# Proven to Raise Achievement for Struggling Students Grades 2 12

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1 Nevada State Standards Proven to Raise Achievement for Struggling Students Grades 2 12 RTI tel epsbooks.com fax

2 Standard 1.0 Grade 2 Numbers, Number Sense, and Computation: To solve problems, communicate, reason, and make connections within and beyond the field of mathematics, students will accurately calculate and use estimation techniques, number relationships, operation rules, and algorithms; they will determine the reasonableness of answers and the accuracy of solutions.

4 Standard 2.0 Patterns, Functions, and Algebra: To solve problems, communicate, reason, and make connections within and beyond the field of mathematics, students will use various algebraic methods to analyze, illustrate, extend, and create numerous representations (words, numbers, tables, and graphs) of patterns, functions, and algebraic relations as modeled in practical situations. Recognize, describe, extend, create, and use complex repeating and increasing patterns with symbols, objects, manipulatives, and numbers to solve problems, using calculators and computers when available. Use variables and number relationships to identify missing terms in open sentences. Create, model, explain, and solve addition and subtraction number sentences to describe situations involving equality and inequality. Recognize, describe, extend, create, and use repeating and increasing patterns using symbols, objects, and manipulatives to solve problems. Use variables and open sentences to express relationships. Create, model, explain, and solve problems using addition and subtraction. Distinguish between repeating and growing patterns; create and describe patterns such as repeating patterns and growing patterns using number, shape, size, color, and letter. Describe functions related to coin trades and measurement trades (e.g., five pennies make one nickel; four cups make one quart). Skip count forward and backward by twos, fives, and tens up 100. Construct and solve open sentences that have variables. Recognize, describe, and extend patterns such as sequences of sounds and shapes or simple numeric patterns and translate from one representation to another Use concrete, pictorial, and verbal representations to develop an understanding of invented and conventional symbolic notations Understand various meanings of addition and subtraction of whole numbers and the relationship between the two operations. Use the commutative and associative rules for addition to simplify mental calculations and to check results.

5 Standard 3.0 Measurement: To solve problems, communicate, reason, and make connections within and beyond the field of mathematics, students will use appropriate tools and techniques of measurement to determine, estimate, record, and verify direct and indirect measurements. Identify and use the correct unit of measure for time, temperature, length, weight, capacity, volume, and area and describe and define various attributes. Identify and use the correct unit of measure to compare differences in objects that are greater than, less than, and /or equal to a given unit. Create possible combinations of bills and coins to equal a given amount. Use a calendar to identify days, weeks, months, year(s), and elapsed time to solve problems. Read time to the nearest five minute interval. Compare and order objects by various measurable attributes, such as time, temperature, length, weight, capacity, volume, and area and describe and define these various attributes. Compare objects that are greater than, less than, and /or equal to a given unit of measure such as inch, yard, centimeter, and meter. Determine the value of any given set of coins and bills. Recite and use the months of the year in order and use a calendar to identify days, weeks, months, and year. Read time to the nearest quarter hour and distinguish between A.M. and P.M. Compare the length, weight, and volume of two or more objects by using direct comparison. Make and use estimates of measurement, including time and weight. Measure the length of objects by repeating a nonstandard or standard unit. Tell time at half-hour intervals on analog and digital clocks using a.m. and p.m. and relate time to events. Make combinations of coins up to 50 cents. Tell time at half-hour intervals on analog and digital clocks using a.m. and p.m. and relate time to events. Demonstrate an understanding of such attributes as length, area, and weight; select the appropriate type of unit for measuring each attribute using both the US Customary and metric systems. Carry out simple unit conversions within a system of measurement such as hours to minutes and cents to dollars (e.g., one hour = 60 minutes).

