In implementing the Algebra process and content performance indicators, it is expected that students will identify and justify mathematical relationships. The intent of both the process and content performance indicators is to provide a variety of ways for students to acquire and demonstrate mathematical reasoning ability when solving problems. Local curriculum and local/state assessments must support and allow students to use any mathematically correct method when solving a problem.
Throughout this document the performance indicators use the words investigate, explore, discover, conjecture, reasoning, argument, justify, explain, proof, and apply. Each of these terms is an important component in developing a studentís mathematical reasoning ability. It is therefore important that a clear and common definition of these terms be understood. The order of these terms reflects different stages of the reasoning process.
Investigate/Explore - Students will be given situations in which they will be asked to look for patterns or relationships between elements within the setting.
Discover - Students will make note of possible patterns and generalizations that result from investigation/exploration.
Conjecture - Students will make an overall statement, thought to be true, about the new discovery.
Reasoning - Students will engage in a process that leads to knowing something to be true or false.
Argument - Students will communicate, in verbal or written form, the reasoning process that leads to a conclusion. A valid argument is the end result of the conjecture/reasoning process.
Justify/Explain - Students will provide an argument for a mathematical conjecture. It may be an intuitive argument or a set of examples that support the conjecture. The argument may include, but is not limited to, a written paragraph, measurement using appropriate tools, the use of dynamic software, or a written proof.
Proof - Students will present a valid argument, expressed in written form, justified by axioms, definitions, and theorems.
Apply - Students will use a theorem or concept to solve an algebraic or numerical problem.
Use a variety of problem solving strategies to understand new mathematical content
A.PS.2 Recognize and understand equivalent representations of a problem situation or a mathematical concept
Observe and explain patterns to formulate generalizations and conjectures
A.PS.4 Use multiple representations to represent and explain problem situations (e.g., verbally, numerically, algebraically, graphically)
Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic)
A.PS.6 Use a variety of strategies to extend solution methods to other problems A.PS.7 Work in collaboration with others to propose, critique, evaluate, and value alternative approaches to problem solving
A.PS.8 Determine information required to solve a problem, choose methods for obtaining the information, and define parameters for acceptable solutions A.PS.9 Interpret solutions within the given constraints of a problem A.PS.10 Evaluate the relative efficiency of different representations and solution methods of a problem
A.RP.1 Recognize that mathematical ideas can be supported by a variety of strategies
Use mathematical strategies to reach a conclusion and provide supportive arguments for a conjecture
A.RP.3 Recognize when an approximation is more appropriate than an exact answer
A.RP.4 Develop, verify, and explain an argument, using appropriate mathematical ideas and language A.RP.5 Construct logical arguments that verify claims or counterexamples that refute them A.RP.6 Present correct mathematical arguments in a variety of forms A.RP.7 Evaluate written arguments for validity
A.RP.8 Support an argument by using a systematic approach to test more than one case A.RP.9
Devise ways to verify results or use counterexamples to refute incorrect statements
A.RP.10 Extend specific results to more general cases A.RP.11 Use a Venn diagram to support a logical argument A.PR.12 Apply inductive reasoning in making and supporting mathematical conjectures
Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problem
Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, Venn diagrams, and other diagrams
A.CM.3 Present organized mathematical ideas with the use of appropriate standard notations, including the use of symbols and other representations when sharing an idea in verbal and written form A.CM.4
Explain relationships among different representations of a problem
A.CM.5 Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid A.CM.6 Support or reject arguments or questions raised by others about the correctness of mathematical work
A.CM.7 Read and listen for logical understanding of mathematical thinking shared by other students A.CM.8
Reflect on strategies of others in relation to oneís own strategy
A.CM.9 Formulate mathematical questions that elicit, extend, or challenge strategies, solutions, and/or conjectures of others
A.CM.10 Use correct mathematical language in developing mathematical questions that elicit, extend, or challenge other studentsí conjectures A.CM.11 Represent word problems using standard mathematical notation A.CM.12 Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and rationale A.CM.13 Draw conclusions about mathematical ideas through decoding, comprehension, and interpretation of mathematical visuals, symbols, and technical writing
Understand and make connections among multiple representations of the same mathematical idea
A.CN.2 Understand the corresponding procedures for similar problems or mathematical concepts
A.CN.3 Model situations mathematically, using representations to draw conclusions and formulate new situations
Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics A.CN.5 Understand how quantitative models connect to various physical models and representations
A.CN.6 Recognize and apply mathematics to situations in the outside world A.CN.7 Recognize and apply mathematical ideas to problem situations that develop outside of mathematics A.CN.8 Develop an appreciation for the historical development of mathematics
A.R.1 Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts A.R.2 Recognize, compare, and use an array of representational forms A.R.3 Use representation as a tool for exploring and understanding mathematical ideas
A.R.4 Select appropriate representations to solve problem situations A.R.5 Investigate relationships between different representations and their impact on a given problem
A.R.6 Use mathematics to show and understand physical phenomena (e.g., find the height of a building if a ladder of a given length forms a given angle of elevation with the ground) A.R.7 Use mathematics to show and understand social phenomena (e.g., determine profit from student and adult ticket sales) A.R.8
Use mathematics to show and understand mathematical phenomena (e.g., compare the graphs of the functions represented by the equations and )
Students will understand numbers, multiple ways of representing numbers, relationships among numbers, and number systems.
