grade 4 mathematics curriculum guide - government of

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.
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.
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.
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.
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.
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.
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).
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.
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.
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).
• 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.
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.
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.
• 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.
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.
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.
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
Number (continued)
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]
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]
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]
Number (continued)
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]
Number (continued)
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]
Number (continued)
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]
Number (continued)
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]
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]
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]
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.
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
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.
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]
Shape and Space (Measurement) (continued)
Grade 3 Grade 4 Grade 5 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).
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]
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]
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]
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]
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
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
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
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
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
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
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
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)
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,

grade 4 mathematics curriculum guide - government of

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