OVERVIEW

MATHEMATICS

Grades 1 - 5

Developmentally appropriate mathematics curriculum for Grades 1 - 5 must encourage the exploration of a wide variety of mathematical ideas and promote in-depth levels of understanding by focusing on the key concepts and processes. Programs should fit the needs of the learner. Student success depends largely on the quality of the foundation that is established during the first years of school. The mathematics curriculum for Grades 1 - 5 must:

Help children develop conceptual understanding of number, space, and situational problems by designing explorations and investigations that make use of everyday objects and specially designed materials (e.g., base-10 blocks).

Actively involve children in doing mathematics with extensive and thoughtful use of manipulatives (concrete materials) in an environment that encourages children to develop, discuss, test, and apply ideas.

Develop newly introduced mathematics concepts by beginning instruction with concrete experiences, progressing to pictorial representations and culminating with abstract symbols.

Require appropriate reasoning and problem-solving experiences from the outset, instilling in students a sense of confidence in their ability to think and communicate mathematically, to detect patterns, and to analyze data.

Emphasize the power of mathematics in helping children understand and interpret their world and solve problems that occur in it.

Include a broad range of content by incorporating an informal approach to measurement, geometry, data analysis, and patterns (algebra). This helps students see the usefulness of mathematics and establishes a foundation for further study.

Provide appropriate and ongoing use of technology by enabling children to explore number ideas and patterns, to focus on problem-solving processes, and to investigate realistic applications. Calculators do not replace the need for students to be fluent with basic facts, have efficient computation strategies, be able to compute mentally, and do paper-and-pencil computation.

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).

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 1

The following concepts and skills are required by all students completing first 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 – With Concrete Objects Patterns – Repeating, Sorting

Number Sense - Counting to 100 Number Sense – Counting to 10

Number Operations and Computation –

Addition & Subtraction Facts through 10

Geometry and Measurement - Geometry and Spatial Sense -

Congruent Shapes, Shapes

Time, Money, Calendar Measurement - Comparisons

Data Analysis - Concrete, Picture Graphs Data Analysis - Concrete Graphs

First Grade Suggested Materials Kit:

snap cubes, keys, fabric, macaroni, buttons, pattern blocks, children’s books, counters, beans, base-10 blocks, dominoes, calculators, geoboards, tangrams, attribute blocks, straws, containers, balance scales, rulers, coins, clocks, graph mats, painted beans or two-color counters, fraction circles, fraction squares

Standard 1: Patterns - The student will use a variety of problem-solving approaches to model, describe and extend patterns.

1. Describe, extend and create a variety of patterns using concrete objects (e.g., sort a bag of objects by attributes and orally communicate the pattern for each grouping).

2. Describe and extend number patterns in a variety of situations (e.g., addition charts, skip counting, calendars).

Standard 2: Number Sense - The student will read, write and model numbers and number relationships to 100.

*1. Use concrete models of tens and ones to develop the concept of place value.

2. Compare and Order Objects

a. Compare objects by size and quantity (e.g., more than, less than, equal to).

b. Use ordinal numbers first through tenth to order objects.

3. Read and write numerals to 100.

4. Count as many as 100 objects by ones, twos, fives, and tens.

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).

*5. Investigate concepts of fractional parts (e.g., halves, thirds, fourths).

Standard 3: Number Operations and Computation - The student will use models to construct addition and subtraction facts with whole numbers through 10.

1. Develop and apply the concepts of addition and subtraction.

*a. Use models to construct addition and subtraction facts through 10 (e.g., counters, cubes).

b. Perform addition by joining sets of objects and subtraction by separating and by comparing sets of objects.

c. Demonstrate fluency with basic addition and subtraction facts (i.e., memorize and apply addition and subtraction facts) through 10. Compute efficiently and accurately with single-digit numbers.

*d. Recognize and apply the commutative and identity properties of addition using models and manipulatives to develop computational skills (e.g., 2 + 4 = 4 + 2,

3 + 0 = 3).

2. Write addition and subtraction number sentences for problem-solving situations.

3. Acquire strategies for making computations (e.g., use estimation, number sense to judge reasonableness, counting on).

Standard 4: Geometry and Measurement - The student will use geometric properties and relationships to recognize and describe shapes and use measurement skills to tell time, identify money and develop calendar concepts.

1. Geometric Properties and Relationships

a. Sort and identify congruent shapes.

b. Identify two-dimensional geometric shapes and objects in everyday situations (e.g., the face of a round clock is a circle, a desktop is a rectangle).

c. Use language to describe relationships of objects in space (e.g., above, below, behind, between).

2. Measurement

a. Tell time on digital and analog clocks to the hour and half-hour.

b. Identify and name the value of pennies, dimes, nickels, and quarters.

c. Use the calendar to develop the concepts of days, weeks, and months.

Standard 5: Data Analysis - The student will demonstrate an understanding of data collection and display.

1. Organize, describe, and display data using concrete objects, pictures, grids, and numbers.

2. Formulate and solve problems that involve collecting and analyzing data common to children’s lives (e.g., color of shoes, numbers of pets, favorite foods).

MATHEMATICS CONTENT STANDARDS

Grade 2

The following concepts and skills are required by all students completing second 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 - With Symbols Patterns – With Concrete Objects

Number Sense - Number Sense - Counting to 100

Place Value through 3 Digits, Estimation

Number Operations and Computation - Number Operations and Computation-

Addition & Subtraction Facts to 18, Addition & Subtraction Facts through 10

Add & Subtract with 2 digits

Geometry and Measurement - Geometry and Measurement –

Symmetry, Standard & Nonstandard Congruent Shapes, Time, Money,

Units, Time, Money Calendar

Data Analysis – Charts, Bar Graphs, Tables Data Analysis - Concrete, Picture Graphs

Second Grade Suggested Materials Kit:

snap cubes, keys, fabric, macaroni, buttons, cans, objects from nature, pattern blocks, children’s books, links, rods, counters, beans, base-10 blocks, dominoes, calculators, geoboards, tangrams, attribute blocks, straws, containers, balance scales, rulers, tape measures, cups, spoons, coins, clocks, graph mats, painted beans or two-color counters

Standard 1: Patterns - The student will use a variety of problem-solving approaches to extend and create patterns.

1. Describe, extend, and create patterns using symbols, shapes or designs (e.g., repeating and growing patterns made up of sets of shapes or designs, create patterns by combining different shapes and taking them apart).

2. Formulate and record generalizations about number patterns in a variety of situations (e.g., addition and subtraction patterns, build a table showing the cost of one pencil at 10 cents, 2 pencils at 20 cents)

Standard 2: Number Sense - The student will use numbers and number relationships to acquire basic facts.

*1. Place Value

a. Use concrete models of hundreds, tens, and ones to develop the concepts of place value.

b. Demonstrate (using concrete objects, pictures, and numerical symbols) fractional parts including halves, thirds, and fourths.

2. Reading and Writing Numbers

a. Link place value concepts to the reading and writing of numbers (e.g., base-10 blocks).

b. Represent a number in a variety of ways (e.g., write the calendar day in different ways, write 15 as 8 + 7, write 25 as 2 tens + 5 ones or as 1 ten + 15 ones).

c. Write a number sentence to compare numbers less than 100 (e.g., 5 is more than 2, 3 is less than 7, page 51 comes after 50, and 51 is between 50 and 60).

