Students will: apply multiple representations (e.g., drawings, manipulatives, base-10 blocks, number lines, expanded form, symbols) to represent whole numbers (0 to 99,999); apply multiple representations (e.g., drawings, manipulatives, base-10 blocks, number lines, symbols) to describe commonly used fractions through tenths and decimals through hundredths; apply these numbers to represent real-world problems; and explain how the base 10 number system relates to place value.

Students will read, write, and rename whole numbers, fractions, and decimals, and apply to real-world and mathematical problems.

Students will compare (<, >, =) and order whole numbers, commonly used fractions and decimals, and explain the relationships (equivalence, order) between and among them.

Students will apply and describe appropriate strategies for estimating quantities of objects and computational results.

Students will analyze real-world problems to identify the appropriate representations using mathematical operations, and will apply operations to solve real-world problems with the following constraints: add and subtract whole numbers with four digits or less; multiply whole numbers with two digits or less; divide whole numbers with three digits or less by single-digit divisors (with or without remainders); add and subtract fractions with like denominators less than or equal to 10; and add and subtract decimals through hundredths.

Students will skip-count forward and backward by 2s, 3s, 4s, 5s, 10s, 20s, 25s, 50s, 100s, 1,000, and 10,000s.

Students will identify and determine odd numbers, even numbers, multiples of a number, and factors of a number, and will apply these numbers to solve real-world problems.

Students will use the commutative properties of addition and multiplication, the associative properties of addition and multiplication, the identity properties of addition and multiplication and the zero property of multiplication in written and mental computation.

Students will apply standard units to measure length (nearest quarter-inch or nearest centimeter) and to determine; weight (ounce, pound; gram, kilogram); perimeter; area (figures that can be divided into rectangular shapes); time (nearest five minutes); and temperature (Fahrenheit and Celsius).

Students will choose and use appropriate tools (e.g., thermometer, scales, balances, clock, meter stick, yardstick, ruler) for specific measurement tasks.

Students will use nonstandard and standard units of measurement to identify measurable attributes of an object (length and width) using appropriate units of measurement.

Students will use measurements to describe and compare attributes of objects to include length (in, ft, yd, mile; cm, m, km), width, height, money (cost), temperature and weight (oz, lb, ton; g, kg); sort objects and compare attributes of objects.

Students will use nonstandard and standard units to measure angles (as compared to 90Âº).

Students will estimate weight, length, perimeter, area, angle measures and time using appropriate units of measurement.

Students will describe, define, give examples of and use to solve real-world and mathematical problems nonstandard and standard (U.S. Customary, metric) units of measurement (e.g., weight - oz., lbs., tons, g, kg; length â€“ in., ft., yd., mile, cm, m, km; area in square units) and money.

Students will determine elapsed time to the nearest quarter hour.

Students will convert units within the same measurement system, including money, time (seconds, minutes, hours, days, weeks, months, years), weight (ounces, pounds) and length (inches, feet, yards).

Students will describe and provide examples of basic geometric elements and terms [points, segments, lines (perpendicular, parallel, intersecting), rays, angles (acute, right, obtuse), sides, edges, faces, bases, vertices] and will apply these elements to solve real-world and mathematical problems.

Students will describe and provide examples of basic two-dimensional shapes [circles, triangles (right, equilateral), squares, rectangles, trapezoids, rhombuses, pentagons, hexagons, octagons] and will apply these shapes to solve real-world and mathematical problems.

Students will describe and provide examples of basic three-dimensional objects (spheres, cones, cylinders, pyramids, cubes, triangular, and rectangular prisms) and will apply the attributes to solve real-world and mathematical problems.

Students will explore two-dimensional representations of three-dimensional objects (nets).

Students will identify and describe congruent and similar figures in real-world and mathematical problems.

Students will describe and provide examples of line symmetry in real-world and mathematical problems or will apply one or two lines of symmetry to construct a simple geometric design.

Students will identify basic two-dimensional shapes in different orientations using 90 degree rotations (turns) around a point of rotation, reflections (flips), and translations (slides) within a plane.

Students will identify and graph ordered pairs on a positive coordinate system scaled by ones or locate points on a grid.

Students will analyze and make inferences from data displays (drawings, tables/charts, tally tables, pictographs, bar graphs, circle graphs, line plots, Venn diagrams).

Students will collect data.

Students will construct data displays (pictographs, bar graphs, line plots, Venn diagrams, tables).

Students will determine the median, mode (for a data set with no more than one mode) and range of a set of data.

Students will pose questions that can be answered by collecting data.

Students will determine all possible outcomes of an activity/event with up to six possible outcomes.

Students will determine the likelihood of an event and the probability of an event (expressed as a fraction).

Students will describe and give examples of the probability of an unlikely event (near zero) and a likely event (near one).

