
Curriculum
Outcomes
[C] Communication [PS] Problem Solving
[CN] Connections [R] Reasoning
[ME] Mental Mathematics [T] Technology
and Estimation [V] Visualization
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34 GRADE 3 MATHEMATICS CURRICULUM GUIDE - INTERIM
Outcomes
PATTERNING
ElaborationsStrategies for Learning and Teaching
Students will be expected to
Strand: Patterns and Relations (Patterns)
3PR1 Demonstrate anunderstanding of increasingpatterns by:
describing
extending
comparing
creating
patterns using manipulatives,diagrams, sounds and actionsand numbers to 1000.
[C, CN, PS, R, V]
In Grade 2, students described, extended, compared and createdrepeating patterns and increasing patterns. Grade 3 students will review
and learn more about increasing shape/number patterns, as well as
explore decreasing patterns. They will begin with building patterns and
talking about them in a logical step-by-step process. Increasing patterns
are sometimes referred to as growing patterns a pattern where the size
of the elements increase in a predictable way. An element is any single
item or step of a pattern. E.g.
28, 31, 34, 37... the pattern begins at 28 and increases by 3. Each
number in the pattern is an element.
... in this example, each figure (group of
triangles) is an element
It is common for students to confuse a repeating pattern with an
increasing or decreasing pattern. Increasing and decreasing patterns do
not have a core. Students will be familiar with the mathematical term
core from working with repeating patterns in Grade 2. Ask students to
look for a core first. The core is the shortest part of the pattern that
repeats. If they cannot find a core, then the pattern is not a repeating
pattern and it must be an increasing or decreasing pattern.
Students need sufficient time to explore increasing patterns throughvarious manipulatives, such as link-its, tiles, flat toothpicks, counters,
pattern blocks, base ten blocks, bread tags, stickers, buttons, etc., to
realize they increase or decrease in a predictable way. Later, students
will connect patterns to numbers, and work with patterns found in the
hundreds chart or record patterns in a T-chart.Achievement Indicator:
3PR1.1 Describe a given
increasing pattern by stating a
pattern rule that includes the
starting point and a description
of how the pattern continues; e.g.,for 42, 44, 46 the pattern
rule is start at 42 and add 2 each
time.
Give students the first three or four
elements of an increasing pattern.
Ask them to determine the pattern
rule and explain how the pattern
continues. A pattern rule tells howto make the pattern and can be
used to extend an increasing or
decreasing pattern. Both have a
starting point and a change that
happens each time.
(continued)
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35GRADE 3 MATHEMATICS CURRICULUM GUIDE - INTERIM
Suggested Assessment Strategies Resources/Notes
PATTERNING
General Outcome: Use Patterns to Describe the World and to Solve Problems
PortfolioAsk students to complete a concept map. Use it to inform instructionby determining what students already know about growing patterns.Note any misconceptions to clarify throughout the unit.
Ask students to place the concept map in a portfolio. After further
instruction, ask students to complete another allowing them the
opportunity to compare it to the first. This allows students to assess
their own development. This strategy can also be used to determinetheir growth and understanding of other concepts. (3PR1)
Performance/Student-Teacher Dialogue
Calculator Activity In Grade 3, students can benefit fromexperiences working with calculators and examining patterns. Askstudents press 0 on a calculator. Ask them to select a number from 1to 9. E.g., 3. Press + followed by 3, then press =. The calculator willadd 3 to the previous sum. Record the number displayed. Press =again. Record the new number. Continue pressing = and recording
the new number displayed. After several entries, ask the students topredict the next few numbers. Ask: What are some other numbersthat are and are not part of the Add 3 pattern? Is there a rule wecan use to predict the numbers? If so, give the rule. Ask students toexplore several different numbers from 1 to 9 and see what happensif they start with 0 and then continue to add the chosen number.(Navigating through Algebra in Grades 3-5, 2001, p. 15) (3PR1.1)
Math Makes Sense 3Launch
TG pp. 2 3
Lesson 1: Exploring Increasing
Patterns
3PR1
TG pp. 4 6
Additional Activities:
Missing Figures
TG p. ix and 41
Game:Whats the Pattern Rule?