6 Standard 4.0 Spatial Relationships and Geometry: To solve problems, communicate, and make connections within and beyond the field of mathematics, students will identify, represent, verify, and apply spatial relationships and geometric properties. Identify, sort, sketch, describe, compare, and contrast plane geometric figures regardless of position. Describe the location of objects and, when given directions, place objects in position relative to each other. Identify and describe similar and congruent two-dimensional figures regardless of how they are positioned relative to each other. Identify symmetry in figures in the environment and create figures and designs that have more than one line of symmetry. Describe, sketch, model, and build two- and three- dimensional figures. Identify, name, sort, sketch, describe, and compare circles, triangles, and rectangles including squares, regardless of position. Describe the location of objects and place objects in position using vocabulary such as before, far, below, and left. Compare the size of similar two-dimensional figures and identify shapes that are congruent. Identify symmetry in figures in the environment and create figures and designs that have a line of symmetry. Identify, name, sort, describe, compare, and contrast two- and three-dimensional figures. Identify, describe, draw, and compare two-dimensional shapes, including both polygonal (up to six sides) and curved figures such as circles. Classify familiar two- and three-dimensional shapes by common attributes such as shape of curved and straight lines, number and shape of faces, edges, and vertices. Match and construct congruent and symmetric shapes. Relate geometric ideas to numbers (e.g., seeing rows in an array as a model of repeated addition). Describe, name, and interpret relative positions in space and apply ideas about relative position. Develop understanding of the relative position and magnitude of whole numbers and of ordinal and cardinal numbers and their connections Apply techniques such as reflections (flips), rotations (turns), and translations (slides) for determining if two shapes are congruent.

7 Standard 5.0 S Data Analysis: To solve problems, communicate, reason, and make connections within and beyond the field of mathematics, students will collect, organize, display, interpret, and analyze data to determine statistical relationships and probability projections. Collect, organize, record, explain, and analyze data using concrete materials and surveys. Collect, organize, record, and explain classification of data using concrete materials. Classify data using tallies, charts, tables, bar graphs, pictographs, and interpret the representations.

10 Standard 2.0 Patterns, Functions, and Algebra: To solve problems, communicate, reason, and make connections within and beyond the field of mathematics, students will use various algebraic methods to analyze, illustrate, extend, and create numerous representations (words, numbers, tables, and graphs) of patterns, functions, and algebraic relations as modeled in practical situations. Recognize, describe, extend, and create repeating, increasing and decreasing patterns using numbers; use number patterns and their extensions to solve complex problems using complicated patterns. Identify missing symbols (+, -, X, >, <, =) and missing numbers in open number sentences involving number facts in addition, subtraction, and multiplication. Recognize, describe, extend, and create repeating and increasing patterns using numbers; use number patterns and their extensions to solve problems. Identify missing symbols (+, -, >, <, =) and missing numbers in open number sentences involving number facts in addition and subtraction. Create, describe, extend, and explain symbolic (geometric) patterns and addition and subtraction patterns; describe patterns in a variety of ways. Use boxes or other symbols to represent unknowns or quantities that vary in expressions and in equations or inequalities (=, < and >). Select appropriate operational and relational symbols to make an expression true. Determine values of variables in simple equations involving addition, subtraction, or multiplication. Know and express the relationships among linear units of measure, i.e., unit conversions (e.g., 3 feet=1 yard; 12 inches=1 foot). Extend and recognize a linear pattern by its rules. Distinguish between repeating and growing patterns; create and describe patterns such as repeating patterns and growing patterns using number, shape, size, color, and letter. Use the commutative and associative rules for addition to simplify mental calculations and to check results.

11 Standard 3.0 Measurement: To solve problems, communicate, reason, and make connections within and beyond the field of mathematics, students will use appropriate tools and techniques of measurement to determine, estimate, record, and verify direct and indirect measurements. Measure to a required degree of accuracy and record the measurements, evaluating for error and describing the appropriateness of self-selected units of measure. Estimate measurements and select and use appropriate measuring devices with standard units to measure length, surface area, liquid volume, capacity, temperature, and weight. Read, write, and use money notation and determine possible combinations of coins and bills to equal given amounts, and apply to practical situations. Read time to the nearest minute using analog and digital clocks and determine elapsed time, applying it to practical situations. Measure to a required degree of accuracy, record the measurement, and evaluate it for error, describing the appropriateness of selected units of measure. Estimate measurements and use measuring devices with standard and non-standard units to measure length, area of a region, liquid volume, capacity, temperature, and weight, communicating the concepts of more, less, and equivalent. Read, write, and use money notation and determine possible combinations of coins and bills to equal given amounts. Read time to the nearest minute using analog and digital clocks and determine elapsed time. Demonstrate an understanding of such attributes as length, area, and weight; select the appropriate type of unit for measuring each attribute using both the US Customary and metric systems. Carry out simple unit conversions within a system of measurement such as hours to minutes and cents to dollars (e.g., one hour = 60 minutes). Identify time to the nearest five minutes on analog and digital clocks using a.m. and p.m. Estimate and find area and perimeter of a rectangle and triangle. Understand the need for measuring with standard units and become familiar with standard units in the customary and metric systems Understand that measurements are approximations and understand how differences in units affect precision Understand such attributes as length, area, weight, volume, and size of angle and select the appropriate type of unit for measuring each attribute Select and apply appropriate standard units and tools to measure length, area, volume, weight, time, temperature, and the size of angles