Students will extend patterns (e.g., 108, 208, 308, 408,...; square, circle, circle, triangle, square, circle, circle, triangle,...) from real world and mathematical problems; compare simple patterns (numbers, pictures, words; e.g., triangle, square, triangle, square, triangle, square; triangle, circle, circle, triangle, circle, circle); and describe rules for simple number patterns (e.g., 1, 3, 5, 7, ...; 5, 10, 15, 20, ...; 30, 27, 24, 21,...).

Students will describe functions (input-output) through pictures, tables, and words; and will analyze functions from a table based on real-world and mathematical problems.

Students will determine the value of an output given a function rule and an input value.

Students will model real-world and mathematical problems with simple number sentences (equations and inequalities) with a variable or a missing value (e.g., 4 = 7 - __, N + 5 > 14, 1/2 + N = 1) and apply simple number sentences to solve mathematical real-world problems.

Students will: apply multiple representations (e.g., drawings, manipulatives, base-10 blocks, number lines, expanded form, symbols) to represent whole numbers (0 to 99,999,999); apply multiple representations (e.g., drawings, manipulatives, base-10 blocks, number lines, symbols) to describe commonly-used fractions, mixed numbers, and decimals through thousandths; apply these numbers to represent real-world problems; and explain how the base-10 number system relates to place value.

Students will read, write, and rename whole numbers, fractions, and decimals, and apply to real-world and mathematical problems.

Students will compare (<, >, =) and order whole numbers), fractions and decimals, and explain the relationships (equivalence, order) between and among them.

Students will apply and describe appropriate strategies for estimating quantities of objects and computational results in real-world problems.

Students will analyze real-world problems to identify the appropriate representations using mathematical operations, and will apply operations to solve real-world problems with the following constraints: add, subtract, multiply, and divide whole numbers (less than 100,000,000), using technology where appropriate; add and subtract fractions with like denominators through 16, with sums less than or equal to one; and add and subtract decimals through hundredths.

Students will skip-count forward and backward.

Students will multiply decimals through tenths.

Students will identify and determine composite numbers, prime numbers, multiples of a number, factors of numbers, and least common multiples (LCM), and will apply these numbers to solve real-world problems.

Students will use the commutative properties of addition and multiplication, the associative properties of addition and multiplication, the identity properties of addition and multiplication and the zero property of multiplication in written and mental computation.

Students will apply standard units to measure length (to the nearest eighth-of-an-inch or nearest centimeter) and to determine: weight (ounce, pound; gram, kilogram); perimeter; area (figures that can divided into rectangular shapes); time (nearest minute); temperature (Fahrenheit and Celsius); and angle measures (nearest degree).

Students will choose and use appropriate tools (e.g., protractor, meter stick, ruler) for specific tasks and apply skills to solve real-world and mathematical problems.

Students will use measurements to identify, describe, sort and compare attributes of objects and apply these to solve real-world and mathematical problems.

Students will measure volume of rectangular prisms, liquid capacity, and money using standard units and apply these skills to solve real-world and mathematical problems.

Students will estimate weight, length, perimeter, area, angle measures and time using appropriate units of measurement.

Students will determine elapsed time.

Students will describe, define, give examples of and use to solve real-world and mathematical problems nonstandard and standard (U.S. Customary, metric) units of measurement.

Students will convert units within the same measurement system [U.S. customary (inches, feet, yards, miles; ounces, pounds, tons), metric (millimeters, centimeters, meters, kilometers; grams, kilograms), money, or time] and use the units to solve problems.

Students will describe and provide examples of basic geometric elements and terms [points, segments, lines (perpendicular, parallel, intersecting), rays, angles (acute, right, obtuse), sides, edges, faces, bases, vertices, radius, diameter] and will apply these elements to solve real-world and mathematical problems.

Students will describe and provide examples of basic two-dimensional shapes [circles, triangles (right, equilateral), all quadrilaterals, pentagons, hexagons, octagons] and will apply these shapes to solve real-world and mathematical problems.

Students will describe and provide examples of basic three-dimensional objects (spheres, cones, cylinders, pyramids, cubes, triangular and rectangular prisms) will identify three-dimensional objects from two-dimensional representations (nets) and will apply the attributes to solve real-world and mathematical problems.

Students will identify and describe congruent and similar figures in real-world and mathematical problems.

Students will describe and provide examples of line symmetry in real-world and mathematical problems or will apply line symmetry to construct a geometric design.

Students will identify or draw 90 degree rotations, reflections, or translations of basic shapes within a plane.

Students will identify and graph ordered pairs on a positive coordinate system scaled by ones, twos, threes, fives, or tens; locate points on a grid; and apply graphing in the coordinate system to solve real-world problems.

Students will analyze and make inferences from data displays (drawings, tables/charts, tally tables, pictographs, bar graphs, circle graphs, line plots, Venn diagrams, line graphs).

