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Geometry and measurement |
• How do I analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships?
• How do I specify locations and describe spatial relationships using coordinate geometry and other representational systems?
• How do I apply transformations and use symmetry to analyze mathematical situations?
• How do I use visualization, spatial reasoning, and geometric modeling to solve problems?
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*Identify, compare, and analyze attributes of two- and three-dimensional shapes and develop vocabulary to describe the attributes
*Classify two- and three-dimensional shapes according to their properties and develop definitions of classes of shapes such as triangles and pyramids
*Investigate, describe, and reason about the results of subdividing, combining, and transforming shapes
*Explore congruence and similarity
*Make and test conjectures about geometric properties and relationships and develop logical arguments to justify conclusions
*Seeing 3-D rectangular arrays of cubes in terms of congruent layers
*Developing, using, describing, and justifying methods of determining volume
*Exploring volume relationships among different containers and among solids, particularly those with the same base and height
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*Determine the area of foot
*Create a candy box with set variables for sides (volume)
*Design a garden plot
*Show knowledge of how to use a formula to determine area and perimeter
*Figure out circumference and area of a circle using multiple methods
*Label parts of a circle
*Measure various circles and apply formulas for solution of circumference, diameter, and area
*Measure classroom and personal bedroom to determine area, perimeter and volume
*Write and evaluation geometric explorations
*Evaluate geometric procedures
-Applying multiplication to find the number of cubes in the box.
-Determining how many cubes fit in a rectangular box
-Seeing and using cubic centimeters as a unit for measuring volume (including nonrectangular solids)
-Using geometric solids to design models and to determine their volume |
*Informal observation of student work
*Assessment of student constructions, writing about unit, and evaluation of methods for solutions
*Participation in class discussions
*One-to-one work with students
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Investigations: Picturing Polygons, Containers and Cubes, Measurement Benchmarks
Marilyn Burns
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Number and Operations: Multiplication |
• How do I understand meanings of operations and how they relate to one another?
• Can I compute fluently and make reasonable estimates?
• Have I mastered my multiplication facts?
• Can I solve multiplication problems using multiple strategies? |
*Identify and use relationships between operations, such as division as the inverse of multiplication, to solve problems
*Develop fluency with basic number combinations for multiplication and division and use these combinations to mentally compute related problems
*Develop and use strategies to estimate computations involving decimals in situations relevant to students' experience, including money
*Review of multiplication facts through the 12's tables
*Multiplication by 10's, 100's, 1000's
*Double-digit and multi-digit multiplication
*Strategies for determining factors, multiples, prime and composite numbers
*Exploration of different ways of multiplying (lattice, expanded, standard algorithm)
*Estimation
*Exploration of arrays |
*Use manipulative to express a multiplication problem
*Demonstrate knowledge of when to use multiplication to solve a problem
*Write multiples of a number
*Solve multiplication problems from single digit through multi-digit problems
*Demonstrate understanding of the connection between patterns and multiplication
*Solve and create multiplication word problems
*Estimate the product of a problem
*Demonstrate knowledge of multiple strategies
*Use games and flash cards to reinforce multiplication facts
*Construct multiplication grid |
*Informal observation of student work
*Multiplication think board
*Assessment of daily work
*Pre-test, mid-assessment and unit test
*Participation in class discussions
*One-to-one work with students
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Investigations: Mathematical Thinking at Grade Five, Building on Numbers You Know
First Steps Math
Everyday Math
Flash cards
Multiplication Games |
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Algebra and Patterns |
• How do I understand patterns, relations, and functions?
• How can I represent and analyze mathematical situations and structures using algebraic symbols?
• How can I use mathematical models to represent and understand quantitative relationships?
• How can I analyze change in various contexts?
• How can I use an understanding of patterns to solve problems?
• How do I identify and extend a pattern? |
*Describe, extend, and make generalizations about geometric and numeric patterns
*Represent and analyze patterns and functions, using words, tables, and graphs
*Identify such properties as communitivity, associativity, and distributivity and use them to compute with whole numbers
*Represent the idea of a variable as an unknown quantity using a letter or a symbol
*Express mathematical relationships using equations
*Model problem situations with objects and use representations such as graphs, tables, and equations to draw conclusions
*Investigate how a change in one variable relates to a change in a second variable
*Identify and describe situations with constant or varying rates of change and compare them
*Growth patterns (Pascal's triangle, Fibonacci)
*Fixed patterns
*Patterns within a hundred's chart
*Skills for assessing patterns
*Numeric representation of patterns
*Patterns within patterns
*Distinction between growing patterns and fixed patterns |
*Extend a visual, growing pattern
*Create a pattern, present with a riddle to give clues to its pattern
*Analyze, identify, and describe numerical and visual patterns
*Use generalizations about patterns to make prediction
*Use patterns and functions to solve problems
*Explore tile patterns
*Analyze and chart tile patterns
*Build designs that change in a regular way
build designs that grow according to number patterns
*Predict later steps of number patterns and designs
*Make tables and graphs to display number patterns
*Investigate changes in the number of new tiles and the total number of tiles
*Use the language of speed and motion to describe number patterns |
*Informal observation of student work
*Pre-assessment, post-assessment
*Homework
*Participation in class discussions
*One-to-one work with students |
Investigations: Patterns of Change
Marilyn Burns' units on patterning
Pattern blocks
Colored squares |
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Measurement |
• How do I understand measurable attributes of objects and the units, systems, and processes of measurement?
