PASSmath

**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 PROCESS 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 <span style="color: rgb(0, 0, 0)">**Geometry and Measurement -** Geometry and Measurement - <span style="color: rgb(0, 0, 0)"> **Classify 2- and 3-Dimensional Shapes,** Lines, Angles, <span style="color: rgb(0, 0, 0)"> **Perimeter, Area, Customary & Metric** Customary & Metric Measurements <span style="color: rgb(0, 0, 0)"> **Measurements** <span style="color: rgb(0, 0, 0)">**Data Analysis and Probability** Data Analysis and Probability - <span style="color: rgb(0, 0, 0)"> **Interpret Tables and Graphs, Range, Mean,** Interpret Graphs, Probability <span style="color: rgb(0, 0, 0)"> **Probability** <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)">**Fifth Grade Suggested Materials Kit:** <span style="color: rgb(0, 0, 0)">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 <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)">**Standard 1: Patterns and Algebraic Reasoning** - **The student will use algebraic methods to describe patterns and solve problems in a variety of contexts.** <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)"> 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. <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)"> 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. <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)">**Standard 2: Number Sense - The student will demonstrate an understanding of the basic concepts and properties of real numbers.** <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)"> 1. Fractions, Decimals and Percents <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)"> a. Solve problems using decimal numbers to the 1000ths place. <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)">**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). <span style="color: rgb(0, 0, 0)"> b. Compare, convert, and order common fractions and decimals to the 100ths place to solve problems. <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)"> 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). <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)"> 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. <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)"> 2. Basic Number Theory Concepts <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)">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. <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)"> 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). <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)">**Standard 3: Number Operations and Computation - The student will estimate and compute with whole numbers, decimals and fractions.** <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)"> 1. Estimation <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)"> a. Use estimation skills to determine solutions to problems involving decimals. <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)"> b. Apply estimation skills to solve problems involving common percents and equivalent fractions. <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)"> 2. Whole Numbers, Decimals, and Fractions <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)"> a. Add and subtract decimal numbers with the same and different place values (e.g., 3.72 + 1.4) to solve problems. <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)">b. Multiply and divide whole numbers and decimal numbers with 1- or 2-digit multipliers or divisors to solve problems. <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)">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]). <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)">**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.** <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)"> 1. Identify and describe the basic properties of figures (e.g., two or three-dimensionality, symmetry, number of faces, types of angles). <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)"> 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). <span style="color: rgb(0, 0, 0)">*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. <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)"> *4. Use the appropriate units and tools to estimate and measure temperature, distance, length, weight, and angles. <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)"> 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). <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)">**Standard 5: Data Analysis and Probability - The student will use data analysis, statistics and probability to interpret data in a variety of contexts.** <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)"> 1. Data Analysis <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)"> a. Analyze data to create and interpret tables and graphs. <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)"> 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). <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)"> 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). <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)"> *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?). <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)"> e. Determine the range (spread) and the mean (average) of a set of data. <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)"> 2. Probability <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)"> 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). <span style="color: rgb(0, 0, 0)"> <span style="color: rgb(0, 0, 0)"> b. List permutations and combinations of up to five items.
 * MATHEMATICS CONTENT STANDARDS**