MATHEMATICS PROCESS STANDARDS Grades 1-5 The National Council of Teachers of Mathematics (NCTM) has identified five process standards: Problem Solving, Communication, Reasoning and Proof, Connections, and Representation. Using these processes students are actively involved in deepening mathematical understandings which lead to increasingly sophisticated abilities required to meet mathematical challenges. Following is an outline of the five process standards and associated objectives. NOTE: When examples are given there is a progression in levels of difficulty from basic to more complex skills. Process Standard 1: Problem Solving 1. Use problem-solving approaches (e.g., act out situations, represent problems with drawings and lists, use concrete, pictorial, graphical, oral, written, and/or algebraic models, understand a problem, devise a plan, carry out the plan, look back). 2. Formulate problems from everyday and mathematical situations (e.g., how many forks are needed?, how many students are absent?, how can we share/divide these cookies?, how many different ways can we find to compare these fractions?). 3. Develop, test, and apply strategies to solve a variety of routine and nonroutine problems (e.g., look for patterns, make a table, make a problem simpler, process of elimination, trial and error). 4. Verify and interpret results with respect to the original problem (e.g., students explain verbally why an answer makes sense, explain in a written format why an answer makes sense, verify the validity of each step taken to obtain a final result). 5. Distinguish between necessary and irrelevant information in solving problems (e.g., play games and discuss “best” clues, write riddles with sufficient information, identify unnecessary information in written story problems). Process Standard 2: Communication 1. Express mathematical ideas coherently and clearly to peers, teachers, and others (e.g., with verbal ideas, models or manipulatives, pictures, or symbols). 2. Extend mathematical knowledge by considering the thinking and strategies of others (e.g., agree or disagree, rephrase another student’s explanation, analyze another student’s explanation). 3. Relate manipulatives, pictures, diagrams, and symbols to mathematical ideas. 4. Represent, discuss, write, and read mathematical ideas and concepts. Start by relating everyday language to mathematical language and symbols and progress toward the use of appropriate terminology (e.g., “add more” becomes “plus”, “repeated addition” becomes “multiplication”, “fair share” becomes “divide”, “balance the equation” becomes “solve the equation”). Process Standard 3: Reasoning 1. Explain mathematical situations using patterns and relationships (e.g., identify patterns in situations, represent patterns in a variety of ways, extend patterns to connect with more general cases). 2. Demonstrate thinking processes using a variety of age-appropriate materials and reasoning processes (e.g., manipulatives, models, known facts, properties and relationships, inductive [specific to general], deductive [general to specific], spatial, proportional, logical reasoning [“and” “or” “not”] and recursive reasoning). 3. Make predictions and draw conclusions about mathematical ideas and concepts. Predictions become conjectures and conclusions become more logical as students mature mathematically. Process Standard 4: Connections 1. Relate various concrete and pictorial models of concepts and procedures to one another (e.g., use two colors of cubes to represent addition facts for the number 5, relate patterns on a hundreds chart to multiples, use base-10 blocks to represent decimals). 2. Link concepts to procedures and eventually to symbolic notation (e.g., represent actions like snap, clap, clap with symbols A B B, demonstrate 3 • 4 with a geometric array, divide a candy bar into 3 equal pieces that represent one piece as ). 3. Recognize relationships among different topics within mathematics (e.g., the length of an object can be represented by a number, multiplication facts can be modeled with geometric arrays, can be written as .5 and 50%). 4. Use mathematical strategies to solve problems that relate to other curriculum areas and the real world (e.g., use a timeline to sequence events, use symmetry in art work, explore fractions in quilt designs and to describe pizza slices). Process Standard 5: Representation 1. Create and use a variety of representations appropriately and with flexibility to organize, record, and communicate mathematical ideas (e.g., dramatizations, manipulatives, drawings, diagrams, tables, graphs, symbolic representations). 2. Use representations to model and interpret physical, social, and mathematical situations (e.g., counters, pictures, tally marks, number sentences, geometric models; translate between diagrams, tables, charts, graphs).