TG pp.18
This game may be used repeatedly
during this unit as extra practice to
reinforce 3PR1
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36 GRADE 3 MATHEMATICS CURRICULUM GUIDE - INTERIM
Outcomes
PATTERNING
ElaborationsStrategies for Learning and Teaching
Students will be expected to
Strand: Patterns and Relations (Patterns)
3PR1 Continued As students describe increasing shape patterns, help them recognize thateach element has a numeric value. E.g.
Other numeric patterns include:
2, 4, 8, 16, The pattern rule is: Start at 2. Double each time.
3, 4, 6, 9, 13, The pattern
rule is: Start at 3. Add 1 and
increase the number added
by 1 more each time.
103, 108, 113, 118, 123, The pattern rule is: Start at 103. Add 5
each time.
Note: A pattern rule must have a starting point. E.g., if a student
describes the pattern 3, 7, 11, 15, as an add 4 pattern without
indicating that it starts at 3, the pattern rule is incomplete.
Leaping Lizards - Take students to an open area such as a gym orplayground to jump like Leaping Lizards while skip-counting an
increasing pattern. They jump 8 times in a row, stop to feel their hearts
beating, then jump 8 more times.
Name the Rule - Tell students the following story: On Earth Day, Mr.
Hann and his students planted a vegetable garden in the school yard.
He put 2 plants in the first section, 4 in the second section and 6 in the
third section. Ask students to:
create the pattern with blocksdescribe the rulepredict what comes nextextend the pattern. E.g., How many plants will they put in thetenth section?
Achievement Indicator:
3PR1.1 Continued
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37GRADE 3 MATHEMATICS CURRICULUM GUIDE - INTERIM
Suggested Assessment Strategies Resources/Notes
PATTERNING
General Outcome: Use Patterns to Describe the World and to Solve Problems
Performance10-frame Patterns build an increasing pattern by placing counterson the 10-frame and ask students to identify how the pattern is
growing. E.g., Pattern 5, 10, 15, 20
These 10-frames show that the numbers increase by 5 because
another full row of 5 is filled each time.
(3PR1.1)Headband Guess my Pattern - Students play with a partner. Oneplayer will wear a headband with a number pattern strip pickedfrom a bag. The player wearing the headband cannot see the numberpattern but must ask his/her partner questions to figure out thepattern. They must ask questions to find out the starting number,the pattern rule, and a missing term or three additional terms. (Or
the start number can be given.) Examples of questions students
might ask their partner:
Does the pattern start with an even or odd number?
Is it a multiple of 10?
Does it have 1 digit, 2 digits, 3 digits?Is it greater than 10?
Is the pattern increasing? Or decreasing?
Is the rule (add or subtract) by 2, 5, 6, etc.
Does the pattern increase by 5s?
Does it increase by more (less) than 5?
(3PR1.1)
Paper and Pencil
Give students a number pattern and ask them to write the patternrule. Check that students have included a starting point and how the
pattern continues. (3PR1.1)
Math Makes Sense 3Lesson 1 (Contd): Exploring
Increasing Patterns
3PR1
TG pp. 4 6
Additional Reading:
Navigating through Algebra in
Grades 3-5( 2001)
Small, Marion (2008)Making
Math Meaningful for Canadian
Students K-8. Chapter 20.
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Outcomes
PATTERNING
ElaborationsStrategies for Learning and Teaching
Students will be expected to
Strand: Patterns and Relations (Patterns)
3PR1 Continued
Students are given the beginning of a pattern (at least three elements)
then asked to extend the pattern by three more elements. They should
always look backward to the beginning of the pattern to see that their
idea works for the rest of their pattern.
50, 100, 150, 200,
6, 13, 20, 27,
5, 8, 12, 17,
2, 2, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6,
Ask students to work on word problems in pairs:
1. Ms. Mercers class planted a special seed. On Monday the plantis 2 cm high. On Tuesday the plant has doubled its height and is 4
cm high. Each day the plant doubles its height from the day before.
How high will the plant be on Friday? Students can make a table
showing each day of the week and how tall the plant is on each day
(they can also use manipulatives to make the pattern.)