Students will collect data (e.g., tallies, surveys) and explain how the skills apply in real-world and mathematical problems.

Students will construct data displays (pictographs, bar graphs, line plots, line graphs, Venn diagrams, tables).

Students will determine and apply the mean, median, mode, and range of a set of data.

Students will describe and give examples of the process of using data to answer questions (e.g., pose a question, plan, collect data, organize and display data, interpret data to answer questions).

Students will determine all possible outcomes of an activityevent with up to 12 possible outcomes.

Students will determine the likelihood of an event and the probability of an event (expressed as a fraction).

Students will extend patterns, find the missing term(s) in a pattern or describe rules for patterns (numbers, pictures, tables, words) from real-world and mathematical problems.

Students will describe functions (input-output) through pictures, tables, or words and will construct tables to analyze functions based on real-world or mathematical problems.

Students will determine an output value or an input value for a function rule given the other value.

Students will model verbal descriptions of real-world and mathematical problems using a variable or a missing value in an expression.

Students will model real-world and mathematical problems with simple number sentences (equations and inequalities) with a variable or missing value (e.g., 4 = 2 x N, __ + 5 > 14) and apply number sentences to solve mathematical and real-world problems.

Students will provide examples of and identify fractions, decimals, and percents.

Students will describe and provide examples of representations of numbers (whole numbers, fractions in simplest form, mixed numbers, decimals, percents) and operations in a variety of equivalent forms using models, diagrams, and symbols (e.g., number lines, 10 by 10 grids, rectangular arrays, number sentences), based on real-world and mathematical problems.

Students will convert between any two of the following numbers: fractions, decimals, and percents (less than or equal to 100%); and will compare and order these numbers.

Students will estimate to solve real-world and mathematical problems with whole numbers, fractions, decimals, and percents, checking for reasonable and appropriate computational results.

Students will add, subtract, multiply, divide whole numbers, fractions, and decimals to solve real-world problems and apply order of operations to simplify numerical expressions.

Students will explain how operations (addition and subtraction; multiplication and division) are inversely related.

Students will describe and apply ratios to solve real-world problems.

Students will identify and apply prime numbers, composite numbers, prime factorization, factors, multiples and divisibility to solve real-world and mathematical problems (e.g. prime factorization to determine a least common multiple [LCM] or greatest common factor [GCF]).

Students will identify the use of properties (commutative properties of addition and multiplication, the associative properties of addition and multiplication and the identity properties for addition and multiplication) to simplify numerical expressions.

Students will measure lengths (to the nearest eighth of an inch or the nearest centimeter) and will determine and use in real-world and mathematical problems: area and perimeter of triangles; area and perimeter of quadrilaterals (rectangles, squares); (using the Pythagorean theorem will not be required as a strategy); and area and perimeter of compound figures composed of triangles and quadrilaterals.

Students will estimate measurements in standard units including fractions and decimals.

Students will explain how measurements and measurement formulas are related or different (perimeter and area of rectangles).

Students will convert units within the same measurement system and use these units to solve real-world problems.

Students will describe and provide examples of the basic geometric elements (points, rays, lines, segments, angles [acute, right, obtuse] planes, radius, diameter, circumference).

Students will describe, and provide examples of the elements (e.g., sides, vertices, angles, congruent parts) of two-dimensional figures (circles, triangles, quadrilaterals, regular polygons), and will apply these elements and figures to solve real-world and mathematical problems.

Students will describe, provide examples of, and identify elements (e.g., vertices, angles, faces, edges, congruent parts) of common three-dimensional figures (spheres, cones, cylinders, prisms, and pyramids).

Students will identify and describe congruent figures, and will apply congruent figures to solve real-world and mathematical problems.

Students will describe, provide examples of, and apply line symmetry to real-world and mathematical problems.

Students will: reflect figures across a horizontal or vertical line in the first quadrant; translate figures in a plane in the first quadrant; and determine the coordinates of the image after transformation in the first quadrant.

Students will identify rotations of figures in the plane (90 degrees and 180 degrees).

Students will identify and graph ordered pairs on a positive coordinate system (Quadrant I), correctly identifying the origin, axes, and ordered pairs; and will apply graphing in the coordinate system to solve real-world and mathematical problems.

Students will analyze and make inferences from data displays (drawings, tables/charts, pictographs, bar graphs, circle graphs, line plots, Venn diagrams, line graphs, stem-and-leaf plots).

Students will explain how different representations of data (e.g., tables, graphs, diagrams, plots) are related.

Students will determine and construct appropriate data displays (bar graphs, line plots, Venn diagrams, tables, line graphs), and will explain why the type of display is appropriate for the data.

Students will determine and apply the mean, median, mode, and range of a set of data.

