• Use pictures, diagrams, or patterns

• Use trial and error (guess)

• Use a simpler but related problem

• Use proportional reasoning, ratios, and rates to solve problems

• Solve by working backwards

• Distinguish similarities and differences among a wide variety of problem types and problem-solving strategies

• Use mathematical sentences to solve problems

     Make and Evaluate Conjectures and Arguments

• Distinguish relevant from irrelevant information

• Communicate the effects of changing the parameters (e.g. variables, constraints, conditions) of a problem statement

• Design a general solution and analyze cases in which the general solution does not apply

• Communicate solution processes in a variety of ways (words, numbers, symbols, pictures, charts, graphs, tables, diagrams, and models)

• Communicate solutions clearly and logically, using appropriate mathematical notation, terms, and language

• Understand that there is no one right way to solve mathematical problems, but that different methods have different advantages and disadvantages

• Support solutions with written and/or algebraic evidence

• Communicate solutions with peers for clarification

• Develop formulas (e.g. surface area, volume, etc.)

• Distinguish patterns in a number sequence (include sequences with integral terms)

• Apply strategies and results from simpler problems to more complex situations

     Justify Conclusions Involving Simple and Compound Statements

• Demonstrate an understanding of how numbers may satisfy one or more conditions                        

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    Understand, represent, and use numbers in a variety of equivalent forms

• Read and write integers, rational, and irrational numbers.

• Communicate the relationship between the subsets of the real number system

• Know how to approximate integers and rational numbers, using scientific notation (positive and negative powers of 10) and explain the process

• Understand the relationship between terminating and repeating decimals

• Communicate the equivalent relationships among representations of rational numbers (fractions, decimals, and percents) and use these representations in estimation, computation, and applications

• Understand and explain a number raised to the zero power

• Use scientific notation to express and compare very large and very small numbers in real-life situations

• Understand the meaning of the absolute value symbol

• Know how to calculate fraction, decimal, and percent equivalents

     Understand and Apply Ratios, Proportions and Percents

• Use a variety of manipulatives (algebra tiles, graph paper, cubes) to demonstrate that percent is part of 100

• Understand the relationships among ratio, proportion, and percent

• Solve real-life problems dealing with scale drawings and similar polygons

• Know how to find the percent of a number; calculate the percent of increases and decreases, rate, commissions, taxes, and simple interest

     Number Theory

• Know what prime and composite mean

• Distinguish prime and composite numbers

• Use factor trees and repeated division to identify prime factors, least common multiple, and greatest common factor

• Know the rules of divisibility (2, 3, 4, 5, 6, 8, 9, 10) and use them to factor numbers

• Communicate the rules of divisibility as they are discovered during prime factoring

     Recognize Order Relations for Decimals, Integers, and Rational Numbers

• Distinguish integers, rational, and irrational numbers and communicate their similarities and differences

• Use symbols (<, >, =, ≤, ≥) when recognizing numerical relationships

• Develop techniques for ordering fractions and decimals including percents and scientific notation.

• Know whole number perfect squares from 0 to 225

• Know that two consecutive whole numbers between which the square root of a whole number from 0 to 225 lies

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• Solve operations on integers, decimals, and rational numbers

• Know how to raise rational numbers to whole number powers

• Calculate the absolute value of real numbers expanded to include numerical expressions beyond a single value (e.g., | - 5 + 3|)

• Solve one and two-step word problems fractions, decimals, and integers

     Order of Operations

• Evaluate a problem using the order of operations including multiple grouping symbols and roots

     Properties & Identity Elements

• Understand that integers consist of zero and natural numbers and their additive inverses

• Know how to simplify numerical expressions

• Solve word problems and equations by applying properties of real numbers

• Know the associative, distributive, and commutative properties and the inverse and identity elements

• Communicate why certain properties hold true or do not hold true under specific operations

• Understand the inverse relationships between addition and subtraction, multiplication and division

• Understand the inverse relationships between and exponentiation and root extraction

• Know the existence of closure under the operations with integers

• Formulate properties (commutative, associative, etc.) involving operations with integers by experimenting with integers under the basic operations

      Algorithms

• Solve and communicate the rules for the operational algorithms relative to real numbers

• Solve and communicate the use of absolute value in operational algorithms

• Know facts and algorithms involving real numbers

     Ratios and Proportions

• Use ratios and proportions to solve problems (e.g. a change of scale in drawings or maps, recipes, percents, etc.)