2. Lilys new puppy, Pokey, is growing fast. When Lily first got
Pokey he weighed only 1 kg. After 1 month Pokey weighed 7
kg. After 2 months, Pokey weighed 12 kg. After 3 months Pokey
weighed 16 kg. Lily saw a pattern. Find a pattern to tell how much
Pokey weighed after 5 months.
Ask students to complete a table like the one below:
Pattern: Start at 2 kg. Add 4 kg and then 1 kg less each time.
Achievement Indicator:
3PR1.2 Identify the pattern rule
of a given increasing pattern, and
extend the pattern for the next
three elements.
1st element 2nd element3rd element
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Suggested Assessment Strategies Resources/Notes
PATTERNING
General Outcome: Use Patterns to Describe the World and to Solve Problems
JournalProvide students with a choice of3 increasing patterns. Students areasked to choose one pattern to extendfor the next three elements andexplain the rule.
(3PR1.2)
Paper and Pencil
Extending Patterns Ask students to complete a chart similar to theone below:
Ask them to extend each pattern three times and record each numberpattern. (3PR1.2)
Portfolio
Provide 1 cm grid paper for the students. Present a pattern such asthe one below. Students will use coloured pencils to continue thepattern. Next, ask the students to create their own growing patterns.
(3PR1.2)
Ask students: How many tiles are needed to make the next 3 figures?
( 3PR1.2)
Math Makes Sense 3Lesson 1 (Contd): Exploring
Increasing Patterns
3PR1
TG pp. 4 6
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Outcomes
PATTERNING
ElaborationsStrategies for Learning and Teaching
Students will be expected to
Strand: Patterns and Relations (Patterns)
3PR1 Continued
Students are provided with a variety of increasing patterns which
contain errors. Students determine what the pattern is and then explain
the error. E.g., 3, 7, 11, 15, 19, 23, 26, 31, 35, 39. The pattern rule
is: Start at 3. Add 4 each time.
Therefore, 26 is an error since it
is only adding on 3 not 4 and 31
is a second error since it is adding
5 and not 4.
Hint: To help students visualize this pattern they can shade numbers ona hundreds chart and look for the mistake:
Students can see that 26 does not fit the number pattern. It is an error.
In the following example, the shape pattern rule is: Start with 1 counter.
Add 1 to each row and column each time.
Therefore the fourth element is a error. There should be 4 counters inthe column not 3.
3PR1.3 Identify and explain
errors in a given increasing
pattern.
Achievement Indicator:
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Suggested Assessment Strategies Resources/Notes
PATTERNING
General Outcome: Use Patterns to Describe the World and to Solve Problems
PerformanceProvide the start of an increasing pattern. Ask studenta to continuethe pattern for the next 3 elements and to describe the pattern
rule. (3PR1.2)
Paper and Pencil
Give students number patterns such as those below and ask them tofind and circle the error.
475, 575, 685, 775
233, 243, 253, 262
25, 28, 32, 34
7, 12, 15, 19 (3PR1.3)
Journal
Present students with the following growing pattern. Ask them to
find the error and explain how they know.
(3PR1.3)
Math Makes Sense 3Lesson 1 (Contd): Exploring
Increasing Patterns
3PR1
TG pp. 4 6
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Outcomes
PATTERNING
ElaborationsStrategies for Learning and Teaching
Students will be expected to
Strand: Patterns and Relations (Patterns)
3PR1 Continued
Since patterns increase in a predictable way, to determine a missing step
students will look at the pattern that comes before and after. They must
identify the pattern rule.
15, 26, 37, 48, ___, 70, 81 Start at 15. Add 11 each time.
5, 6, 8, 11, ___, 20, 26, 33, 41 Start at 5. Add 1, and then increase the
number added by 1 more each time.
13, 26, ___, 52, 65, 78, 91 Start at 13. Add 13 each time.
Ask students to practice finding missing elements by making patterns,
covering a step and asking a partner Whats missing?
Literature Connection - Read the following Skip Count Cheerleaders
chants from Riddle-iculous MATHby Joan Holub. Ask students to fill in
the missing element as they chant:
2, 4, 6, 8,
Who do we appreciate?
8, 10, 12, _?_
Our soccer coach, Ms. Morteen.