3. Develop and use strategies of estimation (e.g., compose, decompose and regroup numbers, use knowledge of 10 to estimate quantities and sums [two numbers less than 10 can not add up to more than 20], use body parts to estimate measurements).

4. Determine whether a number is odd or even.

Standard 3: Number Operations and Computation - The student will compute with whole numbers less than 100.

1. Develop and apply the concepts of addition and subtraction.

a. Demonstrate fluency with basic addition and subtraction facts (i.e., memorize and apply addition and subtraction facts) and fact families to 18.

b. Solve two-digit addition and subtraction problems with and without regrouping using a variety of techniques (e.g., concrete, paper and pencil, mental math).

*c. Develop operation sense by applying the associative property of addition (e.g., 3 + (2 +1) = (3 + 2) + 1 ).

d. Describe the relationship between addition and subtraction (e.g., addition and subtraction are inverse operations).

2. Use mental strategies (or decomposition strategies) for addition and subtraction (e.g., make a group of 10 objects and 2 objects from a group of 7 objects and 5 objects).

3. Complete addition number sentences with a missing addend and use to solve everyday problems.

Standard 4: Geometry and Measurement - The student will use geometric properties and relationships to recognize and describe shapes and use appropriate units of measure in a variety of situations.

1. Geometric Properties and Relationships

a. Sort and classify symmetric and congruent figures.

b. Identify two-dimensional geometric shapes in everyday situations (e.g., a stop sign is an octagon).

2. Measurement

a. Measure objects with nonstandard and standard units (e.g., use a human foot [nonstandard] then a ruler [standard] to measure length to the nearest inch).

b. Select and use appropriate units of measurement in problem solving and everyday situations.

3. Time and Money

a. Tell time on digital and analog clocks to the quarter-hour.

b. Identify and count money; connect coins and bills with place value.

Standard 5: Data Analysis - The student will demonstrate an understanding of data collection, display and interpretation.

1. Collect, sort, organize, and display data in charts, bar graphs, and tables (e.g., collect data on teeth lost and display results in a chart).

2. Summarize and interpret data in charts, bar graphs, and tables.

*3. Make predictions and estimates to describe data (e.g., predict what data on teeth lost might look like for younger children and/or older children).

MATHEMATICS CONTENT STANDARDS

Grade 3

The following concepts and skills are required by all students completing third 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 - Rules Patterns - With Symbols

Number Sense - Number Sense -

Place Value through 4 Digits, Place Value through 3 Digits, Estimation

Fraction Concepts

Number Operations and Computation- Number Operations and Computation-

Addition & Subtraction with Multidigits, Addition & Subtraction Facts to 18,

Money, Basic Multiplication Facts Add & Subtract with 2 digits

Geometry and Measurement – Geometry and Measurement – Symmetry,

2- and 3-Dimensional Shapes, Length, Standard & Nonstandard Units, Time,

Weight, Estimation, Time Money

Data Analysis and Probability- Bar Graphs, Data Analysis - Charts, Bar Graphs,

Pictographs, Probability Tables

Third Grade Suggested Materials Kit:

snap cubes, pattern blocks, 1-inch color tiles, centimeter grid paper, hundreds charts, children’s books, links, rods, counters, beans, base-10 blocks, dominoes, calculators, geoboards, tangrams, attribute blocks, mirrors, flexible straws, egg cartons, containers, balance scales, rulers, tape measures, cups, spoons, coins, clocks, place value mats, graph mats

Standard 1: Patterns and Algebraic Reasoning - The student will use a variety of problem-solving approaches to extend and create patterns.

1. Describe (orally or in written form), create, extend and predict patterns using numbers (e.g., 3, 6, 9, 12 . . . , use a function machine to generate input and output values for a table, show multiplication patterns on a hundreds chart).

2. Analyze tables to formulate generalizations about patterns in a variety of situations (e.g., list the multiples of 5 in a table to show that multiples of 5 have a 0 or 5 in the ones place; given pairs of numbers with a common relationship, determine the rule and generate additional pairs with the same relationship).

Standard 2: Number Sense - The student will use numbers and number relationships to acquire basic facts.

1. Place Value

a. Model the concept of place value through 4 digits (e.g., base-10 blocks, bundles of 10s, place value mats).

b. Read, model and write whole numbers up to 4 digits (e.g., base-10 blocks, expanded form).

2. Whole Numbers and Fractions

a. Compare and order whole numbers up to 4 digits.

b. Compare and order fractions including halves, thirds and fourths using a model (e.g., fraction circles, pictures, egg cartons, fraction strips).

Standard 3: Number Operations and Computation - The student will estimate and compute with whole numbers.

1. Estimate, find the sum and difference, with and without regrouping, of 3- and 4-digit numbers to solve application problems.

2. Multiplication Concepts

a. Demonstrate fluency with basic multiplication facts and fact families (i.e., memorize and apply multiplication facts).

*b. Develop multiplication algorithms (e.g., use physical materials to show 4 groups of 3 objects, show multiplication as repeated addition).

c. Estimate the product of 2-digit numbers by rounding to the nearest multiple of 10 to solve application problems.

*d. Recognize and apply the commutative and identity properties of multiplication using models and manipulatives to develop computational skills (e.g., 3 • 5 = 5 • 3, 7 • 1 = 7).

3. Solve problems involving money that require addition and subtraction.

Standard 4: Geometry and Measurement - The student will use geometric properties and relationships to recognize and describe shapes and use customary and metric measurements to solve problems.

1. Spatial Reasoning and Coordinate Locations

a. Describe and compare two- and three-dimensional shapes (e.g., count the edges and faces of a cube, combine or divide basic shapes to form new shapes, identify and draw congruent shapes).

b. Identify locations on a grid with ordered pairs (e.g., give the location of a ship on a grid by selecting D, 1).

2. Measurement

a. Solve problems with customary units involving length using half-inch and quarter-inch measurements and weight using pound and ounce.

b. Solve problems with metric units involving length using meter and centimeter and mass using gram and kilogram.

c. Use manipulatives to develop the concept of perimeter and area (e.g., cover a shape with pattern blocks to find area).

*3. Develop and use strategies to estimate measurements (e.g., use parts of the body as benchmarks for measuring length).

4. Tell time on digital and analog clocks to 5 minutes and use information to solve problems involving time and temperature (e.g., read a thermometer).

Standard 5: Data Analysis and Probability - The student will demonstrate an understanding of data collection, display and interpretation.

1. Data Analysis

*a. Pose questions, collect, record, and interpret data to help answer questions (e.g., Which was the most popular booth at our carnival?).

b. Read graphs and charts; identify the main idea, draw conclusions, make predictions based on the data (e.g., predict how many children will bring their lunch based on a menu).

c. Construct a bar graph or pictograph with labels and a title from a set of data.

2. Probability

a. Describe the probability (more, less, or equally likely) of chance events.

b. List arrangements (permutations) and combinations of up to three items (e.g., possible ways to arrange scoops of chocolate, strawberry and vanilla ice cream on a cone).