• How do I apply appropriate techniques, tools, and formulas to determine measurements? |
*Attributes such as length, area, weight, and size of angle and select the appropriate type of unit for measuring each attribute
*The need for measuring with standard units and become familiar with standard units in the customary and metric systems
*Understand how to carry out simple unit conversions, such as from centimeters to meters, within a system of measurement
*Understand that measurements are approximations and how differences in units affect precision
*Understand what happens to measurements of a two-dimensional shape such as its perimeter and area when the shape is changed in some way
*Understand how to estimate the perimeters, areas, and volumes of irregular shapes
*Understand how to select and apply appropriate standard units and tools to measure length, area, weight, time, temperature, and the size of angles
*Understand how to select and use benchmarks to estimate measurements
*Understand how to develop, understand, and use formulas to find the area of rectangles and related triangles and parallelograms
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*Use U.S. standard and metric tools for measuring length, weight, volume and time
*Find and use benchmarks to estimate measures
*Determine when precise measurement is required and when estimates are good enough
*Recognize and explain possible sources of measurement error
*Compare lengths expressed in different ways, such as meters and centimeters, meters and decimal fractions of a meter, and meters and fractions of a meter
*Keep track of and calculate total measurements
*Estimate and measure using non-standard metric and standard units: measure the classroom (windows, surfaces, doors, furniture, steps)
*Use information from classroom measurement to create a scale map of classroom
*Create a treasure map in an imaginary place using set criteria |
*Informal observation of student work
*Assessment of classwork
*Participation in class discussions
*Evaluation of collaborative work in partnership
*One-to-one work with students |
Investigations: Measurement Benchmarks
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Number and Operations: Division |
• How do I understand numbers, ways of representing numbers, relationships among numbers, and number systems?
• How do I understand meanings of operations and how they relate to one another?
• Can I compute fluently and make reasonable estimates?
• Can I master long division problems?
• When do I use division?
• How do I know when a problem is correct?
• How can I remember the steps in a long division problems?
• What are the parts of division problem? |
*Division as the inverse of multiplication
*Parts of a division problem (divisor, dividend, quotient)
*Procedures for solving a division problem
*Pneumonic devices for remembering steps in a division problem (DMSB)
*Division with multiples of ten
*Use of zero as place keeper in the quotient
*Standard algorithm for division
*Problems with single digit divisors through multi-digit divisors with/without remainders
*Remainders expressed in whole numbers, fractions, and decimals
*Estimation |
*Represent division problems using tile squares
*Graph paper representation of division problems (Base 10)
*Demonstrate knowledge of the connection between multiplication and division (piece X piece = whole, whole/piece = piece)
*Label parts of a division problem
*Use standard algorithm to solve a division problem
*Estimate the answer of division problem
*Write remainder of a division problem in three ways
*Use multiplication to check a division problem
*Solve multi-divisor division problems
*Solve and create division word problems |
*Informal observation of student work
*Assessment of classwork
*Participation in class discussions
*Evaluation of collaborative work in partnership
*One-to-one work with students |
Investigations: Building on Numbers You know
Marilyn Burns
Krypto
Heath Math series
Visual Mathematics
Scott Foresman series
Everyday Math |
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Number and Operations: Fractions, Decimals and Percents |
• How can I describe a portion of a group that shares a characteristic?
• How can I find fraction and percent equivalents?
• How can I interpret and make sense of common uses of percents?
• How can I understand, interpret and use unit fractions?
• What are the parts of a fraction?
• How do I add, subtract, multiply and divide fractions?
• When I divide a fraction why do the fractional pieces change in size?
• How do I compare fractions, decimals and percents?
• How do division and fractions relate?
• How can I make sense of word problems involving fractions, decimals, and/or percents? |
*Label parts of a fraction (numerator and denominator)
*Comparison of fractions (greater than, less than, cross multiplication for comparison, circle comparison)
*Equivalent fractions
*Expression of fractions as division problem
*Expression of a fraction as parts of a whole
*Qualities of denominator and numerator
*Addition and subtraction of common and uncommon fractions
*Multiplication and division of fractions by fractions, whole numbers and mixed numbers
*Estimation
*Mixed numbers and improper fractions
*Understanding of multiplication as a fraction of a fraction (e.g. 2/3 of 3/5)
*Lowest terms
*Least common denominators
*Relationship between fractions, decimals and percents
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*Use fractions to describe how many in a group share a particular characteristic
*Find equivalent fractions
*Represent, identify and order fractions and percents, using 1/2 and 1 as references
*Interpret common uses of percent
*Build on knowledge of unit fractions to use fractions with numerators greater than 1
*Represent fractions as rotation around a circle
*Mark strips into fractional parts
*Find equivalent fractions
*Order fractions
*Add fractions
*Solve addition and subtraction problems with common/uncommon denominators
*Solve multiplication problems with whole and mixed numbers
*Solve and create word problems using fractions
*Simplify answers
*Convert improper fractions to mixed numbers and vice versa
*Represent and add decimals on grids
*Read and write decimals
*Order decimals
*Divide to find decimal equivalents of fractions
*Plan and conduct surveys
*Organize and represent data as fractions, percents and in circle graphs
*Interpret common uses of fractions, decimals and percents
*Understand equivalencies between fractions, decimals and percents
*Convert fractions, decimals and percents
*Create multiple types of graph data using percents |
*Students make the connection between manipulative and algorithm
*Informal observation of student work
*Analysis of student constructions
*Assessment of completed fraction sheets, pre-test, and unit test
*Participation in class discussions
*One-to-one work with students |
Investigations: Name That Portion
Fraction bars, fractional pieces
Food
Fraction puzzles
Marilyn Burns
Visual math
Scott Foresman |