Identify and apply the properties of real numbers (closure, commutative, associative, distributive, identity, inverse) Note: Students do not need to identify groups and fields, but students should be engaged in the ideas.
Students will understand meanings of operations and procedures, and how they relate to one another.
A.N.2 Simplify radical terms (no variable in the radicand) A.N.3 Perform the four arithmetic operations using like and unlike radical terms and express the result in simplest form A.N.4 Understand and use scientific notation to compute products and quotients of numbers greater than 100% A.N.5 Solve algebraic problems arising from situations that involve fractions, decimals, percents (decrease/increase and discount), and proportionality/direct variation A.N.6 Evaluate expressions involving factorial(s), absolute value(s), and exponential expression(s) A.N.7 Determine the number of possible events, using counting techniques or the Fundamental Principle of Counting A.N.8 Determine the number of possible arrangements (permutations) of a list of items
Variables and Expressions
A.A.1 Translate a quantitative verbal phrase into an algebraic expression A.A.2 Write verbal expressions that match given mathematical expressions
|A.A.3||Distinguish the difference between an algebraic expression and an algebraic equation|
|A.A.4||Translate verbal sentences into mathematical equations or inequalities|
|A.A.5||Write algebraic equations or inequalities that represent a situation|
|A.A.6||Analyze and solve verbal problems whose solution requires solving a linear equation in one variable or linear inequality in one variable|
|A.A.7||Analyze and solve verbal problems whose solution requires solving systems of linear equations in two variables|
|A.A.8||Analyze and solve verbal problems that involve quadratic equations|
|A.A.9||Analyze and solve verbal problems that involve exponential growth and decay|
|A.A.10||Solve systems of two linear equations in two variables algebraically (See A.G.7)|
|A.A.11||Solve a system of one linear and one quadratic equation in two variables, where only factoring is required Note: The quadratic equation should represent a parabola and the solution(s) should be integers.|
Variables and Expressions
A.A.12 Multiply and divide monomial expressions with a common base, using the properties of exponents Note: Use integral exponents only. A.A.13 Add, subtract, and multiply monomials and polynomials A.A.14 Divide a polynomial by a monomial or binomial, where the quotient has no remainder A.A.15 Find values of a variable for which an algebraic fraction is undefined. A.A.16 Simplify fractions with polynomials in the numerator and denominator by factoring both and renaming them to lowest terms A.A.17 Add or subtract fractional expressions with monomial or like binomial denominators A.A.18 Multiply and divide algebraic fractions and express the product or quotient in simplest form A.A.19 Identify and factor the difference of two perfect squares A.A.20 Factor algebraic expressions completely, including trinomials with a lead coefficient of one (after factoring a GCF)
Equations and Inequalities
A.A.21 Determine whether a given value is a solution to a given linear equation in one variable or linear inequality in one variable A.A.22 Solve all types of linear equations in one variable A.A.23 Solve literal equations for a given variable A.A.24 Solve linear inequalities in one variable A.A.25 Solve equations involving fractional expressions Note: Expressions which result in linear equations in one variable. A.A.26 Solve algebraic proportions in one variable which result in linear or quadratic equations A.A.27 Understand and apply the multiplication property of zero to solve quadratic equations with integral coefficients and integral roots A.A.28 Understand the difference and connection between roots of a quadratic equation and factors of a quadratic expression
Patterns, Relations, and Functions
A.A.29 Use set-builder notation and/or interval notation to illustrate the elements of a set, given the elements in roster form A.A.30 Find the complement of a subset of a given set, within a given universe A.A.31 Find the intersection of sets (no more than three sets) and/or union of sets (no more than three sets)
A.A.32 Graph the Explain slope as a rate of change between dependent and independent variables A.A.33 Determine the slope of a line, given the coordinates of two points on the line A.A.34 Write the equation of a line, given its slope and the coordinates of a point on the line A.A.35 Write the equation of a line, given the coordinates of two points on the line A.A.36 Write the equation of a line parallel to the x- or y-axis A.A.37 Determine the slope of a line, given its equation in any form A.A.38 Determine if two lines are parallel, given their equations in any form A.A.39 Determine whether a given point is on a line, given the equation of the line A.A.40 Determine whether a given point is in the solution set of a system of linear inequalities A.A.41 Determine the vertex and axis of symmetry of a parabola, given its equation (See A.G.10 )
Find the sine, cosine, and tangent ratios of an angle of a right triangle, given the lengths of the sides
|A.A.43||Determine the measure of an angle of a right triangle, given the length of any two sides of the triangle|
|A.A.44||Find the measure of a side of a right triangle, given an acute angle and the length of another side|
|A.A.45||Determine the measure of a third side of a right triangle using the Pythagorean theorem, given the lengths of any two sides|
A.G.1 Find the area and/or perimeter of figures composed of polygons and circles or sectors of a circle Note: Figures may include triangles, rectangles, squares, parallelograms, rhombuses, trapezoids, circles, semi-circles, quarter-circles, and regular polygons (perimeter only). A.G.2 Use formulas to calculate volume and surface area of rectangular solids and cylinders
Students will apply coordinate geometry to analyze problem solving situations.