Blueprints for each Criterion-Referenced Test reflect the degree of representation given on the test to each PASS standard and objective. To access the current blueprint (when available) go to the State Department of Education Web site at < http://sde.state.ok.us >, click on site index, then click “s” to go to student assessment, then click on “Student Tests & Materials” then scroll down to “alignment blueprints.”

MATHEMATICS CONTENT STANDARDS

Grade 4

The following concepts and skills are required by all students completing fourth 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 –

Extend Rules, Functions Rules

Number Sense - Number Sense-

Place Value through 6 Digits, Place Value through 4 Digits,

Decimals to the 100ths place, Fractions Fraction Concepts

Number Operations and Computation- Number Operations and Computation-

Estimation, Basic Division Facts, Addition & Subtraction with

Fraction Concepts Multidigits, Basic Multiplication Facts

Geometry and Measurement - Lines, Angles, Geometry and Measurement –

Customary & Metric Measurements 2- and 3-Dimensional Shapes,

Length, Weight, Estimation, Time

Data Analysis and Probability - Data Analysis and Probability –

Interpret Graphs, Probability Bar Graphs, Pictographs, Probability

Fourth Grade Suggested Materials Kit:

snap cubes, number cubes, pattern blocks, 1-inch color tiles, grid paper, hundreds charts, cereal and shoe boxes, children’s books, journals, rods, counters, beans, base-10 blocks, calculators, geoboards, dot paper, clay, toothpicks, mirrors, flexible straws, pipe cleaners, egg cartons, containers, balance scales, rulers, tape measures, thermometers, cups, spoons, coins, clocks, graph mats, spinners, painted beans or two-color counters

Standard 1: Patterns and Algebraic Reasoning - The student will use a variety of problem-solving approaches to analyze, extend and create patterns.

1. Discover, describe, extend, and create a wide variety of patterns using tables, graphs, rules, and models (e.g., use 1-inch tiles to demonstrate that doubling the length of the side of a square more than doubles the area, explore the characteristics of odd and even numbers, extend patterns of geometric shapes).

2. Elementary Function Concepts

a. Use a variety of techniques to generalize number patterns (e.g., use function machines and “t-tables” to demonstrate "What is the rule?").

b. Solve simple open sentences involving operations on whole numbers (with a variable, e.g., a + 17 = 23).

Standard 2: Number Sense - The student will use numbers and number relationships to acquire basic number facts.

1. Place Value

a. Apply the concept of place value through 6 digits (e.g., write numbers in expanded form, play a trading game involving place value).

b. Read, write and rename whole numbers through 6 digits and decimal numbers to the hundredths (e.g., money, numerals to words).

2. Compare and order whole numbers and decimals to the hundredths place (e.g., pictures of shaded regions of two-dimensional figures, use >, < , = symbols).

3. Fractions

a. Use 0, 12, and 1 or 0, 0.5, and 1, as benchmarks and place additional fractions

and decimals on a number line (e.g., , , 0.7, 0.4).

b. Create physical and pictorial models of equivalent and nonequivalent fractional

parts to be compared, added or subtracted (e.g., egg cartons, fraction strips, circles, and squares).

Standard 3: Number Operations and Computation - The student will estimate and compute with whole numbers.

1. Estimate and find the product of 2- and 3-digit numbers to solve application problems.

2. Division Concepts

a. Demonstrate fluency with basic division facts and fact families (i.e., memorize and apply division facts).

*b. Develop division algorithms (e.g., use physical materials to show 12 objects arranged in 3 groups, show division as repeated subtraction and as the inverse of multiplication).

c. Estimate and find the quotient (with and without remainders) with a 1-digit divisor and a 2- or 3-digit dividend to solve application problems.

3. Apply a variety of estimation and mental math techniques to simplify computations (e.g., use rounding to estimate 82 - 58 is about 80 - 60 or 20, use 30 • 5 to find the product of 300 • 5).

*4. Develop operation sense by applying the associative property of multiplication (e.g., 6 • (2 • 3) = (6 • 2) • 3).

Standard 4: Geometry and Measurement - The student will use geometric properties and relationships to analyze shapes and use standard units of customary and metric measurements to solve problems.

1. Basic Characteristics of Lines and Angles

a. Identify, draw, and construct models of intersecting, parallel, and perpendicular lines (e.g., use spaghetti, straws, toothpicks).

b. Identify and compare angles equal to, less than, or greater than 90 degrees (e.g., use right angles to determine the approximate size of other angles; make a variety of angles using flexible straws and compare).

*2. Identify basic characteristics of the rectangular coordinate system and find the distance between horizontal and vertical lines of a rectangular coordinate system (e.g., the x-axis is the horizontal axis).

3. Spatial Reasoning

a. Describe the effects on two- and three-dimensional objects when they slide (translate), flip (reflect), and turn (rotate) (e.g., tessellations).

b. Predict and verify the effects of combining, subdividing, and changing two- and three-dimensional figures (e.g., folding paper, tiling, and rearranging pieces of solids).

4. Measurement

a. Establish benchmarks for customary and metric units and estimate the measures of a variety of objects and compare units (e.g., mass: the mass of a raisin is about 1 gram, length: the width of a finger is about 1 centimeter).

b. Select appropriate customary and metric units of measure to solve application problems involving length, weight, mass, and volume.

c. Solve application problems involving money, time and temperature (e.g., elapsed time).

Standard 5: Data Analysis and Probability - The student will demonstrate an understanding of data collection, display and interpretation.

1. Data Analysis

a. Examine data displays such as tallies, tables, charts and graphs and use the observations to pose and answer questions (e.g., choose a table in social studies of population data and write problems).

b. Collect, organize and record data in tables and graphs (e.g., bar, pictograph, line plots).

*2. Investigate and record probabilities by experimenting with devices that generate random outcomes (e.g., coins, number cubes, spinners).

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.

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.

OVERVIEW

Grades 6 - 8

Students in the middle grades will expand and deepen their knowledge of numbers, computation, estimation, measurement, geometry, statistics, probability, patterns and functions, and the fundamental concepts of algebra by focusing on meaningful mathematics in each of these areas.

Instruction in the middle grades should include activities in which the students actively work to pose and solve problems both individually and together. Learning tools such as concrete models, fraction manipulatives, algebra tiles, geoboards, calculators and computers are beneficial and should be available to all students.

MATHEMATICS PROCESS STANDARDS

Grades 6 - 8

The National Council of Teachers of Mathematics (NCTM) has identified five process standards: Problem Solving, Reasoning and Proof, Communication, Connections, and Representation. Active involvement by students using these processes is likely to broaden mathematical understandings and lead to increasingly sophisticated abilities required to meet mathematical challenges in meaningful ways.

Process Standard 1: Problem Solving

1. Develop and test strategies to solve practical, everyday problems which may have single or multiple answers.

2. Use technology to generate and analyze data to solve problems.

3. Formulate problems from situations within and outside of mathematics and generalize solutions and strategies to new problem situations.

4. Evaluate results to determine their reasonableness.

5. Apply a variety of strategies (e.g., restate the problem, look for a pattern, diagrams, solve a simpler problem, work backwards, trial and error) to solve problems, with emphasis on multistep and nonroutine problems.