MATHEMATICS CONTENT STANDARDS Grade 5 The following concepts and skills are required by all students completing fifth grade. The Major Concepts should be taught in depth using a variety of methods and applications so that all students have accessibility to and an understanding of these concepts. Maintenance Concepts have been taught previously and are a necessary foundation for success in mathematics at this level. MAJOR CONCEPTSMAINTENANCE CONCEPTS Patterns and Algebraic Reasoning - Patterns and Algebraic Reasoning – Algebraic Expressions and Equations Extend Rules, Functions Number Sense - Number Sense – Fractions, Decimals, & Percents Place Value through 6 Digits, Basic Number Concepts Decimals to the 100ths place, Fractions Number Operations and Computation - Number Operations and Computation - Multiplication and Division Algorithms, Estimation, Basic Division Facts, Decimals - Add, Subtract, Multiply, Divide Fraction Concepts Geometry and Measurement - Geometry and Measurement - Classify 2- and 3-Dimensional Shapes, Lines, Angles, Perimeter, Area, Customary & Metric Customary & Metric Measurements Measurements Data Analysis and Probability Data Analysis and Probability - Interpret Tables and Graphs, Range, Mean, Interpret Graphs, Probability Probability Fifth Grade Suggested Materials Kit: snap cubes, rods, 1-inch color tiles, calculators, boxes, pawns, number cubes, balance scale, fraction strips, tangrams, protractors, double-sided measuring tapes, spinners, geometric solids, squares, circles, base-10 blocks, 10 x 10 grid paper, pattern blocks, fraction and decimal towers, geoboards, computer tessellation software Standard 1: Patterns and Algebraic Reasoning - The student will use algebraic methods to describe patterns and solve problems in a variety of contexts. 1. Describe rules that produce patterns found in tables, graphs, and models, and use variables (e.g., boxes, letters, pawns, number cubes, or other symbols) to solve problems or to describe general rules in algebraic expression or equation form. 2. Use algebraic problem-solving techniques (e.g., use a balance to model an equation and show how subtracting a number from one side requires subtracting the same amount from the other side) to solve problems. Standard 2: Number Sense - The student will demonstrate an understanding of the basic concepts and properties of real numbers. 1. Fractions, Decimals and Percents a. Solve problems using decimal numbers to the 1000ths place. Note: Asterisks (*) have been used to identify standards and objectives that must be assessed by the local school district. All other skills may be assessed by the Oklahoma School Testing Program (OSTP). b. Compare, convert, and order common fractions and decimals to the 100ths place to solve problems. c. Represent with models the connection between fractions, decimals, and percents and be able to convert from one representation to another (e.g., use 10 x 10 grids, base-10 blocks; limit fractions to halves, fourths, fifths, and tenths). d. Explain verbally with manipulatives and diagrams 25%, 50%, 75%; use these percents to solve problems and relate them to their corresponding fractions and decimals. 2. Basic Number Theory Concepts a. Apply the basic properties of arithmetic: commutative, associative, distributive, and identity (e.g., show 2 (5 +1) = (2 • 5) + (2 • 1), given (5 + 1) + (5 + 1) regroup to show this equals (5 + 5) + (1 + 1) concluding with (2 • 5) + (2 • 1)) to solve problems. b. Identify and apply factors, multiples, prime, and composite numbers in a variety of problem-solving situations (e.g., build rectangular arrays for numbers 1-100 and classify as prime or composite). Standard 3: Number Operations and Computation - The student will estimate and compute with whole numbers, decimals and fractions. 1. Estimation a. Use estimation skills to determine solutions to problems involving decimals. b. Apply estimation skills to solve problems involving common percents and equivalent fractions. 2. Whole Numbers, Decimals, and Fractions a. Add and subtract decimal numbers with the same and different place values (e.g., 3.72 + 1.4) to solve problems. b. Multiply and divide whole numbers and decimal numbers with 1- or 2-digit multipliers or divisors to solve problems. c. Add and subtract fractions and mixed numbers to solve problems using a variety of methods (e.g., use fraction strips, find the least common denominator [LCD]). Standard 4: Geometry and Measurement - The student will apply geometric properties and relationships and use measurements within the metric and customary systems to solve problems in a variety of contexts. 