Students will identify similar figures and apply similar figures to solve real-world and mathematical problems.

Students will describe or determine (e.g., tables, tree diagrams) the sample space of an event for a real-world or mathematical situation.

Students will determine single event probabilities based on the results of an experiment and will make inferences based on the data.

Students will explore the theoretical probability of simple events.

Students will extend, describe rules for patterns and find a missing term in a pattern from real-world and mathematical problems.

Students will create tables for functions and will apply the tables to solve real-world problems.

Students will describe, define, provide examples of, and apply to real-world and mathematical problems functions using tables, graphs and verbal rules.

Students will explain how tables, graphs and patterns relate to each other.

Students will explain how the change in one quantity affects change in another quantity (e.g., in tables or graphs, input/output tables).

Students will substitute values for variables (up to two different variables) and evaluate algebraic expressions.

Students will describe, define and provide examples of variables and expressions with a missing value based on real-world and mathematical problems.

Students will model and solve real-world and mathematical problems with simple equations and inequalities (e.g., 8x = 4, x+2> 5).

Students will provide examples of and identify integers, fractions, decimals, percents, and pi.

Students will describe and provide examples of representations of numbers (whole numbers, fractions, decimals, percents, integers, square roots, and pi) and operations in a variety of equivalent forms using models, diagrams, and symbols (e.g., number lines, 10 by 10 grids, rectangular arrays, number sentences), based on real-world and mathematical problems.

Students will convert among whole numbers, fractions, decimals, percents and pi, and will compare and order these numbers.

Students will estimate to solve real-world and mathematical problems with fractions, decimals, and percents, checking for reasonable and appropriate computational results.

Students will add, subtract, multiply and divide whole numbers, fractions and decimals to solve real-world problems and apply order of operations (including positive whole number exponents) to simplify numerical expressions.

Students will explain how operations (addition and subtraction; multiplication and division) are inversely related.

Students will add and subtract integers.

Students will apply ratios and proportional reasoning to solve real-world problems (e.g., percents, sales tax, discounts, rate).

Students will identify and apply prime numbers, composite numbers, prime factorization, factors, multiples, and divisibility to solve real-world and mathematical problems (e.g., prime factorization to determine a least common multiple [LCM] or greatest common factor [GCF]).

Students will identify the use of properties (commutative properties of addition and multiplication, the associative properties of addition and multiplication and the identity properties for addition and multiplication) to justify a given step in solving problems.

Students will measure lengths (to the nearest eighth of an inch or the nearest centimeter) and will determine and use in real-world and mathematical problems: area and perimeter of triangles; area and perimeter of quadrilaterals (rectangles, squares, trapezoids) (using the Pythagorean theorem will not be required as a strategy); area and circumference of circles; and area and perimeter of compound figures composed of triangles, quadrilaterals and circles.

Students will estimate measurements of regular and irregular polygons and circles in standard units.

Students will explain how measurements and measurement formulas are related or different (e.g., perimeter and area of rectangles).

Students will find the measures of angles by estimation and measurement with a protractor or angle ruler.

Students will convert units within the same measurement system and use these units to solve real-world problems.

Students will describe, provide examples of and identify (using correct notation, label and name) the basic geometric elements (e.g., points, segments, rays, lines, angles and planes), in real-world and mathematical problems.

Students will describe and provide examples of the elements (e.g., sides, vertices, angles, congruent parts) of two-dimensional figures (circles, triangles [acute, right, obtuse, scalene, isosceles, equilateral], quadrilaterals [square, rectangle, rhombus, parallelogram, trapezoid], regular polygons), and will apply these elements and figures to solve real-world and mathematical problems.

Students will describe, provide examples of, and identify elements (e.g., vertices, angles, faces, edges, congruent parts) of common three-dimensional figures (spheres, cones, cylinders, prisms, and pyramids).

Students will describe and provide examples of congruent and similar figures, and will apply congruent and similar figures to solve real-world and mathematical problems.

Students will translate (slide) and reflect (flip) figures in a coordinate plane.

Students will identify rotations (clockwise or counterclockwise) of figures about the origin in the plane (90 degrees, 180 degrees, 270 degrees).

Students will identify and graph ordered pairs on a coordinate system, correctly identifying the origin, axes, and ordered pairs; and will apply graphing in the coordinate system to solve real-world and mathematical problems.

Students will analyze and make inferences from data displays (drawings, tables/charts, pictographs, bar graphs, circle graphs, line plots, Venn diagrams, line graphs, stem-and-leaf plots, scatter plots).

Students will explain how different representations of data (e.g., tables, graphs, diagrams, plots) are related.

Students will read/interpret, analyze and make inferences from box and whisker plots of data and make predictions and draw conclusions from the data.