6. Use oral, written, concrete, pictorial, graphical, and/or algebraic methods to model mathematical situations.

Process Standard 2: Communication

1. Discuss, interpret, translate (from one to another) and evaluate mathematical ideas (e.g., oral, written, pictorial, concrete, graphical, algebraic).

2. Reflect on and justify reasoning in mathematical problem solving (e.g., convince, demonstrate, formulate).

3. Select and use appropriate terminology when discussing mathematical concepts and ideas.

Process Standard 3: Reasoning

1. Identify and extend patterns and use experiences and observations to make suppositions.

2. Use counter examples to disprove suppositions (e.g., all squares are rectangles, but are all rectangles squares?).

3. Develop and evaluate mathematical arguments (e.g., agree or disagree with the reasoning of other classmates and explain why).

4. Select and use various types of reasoning (e.g., recursive [loops], inductive [specific to general], deductive [general to specific], spatial, and proportional).

Process Standard 4: Connections

1. Apply mathematical strategies to solve problems that arise from other disciplines and the real world.

2. Connect one area or idea of mathematics to another (e.g., relate equivalent number representations to each other, relate experiences with geometric shapes to understanding ratio and proportion).

Process Standard 5: Representation

1. Use a variety of representations to organize and record data (e.g., use concrete, pictorial, and symbolic representations).

2. Use representations to promote the communication of mathematical ideas (e.g., number lines, rectangular coordinate systems, scales to illustrate the balance of equations).

3. Develop a variety of mathematical representations that can be used flexibly and appropriately (e.g., base-10 blocks to represent fractions and decimals, appropriate graphs to represent data).

4. Use a variety of representations to model and solve physical, social, and mathematical problems (e.g., geometric objects, pictures, charts, tables, graphs).

MATHEMATICS CONTENT STANDARDS

Grade 6

The following concepts and skills are required by all students completing sixth 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

Algebraic Reasoning - Patterns and Algebraic Reasoning-

Algebraic Expressions Algebraic Expressions and Equations

Number Sense - Number Sense -

Fractions, Order of Operations Fractions, Decimals, & Percents

Decimals to the 1000ths place

Number Operations and Computation- Multiplication and Division

Algorithms, Decimals – Add,

Subtract, Multiply, Divide

Geometry - Geometry and Measurement -

Angles, Congruency, Similarity, Classify 2- and 3-Dimensional

Basic Transformations Shapes, Perimeter, Area

Measurement –Circles, Customary & Metric Measurements

Customary & Metric Estimates

Data Analysis and Statistics - Data Analysis and Probability

Interpret Data and Graphs, Median, Interpret Tables and Graphs, Range,

Mode Mean, Probability

Standard 1: Algebraic Reasoning - The student will use algebraic methods to describe patterns and simplify algebraic expressions in a variety of contexts.

1. Extend and create patterns from tables, graphs, rules and number properties and generalize patterns algebraically (e.g., recursive patterns like the Fibonacci numbers, number sequences, prime and composite numbers).

2. Use substitution and order of operations to simplify and evaluate algebraic expressions (e.g., if x = 5 evaluate 2x + 3).

Standard 2: Number Sense - The student will use numbers and number relationships to solve problems.

1. Multiply and divide fractions and mixed numbers to solve problems using a variety of methods.

2. Convert, compare and order decimals (terminating and nonterminating), fractions and percents using a variety of methods.

3. Estimate solutions to single and multi-step problems using whole numbers, decimals, fractions, and percents and assess whether solutions are reasonable (e.g., + is about 2, 0.9 + 0.3 is about 1).

*4. Build and explore multiples and their patterns to develop the concept of exponents.

5. Simplify numerical expressions with exponents and parentheses using order of operations.

Standard 3: Geometry - The student will use geometric properties and relationships to recognize, describe and analyze shapes and representations in a variety of contexts.

1. Angles

a. Compare, estimate and determine the measurement of angles.

b. Find the complement and supplement of an angle.

2. Differentiate between congruent and similar figures.

*3. Describe the effect of performing basic transformations on objects and figures in a variety of contexts. (e.g., explore reflection [flip] with mirrors, explore rotation [turn] and translation [slide] by designing the layout of different shapes of floor tile).

Standard 4: Measurement - The student will use measurements within the metric and U.S. customary system to solve problems in a variety of contexts.

*1. Collect data and develop formulas to find the circumference and area of circles (e.g., use string the length of the diameter of various circular lids to approximate the circumference and develop the concept of pi).

2. Compare and convert units within the same measurement system; express conversions using appropriate unit labels (e.g., square inches to square feet, centimeters to millimeters, hours to minutes); and compute measurements of combined units (e.g., 9’8” + 3’6” = ?' and ?" , 150 minutes = ? hours and ? minutes).

3. Find reasonable estimates for measurements using measurements in standard and metric units.

Standard 5: Data Analysis and Statistics - The student will use data analysis and statistics to interpret data in a variety of contexts.

1. Collect, organize, and interpret data to solve problems (e.g., data from student experiments, tallies, Venn diagrams, tables, circle and bar graphs, spreadsheets).

2. Construct and interpret graphs of statistical data (e.g., explain how different representations lead to different interpretations and may distort information).

3. Find the median and mode for a set of data in a variety of contexts.

MATHEMATICS CONTENT STANDARDS

Grade 7

The following concepts and skills are required by all students completing seventh 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

Algebraic Reasoning - Algebraic Reasoning -

Number Properties, Equations Algebraic Expressions

Number Sense - Number Sense -

Integers, Ratio, Proportion, Percent Fractions, Order of Operations

Geometry - Geometry -

Geometric Figures, Rectangular Angles, Congruency, Similarity,

Coordinate System Basic Transformations

Measurement – Area, Perimeter, Measurement - Circles,

Customary & Metric Measurements Customary & Metric Estimates

Data Analysis and Probability –

Predictions from Data, Data Analysis and Statistics -

Simple Probabilities Interpret Data and Graphs, Median,

Mode

Standard 1: Algebraic Reasoning - The student will use number properties to simplify and solve simple linear equations.

1. Identify and apply the commutative, associative, distributive, inverse and identity properties (e.g., n + 0 = n, 2(x + 3) = 2x + 6).

2. Use a variety of methods to model and solve one-step linear equations (e.g., use properties of equality, graph ordered pairs with paper and pencil, use graphing calculators).

Standard 2: Number Sense - The student will use numbers and number relationships to acquire basic facts and determine the reasonableness of results.

1. Integers

a. Compare and order positive and negative integers and describe their use in real-life situations (e.g., temperature, sea level, stock market fluctuations, football yardage).

b. Use the basic operations on integers to solve problems.

2. Ratio, Proportion and Percents

*a. Demonstrate the concept of ratio and proportion with models (e.g., similar geometric shapes, scale models).

b. Set up equivalent ratios, estimate and solve problems using ratio, proportions, and percents including percents greater than 100 and less than 1 (e.g., determine missing sides of similar figures, heart rate per minute, cost per pound, pay to hours worked overtime).

c. Solve percent application problems (e.g., discounts, tax, finding the missing value of percent/part/whole).