• Calculate the unit cost of items to compare prices

• Use ratios to show that the lengths of corresponding sides are proportional to determine if triangles are similar

     Operations using Roots and Powers

• Use operations applying to powers of a common base

• Evaluate a number which has been raised to a positive integral exponent

• Evaluate a number which has been raised to a zero exponent

• Know that the square of any nonzero integer is a positive number

• Understand that every positive number has two square roots (introduce the ± symbol)

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• Identify and construct two-dimensional patterns for three- dimensional models

• Know elements of three-dimensional geometric objects

     Maps and Scale Drawings

• Select appropriate units of measure and use proportional reasoning to convert measures (e.g. standard and metric)

• Construct scale drawings and models with reasonable measurement accuracy

     Coordinate Planes

• Know how to locate a point, using ordered pairs of integers on the coordinate plane

• Evaluate geometric measurements and computations on coordinate axes as they are applied to parallel lines, congruent and similar figures

• Know how to locate the quadrant in which an ordered pair of integers is located

• Develop geometric ideas such as measurement formulas (area, perimeter, distance), using geoboards and graph paper

     One and Two Dimensional Graphs

• Use a number line graph to represent the solution of a problem with one unknown

• Use two-dimensional graphs, including the coordinate plane, to represent the solution of a problem

     Variables to Represent Relationships

• Use variables and appropriate operations to write an expression or equation

• Use variables and appropriate operations to write an inequality, or system of equations or inequalities that represent a verbal description (three less than a number, half as large as area A)

• Know how to evaluate single-variable expressions, formulas

• Know how to solve single-variable, first-degree equations

• Know how to use variables to represent properties (e.g. commutative, associative, distributive, identity, inverse)

     Use Concrete Models and Diagrams to Describe  Real-World Processes

• Evaluate, formulate, and solve geometric problems through modeling (e.g. perimeter, area, volume, surface area)

     Develop and Explore Models that Do and Do Not Rely on Chance

• Construct an appropriate sample space (board games, spinners, dice, coins).

• Analyze a situation in terms of the range of probabilities (certainty, impossibility, sometimes)

• Consider the reliability of sampling procedures

      Investigate Both Two- and Three-Dimensional Transformations

• Understand similarity and rotational and bilateral symmetry in two- and three-dimensional figures

• Understand and use coordinate grids to plot simple figures and to determine lengths and areas related to them

• Demonstrate an understanding of similar and congruent

• Use the concepts of similar and congruent on a coordinate plane

• Use the concepts of similar and congruent to determine image under simple transformations (translation, rotation, reflection, dilation) in the coordinate plane

      Use Appropriate Tools to Construct and Verify Geometric Relationships

• Use compasses, rulers, and pro-tractors to identify and construct basic elements of geometric figures, (altitudes, midpoints, diagonals, angle bisectors, and perpendicular bisectors; and central angles, radii, diameters, and chords of circles)

• Know how to identify the properties of congruent and similar triangles

• Know how to identify corresponding sides in similar or congruent triangles

• Analyze vertical angles to determine if they have equal measure

      Develop Procedure for Basic Geometric Constructions

• Construct an angle with a given measure

• Use a compass and a straightedge to bisect an angle

• Construct the perpendicular bisector of a line segment

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• Know how to measure the distance of objects, using scientific notation (e.g. shuttle from the Earth)

• Know how to solve distance problems in miles per hour

• Use measurement in everyday situations

• Know how to make an appropriate estimate relating to size, quantity, temperature, capacity, and the passage of time

     Select Appropriate Standard and Nonstandard Measurement Units and Tools to   Measure to a Desired Degree of Accuracy 

• Understand the uses of units, square units, and cubic units

• Know how to find the measure of angles using a protractor

• Know how to measure to a certain degree of accuracy, including determining significant digits and appropriate units

     Develop Measure Skills and Informally Derive and Apply Formulas in direct Measurement Activities

• Know and apply formulas for perimeter and area of polygons, volume of rectangular solids, circumference, and area of circles

• Develop and use formulas for surface area of a solid, volume of right circular cylinders, spheres, cones, and pyramids

• Understand length, area, and volume and make relationships between the measurements

• Use the Pythagorean theorem to find the measure of the sides of a right triangle

• Use trigonometric ratios to find the measure of the sides and angles of a right triangle

      Use Statistical Methods and Measures of Central Tendencies to Display, Describe, and Compare Data

• Know how to interpret graphs, tables, scales, and charts by making comparisons and calculations

• Know the three measures of central tendency (mean, median, mode)

• Use appropriate statistical measures to compare data

• Determine which measures of central tendency (mean, median, mode) best represent the sets of data

• Organize, analyze, and display collected data, using appropriate tables, charts, or graphs including histograms, broken line, circle graphs, stem and leaf plots, and box and whisker plots

       Explore and Produce Graphic Representations of Data

• Use appropriate tools (e.g. graphing calculators and computer spreadsheets) to organize and analyze data

• Construct histograms and frequency polygons to represent data

       Develop Critical Judgment for the Reasonableness of Measurement

• Select, use, and communicate a method for comparing weights, capacities, geometric measures, times, and temperatures within and between measurement systems (e.g., miles per hour and feet per second, about how many inches in a given number of meters)

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• Know how to estimate the results of a problem prior to arriving at a solution.

