Mathematics Standards for Grade 8

GRADE 8

Standards	Clarification and Examples
Number Sense - Grade 8
1. Students understand the real number system as a coherent set of elements, operations, and properties.
1.1 Students identify and apply field properties (including closure), axioms of equality and inequality, and properties of order that are valid for the set of real numbers and its subsets.
1.2 Students describe relationships among the subsets of the set of real numbers
2. Students understand and use operations such as opposite, reciprocal, raising to a power, and taking a root.	Note: This would be assessed in conjunction with Symbols and Algebra 1 and 2 and their sub-standards.
2.1 Students use identities, opposites, reciprocals, integral and rational powers to simplify expressions and justify steps in an algebraic process.
Symbols and Algebra - Grade 8
1. Students write and solve linear equations and justify the process.
1.1 Students simplify expressions and simplify and solve linear equations and inequalities in one variable, using inverse operations, integers, and fractions and the properties of the real number system to justify each step and graph the solutions of inequalities on a number line	solve linear equations such as 3(2x - 5) + 4(x - 2) = 12x + 17
1.2 Students solve linear equations and inequalities involving absolute value and quadratic inequalities and graph their solutions on a number line	solve absolute value inequalities such as \|3x - 4\| < 5 solve quadratic inequalities such as x² - 7x -1 > 5
1.3 Students solve word problems including those involving direct and inverse variation
2. Students apply the operations and properties of the real number system to simplify polynomial expressions and justify the process used.
2.1 Students expand or combine polynomial expressions and justify the steps by referring to a particular property
2.2 Students apply basic factoring techniques to second- and simple third-degree polynomials (common factor, difference of squares, perfect square trinomials, sum and difference of cubes, and general trinomials).	(3x - 2)² - (y + 4)²x⁶ - 5x³ – 6
2.3 Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing to lowest terms
2.4 Students apply factoring techniques and properties of fractions to add and subtract rational expressions and simplify the sums and differences
2.5 Students use properties of fractions to multiply rational expressions and reduce them to lowest terms
2.6 Students simplify square roots of monomials that are not perfect squares	square-root of 32x⁸ square-root of 18x⁵
2.7 Students solve equations containing rational expressions and equations containing radical expressions algebraically and representing the solutions graphically
3. Students formulate and solve simple factorable polynomial equations of higher degree and systems of linear equations and inequalities.
3.1 Students solve simple higher degree polynomial equations in one variable by setting equal to zero and factoring	a² - 64 = 0 x⁴ - 7x³ + 3x² - 21 = 0
3.2 Students formulate higher degree polynomial equations in response to problems involving appropriate situations, and interpret the solutions in terms of the problem situation
3.3 Students solve systems of two linear equations or inequalities in two variables, both algebraically and represent them graphically
3.4 Students formulate systems of two linear equations or inequalities in two variables in response to appropriate problem situations and interpret the solution in terms of the problem situation.
Functions - Grade 8
1. Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions.
1.1 Students determine the domain and range of a relation defined by a graph, a set of ordered pairs, or a symbolic expression
1.2 Students determine whether a relation defined by a graph, a set of ordered pairs, or a symbolic expression is a function and justify the conclusion
2. Students determine and interpret the slope and intercept(s) of a line
2.1 Students interpret and use linear functions as a mathematical representation of proportional relationships and of non-proportional relationships having a constant rate of change
2.2 Students write the equation of a line, given two points on the line, one point and the slope, or the slope and y-intercept
2.3 Students find the slope of a line given the coordinates of two points on the line, a graph or an equation in any form
2.4 Students interpret the meaning of the slope as a constant rate of change or in the context of a verbal problem	The cost of mailing a letter increases 24 cents for each additional half ounce
2.5 Students identify the solution of a linear equation by locating the x-intercept in a graph of the corresponding linear function
Measurement and Geometry - Grade 8
1. Students use coordinate geometry to determine attributes of lines and line segments.
1.1 Students determine, algebraically, the length, midpoint and slope of a line segment on a coordinate graph
1.2 Students find the distance between two points, between a point and a vertical or horizontal line and between two vertical or horizontal lines
1.3 Students use slope to determine whether two lines are perpendicular or parallel
2. Students model situations geometrically to answer questions about length, area, or angle measure.
2.1 Students draw diagrams to interpret practical situations geometrically (Draft 1.0 Grade 8 Measurement and Geometry 2.1 with modification to add other examples in which a pictorial representation might be helpful)	The amount of wood needed to frame a 3' x 2.5' portrait. The number of plants that would fit in a garden plot if the plants were to be placed 6' apart. Relative distances that two cars would travel. Fractions of a total represented by various components. Problems involving falling objects, etc.
2.2 Students write and solve equations involving perimeter, area, or angle measure
Statistics, Data Analysis, and Probability - Grade 8
1. Students make inferences and predictions based on the analysis of a set of data.
1.1 Students draw an approximation of the line of best fit for a set of bi-variate data, write its equation, and use the graph or the equation to make predictions	Best fit is done "by eye" at this grade level. Students should at least get the sign of the slope correct in their approximations. The critical aspect is that the equation agrees with the estimated line
Mathematical Reasoning - Grade 8
1. Students will use and know simple aspects of a logical argument
1.1 Students explain the difference between inductive and deductive reasoning and identify and provide examples of each
1.2 Students identify the hypothesis and conclusion in logical deduction
1.3 Students use counterexamples to show that an assertion is false, and recognize that a single counterexample is sufficient to refute the assertion
2. Students use properties of the number system to judge the validity of results, to justify each step of a procedure and to prove or disprove statements
2.1 Students use properties of numbers to construct simple valid arguments (direct and indirect) of, or formulate counter-examples to, claimed assertions
2.2 Students judge the validity of an argument based on whether the properties of the real number system and order of operations have been applied correctly at each step
2.3 Students prove general algebraic statements for all real numbers by justification of steps until an obvious identity is achieved, or disprove by specific counterexample or contradiction	Prove or disprove that the statement (a + b)² = a² + b² is true for all real numbers
2.4 Given a specific algebraic statement involving linear, quadratic or absolute value expressions, equations or inequalities, students determine if the statement is true sometimes, always or never