12 Standard 4.0 Spatial Relationships and Geometry: To solve problems, communicate, and make connections within and beyond the field of mathematics, students will identify, represent, verify, and apply spatial relationships and geometric properties. Describe, sketch, compare, and contrast plane geometric figures in great detail. Demonstrate and describe a sequence of transformations (motions) of geometric figures as slides, rotations, and/or flips. Describe, sketch, model, build, compare, and contrast two- and three-dimensional geometric figures, with great detail. Describe, sketch, compare, and contrast plane geometric figures. Demonstrate and describe the transformation (motion) of a geometric figure as a slide, rotation, or a flip. Describe, sketch, model, build, compare, and contrast two- and three-dimensional geometric figures. Compare and analyze attributes and other features (e.g., number and shape of sides, faces, corners, right angles) of two-dimensional geometric shapes, especially the attributes of triangles (isosceles, equilateral, right) and quadrilaterals (rectangle, square, parallelogram). Describe, compare, and classify two-dimensional shapes such as circles and polygons, especially triangles and quadrilaterals. Identify angles as right, acute (less than a right angle), or obtuse (greater than a right angle). Identify parallel, perpendicular, and intersecting lines. Identify lines of symmetry in two-dimensional shapes. Describe, name, and interpret relative positions in space and apply ideas about relative position. Develop understanding of the relative position and magnitude of whole numbers and of ordinal and cardinal numbers and their connections Apply techniques such as reflections (flips), rotations (turns), and translations (slides) for determining if two shapes are congruent.

13 Standard 5.0 Data Analysis: To solve problems, communicate, reason, and make connections within and beyond the field of mathematics, students will collect, organize, display, interpret, and analyze data to determine statistical relationships and probability projections. Collect, organize, display, describe, and interpret simple data using number lines, pictographs, bar graphs, and frequency tables, by hand and with computers when they are available. Use concepts of probability to make predictions about future events. Collect, organize, display, and describe simple data using number lines, pictographs, bar graphs, and frequency tables, by hand and with computers when they are available. Use concepts of probability such as impossible, unlikely, likely, and certain to make predictions about future events. Organize data using observations, measurements, surveys, or experiments. Identify the main idea, and make predictions from various representations of data sets in the forms of tables, bar graphs (horizontal and vertical forms), pictographs, and tallies.

14 Standard 1.0 Numbers, Number Sense and Computation: To solve problems, communicate, reason, and make connections within and beyond the field of mathematics, students will accurately calculate and use estimation techniques, number relationships, operation rules, and algorithms; they will determine the reasonableness of answers and the accuracy of solutions. Immediately recall and use multiplication and corresponding division facts. Multiply and divide multi-place numbers. Generate and solve complex addition, subtraction, multiplication, division problems involving whole numbers and order of operations in practical situations. Compare and order negative numbers within the context of practical situations and plot rational numbers on a number line. Estimate, using a variety of methods, to determine and justify the reasonableness of an answer. Model, draw, identify, compare, add, and subtract decimals and fractions with like and unlike denominators to solve problems. Grade 5 Estimate, round, and manipulate very large (e.g., billions) and very small (e.g., thousandths) numbers; demonstrate an understanding of place value to billions and thousandths. Represent and compare very large (billions) and very small (thousandths) positive numbers in various forms such as expanded notation without exponents. Find and position integers, fractions, mixed numbers, and decimals (both positive and negative) on the number line. Compare and order integers (including negative integers) and positive fractions, mixed numbers, decimals, and percents. Apply the number theory concepts of common factor, common multiple, and divisibility rules for 2, 3, 5, and 10 to the solution of problems. Demonstrate an understanding of the concepts of prime and composite