• Know how to round whole numbers, decimals, and fractions.

• Know how to estimate the outcomes of problems/ experiments, complete the task, and compare the results with the prediction.

      Use Estimation to Solve Problems for Which Exact Answers are Inappropriate

• Know how to recognize when an estimate is appropriate.

     Estimate the Probability of Events

• Understand that the larger a well-chosen sample is, the more likely it is to represent the whole, and that there are many ways of choosing a sample that can make it unrepresentative of the whole.

• Know how to use combinations and permutations.

• Conduct and predict outcomes of experiments with independent events.

• Understand the terms relative frequency, cumulative frequency, and cumulative relative frequency.

     Use Simulation Techniques to Estimate Probabilities

• Conduct a variety of simulations to represent an experiment that can not be determined by theoretical probability or is not practical to determine experimentally.

     Determine probabilities of Independent and Mutually Exclusive Events

• Understand and use empirical and theoretical probability, using the formula P(E) = f/n.

• Know how to develop and explore combinations and permutations.

• Know how to express probabilities as fractions, percents, or decimals.

• Know how to predict the results of a series of trials once the probability for one trial is known.

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• Know how to identify, describe, represent, extend, and create patterns (numerical and geometric).

• Know how to describe functions and generalize by the use of rules and algebraic expressions.

     Describe and Represent Patterns and Functional Relationships, Using Tables, Charts, Graphs, Algebraic Expressions, Rules, and Verbal Descriptions

• Know how to describe and represent numerical and geometric patterns and functions, using equations, graphs, and tables.

• Know how to organize and analyze data resulting in function applications through use of a table of values, sentence, formula, graph, and prediction.

     Develop Methods to solve Basic Linear and Quadratic Equations

• Solve multi-step equations in one variable.

• Solve one- and two-step equations.

• Use five basic properties of equality in solving equations with one variable.

• Understand the addition, subtraction, multiplication, and division properties as they pertain to problem solving situations with inequalities.

• Know how to model and solve multi-step problems involving rate, average speed, distance and time, or direct variation.

• Use algebraic expressions, equations, and inequalities to model linear and nonlinear situations, including direct and inverse variation, exponential growth, and quadratic behavior.

• Understand fundamental ideas of the quadratic equation and its graph.

      Develop an Understanding of Functions and Functional Relationships

• Evaluate a situation and determine if the quantities vary directly or indirectly, and represent that variation graphically, in a table and in an equation.

• Use a variety of representations to describe a functional relationship.

• Know how to identify the input and the output in a relationship between two variables and determine whether the relationship is a function.

• Know how to identify and justify proportional relationships

      Verify Results of Substituting Variables

• Solve and equation and check the solution set by substitution.

• Understand that an equation containing a variable may be true for just one value of the variable.

     Apply the Concept of Similarity in Relevant Situations

• Demonstrate an understanding of congruence between two geometric figures and what congruence means about the relationships between the sides and angles of the two figures

• Understand the difference between similarity and congruence.

• Know how to identify similar and congruent triangles and other polygons and their corresponding parts.

     Use Properties of Polygons to Classify Them

• Know how to apply the relationship between the interior and exterior angles of a polygon

• Use the sum of the number of degrees of measure of triangles, quadrilaterals, hexagons, etc. to solve problems.

• Know how to classify triangles according to angle size and/or length of sides.

     Explore Relationships Involving Points, Lines, Angles, and Planes

• Understand and use proper terminology, symbols, definitions, and formulas for undefined and defined terms.

• Know how to name, define, and measure angles and angle pairs such as complementary, supplementary, alternate interior and exterior, and vertical angles

     Develop and Apply the Pythagorean Principle in the Solution of Problems

• Use the Pythagorean theorem in the solution of problems (include rational and irrational numbers)

     Explore and Develop Basic Concepts of Right Triangle Trigonometry

• Understand the relationships of the sides of a right triangle.

• Explore and develop the concept that corresponding angles of similar triangles have the same measure.

• Develop and apply the formulas for sine, cosine, and tangent.

     Use Patterns and Functions to Represent and Solve Problems

• Use patterns and functions to solve problems.

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