5, 10, 15, 20,
Who do we all like, and plenty?
20, 25, 30, _?_
Our lunch lady, Mrs. Dive.
20, 30, 40, 50,
Who do we all think is nifty?
50, 60, 70, _?_
Our principal, Mr. Grady.
Achievement Indicator:
3PR1.4 Identify and apply a
pattern rule to determine missing
elements for a given pattern.
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Suggested Assessment Strategies Resources/Notes
PATTERNING
General Outcome: Use Patterns to Describe the World and to Solve Problems
PerformancePattern BANG! Have a variety of cards in a paper bag, such as theexamples below:
Include 1 BANG card for every 4 or 5 question cards.
Give each small group a bag. Students take turns drawing a card out
and answering the question. If the student answers correctly, she/he
gets to keep the card, (group members can help each other with the
answer). They then pass the bag to the next player. If a student pulls
out a BANG card, she/he must put all of her/his cards back into the
bag (leaving the BANG card out). They continue playing until there
are no cards left in the bag and whoever has the most cards wins.
(3PR1.1, 3PR1.2, 3PR1.3, 3PR1.4)
Ask each student to make a growing pattern using manipulatives.Next she/he covers one element of the pattern to reveal it to aclassmate. The classmate will then recreate the pattern putting inthe missing element. The initial pattern is uncovered and the twopatterns compared.
(3PR1.8, 3PR1.4)
Paper and Pencil
In pairs, ask students to make up their own chants and riddles which
include including a missing element, to put into a class Riddle-iculousMATHbook.
(3PR1.4)
Math Makes Sense 3Lesson 1 (Contd): Exploring
Increasing Patterns
3PR1
TG pp. 4 6
Childrens Literature(not
provided):
Holub, Joan. Riddle-iculous Math
ISBN: 9780807549964
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Outcomes
PATTERNING
ElaborationsStrategies for Learning and Teaching
Students will be expected to
Strand: Patterns and Relations (Patterns)
3PR1 Continued
Students identify the pattern rule and then describe how they discovered
that rule. E.g., 3, 6, ___, 12, 15
The rule is: Start at 3. Add 3 each time.
Possible strategies to determine missing elements include use of:
Number lines
Hundreds chart
Pictures
Manipulatives
Skip counting
It is important to accept other possible strategies that students use and
to discuss them.
Achievement Indicator:
3PR1.5 Describe the strategy used
to determine missing elements in
a given increasing pattern.
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Suggested Assessment Strategies Resources/Notes
PATTERNING
General Outcome: Use Patterns to Describe the World and to Solve Problems
Paper and Pencil/PortfolioWanted Poster students will make wanted posters, asking readers tofind the missing element of an increasing pattern. See sample below:
Students create a number
or shape pattern, leaving
one element out. They
will include a hint flap,
which tells the pattern
rule for those who need
to use it. Also, they will
create a pull tab, which
will give the missingelement so that the
detectives can check to
see if they are correct.
Note: To make pull
tab, tape half of an
envelope to the back of
the poster for sliding
the tab card in and out
(be sure the tab is longer than the envelope).
(3PR1.5, 3PR1.4)Student-Teacher Dialogue
Where is the Birthday Party today? (Can be included in MorningRoutine). Present students with a pattern of numbers on a displayof houses. Add six extra houses with no number. Ask students to tellwhat the pattern is. Tell students that the party will be at a certainhouse (pick an extended number from the pattern). Ask students topick out the location of the house and describe the strategy (pattern
rule) they used. Examples of streets could be:
Street # 170, 180, 190, House 4, House 5, House 6 Rule: Start at170. Go up by 10 each time. The party is at house number 200.
Street # 31, 36, 41, House 4, House 5, House 6 . Rule: Start at31. Increase by 5 each time. We are looking for the house whichwould have the street number 61.Street # 101, 104, 107, House 4, House 5, House 6. Rule: Startat 101. Increase by 3 each time. The party is at house number116. (3PR1.1, 3PR1.5, 3PR1.9)
Math Makes Sense 3Lesson 1 (Contd): Exploring
Increasing Patterns
3PR1
TG pp. 4 6
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Outcomes
PATTERNING
ElaborationsStrategies for Learning and Teaching
Students will be expected to
Strand: Patterns and Relations (Patterns)
3PR1 ContinuedAchievement Indicator:
3PR1.6 Create a concrete,
pictorial or symbolic
representation of an increasing
pattern for a given pattern rule.