15 Immediately recall and use multiplication and corresponding division facts using factors of 0 through 12. Multiply and divide multi-place numbers by two-digit numbers including multiples of 10. Generate and solve addition, subtraction, multiplication, and division problems involving whole numbers and order of operations in practical situations. Compare and order negative numbers within the context of practical situations and plot integer values on a number line. Estimate to determine the reasonableness of answer by identifying and using the correct place value position. Model, draw, identify, compare, add, and subtract decimals and fractions with like denominators to solve problems. numbers. Know the set of prime numbers to 100. Determine the prime factors of all numbers through 50 and write the numbers as the product of their prime factors by using exponents to show multiples of a factor (e.g., 24=2x2x2x3=23x3). Explain different interpretations of fractions as a ratio of whole numbers, as parts of unit wholes, as parts of a collection, as division of whole numbers by whole numbers, as locations on the number line. Interpret percents as parts out of 100, use % notation, and express a part of a whole as a percentage. Identify and determine common equivalent fractions, mixed numbers, decimals, and percents. Write improper fractions as mixed numbers, and know that a mixed number represents the number of wholes and the part of a whole remaining. Add and subtract fractions with like and unlike denominators and express answers in the simplest form. Add and subtract positive decimals. Use models to show an understanding of multiplication and division of fractions; multiply positive fractions with whole numbers. Simplify fractions in cases when both the numerator and the denominator have 2, 3, 4, 5, or 10 as a common factor. Multiply positive decimals with whole numbers. Add with negative integers, subtract positive integers from negative integers, and verify the reasonableness of the results. Solve problems involving multiplication and division of any whole number. Demonstrate proficiency with division, including division with positive decimals and long division with multi-digit divisors. Demonstrate an understanding of and compute (positive integer) powers of ten (e.g., 102, 105); compute examples as repeated multiplication. Demonstrate an understanding of how parentheses affect expressions involving addition, subtraction, and multiplication, and use that understanding to solve problems, e.g., 3 x (4 + 2) = 3 x 6. Estimate sums and differences of whole numbers, positive fractions, and positive decimals. Estimate products of whole numbers and products of positive decimals with whole numbers. Use a variety of strategies and judge reasonableness of answers.

16 Standard 2.0 Patterns, Functions, and Algebra: To solve problems, communicate, reason, and make connections within and beyond the field of mathematics, students will use various algebraic methods to analyze, illustrate, extend, and create numerous representations (words, numbers, tables, and graphs) of patterns, functions, and algebraic relations as modeled in practical situations. Create tables and charts to identify, describe, and extend number patterns and relationships. Use variables in open sentences to describe a wide variety of functions and relationships. Solve simple equations and inequalities using whole numbers, decimals, and common fractions. Generate complex number sequences given the first term of the sequence and any computation rule. Identify, describe, and explain number patterns and relationships, including triangular numbers, perfect squares, arithmetic and geometric sequences, using concrete materials, paper and pencil, and calculators. Use variables in open sentences to describe simple functions and relationships. Solve simple whole numbers equations and inequalities using a variety of methods. Generate number sequences given the first term of the sequence and any simple computation rule. Analyze and determine the rules for extending symbolic, arithmetic, and geometric patterns. Replace variables with given values, evaluate and simplify. Use the properties of equality to solve problems with whole numbers (e.g., if x + 7 = 13, then x = 13 7, therefore x = 6; if 3 x = 15, then 1/3 x 3 x = 1/3 x 15, therefore = 5). Interpret and evaluate mathematical expressions that use parentheses; use parentheses to indicate which operation to perform first when writing expressions containing more than two terms and different operations. Interpret graphs that represent the relationship between two variables in everyday situations. Use the properties of equality to solve problems with whole numbers (e.g., if x + 7 = 13, then x = 13 7, therefore x = 6; if 3 x = 15, then 1/3 x 3 x = 1/3 x 15, therefore = 5). Use, the commutative, associative, and identity properties of operations on whole numbers in problem situations. Use the relationship between multiplication and division to simplify computations and check results.