1. Identify and describe the basic properties of figures (e.g., two or three-dimensionality, symmetry, number of faces, types of angles). 2. Find the perimeter of simple polygons and area of a rectangle (e.g., use 1-inch tiles to build rectangles of different perimeters and areas). *3 Use nonstandard units (beans, rice, candies) and standard units (centimeter cubes, 1-inch cubes) to find the volume of rectangular solids and estimate the volume of other solids. *4. Use the appropriate units and tools to estimate and measure temperature, distance, length, weight, and angles. 5. Convert basic measurements of volume, weight and distance within the same system for metric and customary units (e.g., inches to feet, hours to minutes, centimeters to meters). Standard 5: Data Analysis and Probability - The student will use data analysis, statistics and probability to interpret data in a variety of contexts. 1. Data Analysis a. Analyze data to create and interpret tables and graphs. b. Justify the selection of the type of table or graph (e.g., a line graph may be more appropriate than a bar graph when displaying the height of a person over time). c. Compare and translate between displays of data (e.g., multiple sets of data on the same graph, Venn diagrams, a combination of diagrams, charts, tables, graphs). *d. Formulate questions, design investigations, consider samples, and collect, organize, and analyze data using observation, measurement, surveys, or experiments (e.g., how far can 5th graders throw a softball based on where it first hits the ground?). e. Determine the range (spread) and the mean (average) of a set of data. 2. Probability a. Determine the probability of events occurring in familiar contexts or experiments and express probabilities as fractions (e.g., find the fractional probability of an event given a biased spinner). b. List permutations and combinations of up to five items.
Grades 1-5
The National Council of Teachers of Mathematics (NCTM) has identified five process standards: Problem Solving, Communication, Reasoning and Proof, Connections, and Representation. Using these processes students are actively involved in deepening mathematical understandings which lead to increasingly sophisticated abilities required to meet mathematical challenges. Following is an outline of the five process standards and associated objectives.
NOTE: When examples are given there is a progression in levels of difficulty from basic to more complex skills.
Process Standard 1: Problem Solving
1. Use problem-solving approaches (e.g., act out situations, represent problems with drawings and lists, use concrete, pictorial, graphical, oral, written, and/or algebraic models, understand a problem, devise a plan, carry out the plan, look back).
2. Formulate problems from everyday and mathematical situations (e.g., how many forks are needed?, how many students are absent?, how can we share/divide these cookies?, how many different ways can we find to compare these fractions?).
3. Develop, test, and apply strategies to solve a variety of routine and nonroutine problems (e.g., look for patterns, make a table, make a problem simpler, process of elimination, trial and error).
4. Verify and interpret results with respect to the original problem (e.g., students explain verbally why an answer makes sense, explain in a written format why an answer makes sense, verify the validity of each step taken to obtain a final result).
5. Distinguish between necessary and irrelevant information in solving problems (e.g., play games and discuss “best” clues, write riddles with sufficient information, identify unnecessary information in written story problems).
Process Standard 2: Communication
1. Express mathematical ideas coherently and clearly to peers, teachers, and others (e.g., with verbal ideas, models or manipulatives, pictures, or symbols).
2. Extend mathematical knowledge by considering the thinking and strategies of others (e.g., agree or disagree, rephrase another student’s explanation, analyze another student’s explanation).
3. Relate manipulatives, pictures, diagrams, and symbols to mathematical ideas.
4. Represent, discuss, write, and read mathematical ideas and concepts. Start by relating everyday language to mathematical language and symbols and progress toward the use of appropriate terminology (e.g., “add more” becomes “plus”, “repeated addition” becomes “multiplication”, “fair share” becomes “divide”, “balance the equation” becomes “solve the equation”).