A.G.3 Determine when a relation is a function, by examining ordered pairs and inspecting graphs of relations A.G.4 Identify and graph linear, quadratic (parabolic), absolute value, and exponential functions A.G.5 Investigate and generalize how changing the coefficients of a function affects its graph A.G.6 Graph linear inequalities A.G.7 Graph and solve systems of linear equations and inequalities with rational coefficients in two variables (See A.A.10) A.G.8 Find the roots of a parabolic function graphically Note: Only quadratic equations with integral solutions. A.G.9 Solve systems of linear and quadratic equations graphically Note: Only use systems of linear and quadratic equations that lead to solutions whose coordinates are integers. A.G.10 Determine the vertex and axis of symmetry of a parabola, given its graph (See A.A.41) Note: The vertex will have an ordered pair of integers and the axis of symmetry will have an integral value.
Units of Measurement
A.M.1 Calculate rates using appropriate units (e.g., rate of a space ship versus the rate of a snail) A.M.2 Solve problems involving conversions within measurement systems, given the relationship between the units
Students will understand that all measurement contains error and be able to determine its significance.
Error and Magnitude
|A.M.3||Calculate the relative error in measuring square and cubic units, when there is an error in the linear measure|
|A.M.2||Solve problems involving conversions within measurement systems, given the relationship between the units|
Students will collect, organize, display, and analyze data.
Organization and Display of Data
|A.S.1||Categorize data as qualitative or quantitative|
|A.S.2||Determine whether the data to be analyzed is univariate or bivariate|
|A.S.3||Determine when collected data or display of data may be biased|
|A.S.4||Compare and contrast the appropriateness of different measures of central tendency for a given data set|
|A.S.5||Construct a histogram, cumulative frequency histogram, and a box-and-whisker plot, given a set of data|
|A.S.6||Understand how the five statistical summary (minimum, maximum, and the three quartiles) is used to construct a box-and-whisker plot|
|A.S.7||Create a scatter plot of bivariate data|
|A.S.8||Construct manually a reasonable line of best fit for a scatter plot and determine the equation of that line|
Analysis of Data
|A.S.9||Analyze and interpret a frequency distribution table or histogram, a cumulative frequency distribution table or histogram, or a box-and-whisker plot|
|A.S.10||Evaluate published reports and graphs that are based on data by considering: experimental design, appropriateness of the data analysis, and the soundness of the conclusions|
|A.S.11||Find the percentile rank of an item in a data set and identify the point values for first, second, and third quartiles|
Identify the relationship between the independent and dependent variables from a scatter plot (positive, negative, or none)
|A.S.13||Understand the difference between correlation and causation|
|A.S.14||Identify variables that might have a correlation but not a causal relationship|
Students will make predictions that are based upon data analysis.
Predictions from Data
Identify and describe sources of bias and its effect, drawing conclusions from data
|A.S.16||Recognize how linear transformations of one-variable data affect the dataís mean, median, mode, and range|
|A.S.17||Use a reasonable line of best fit to make a prediction involving interpolation or extrapolation|
Students will understand and apply concepts of probability.
|A.S.18||Know the definition of conditional probability and use it to solve for probabilities in finite sample spaces|
|A.S.19||Determine the number of elements in a sample space and the number of favorable events|
Calculate the probability of an event and its complement
|A.S.21||Determine empirical probabilities based on specific sample data|
Determine, based on calculated probability of a set of events, if:
Calculate the probability
|Table of Contents||Prekindergarten||Kindergarten||Grade 1||Grade 2|
|Grade 3||Grade 4||Grade 5||Grade 6||Grade 7|