17 Standard 3.0 Measurement: To solve problems, communicate, reason, and make connections within and beyond the field of mathematics, students will use appropriate tools and techniques of measurement to determine, estimate, record, and verify direct and indirect measurements. Measure and compare lengths, masses, and capacities and convert those measurements within the same measurement system. Estimate measures of length, volume, capacity, quantity, and weight justifying the reasonableness of the estimates. Select the method of measurement and justify the use of estimation or direct measurement. Determine several different combinations of bills and coins that would provide correct change in practical situations. Determine the perimeter and area of given polygons, and describe how changes in dimensions affect changes in area. Convert units of time to equivalent units. Measure, compare, and convert units of length, within the same measurement system, to the nearest fractional/decimal part. Estimate and directly measure length, volume, capacity, and quantity. Select and justify the use of estimation or direct measurement and weight in a given situation. Determine the total cost of purchases and the amount of change in practical situations. Describe the difference between perimeter and area and determine the perimeter of any polygon and the area of right triangles and rectangles, including squares. Identify equivalent periods of time using relationships between and among seconds, minutes, hours, days, months, and years. Apply the concepts of perimeter and area to the solution of problems involving triangles and rectangles. Apply formulas where appropriate. Apply formulas for the areas of triangles, rectangles, and parallelograms; recognize that shapes with the same number of sides but different appearances can have the same area. Solve problems involving proportional relationships and units of measurement. Identify, measure, and describe circles and the relationships of the radius, diameter, circumference, and area (e.g., d=2r) and use these concepts to solve problems. Find volumes and surface areas of rectangular prisms. Know that angles on a straight line add up to 180, interior angles of a triangle add up to 180, angles surrounding a point add up to 360, and interior angles of a quadrilateral add up to 360 ; use these properties to solve problems. Identify angles and triangles, given sides and the angle between them or given two angles and the side between them. Understand the need for measuring with standard units and become familiar with standard units in the customary and metric systems Understand that measurements are approximations and understand how differences in units affect precision Understand such attributes as length, area, weight, volume, and size of angle and select the appropriate type of unit for measuring each attribute Select and apply appropriate standard units and tools to measure length, area, volume, weight, time, temperature, and the size of angles

18 Standard 4.0 Spatial Relationships and Geometry: To solve problems, communicate, and make connections within and beyond the field of mathematics, students will identify, represent, verify, and apply spatial relationships and geometric properties. Draw and classify angles and triangles according to given measurements. Identify, draw, and label circles and elements of circles, describing the relationships between the various elements. Identify transformations as a translation, rotation, reflection, enlargement, or reduction using the formal vocabulary for shapes that have congruence, similarity, and symmetry. Identify and draw shapes that have congruence, similarity, and symmetry using a wide variety of methods. Graph ordered pairs and identify coordinates for a given point in any quadrant. Draw and classify complex two- and three-dimensional figures by their properties including the number of vertices, and edges and the number and shape of the faces. Identify, describe, classify, and construct one- and two dimensional geometric figures including intersecting, perpendicular and parallel lines, line segments, rays, and angles when given measurements and describe the relationships among various elements. Draw and classify angles and triangles as right, acute, or obtuse. Identify and draw circles and elements of circles, describing the relationships between the various elements. Identify a transformation as translation, rotation, reflection, enlargement, or reduction. Identify shapes that have congruence, similarity, and/or symmetry using a variety of methods, including transformational motions and models, drawings, and measurement tools. Graph ordered pairs and identify coordinates for a given point in the first quadrant. Identify, describe, compare and classify two- and threedimensional figures by their properties including the number of vertices, and edges and the number and shape of the faces. Identify, describe, classify and draw one- and two-dimensional geometric figures including intersecting, perpendicular and parallel lines, line segments, rays, and angles with given measurements. Identify polygons based on their properties, including types of interior angles, perpendicular or parallel sides, and congruence of sides (e.g., squares, rectangles, rhombuses, parallelograms, trapezoids, and isosceles, equilateral, and right triangles). Identify, describe, and compare special types of three-dimensional shapes (e.g., cubes, prisms, spheres, cones, and pyramids) based on their properties, such as edges and faces. Identify relationships among points, lines, and planes (e.g., intersecting, parallel, perpendicular). Identify and describe types of symmetry, including line and rotational. Determine if two triangles are congruent by measuring sides or a combination of sides and angles. Describe transformations on two-dimensional shapes. Identify coordinates of points on the Cartesian coordinate plane in the first two quadrants. Describe transformations on two-dimensional shapes. Identify and describe types of symmetry, including line and rotational. Determine if two triangles are congruent by measuring sides or a combination of sides and angles.