3. Exponents

*a. Analyze and develop generalizations of exponential patterns, including zero as an exponent, using manipulatives and calculators (e.g., model getting paid a penny the first day, 2 cents the second day, 4 cents the third day . . .).

b. Build and recognize models of multiples to investigate squares and square roots (e.g., build rectangular arrays for numbers 1 to 100 and note which can be represented as squares).

c. Estimate the square root of a number (e.g., between two consecutive integers).

Standard 3: Geometry - The student will apply the properties and relationships of plane geometry in a variety of contexts.

1. Classifying Geometric Figures

a. Classify triangles according to their sides and angles.

b. Classify quadrilaterals according to their sides and angles (e.g., determine whether all squares are rectangles).

2. Identify and compare bisectors, interior, exterior, and vertical angles (e.g., using graph paper, software, protractors to measure angles between parallel lines with a transversal).

3. Rectangular Coordinate System

a. Locate points on a plane in all four quadrants.

b. Identify geometric transformation of figures (rotations, translations, and reflections).

Standard 4: Measurement - The student will use measurement to solve problems in a variety of contexts.

1. Area and Perimeter

a. Develop area and perimeter concepts (e.g., use grids to estimate the area of irregular shapes).

b. Apply formulas to solve problems involving perimeter (circumference) and area of polygons and circles.

2. Customary and Metric Measurements

a. Select and use appropriate tools for measurements in practical applications and make reasonable estimates of measurements in a particular situation using the appropriate unit.

b. Use estimates to relate customary and metric measurements to each other.

Standard 5: Data Analysis and Probability - The student will use probability to formulate and justify predictions from a set of data.

1. Use data from a sample to predict possible outcomes and compute simple probabilities as fractions, decimals or percents (e.g., use data from lists, tree diagrams, frequency distribution tables, area models).

2. Determine the probability of an event involving “or”, “and”, or “not” (e.g., on a spinner with 1 blue, 2 red and 2 yellow sections, what is the probability of getting a red or a yellow?).

3. Find all possible combinations and permutations involving a limited number of variables.

MATHEMATICS CONTENT STANDARDS

Grade 8

The following concepts and skills are required by all students completing eighth 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

Algebraic Reasoning - Algebraic Reasoning -

Equations, Inequalities Number Properties, Equations

Number Sense - Number Sense -

Rational Numbers, Exponents Integers, Ratio, Proportion, Percent

Geometry and Measurement - Geometry -

Surface Area, Volume, Geometric Figures, Rectangular

Ratio and Proportion, Formulas Coordinate System

Measurement - Area, Perimeter,

Customary & Metric Measurements

Data Analysis and Statistics - Data Analysis and Probability -

Graphical Representations, Sampling, Predictions from Data,

Measures of Central Tendency Simple Probabilities

Standard 1: Algebraic Reasoning - The student will graph and solve linear equations and inequalities in problem-solving situations.

1. Equations

a. Model, write, and solve 2-step linear equations using a variety of methods.

b. Graph and interpret the solution to linear equations on a number line with one variable and on a coordinate plane with two variables.

c. Predict the effect on the graph of a linear equation when the slope changes (e.g., make predictions from graphs, identify the slope in the equation y = mx + b and relate to a graph).

2. Inequalities

a. Model, write, and solve 1-step and 2-step linear inequalities with one variable.

b. Graph the solution to linear inequalities with one variable on a number line.

Standard 2: Number Sense - The student will use numbers and number relationships to solve problems.

1. Rational Numbers and Proportional Reasoning

a. Compare and order rational numbers (positive and negative integers, fractions, decimals) in real-life situations.

b. Use the basic operations on rational numbers to solve problems in real-life situations (e.g., describe the effect of multiplying whole numbers by a fraction or a decimal less than 1).

c. Apply ratios and proportions to solve problems.

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).

2. Exponents

a. Use the rules of exponents, including integer exponents, to solve problems (e.g., 72 • 73 = 7⁵).

b. Represent and interpret large numbers and numbers less than one in exponential and scientific notation.

c. Use estimation strategies (e.g., rounding) to describe the magnitude of large numbers and numbers less than one.

Standard 3: Geometry - The student will use geometric properties to solve problems in a variety of contexts.

1. Construct models, sketch (from different perspectives), and classify solid figures such as rectangular solids, prisms, cones, cylinders, pyramids, and combined forms (e.g., draw a figure that could result from making 1, 2, or 3 cuts in a given solid).

2. Develop the Pythagorean Theorem and apply the formula to find the length of missing sides of a right triangle and the length of other line segments.

Standard 4: Measurement - The student will use measurement to solve problems in a variety of contexts.

1. Estimate and find the surface area and volume in real world settings (e.g., unwrap a box to explore surface area; use rice, 1-inch cubes, centimeter cubes, cups . . . to estimate the volume of boxes, irregular shaped objects, containers).

2. Apply knowledge of ratio and proportion to solve relationships between similar geometric figures (e.g., build a model of a 3-dimensional object to scale).

3. Formulas

a. Select and apply appropriate formulas for given situations:

I. an equation (e.g., d = rt, i = prt)

measurement problems (e.g., p = 2l + 2w, v = lwh)

b. Find the area of a “region of a region” for simple composite figures (e.g., area of a rectangular picture frame).

Standard 5: Data Analysis and Statistics - The student will use data analysis and statistics to interpret data in a variety of contexts.

1. Select and apply appropriate formats (e.g., line plots, bar graphs, stem-and-leaf plots, scatter plots, histograms, circle graphs) to display collected data.

2. Measures of Central Tendency

a. Find the measures of central tendency (mean, median and mode) of a set of data and understand why a specific measure provides the most useful information in a given context.

b. Compute the mean, median, and mode for data sets and understand how additional data in a set may affect the measures of central tendency.

*3. Determine how samples are chosen (random, limited, biased) to draw and support conclusions about generalizing a sample to a population (e.g., is the average height of a men’s college basketball team a good representative sample for height predictions?).

OVERVIEW

High School

The Priority Academic Student Skills (PASS) in mathematics for high school establishes a framework for a curriculum that reflects the needs of all students. Such a curriculum recognizes that they will spend their adult lives in a society increasingly dominated by technology and quantitative methods.

A broadened view of mathematics will include the traditional topics of algebra and geometry but must also include the mathematical processes of problem-solving, communication, reasoning, connections, and representation. Although they are stated separately for emphasis, these process standards should be integrated throughout the high school core curriculum.

A school’s curriculum in mathematics should be organized to permit all students to progress as far into the mathematics proposed here as their achievement with the objectives allows. Schools should use this material to create a curriculum most beneficial to their students. Those students planning to continue their mathematics education should study additional advanced mathematics topics such as trigonometry and calculus.

The curriculum is intended to provide a common body of mathematical ideas accessible to all students. It is recognized that students entering high school differ in many ways, including mathematical achievement, but it is believed these differences are best addressed by extensions of the proposed content rather than by deletions.

The increasing role of technology in instruction will alter the teaching and learning of mathematics. Calculators and computers should be integrated throughout the curriculum so that students will concentrate on the problem-solving process as well as the calculations associated with problems.