Process Standard 3: Reasoning
1. Explain mathematical situations using patterns and relationships (e.g., identify patterns in situations, represent patterns in a variety of ways, extend patterns to connect with more general cases).
2. Demonstrate thinking processes using a variety of age-appropriate materials and reasoning processes (e.g., manipulatives, models, known facts, properties and relationships, inductive [specific to general], deductive [general to specific], spatial, proportional, logical reasoning [“and” “or” “not”] and recursive reasoning).
3. Make predictions and draw conclusions about mathematical ideas and concepts. Predictions become conjectures and conclusions become more logical as students mature mathematically.
Process Standard 4: Connections
1. Relate various concrete and pictorial models of concepts and procedures to one another (e.g., use two colors of cubes to represent addition facts for the number 5, relate patterns on a hundreds chart to multiples, use base-10 blocks to represent decimals).
2. Link concepts to procedures and eventually to symbolic notation (e.g., represent actions like snap, clap, clap with symbols A B B, demonstrate 3 • 4 with a geometric
array, divide a candy bar into 3 equal pieces that represent one piece as
3. Recognize relationships among different topics within mathematics (e.g., the length of an object can be represented by a number, multiplication facts can be modeled with
geometric arrays,
4. Use mathematical strategies to solve problems that relate to other curriculum areas and the real world (e.g., use a timeline to sequence events, use symmetry in art work, explore fractions in quilt designs and to describe pizza slices).
Process Standard 5: Representation
1. Create and use a variety of representations appropriately and with flexibility to organize, record, and communicate mathematical ideas (e.g., dramatizations, manipulatives, drawings, diagrams, tables, graphs, symbolic representations).
2. Use representations to model and interpret physical, social, and mathematical situations (e.g., counters, pictures, tally marks, number sentences, geometric models; translate between diagrams, tables, charts, graphs).
MATHEMATICS CONTENT STANDARDS
Grade 5
The following concepts and skills are required by all students completing fifth grade. The Major Concepts should be taught in depth using a variety of methods and applications so that all students have accessibility to and an understanding of these concepts. Maintenance Concepts have been taught previously and are a necessary foundation for success in mathematics at this level.
MAJOR CONCEPTS MAINTENANCE CONCEPTS
Patterns and Algebraic Reasoning - Patterns and Algebraic Reasoning –
Algebraic Expressions and Equations Extend Rules, Functions
Number Sense - Number Sense –
Fractions, Decimals, & Percents Place Value through 6 Digits,
Basic Number Concepts Decimals to the 100ths place, Fractions
Number Operations and Computation - Number Operations and Computation -
Multiplication and Division Algorithms, Estimation, Basic Division Facts,
Decimals - Add, Subtract, Multiply, Divide Fraction Concepts
Geometry and Measurement - Geometry and Measurement -
Classify 2- and 3-Dimensional Shapes, Lines, Angles,
Perimeter, Area, Customary & Metric Customary & Metric Measurements
Measurements
Data Analysis and Probability Data Analysis and Probability -
Interpret Tables and Graphs, Range, Mean, Interpret Graphs, Probability
Probability
Fifth Grade Suggested Materials Kit:
snap cubes, rods, 1-inch color tiles, calculators, boxes, pawns, number cubes, balance scale, fraction strips, tangrams, protractors, double-sided measuring tapes, spinners, geometric solids, squares, circles, base-10 blocks, 10 x 10 grid paper, pattern blocks, fraction and decimal towers, geoboards, computer tessellation software
Standard 1: Patterns and Algebraic Reasoning - The student will use algebraic methods to describe patterns and solve problems in a variety of contexts.
1. Describe rules that produce patterns found in tables, graphs, and models, and use variables (e.g., boxes, letters, pawns, number cubes, or other symbols) to solve problems or to describe general rules in algebraic expression or equation form.