Students will determine and construct appropriate data displays (bar graphs, line plots, Venn diagrams, tables, line graphs, stem-and-leaf plots), and will explain why the type of display is appropriate for the data.

Students will make decisions about how misleading representations affect interpretations and conclusions about data (e.g., changing the scale on a graph).

Students will determine the mean, median, mode, and range of a set of data, and will identify clusters, gaps, and outliers within the data.

Students will apply counting techniques to determine the size of a sample space for a real-world or mathematical situation.

Students will: determine theoretical probabilities of simple events; determine probabilities based on the results of an experiment; and make inferences from probability data.

Students will tabulate experimental results from simulations and explain how theoretical and experimental probabilities are related.

Students will extend, and describe rules for patterns and find a missing term in a pattern from real-world and mathematical problems.

Students will represent, analyze, and generalize first degree relationships using tables, graphs, and words, and will apply the relationships to solve real-world and mathematical problems.

Students will explain how tables, graphs, patterns, verbal rules and equations relate to each other.

Students will explain how the change in one quantity affects the change in another quantity (e.g., in tables or graphs).

Students will substitute values for variables (up to three different variables) and evaluate algebraic expressions.

Students will describe, define and provide examples of variables and expressions with a missing value based on real-world and mathematical problems.

Students will model and solve real-world and mathematical problems with one- or two-step single variable, first degree equations or inequalities (e.g., 2x+1=9, 3x+3<9). (Statements and solutions use only non-negative numbers.)

Students will provide examples of and identify rational numbers and irrational numbers (square roots and pi only).

Students will describe and provide examples of representations of numbers (rational, square roots, and pi) and operations in a variety of equivalent forms using models, diagrams and symbols (e.g., number lines, 10 by 10 grids, rectangular arrays, number sentences) based on real-world and mathematical problems.

Students will convert, compare and order multiple numerical representations (e.g., fractions, decimals, percentages) of rational numbers and irrational numbers (square roots and pi only).

Students will estimate to solve real-world and mathematical problems with rational numbers, checking for reasonable and appropriate computational results.

Students will add, subtract, multiply, and divide rational numbers to solve real-world problems and apply order of operations (including positive whole number exponents) to simplify numerical expressions.

Students will explain how operations (additions and subtraction; multiplication and division; squaring and taking the square root of a number) are inversely related.

Students will apply ratios and proportional reasoning to solve real-world problems (e.g., percents, constant rate of change, unit pricing, percent of increase or decrease).

Students will identify the use of properties (commutative properties of addition and multiplication, the associative properties of addition and multiplication, the identity properties for addition and multiplication, inverse properties and the distributive property of multiplication over addition and subtraction) to justify a given step in solving problems.

Students will measure lengths (to the nearest sixteenth of an inch or the nearest millimeter) and will determine and use in real-world or mathematical problems: area and perimeter of triangles and quadrilaterals; area and circumference of circles; area and perimeter of compound figures composed of triangles, quadrilaterals and circles; area from circumference or perimeter; and circumference or perimeter from area.

Students will estimate measurements in standard units in real-world and mathematical problems.

Students will evaluate the measures of angles by estimation, measurement with a protractor or angle ruler and determine angle measures in mathematical and/or real-world situations (e.g., supplementary, external, vertical).

Students will apply formulas to determine the volume of right rectangular prisms in real-world problems.

Students will use formulas to find surface area of right rectangular prisms in real-world and mathematical problems.

Students will apply the Pythagorean theorem to determine the length of a hypotenuse.

Students will convert units within the same measurement system and use these units to solve real-world problems.

Students will describe and provide examples of basic geometric elements that include points, segments, rays, lines, angles, and planes and will use these elements in real-world and mathematical problems.

Students will identify and compare properties of two-dimensional figures (circles, triangles, [acute, right, obtuse, scalene, isosceles, equilateral], quadrilaterals [square, rectangle, rhombus, parallelogram, trapezoid], regular/irregular polygons), and will apply these properties and figures to solve real-world and mathematical problems.

Students will compare properties of three-dimensional figures (spheres, cones, cylinders, prisms, pyramids), and will apply these properties and figures to solve real-world and mathematical problems.

Students will: provide examples of congruent and similar figures; apply congruent and similar figures to solve real-world and mathematical problems; and apply proportional reasoning to solve problems involving scale drawings and proportional figures.

Students will describe, provide examples of, and apply to real-world and mathematical problems rotational symmetry (90 degrees, 180 degrees, 360 degrees).

Students will transform (translations, reflections, and dilations with the center of dilation at the origin) figures in a coordinate plane and determine the new coordinates of the image after the transformation.

Students will identify rotations (clockwise or counterclockwise) of figures about the origin in a coordinate plane.

Students will identify and graph ordered pairs on a coordinate system, correctly identifying the origin, axes, and ordered pairs; and will apply graphing in the coordinate system to solve real-world and mathematical problems.