PROCESS STANDARDS

High School

Process Standard 1: Problem Solving

1. Apply a wide variety of problem-solving strategies (identify a pattern, use equivalent representations) to solve problems from within and outside mathematics.

2. Identify the problem from a described situation, determine the necessary data and apply appropriate problem-solving strategies.

Process Standard 2: Communication

1. Use mathematical language and symbols to read and write mathematics and to converse with others.

2. Demonstrate mathematical ideas orally and in writing.

3. Analyze mathematical definitions and discover generalizations through investigations.

Process Standard 3: Reasoning

1. Use various types of logical reasoning in mathematical contexts and real-world situations.

2. Prepare and evaluate suppositions and arguments.

3. Verify conclusions, identify counterexamples, test conjectures, and justify solutions to mathematical problems.

4. Justify mathematical statements through proofs.

Process Standard 4: Connections

1. Link mathematical ideas to the real world (e.g., statistics helps qualify the confidence we can have when drawing conclusions based on a sample).

2. Apply mathematical problem-solving skills to other disciplines.

3. Use mathematics to solve problems encountered in daily life.

4. Relate one area of mathematics to another and to the integrated whole (e.g., connect equivalent representations to corresponding problem situations or mathematical concepts).

Process Standard 5: Representation

1. Use algebraic, graphic, and numeric representations to model and interpret mathematical and real world situations.

2. Use a variety of mathematical representations as tools for organizing, recording, and communicating mathematical ideas (e.g., mathematical models, tables, graphs, spreadsheets).

3. Develop a variety of mathematical representations that can be used flexibly and appropriately.

MATHEMATICS CONTENT STANDARDS

Algebra I

The following skills are required of all students completing Algebra I. Major Concepts should be taught in depth using a variety of methods and applications (concrete to the abstract). Maintenance Concepts have been taught previously and are a necessary foundation for this course. The major concepts are considered minimal exit skills and districts are strongly encouraged to exceed these skills when building an Algebra I curriculum. Visual and physical models, calculators, and other technologies are recommended when appropriate and can enhance both instruction and assessment.

MAJOR CONCEPTS MAINTENANCE CONCEPTS

Number Sense and Algebraic Operations - Number Sense & Algebraic Reasoning-

Polynomials, Exponents, Expressions Equations, Inequalities, Exponents,

Rational Numbers

Relations and Functions - Geometry

Linear Functions & Slope Volume, Surface Area, Ratio,

Formulas Proportion, Formulas

Data Analysis, Statistics and Probability- Data Analysis and Statistics -

Tables, Graphs, Charts, Scatter Plots Graphical Representations,

Measures of Central Tendency

Standard 1: Number Sense and Algebraic Operations - The student will use expressions and equations to model number relationships.

1. Equations and Formulas

a. Translate word phrases and sentences into expressions and equations and vice versa.

b. Solve literal equations involving several variables for one variable in terms of the others.

c. Use the formulas from measurable attributes of geometric models (perimeter, circumference, area and volume), science, and statistics to solve problems within an algebraic context.

d. Solve two-step and three-step problems using concepts such as rules of exponents, rate, distance, ratio and proportion, and percent.

2. Expressions

a. Simplify and evaluate linear, absolute value, rational and radical expressions.

b. Simplify polynomials by adding, subtracting or multiplying.

c. Factor polynomial expressions.

Standard 2: Relations and Functions - The student will use relations and functions to model number relationships.

1. Relations and Functions

a. Distinguish between linear and nonlinear data.

b. Distinguish between relations and functions.

c. Identify dependent and independent variables, domain and range.

d. Evaluate a function using tables, equations or graphs.

2. Linear Equations and Graphs

a. Solve linear equations by graphing or using properties of equality.

b. Recognize the parent graph of the functions y = k, y = x, y = |x|, and predict the effects of transformations on the parent graph.

c. Slope

I. Calculate the slope of a line using a graph, an equation, two points or a set of data points.

II. Use the slope to differentiate between lines that are parallel, perpendicular, horizontal, or vertical.

III. Interpret the slope and intercepts within the context of everyday life (e.g., telephone charges based on base rate [y-intercept] plus rate per minute [slope]).

d. Develop the equation of a line and graph linear relationships given the following: slope and y-intercept, slope and one point on the line, two points on the line, x-intercept and y-intercept, a set of data points.

e. Match appropriate equations to a graph, table, or situation and vice versa.

3. Linear Inequalities and Graphs

a. Solve linear inequalities by graphing or using properties of inequalities.

b. Match appropriate inequalities (with 1 or 2 variables) to a graph, table, or situation and vice versa.

4. Solve a system of linear equations by graphing, substitution or elimination.

* 5. Nonlinear Functions

a. Match exponential and quadratic functions to a table, graph or situation and vice versa.

b. Solve quadratic equations by graphing, factoring, or using the quadratic formula.

Standard 3: Data Analysis, Probability and Statistics - The student will use data analysis, probability and statistics to formulate and justify predictions from a set of data.

1. Data Analysis

a. Translate from one representation of data to another and understand that the data can be represented using a variety of tables, graphs, or symbols and that different modes of representation often convey different messages.

b. Make valid inferences, predictions, and/or arguments based on data from graphs, tables, and charts.

c. Solve two-step and three-step problems using concepts such as probability and measures of central tendency.

Collect data involving two variables and display on a scatter plot; interpret results using a linear model/equation and identify whether the model/equation is a line best fit for the data.

MATHEMATICS CONTENT STANDARDS

Geometry

The following skills are required of all students completing Geometry. Major Concepts should be taught in depth using a variety of methods and applications (concrete to the abstract). Maintenance Concepts have been taught previously and are a necessary foundation for this course. The major concepts are considered minimal exit skills and districts are strongly encouraged to exceed these skills when building a Geometry curriculum. Visual and physical models, calculators, and other technologies are recommended when appropriate and can enhance both instruction and assessment.

MAJOR CONCEPTS MAINTENANCE CONCEPTS

Logical Reasoning Ratios, Proportions

Properties Perimeter, Area, Surface Area, Volume

Coordinate Geometry Equations

Angles and Triangles Formulas

Standard 1: Logical Reasoning - The student will use deductive and inductive reasoning to solve problems.

1. Identify and use logical reasoning skills (inductive and deductive) to make and test conjectures, formulate counter examples, and follow logical arguments.

2. State, use, and examine the validity of the converse, inverse, and contrapositive of “if-then” statements.

* 3. Compare the properties of Euclidean geometry to non-Euclidean geometries (for example, elliptical geometry, as shown on the surface of a globe, does not uphold the parallel postulate).

Standard 2: Properties of 2-Dimensional Figures - The student will use the properties and formulas of geometric figures to solve problems.

* 1. Use geometric tools (for example, protractor, compass, straight edge) to construct a variety of figures.

2. Line and Angle Relationships

a. Use the angle relationships formed by parallel lines cut by a transversal to solve problems.

b. Use the angle relationships formed by two lines cut by a transversal to determine if the two lines are parallel and verify, using algebraic and deductive proofs.

c. Use relationships between pairs of angles (for example, adjacent, complementary, vertical) to solve problems.