2. Use algebraic problem-solving techniques (e.g., use a balance to model an equation and show how subtracting a number from one side requires subtracting the same amount from the other side) to solve problems.
Standard 2: Number Sense - The student will demonstrate an understanding of the basic concepts and properties of real numbers.
1. Fractions, Decimals and Percents
a. Solve problems using decimal numbers to the 1000ths place.
Note: Asterisks (*) have been used to identify standards and objectives that must be assessed by the local school district. All other skills may be assessed by the Oklahoma School Testing Program (OSTP).
b. Compare, convert, and order common fractions and decimals to the 100ths place to solve problems.
c. Represent with models the connection between fractions, decimals, and percents and be able to convert from one representation to another (e.g., use 10 x 10 grids, base-10 blocks; limit fractions to halves, fourths, fifths, and tenths).
d. Explain verbally with manipulatives and diagrams 25%, 50%, 75%; use these percents to solve problems and relate them to their corresponding fractions and decimals.
2. Basic Number Theory Concepts
a. Apply the basic properties of arithmetic: commutative, associative, distributive, and identity (e.g., show 2 (5 +1) = (2 • 5) + (2 • 1), given (5 + 1) + (5 + 1) regroup to show this equals (5 + 5) + (1 + 1) concluding with (2 • 5) + (2 • 1)) to solve problems.
b. Identify and apply factors, multiples, prime, and composite numbers in a variety of problem-solving situations (e.g., build rectangular arrays for numbers 1-100 and classify as prime or composite).
Standard 3: Number Operations and Computation - The student will estimate and compute with whole numbers, decimals and fractions.
1. Estimation
a. Use estimation skills to determine solutions to problems involving decimals.
b. Apply estimation skills to solve problems involving common percents and equivalent fractions.
2. Whole Numbers, Decimals, and Fractions
a. Add and subtract decimal numbers with the same and different place values (e.g., 3.72 + 1.4) to solve problems.
b. Multiply and divide whole numbers and decimal numbers with 1- or 2-digit multipliers or divisors to solve problems.
c. Add and subtract fractions and mixed numbers to solve problems using a variety of methods (e.g., use fraction strips, find the least common denominator [LCD]).
Standard 4: Geometry and Measurement - The student will apply geometric properties and relationships and use measurements within the metric and customary systems to solve problems in a variety of contexts.
1. Identify and describe the basic properties of figures (e.g., two or three-dimensionality, symmetry, number of faces, types of angles).
2. Find the perimeter of simple polygons and area of a rectangle (e.g., use 1-inch tiles to build rectangles of different perimeters and areas).
*3 Use nonstandard units (beans, rice, candies) and standard units (centimeter cubes, 1-inch cubes) to find the volume of rectangular solids and estimate the volume of other solids.
*4. Use the appropriate units and tools to estimate and measure temperature, distance, length, weight, and angles.
5. Convert basic measurements of volume, weight and distance within the same system for metric and customary units (e.g., inches to feet, hours to minutes, centimeters to meters).
Standard 5: Data Analysis and Probability - The student will use data analysis, statistics and probability to interpret data in a variety of contexts.
1. Data Analysis
a. Analyze data to create and interpret tables and graphs.
b. Justify the selection of the type of table or graph (e.g., a line graph may be more appropriate than a bar graph when displaying the height of a person over time).
c. Compare and translate between displays of data (e.g., multiple sets of data on the same graph, Venn diagrams, a combination of diagrams, charts, tables, graphs).
*d. Formulate questions, design investigations, consider samples, and collect, organize, and analyze data using observation, measurement, surveys, or experiments (e.g., how far can 5th graders throw a softball based on where it first hits the ground?).
e. Determine the range (spread) and the mean (average) of a set of data.
2. Probability
a. Determine the probability of events occurring in familiar contexts or experiments and express probabilities as fractions (e.g., find the fractional probability of an event given a biased spinner).
b. List permutations and combinations of up to five items.