19 Standard 5.0 Data Analysis: To solve problems, communicate, reason, and make connections within and beyond the field of mathematics, students will collect, organize, display, interpret, and analyze data to determine statistical relationships and probability projections. Collect, organize, and interpret data using a variety of graphic representations. Use data and graphs to draw conclusions and/or make predictions using data in a variety of written and oral forms, with and without technology. Conduct simple and compound probability experiments using concrete materials and represent the results in fractional forms. Solve and analyze probability problems using a variety of methods. Use measures of central tendency in practical problem situations. Select a type of graph to represent a given set of data and provide written and oral justification of selection, including discussion of limitations of graphs not selected. Collect, organize, read, and interpret data using graphic representations including tables, line plots, stem and leaf plots, scatter plots, and histograms. Use data and graphs to formulate and explain conclusions and predictions with and without technology. Conduct simple probability experiments using concrete materials and represent the results in fractional form. Solve probability problems using a variety of methods including constructing sample spaces and tree diagrams. Model and compute measures of central tendency including mean, median, and mode. Describe the limitations of various graph formats and select a type of graph to accurately represent the given data; justify the selection. Organize data using observations, measurements, surveys, or experiments and identify appropriate ways to display the data. Make predictions from various representations of data sets, including tables and bar graphs (where symbols or scales represent multiple units), line graphs, and line plots. Interpret line plots, bar graphs, and circle graphs. Define and apply the concepts of mean, median, and mode. Make predictions from various representations of data sets, including tables and bar graphs (where symbols or scales represent multiple units), line graphs, and line plots.

21 Standard 2.0 Patterns, Functions, and Algebra: To solve problems, communicate, reason, and make connections within and beyond the field of mathematics, students will use various algebraic methods to analyze, illustrate, extend, and create numerous representations (words, numbers, tables, and graphs) of patterns, functions, and algebraic relations as modeled in practical situations. Use inductive reasoning to find a missing term in numeric, arithmetic, and geometric sequences and to generalize basic patterns and formulas to the nth term, with and without calculators. Identify, model, describe, and evaluate complex relationships including functions using diagrams, written, oral, graphic, and symbolic language. Solve a complex equation or formula for any variable. Describe and demonstrate how a change in one variable of a complex mathematical relationship affects the remaining variables. Solve complex linear equations and inequalities. Add and subtract polynomials describing the connection between the algebraic and arithmetic processes. Use inductive reasoning to find a missing term in numeric, arithmetic, and geometric sequences and to generalize basic patterns to the nth term, with and without calculators. Identify, describe, model, and evaluate relationships including patterns, sequences, and functions using oral, written, and symbolic language, with and without technology. Solve an equation or a formula for any variable. Describe how a change in one variable of a mathematical relationship affects the remaining variables using various tools and methods. Model, identify, and solve simple linear equations and inequalities and relate that process to the order of operations, using formal and informal methods. Add and subtract binomials describing the connection between the algebraic process and the arithmetic process. Solve linear equations and inequalities with one or two variables using algebraic methods. Use linear equations to analyze problems. Identify the roles of variables within an equation. Distinguish between numerical and algebraic expressions, equations, and inequalities