3. Polygons and Other Plane Figures

a. Identify, describe, and analyze polygons (for example, convex, concave, regular, pentagonal, hexagonal, n-gonal).

b. Apply the interior and exterior angle sum of convex polygons to solve problems, and verify using algebraic and deductive proofs.

c. Develop and apply the properties of quadrilaterals to solve problems (for example, rectangles, parallelograms, rhombi, trapezoids, kites).

d. Use properties of 2-dimensional figures and side length, perimeter or circumference, and area to determine unknown values and correctly identify the appropriate unit of measure of each.

4. Similarity

a. Determine and verify the relationships of similarity of triangles, using algebraic and deductive proofs.

b. Use ratios of similar 2-dimensional figures to determine unknown values, such as angles, side lengths, perimeter or circumference, and area.

5. Congruence

a. Determine and verify the relationships of congruency of triangles, using algebraic and deductive proofs.

b. Use the relationships of congruency of 2-dimensional figures to determine unknown values, such as angles, side lengths, perimeter or circumference, and area.

6. Circles

a. Find angle measures and arc measures related to circles.

b. Find angle measures and segment lengths using the relationships among radii, chords, secants, and tangents of a circle.

Standard 3: Triangles and Trigonometric Ratios - The student will use the properties of right triangles and trigonometric ratios to solve problems.

1. Use the Pythagorean Theorem and its converse to find missing side lengths and to determine acute, right, and obtuse triangles, and verify using algebraic and deductive proofs.

2. Apply the 45-45-90 and 30-60-90 right triangle relationships to solve problems, and verify using algebraic and deductive proofs.

3. Express the trigonometric functions as ratios and use sine, cosine, and tangent ratios to solve real-world problems.

* 4. Use the trigonometric ratios to find the area of a triangle.

Standard 4: Properties of 3-Dimensional Figures - The student will use the properties and formulas of geometric figures to solve problems.

1. Polyhedra and Other Solids

a. Identify, describe, and analyze polyhedra (for example, regular, decahedral).

b. Use properties of 3-dimensional figures; side lengths, perimeter or circumference, and area of a face; and volume, lateral area, and surface area to determine unknown values and correctly identify the appropriate unit of measure of each.

2. Similarity and Congruence

a. Use ratios of similar 3-dimensional figures to determine unknown values, such as angles, side lengths, perimeter or circumference of a face, area of a face, and volume.

b. Use the relationships of congruency of 3-dimensional figures to determine unknown values, such as angles, side lengths, perimeter or circumference of a face, area of a face, and volume.

3. Create a model of a 3-dimensional figure from a 2-dimensional drawing and make a 2-dimensional representation of a 3-dimensional object (for example, nets, blueprints, perspective drawings).

Standard 5: Coordinate Geometry - The student will solve problems with geometric figures in the coordinate plane.

1. Use coordinate geometry to find the distance between two points; the midpoint of a segment; and to calculate the slopes of parallel, perpendicular, horizontal, and vertical lines.

2. Properties of Figures

a. Given a set of points determine the type of figure formed based on its properties.

b. Use transformations (reflection, rotation, translation) on geometric figures to solve problems within coordinate geometry.

MATHEMATICS CONTENT STANDARDS

Algebra II

The following skills are required of all students completing Algebra II. Major Concepts should be taught in depth using a variety of methods and applications (concrete to the abstract). Maintenance Concepts have been taught previously and are a necessary foundation for this course. The major concepts are considered minimal exit skills and districts are strongly encouraged to exceed these skills when building an Algebra II curriculum. Visual and physical models, calculators, and other technologies are recommended when appropriate and can enhance both instruction and assessment.

MAJOR CONCEPTS MAINTENANCE CONCEPTS

Number Systems and Algebraic Operations – Polynomials

Real and Complex Numbers Exponents

Functions and Relations - Expressions

Quadratic, Polynomial, Exponential, Slope

Logarithmic, Rational Data Displays

Data Analysis, Statistics, and Probability

Relationships, Measures of Central Tendency

And Variability, Sequences and Series

Standard 1: Number Systems and Algebraic Operations - The student will perform operations with rational, radical, and polynomial expressions, as well as expressions involving complex numbers.

1. Rational Exponents

a. Convert expressions from radical notations to rational exponents and vice versa.

b. Add, subtract, multiply, divide, and simplify radical expressions and expressions containing rational exponents.

2. Polynomial and Rational Expressions

a. Divide polynomial expressions by lower degree polynomials.

b. Add, subtract, multiply, divide, and simplify rational expressions, including complex fractions.

3. Complex Numbers

* a. Recognize that to solve certain problems and equations, number systems need to be extended from real numbers to complex numbers.

b. Add, subtract, multiply, divide, and simplify expressions involving complex numbers.

Standard 2: Relations and Functions - The student will use the relationships among the solution of an equation, zero of a function, x-intercepts of a graph, and factors of a polynomial expression to solve problems involving relations and functions.

1. Functions and Function Notation

a. Recognize the parent graphs of polynomial, exponential, and logarithmic functions and predict the effects of transformations on the parent graphs, using various methods and tools which may include graphing calculators.

b. Add, subtract, multiply, and divide functions using function notation.

c. Combine functions by composition.

d. Use algebraic, interval, and set notations to specify the domain and range of functions of various types.

e. Find and graph the inverse of a function, if it exists.

2. Systems of Equations

a. Model a situation that can be described by a system of equations or inequalities and use the model to answer questions about the situation.

b. Solve systems of linear equations and inequalities using various methods and tools which may include substitution, elimination, matrices, graphing, and graphing calculators.

c. Use either one quadratic equation and one linear equation or two quadratic equations to solve problems.

3. Quadratic Equations and Functions

a. Solve quadratic equations by graphing, factoring, completing the square and quadratic formula.

b. Graph a quadratic function and identify the x- and y-intercepts and maximum or minimum value, using various methods and tools which may include a graphing calculator.

c. Model a situation that can be described by a quadratic function and use the model to answer questions about the situation.

4. Identify, graph, and write the equations of the conic sections (circle, ellipse, parabola, and hyperbola).

5. Exponential and Logarithmic Functions

a. Graph exponential and logarithmic functions.

b. Apply the inverse relationship between exponential and logarithmic functions to convert from one form to another

c. Model a situation that can be described by an exponential or logarithmic function and use the model to answer questions about the situation.

6. Polynomial Equations and Functions

a. Solve polynomial equations using various methods and tools which may include factoring and synthetic division.

b. Sketch the graph of a polynomial function.

c. Given the graph of a polynomial function, identify the x- and y-intercepts, relative maximums and relative minimums, using various methods and tools which may include a graphing calculator.

d. Model a situation that can be described by a polynomial function and use the model to answer questions about the situation.

7. Rational Equations and Functions

a. Solve rational equations.

b. Sketch the graph of a rational function.

c. Given the graph of a rational function, identify the x- and y-intercepts, asymptotes, using various methods and tools which may include a graphing calculator.

d. Model a situation that can be described by a rational function and use the model to answer questions about the situation.

Standard 3: Data Analysis and Statistics - The student will use data analysis and statistics to formulate and justify predictions from a set of data.