Students will analyze and make inferences from data displays (drawings, tables/charts, pictographs, bar graphs, circle graphs, line plots, Venn diagrams, line graphs, stem-and-leaf plots, scatter plots, histograms, box-and-whiskers plots).

Students will explain how different representations of data (e.g., tables, graphs, diagrams, plots) are related.

Students will: construct data displays (Venn diagrams, tables, line graphs, stem-and-leaf plots, circle graphs, scatter plots); explain why the type of display is appropriate for the data; and explain how misleading representations affect interpretations and conclusions about data (e.g., changing the scale on a graph).

Students will construct box-and-whiskers plots.

Students will: determine the mean, median, mode, and range of a set of data; identify clusters, gaps, and outliers; and apply these concepts to compare sets of data.

Students will explain how data gathering, bias issues, and faulty data analysis can affect the results of data collection.

Students will apply counting techniques to determine the size of a sample space for a real-world or mathematical situation.

Students will: determine theoretical probabilities of simple events; determine probabilities based on the results of an experiment; and make inferences from probability data.

Students will tabulate experimental results from simulations and explain how theoretical and experimental probabilities are related.

Students will determine theoretical probabilities and represent them using area models.

Students will use variables to describe numerical patterns based on arithmetic sequences in real-world and mathematical problems (e.g., f(N) = 2N+3).

Students will represent, analyze and generalize simple first and second degree relationships using tables, graphs, words and algebraic notations, and will apply the relationships to solve real-world and mathematical problems.

Students will explain how the change in one variable affects the change in another variable (e.g., if rate remains constant, an increase in time results in an increase in distance).

Students will evaluate and simplify algebraic expressions applying the order of operations.

Students will describe, define and provide examples of variables and expressions with a missing value based on real-world and mathematical problems.

Students will model and solve single variable, first-degree real-world and mathematical problems (e.g., 5x+2 = x+22, x-4 < -60).

Students will: apply multiple representations (e.g., drawings, manipulatives, base-10 blocks, number lines, expanded form, symbols) to describe whole numbers (0 to 9,999); apply multiple representations (e.g., drawings, manipulatives, base-10 blocks, number lines, symbols) to describe fractions (halves, thirds, fourths); apply these numbers to represent real-world problems; and explain how the base 10 number system relates to place value.

Students will read, write, and rename whole numbers (0 to 9,999) and apply to real-world and mathematical problems.

Students will compare (<, >, =) and order whole numbers to whole numbers, decimals to decimals (as money only) and fractions to fractions (limited to pictorial representations).

Students will apply and describe appropriate strategies for estimating quantities of objects and computational results (limited to addition and subtraction).

Students will analyze real-world problems to identify appropriate representations using mathematical operations, and will apply operations to solve real-world problems with the following constraints: add and subtract whole numbers with three digits or less; multiply whole numbers of 10 or less; add and subtract fractions with like denominators less than or equal to four; and add and subtract decimals related to money.

Students will skip-count forward and backward by 2s, 5s, 10s, and 100s.

Students will divide two digit numbers by single digit divisors (with or without remainders) in real-world and mathematical problems.

Students will identify and provide examples of odd numbers, even numbers, and multiples of a number, and will apply these numbers to solve real-world problems.

Students will use the commutative properties of addition and multiplication, the identity properties of addition and multiplication, and the zero property of multiplication in written and mental computation.

Students will apply standard units to measure length (to the nearest half-inch or nearest centimeter) and to determine: weight (nearest pound), time (nearest quarter hour); money (identify coins and bills by value); and temperature (Fahrenheit).

Students will use standard units to measure temperature in Fahrenheit and Celsius to the nearest degree.

Students will choose and use appropriate tools (e.g., thermometer, scales, balances, clock, ruler) for specific measurement tasks.

Students will use nonstandard and standard units of measurement to identify measurable attributes of an object (length â€“ in, cm; weight â€“ oz, lb) and make an estimate using appropriate units of measurement.

Students will use units of measurement to describe and compare attributes of objects to include length (in, cm), width, height, money (cost), temperature (F) and weight (oz, lb), and sort objects and compare attributes by shape, size and color.

Students will estimate weight, length, perimeter, area, angle measures and time using appropriate units of measurement.

Students will describe, define, give examples of and use to solve real-world and mathematical problems nonstandard and standard (U.S. Customary, metric) units of measurement to include length (in., cm.), time, money, temperature (Fahrenheit) and weight (oz., lb).

Students will determine elapsed time by half hours.

Students will convert units within the same measurement system including money (dollars, cents), time (minutes, hours, days, weeks, months), weight (ounce, pound) and length (inch, foot).

Students will describe and provide examples of basic geometric elements and terms (sides, edges, faces, bases, vertices, angles) and will apply these elements to solve real-world and mathematical problems.