22 Standard 3.0 Measurement: To solve problems, communicate, reason, and make connections within and beyond the field of mathematics, students will use appropriate tools and techniques of measurement to determine, estimate, record, and verify direct and indirect measurements. Use conversion factors to compare and convert units of measure for length, weight/mass, and volume within the same measurement system (customary or metric); estimate conversions between like units of the two systems to solve problems. Describe the distinction between precision, error of measure, and tolerance in measurement when using an appropriate measurement tool. Determine an appropriate degree of accuracy and measure to that degree of accuracy for a specified measurement situation. Recall and apply formulas to find perimeter, circumference, and area of plane figures and volume and surface area of solid figures; identify the relationship between changes in area and volume and changes in linear measures of figures. Evaluate formulas and algebraic expressions for given values of a variable, using various tools and methods. Apply ratios and proportions in multi-step problems. Compare and convert units of measure for length, weight/mass, and volume within the same measurement system (customary or metric); estimate conversions between like units of the two systems to solve problems. Identify the range of precision, error of measure, and tolerance in measurement when using the appropriate measurement tool and measuring to the required degree of accuracy. Estimate and measure length, weight/mass, and volume to the required degree of accuracy. Derive and apply formulas to find perimeter, circumference, and area of plane figures and volume and surface area of solid figures; identify the relationship between changes in area and volume and changes in linear measures of figures. Evaluate formulas and algebraic expressions for given values of a variable. Apply ratio and proportion to calculate rates and as a method of indirect measure. Understand the need for measuring with standard units and become familiar with standard units in the customary and metric systems Understand that measurements are approximations and understand how differences in units affect precision Understand such attributes as length, area, weight, volume, and size of angle and select the appropriate type of unit for measuring each attribute Given the formulas, convert from one system of measurement to another. Understand the concept of surface area and volume; given the formulas, determine the surface area and volume of rectangular prisms. Use proportions to express relationships between corresponding parts of similar figures.

23 Standard 4.0 Spatial Relationships and Geometry: To solve problems, communicate, and make connections within and beyond the field of mathematics, students will identify, represent, verify and apply spatial relationships and geometric properties. Identify, classify, compare, and draw regular and complicated irregular polygons, with given specifications and determine the sum of the interior angles of convex polygons, developing a rule to describe the sum. Apply the properties of equality and proportionality to solve complex problems involving congruent or similar shapes. Use coordinate geometry and graphs to show multiple-step geometric transformations. Create a variety of models of a three-dimensional figures from two-dimensional drawings and make two-dimensional proportional sketches of three-dimensional objects. Represent, interpret, and generalize relationships defined by equations and formulas (including distance, midpoint, and slope) on a coordinate plane, with and without technology. Form generalizations and validate conclusions about properties of geometric shapes including those associated with parallel lines, perpendicular lines, bisectors, triangles, and polygons and use these generalizations to solve problems. Verify, explain and use both the Pythagorean Theorem and the Triangle Sum Theorem to determine missing sides and angles of triangles in practical situations. Construct, draw, and sketch geometric figures, bisected angles and lines, accurately and efficiently using hand tools, technology, and models. Analyze, and apply the relationship between the number of sides and the sums of the interior and exterior angle measures of polygons. Demonstrate an understanding of the relationships of angles formed by intersecting lines, including parallel lines cut by a transversal. Demonstrate an understanding of conditions that indicate two triangles are similar: the corresponding angles are congruent (AAA similarity); the ratios of two pairs of corresponding sides are equal and the included angles are congruent (SAS similarity); ratios of all pairs of corresponding sides are equal (SSS similarity). Identify coordinates of points on the Cartesian coordinate plane in all four quadrants. Understand and use coordinate graphs to plot simple figures Determine if two shapes are congruent by motions or series of motions (e.g., translations, rotations, and reflections); predict the results of transformations on unmarked planes.

24 Identify, classify, compare, and draw regular and irregular polygons, given specifications; determine the sum of the interior angles of convex polygons. Apply the properties of equality and proportionally to solve problems involving congruent or similar shapes. Use coordinate geometry and models to illustrate change in scale and other geometric transformations. Create a model of a three-dimensional figure from twodimensional drawings and make a two-dimensional drawing of a three-dimensional object. Represent and interpret relationships defined by equations and formulas (including distance, midpoint, and slope) on a coordinate plane with and without technology. Form generalizations and validate conclusions about properties of geometric shapes including those associated with parallel lines, perpendicular lines, bisectors, triangles, and quadrilaterals. Verify, explain and use both the Pythagorean Theorem and the Triangle Sum Theorem to determine missing sides and angles of triangles. Construct, draw, and sketch geometric figures, bisected angles and lines and line segments with given specifications, using hand tools and technology.

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