1. Analysis of Collected Data Involving Two Variables

a. Display data on a scatter plot.

b. Interpret results using a linear, exponential or quadratic model/equation.

c. Identify whether the model/equation is a curve of best fit for the data, using various methods and tools which may include a graphing calculator.

* 2. Measures of Central Tendency and Variability

a. Analyze and synthesize data from a sample using appropriate measures of central tendency (mean, median, mode, weighted average).

b. Analyze and synthesize data from a sample using appropriate measures of variability (range, variance, standard deviation).

c. Use the characteristics of the Gaussian normal distribution (bell-shaped curve) to solve problems.

d. Identify how given outliers affect representations of data.

3. Identify and use arithmetic and geometric sequences and series to solve problems.

GLOSSARY

addend - in the addition problem 3 + 2 + 6 = 11, the addends are 3, 2, and 6.

algorithm - step-by-step procedure for solving a problem.

analog time - time displayed on a timepiece having hour and minute hands.

array - (rectangular) an orderly arrangement of objects into a rectangular configuration (e.g., take six tiles and arrange two long and three wide to form a rectangle).

attribute - characteristics (e.g., size, shape, color, weight).

combinations - a selection of objects without regard to order.

complementary angles - two angles whose measure have a sum of 90 degrees.

complex numbers - numbers of the form a + bi, where a and b are real numbers and i equals the square root of -1.

composite numbers - any positive integer exactly divisible by one or more positive integers other than itself and 1.

congruent - geometric figures having exactly the same size and shape.

conic sections - circles, parabolas, ellipses, and hyperbolas which can all be represented by passing a plane through a hollow double cone.

conjecture - a statement believed to be true but not proved.

cosine - in a right triangle, the cosine of an acute angle is the ratio of the length of the leg adjacent to the angle to the length of the hypotenuse.

dependent events - events that influence each other. If one of the events occurs, it changes the probability of the other event.

domain of a relation - the set of all the first elements or x-coordinates of a relation.

exponential function - an exponential function with base b is defined by y = b^x, where b > 0 and b is not equal to 1.

expression - a mathematical phrase that can include operations, numerals and variables. In algebraic terms: 2m + 3x; in numeric terms: 2.4 - 1.37.

Fibonacci sequence - the sequence of numbers, 1, 1, 2, 3, 5, 8, 13, 21, . . . where each number, except the first two, is the sum of the two preceding numbers.

function - a relation in which each element of the domain is paired with exactly one element of the range.

function machine - an input/output box (often made with milk cartons, boxes, or drawn on the board) to show one number entering and a different number exiting. Students guess the rule that produced the second number (e.g., enter 3, exit 5, rule: add 2).

histogram - a bar graph of a frequency distribution.

imaginary number - any complex number, a + bi, for which a = 0 and b does not = 0.

independent events - events that do not influence one another. Each event occurs without changing the probability of the other event.

integers - . . . -2, -1, 0, 1, 2, . . .

intercepts (x & y) - the x (y)-coordinate of the point where a graph intercepts the x (y)- axis.

inverse operations - operations that undo each other (e.g., addition and subtraction are inverse operations; multiplication and division are inverse operations).

irrational numbers - nonterminating, nonrepeating decimals (e.g., square root of 2, pi).

logarithmic functions - logarithmic function with base b is the inverse of the exponential function, and is defined by x = log_b y (y > 0, b > 0, b not equal to 1).

manipulatives - concrete materials (e.g., buttons, beans, egg and milk cartons, counters, attribute and pattern blocks, interlocking cubes, base-10 blocks, geometric models, geoboards, fractions pieces, rulers, balances, spinners, dot paper) to use in mathematical calculations.

mean - in a set of n numbers, the sum of the numbers divided by n.

median - the middle number in the set, or the mean of the two middle numbers, when the numbers are arranged in order from least to greatest.

mode - a number in a set of data that occurs most often.

multiple - a number that is the product of a given integer and another integer (e.g., 6 and 9 are multiples of 3).

natural numbers - (counting numbers) 1, 2, 3, 4, . . .

nonstandard measurement - a measurement determined by the use of nonstandard units like hands, paper clips, beans, cotton balls, etc.

number sense - involves the understanding of number size (relative magnitude), number representations, number operations, referents for quantities and measurements used in everyday situations, etc.

operation - addition, subtraction, multiplication, division, etc.

order of operations - rules for evaluating an expression: work first within parentheses; then calculate all powers, from left to right; then do multiplications or divisions, from left to right; then do additions and subtractions, from left to right.

ordinal - a number that is used to tell order (e.g., first, fifth).

permutation - an arrangement of a set of objects in a particular order (the letters a, b, c have the following permutations: abc, acb, bac, bca, cab, cba).

prime number - an integer greater than one whose only positive factors are 1 and itself (e.g., 2, 3, 5, 7, 11, 13 . . .).

probability - the study and measure of the likelihood of an event happening.

properties of arithmetic - for all real numbers a, b and c:

commutative property: a + b = b + a and a • b = b • a

associative property: (a + b) + c = a + (b + c) and (a • b) • c = a • (b • c)

distributive property: a(b + c) = (a • b) + (a • c)

identity property: a + 0 = a and a • 1 = a

inverse property: a + (-a) = 0 and a • = 1

proportion - a statement that ratios are equal.

quadrants - the four regions formed by the axes in a coordinate plane.

quadratic equation - an equation of the form ax² + bx + c = 0, where a, b and c are real numbers and a is not equal to 0.

quadratic formula - if ax²+ bx + c = 0, where a, b and c are real numbers and a is not equal to

0, then x = . −b−4

range of a relation - the set of all the second elements or y-coordinates of a relation is called the range.

ratio - the comparison of two quantities by division.

rational numbers - quotients of integers (commonly called fractions - includes both positive and negative numbers).

real numbers - the set of all rational and irrational numbers.

recursive patterns - patterns in which each number is found from the previous number by repeating a process (e.g., Fibonacci numbers).

relation - a set of one or more pairs of numbers.

relative magnitude - the size of an object or number compared to other objects and numbers.

scatter plot - a dot or point graph of data.

sequence - a set of numbers arranged in a pattern.

sine - in a right triangle, the sine of an acute angle is the ratio of the length of the leg opposite the angle to the length of the hypotenuse.

slope of a line - the ratio of the change in y to the corresponding change in x. For any

two points (x₁, y₁) and (x₂, y₂), m = . 21 (x₂ - x₁)

(y - y)

spatial sense - involves building and manipulating mental representations of 2- and 3-dimensional objects and ideas.

standard deviation - measures how much each value in the data differs from the mean of the data.

statistics - the study of data.

stem-and-leaf plot - a frequency distribution made by arranging data in the following way (e.g., student scores on a test were 96, 87, 77, 93, 85, 85, and 75 would be displayed as 9 | 6, 3

8 | 7, 5, 5

7 | 7, 5

supplementary angles - two angles whose measures have a sum of 180 degrees.

supposition - (act of supposing) making a statement or assumption without proof.

tangent - in a right triangle, the tangent is the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle.

transformation - motion of a geometric figure (rotation [turn], translation [slide], and reflection [flip]).

whole numbers - 0, 1, 2, 3, 4, . . .