Students will describe and provide examples of basic two-dimensional shapes (circles, triangles, squares, rectangles, trapezoids, rhombuses, hexagons) and will apply these shapes to solve real-world and mathematical problems.

Students will describe and provide examples of basic three-dimensional objects (spheres, cones, cylinders, pyramids, cubes) and will apply the attributes to solve real-world and mathematical problems.

Students will identify and describe congruent figures in real-world and mathematical problems

Students will describe and provide examples of line symmetry in real-world and mathematical problems or will apply one line of symmetry to construct a simple geometric design.

Students will locate points on a grid representing a positive coordinate system.

Students will analyze and make inferences from data displays (drawings, tables/charts, tally tables, pictographs, bar graphs, circle graphs with two or three sectors, line plots, two-circle Venn diagrams).

Students will collect data.

Students will organize and display data.

Students will determine the mode (of a set of data with no more than one mode) and the range of a set of data.

Students will pose questions that can be answered by collecting data.

Students will describe and give examples of the probability of an unlikely event (near zero) and a likely event (near one).

Students will extend simple patterns (e.g., 2,4,6,8,...; (diamond, triangle, diamond, triangle,...).

Students will describe functions (input-output) through pictures and words.

Students will determine the value of an output given a function rule and an input value.

Students will model real-world and mathematical problems with simple number sentences (equations and inequalities) with a missing value (e.g., 2 + ? = 7, __ < 6) and apply simple number sentences to solve mathematical real-world problems.

Students will compare real numbers using order relations (less than, greater than, equal to) and represent problems using real numbers.

Students will demonstrate the relationships between different subsets of the real number system.

Students will use scientific notation to express very large or very small quantities.

Students will estimate solutions to problems with real numbers (including very large and very small quantities) in both real-world and mathematical problems, and use the estimations to check for reasonable computational results.

Students will solve real-world and mathematical problems to specified accuracy levels by simplifying expressions with real numbers involving addition, subtraction, multiplication, division, absolute value, integer exponents, roots (square, cube), and factorials.

Students will: describe and extend arithmetic and geometric sequences; determine a specific term of a sequence given an explicit formula; determine an explicit rule for the nth term of an arithmetic sequence; and apply sequences to solve real-world problems.

Students will write an explicit rule for the nth term of a geometric sequence.

Students will recognize and solve problems that can be modeled using a finite geometric series, such as home mortgage problems and other compound interest problems.

Students will apply ratios, percents, and proportional reasoning to solve real-world problems (e.g., those involving slope and rate, percent of increase and decrease) and will explain how slope determines a rate of change in linear functions representing real-world problems.

Students will identify real number properties (commutative properties of addition and multiplication, associative properties of addition and multiplication, distributive property of multiplication over addition and subtraction, identity properties of addition and multiplication and inverse properties of addition and multiplication) when used to justify a given step in simplifying an expression or solving an equation.

Students will use equivalence relations (reflexive, symmetric, transitive).

Students will determine the surface area and volume of right rectangular prisms, pyramids, cylinders, cones, and spheres in real-world and mathematical problems.

Students will describe how a change in one or more dimensions of a geometric figure affects the perimeter, area, and volume of the figure.

Students will apply definitions and properties of right triangle relationships (right triangle trigonometry and the Pythagorean theorem) to determine length and angle measures to solve real-world and mathematical problems.

Students will apply special right triangles and the converse of the Pythagorean theorem to solve real-world problems.

Students will analyze and apply spatial relationships (not using Cartesian coordinates) among points, lines, and planes (e.g., betweenness of points, midpoint, segment length, collinear, coplanar, parallel, perpendicular, skew).

Students will use spatial relationships to prove basic theorems.

Students will analyze and apply angle relationships (e.g., linear pairs, vertical, complementary, supplementary, corresponding and alternate interior angles) in real-world and mathematical problems.

Students will use angle relationships to prove basic theorems.

Students will classify and apply properties of two-dimensional geometric figures (e.g., number of sides, vertices, length of sides, sum of interior and exterior angle measures).

Students will know the definitions and basic properties of a circle and will use them to prove basic theorems and solve problems.

Students will solve real-world and mathematical problems by applying properties of triangles (e.g., Triangle Sum theorem and Isosceles Triangle theorems).

Students will use the properties of triangles to prove basic theorems.

Students will classify and apply properties of three-dimensional geometric figures.

Students will describe the intersection of a plane with a three-dimensional figure.

Students will visualize solids and surfaces in three-dimensional space when given two-dimensional representations (e.g., nets, multiple views) and create two-dimensional representations for the surfaces of three-dimensional objects.

Students will apply the concepts of congruence and similarity to solve real-world and mathematical problems.

Students will prove triangles congruent and similar.