Give students various pattern rules to create their own model, picture or
number representation.
To represent concretely they can choose from a variety of manipulatives
(such as pattern blocks, coins or buttons) or they may choose to draw a
picture or use numbers. E.g.,
Start at 2 and double each time.
2, 4, 8, 16,
Examples of other increasing number patterns include:
1, 2, 2, 3, 3, 3, each digit repeats according to its value
2, 4, 6, 8, 10, even numbers skip counting by 2
1, 2, 4, 8, 16, double the previous number
2, 5, 11, 23, double the previous number and add 1
1, 2, 4, 7, 11, 16, successively add 1, then 2, then 3, and so on
2, 2, 4, 6, 10, 16, add the preceding two numbers
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Suggested Assessment Strategies Resources/Notes
PATTERNING
General Outcome: Use Patterns to Describe the World and to Solve Problems
PerformanceStaircase Give students the first 3 frames of a staircase pattern (seebelow). Ask them to use square tiles pattern blocks, base-ten units,or multi-link cubes to build the next three frames of the staircasepattern. Students then predict what each frame will look like beforethey build it.
A growing pattern can be recorded in a table. This allows students
to see the relationship between a concrete/pictorial pattern and the
corresponding number pattern. Ask students make a table and recordthe number of frames, the number of squares added each time and
the number of squares in each frame.
(3PR1.6, 3PR1.7)
Math Makes Sense 3Lesson 2: Exploring Increasing
Patterns
3PR1
TG pp. 7 9
Childrens Literature (not
provided):
Hutchins, Pat. The Doorbell Rang
ISBN: 0688092349
Anno, Mitsumasa.Annos Magic
Seeds
ISBN: 9780698116184
Crews, Donald. Ten Black Dots
ISBN: 978-0688135744
Hong, Lily Toy. Two of Everything
ISBN: 978-0807581575
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Outcomes
PATTERNING
ElaborationsStrategies for Learning and Teaching
Students will be expected to
Strand: Patterns and Relations (Patterns)
3PR1 ContinuedAchievement Indicator:
3PR1.7 Create a concrete,
pictorial or symbolic increasing
pattern; and describe the
relationship, using a pattern rule.
Students may use base ten blocks to concretely create an increasing
pattern with larger numbers. For example,
Pattern Rule: Start at 222. Add 10 each time.
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Suggested Assessment Strategies Resources/Notes
PATTERNING
General Outcome: Use Patterns to Describe the World and to Solve Problems
Performance/Paper and PencilStudents will build a concrete pattern of their choice using objectssuch as pattern blocks, square tiles, base ten blocks, buttons, coins,etc. Next, ask students to create a flip book with each page slightlybigger than the next (stapling the smallest page on the top andlargest page on the bottom). Students draw the increasing patternon the top of each page and label the bottom of each page withthe correct numeric value. When the book is closed the numberpattern will be visible and as they open each page the picture will be
revealed.The last page of the book reveals the pattern rule. Extendthe above activity by having students exchange their flip book with
a partner. Each student will then use manipulatives to concretelycreate the number pattern represented on the outside flaps. Then askstudents to describe to their partner the pattern rule before checkingthe last page. Observe students concrete representations and ability
to describe the pattern rule to each other.
(3PR1.7, 3PR1.8)
Math Makes Sense 3Lesson 2 (Contd): Exploring
Increasing Patterns
3PR1
TG pp. 7 9
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Outcomes
PATTERNING
ElaborationsStrategies for Learning and Teaching
Students will be expected to
Strand: Patterns and Relations (Patterns)
3PR1 Continued
Students should have frequent experiences with solving real-world
problems that interest and challenge them. Ask students to solve the
following problems:
Carrie buys Yummy cat food for her cat, Cleo. One can of Yummycosts 15. How many cans can she buy for 90?
Complete a table
Robert decided to count