Students will identify and describe properties of and apply geometric transformations within a plane to solve real-world and mathematical problems.

Students will apply algebraic concepts or graphing in the coordinate plane to analyze and solve problems (e.g., finding the final coordinates for a specified polygon, midpoints, betweeness of points, parellel and perpendicular lines, the distance between two points, the slope of a segment).

Students will identify definitions, axioms and theorems, explain the necessity for them and of and give examples of them.

Students will recognize that there are geometries, other than Euclidean geometry, in which the parallel postulate is not true.

Students will be able to perform constructions such as a line parallel to a given line through a point not on the line, the perpendicular bisector of a line segment and the bisector of an angle.

Students will analyze and make inferences from a set of data with no more than two variables, and will analyze problems for the use and misuse of data representations.

Students will construct data displays for data with no more than two variables.

Students will represent real-world data using matrices and will use matrix addition, subtraction, multiplication (with matrices no larger than 2x2) and scalar multiplication to solve real-world problems.

Students will describe and compare data distributions and make inferences from the data based on the shapes of graphs, measures of center (mean, median, mode) and measures of spread (range, standard deviation).

Students will know the characteristics of the Gaussian normal distribution (bell-shaped curve).

Students will: identify an appropriate curve of best fit (linear, quadratic, exponential) for a set of two-variable data; determine a line of best fit equation for a set of linear two-variable data; and apply a line of best fit to make predictions within and beyond a given set of two-variable data.

Students will recognize when arguments based on data confuse correlation and causation.

Students will recognize potential for bias resulting from the misuse of sampling methods (e.g., non-random sampling, polling only a specific group of people, using limited or extremely small sample sizes) and explain why these samples can lead to inaccurate inferences.

Students will design simple experiments or investigations to collect data to answer questions of interest.

Students will explain the differences between randomized experiments and observational studies.

Students will: determine theoretical and experimental (from given data) probabilities; make predictions and draw inferences from probabilities; compare theoretical and experimental probabilities; and determine probabilities involving replacement and non-replacement.

Students will recognize and identify the differences between combinations and permutations and use them to count discrete quantities.

Students will represent probabilities in multiple ways, such as fractions, decimals, percentages and geometric area models.

Students will explain how the law of large numbers can be applied in simple examples.

Students will identify multiple representations (tables, graphs, equations) of functions (linear, quadratic, absolute value, exponential) in real-world or mathematical problems.

Students will identify, relate and apply representations (graphs, equations, tables) of a piecewise function (such as long distance telephone rates) from mathematical or real-world information.

Students will demonstrate how equations and graphs are models of the relationship between two real-world quantities (e.g., the relationship between degrees Celsius and degrees Fahrenheit).

Students will recognize and solve problems that can be modeled using an exponential function, such as compound interest problems.

Students will: determine if a relation is a function; determine the domain and range of a function (linear and quadratic); determine the slope and intercepts of a linear function; determine the maximum, minimum, and intercepts (roots/zeros) of a quadratic function; and evaluate a function written in function notation for a specified rational number.

Students will find the domain and range for absolute value functions.

Students will apply and use direct and inverse variation to solve real-world and mathematical problems.

Students will identify the changes and explain how changes in parameters affect graphs of functions (linear, quadratic, absolute value, exponential) (e.g., compare y = x2, y = 2x2, y = (x-4)2, and y = x2+3).

Students will apply order of operations, real number properties (identity, inverse, commutative, associative, distributive, closure), and rules of exponents (integer) to simplify algebraic expressions.

Students will evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified values of their variables.

Students will: add, subtract and multiply polynomial expressions; factor polynomial expressions using the greatest common monomial factor; and factor quadratic polynomials of the form ax2 + bx + c, when a = 1 and b and c are integers.

Students will factor quadratic polynomials, such as perfect square trinomials and quadratic polynomials of the form axx(x squared)+bx+c when a is not equal to 1 and b and c are integers.

Students will add, subtract, multiply and divide simple rational expressions with monomial first-degree denominators and integer numerators (e.g., 3/5x + 4/3y; 9/2a - -7/4b; 3/-5x x -4/7y; 5/2c / 9/-11d), and will express the results in simplified form.

Students will model, solve and graph first degree, single variable equations and inequalities, including absolute value, based in real-world and mathematical problems and graph the solutions on a number line.

Students will solve for a specified variable in a multivariable equation.

Students will model, solve and graph first degree, two-variable equations and inequalities in real-world and mathematical problems.

Students will model, solve and graph systems of two linear equations in real-world and mathematical problems.

Students will write, graph, and solve systems of two linear inequalities based on real-world or mathematical problems and interpret the solution

Students will model, solve and graph quadratic equations in real-world and mathematical problems.

Students will continue to apply to both real-world and mathematical problems U.S. customary and metric systems of measurement.