Standardized Test Preparation

Arizona State Standards

Math - Grade 8

Assessment Exam - AZM2 Math Grade 8

The Number System	eTAP Lesson
Understand that there are irrational numbers, and approximate them using rational numbers.
Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion. Know that numbers whose decimal expansions do not terminate in zeros or in a repeating sequence of fixed digits are called irrational. AZ.M2.8.NS.A.1	Rational & Irrational Numbers Operations with Decimals
Use rational approximations of irrational numbers to compare the size of irrational numbers. Locate them approximately on a number line diagram, and estimate their values. AZ.M2.8.NS.A.2	Real and Imaginary Numbers
Understand that given any two distinct rational numbers, a < b, there exist a rational number c and an irrational number d such that a < c < b and a < d < b. Given any two distinct irrational numbers, a < b, there exist a rational number c and an irrational number d such that a < c < b and a < d < b. AZ.M2.8.NS.A.3	Positive and Negative Numbers
Expressions and Equations	eTAP Lesson
Work with radicals and integer exponents.
Understand and apply the properties of integer exponents to generate equivalent numerical expressions. AZ.M2.8.EE.A.1	Raising Numbers to Various Powers
Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³= p, where pis a positive rational number. Know that √2is irrational. Evaluate square roots of perfect squares less than or equal to 225. Evaluate cube roots of perfect cubes less than or equal to 1000. AZ.M2.8.EE.A.2	Rules of Exponents Roots and Radicals Finding Square Roots and Other Roots
Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and express how many times larger or smaller one is than the other. AZ.M2.8.EE.A.3	Base Ten Exponents
Perform operations with numbers expressed in scientific notation including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities. AZ.M2.8.EE.A.4
Understand the connections between proportional relationships, lines, and linear equations.
Graph proportional relationships interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. AZ.M2.8.EE.B.5	Motion, Distance and Time Relationships
Use similar triangles to explain why the slope mis the same between any two distinct points on a non-vertical line in the coordinate plane. Derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at (0, b). AZ.M2.8.EE.B.6	Slope of a Line
Analyze and solve linear equations, inequalities, and pairs of simultaneous linear equations.
Fluently solve linear equations and inequalities in one variable. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solution. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). Solve linear equations and inequalities with rational number coefficients, including solutions that require expanding expressions using the distributive property and collecting like terms. AZ.M2.8.EE.C.7	Perfect Square Trinomial Quadratic Formula Type of Roots Quadratic Word Problems
Analyze and solve pairs of simultaneous linear equations. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations including cases of no solution and infinite number of solutions. Solve simple cases by inspection. Solve mathematical problems and problems in real-world context leading to two linear equations in two variables. AZ.M2.8.EE.C.8	Why Do We Need Two Equations? Simultaneous Equations The Substitution Method The Addition - Subtraction Method Simultaneous Equations With Strange Solutions
Functions	eTAP Lesson
Define, evaluate, and compare functions.
Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (Function notation is not required in Grade 8.) AZ.M2.8.F.A.1	What is a Function?
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. AZ.M2.8.F.A.2	The Application of Equation Rules to Formulas Other Commonly Used Functions – Cosine and Tangent
Interpret the equation y = mx + b as defining a linear function whose graph is a straight line; give examples of functions that are not linear.For example, the function A = s² giving the area of a square as a function of its side length in not linear because its graph contains the points (1,1), (2,4), and (3,9) which are not on a straight line. AZ.M2.8.F.A.3	Sine-Cosine Relationships Interpolation Other Functions in a Right Triangle – Secant, Cosecant, and Cotangent
Geometry	eTAP Lesson
Understand congruence and similarity.
Verify experimentally the properties of rotations, reflections, and translations. Properties include: lines are taken to lines, line segments are taken to line segments of the same length, angles are taken to angles of the same measure, parallel lines are taken to parallel lines. AZ.M2.8.G.A.1	Transformations
Understand that a two-dimensional figure is congruent to another if one can be obtained from the other by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that demonstrates congruence. AZ.M2.8.G.A.2	Properties of Isometrics
Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. AZ.M2.8.G.A.3	Tessellations Using Translations, Rotations, and Glide Reflections
Understand that a two-dimensional figure is similar to another if, and only if, one can be obtained from the other by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that demonstrates similarity. AZ.M2.8.G.A.4	Symmetry
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so. AZ.M2.8.G.A.5	Tessellations with Polygons
Understand and apply the Pythagorean Theorem.
Understand the Pythagorean Theorem and its converse. AZ.M2.8.G.B.6	The Theorem of Pythagoras Converse
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world context and mathematical problems in two and three dimensions. AZ.M2.8.G.B.7	Distance Calculations
Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. AZ.M2.8.G.B.8	Functions as Graphs in the Coordinate System
Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
Understand and use formulas forvolumes of cones, cylinders and spheres and use them to solve real-world context and mathematical problems. AZ.M2.8.G.C.9	Rectangular Prism Volume Solids with Curved Surfaces
Statistics and Probability	eTAP Lesson
Investigate patterns of association in bivariate data.
Construct and interpret scatter plots for bivariate measurement data to investigate and describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. AZ.M2.8.SP.A.1	Variability Distribution
Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. AZ.M2.8.SP.A.2	Distribution Centers and Variation Probability Data Plots
Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. AZ.M2.8.SP.A.3	Methods of Collecting, Representing, & Displaying Data
Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. AZ.M2.8.SP.A.4	Data Calculations
Investigate chance processes and develop, use, and evaluate probability models.
Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. Understand that the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. Represent sample spaces for compound events using organized lists, tables, tree diagrams and other methods. Identify the outcomes in the sample space which compose the event. Design and use a simulation to generate frequencies for compound events. AZ.M2.8.SP.B.5	What are Statistics? Probability
Standards for Mathematical Practice	eTAP Lesson
Make sense of problems and persevere in solving them.
Mathematically proficient students explain to themselves the meaning of a problem, look for entry points to begin work on theproblem, and plan and choose a solution pathway. While engaging in productive struggle to solve a problem, they continually ask themselves, “Does this make sense?" to monitor and evaluate their progress and change course if necessary. Once they have a solution, they look back at the problem to determine if the solution is reasonable and accurate. Mathematically proficient students check their solutions to problems using different methods, approaches, or representations. They also compare and understand different representations of problems and different solution pathways, both their own and those of others. AZ.M2.8.MP.1	Introduction Basic Operations Adding the Same Number Subtracting the Same Number Dividing the Same Number to Each Side of an Equation, Solving for an Unknown
Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem situations. Students can contextualize and decontextualize problems involving quantitative relationships. They contextualize quantities, operations, and expressions by describing a corresponding situation. They decontextualize a situation by representing it symbolically. As they manipulate the symbols, they can pause as needed to access the meaning of the numbers, the units, and the operations that the symbols represent. Mathematically proficient students know and flexibly use different properties of operations, numbers, and geometric objects and when appropriate they interpret their solution in terms of the context. AZ.M2.8.MP.2	Squares and Square Roots Solving Equations When Several Terms and Procedures are Involved When to do What
Construct viable arguments and critique the reasoning of others.
Mathematically proficient students construct mathematical arguments (explain the reasoning underlying a strategy, solution, or conjecture) using concrete, pictorial, or symbolic referents. Arguments may also rely on definitions, assumptions, previously established results, properties, or structures. Mathematically proficient students make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. Mathematically proficient students present their arguments in the form of representations, actions on those representations, and explanations in words (oral or written). Students critique others by affirming or questioning the reasoning of others. They can listen to or read the reasoning of others, decide whether it makes sense, ask questions to clarify or improve the reasoning, and validate or build on it. Mathematically proficient students can AZ.M2.8.MP.3	Changing Words into Math Setting Up and Solving Equations Solving Quadratic Equations Joe's Method Completing the Square
Model with mathematics.
Mathematically proficient students apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. When given a problem in a contextual situation, they identify the mathematical elements of a situation and create a mathematical model that represents those mathematical elements and the relationships among them. Mathematically proficient students use their model to analyze the relationships and draw conclusions. They interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. AZ.M2.8.MP.4	Visually Representing Numerical Data Splitting up the Whole with Pie Graphs Showing the Overlap with Venn Diagrams
Use appropriate tools strategically.
Mathematically proficient students consider available tools when solving a mathematical problem. They choose tools that are relevant and useful to the problem at hand. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful; recognizing both the insight to be gained andtheir limitations. Students deepen their understanding of mathematical concepts when using tools to visualize, explore, compare, communicate, make and test predictions, and understand the thinking of others. AZ.M2.8.MP.5	Comparing Categories with Bar Graphs Seeking Trends with Line Graphs
Attend to precision.
Mathematically proficient students clearly communicate to others using appropriate mathematical terminology, and craft explanations that convey their reasoning. When making mathematical arguments about a solution, strategy, or conjecture, they describe mathematical relationships and connect their words clearly to their representations. Mathematically proficient students understand meanings of symbols used in mathematics, calculate accurately and efficiently, label quantities appropriately, and record their work clearly and concisely. AZ.M2.8.MP.6	Showing Orderly Data with Histograms
Look for and make use of structure
Mathematically proficient students use structure and patterns to assist in making connections among mathematical ideas or concepts when making sense of mathematics. Students recognize and apply general mathematical rules to complex situations. They are able to compose and decompose mathematical ideas and notations into familiar relationships. Mathematically proficient students manage their own progress, stepping back for an overview and shifting perspective when needed. AZ.M2.8.MP.7	Functions as Graphs in the Coordinate System
Look for and express regularity in repeated reasoning.
Mathematically proficient students look for and describe regularities as they solve multiple related problems. They formulate conjectures about what they notice and communicate observations with precision. While solving problems, students maintain oversight of the process and continually evaluate the reasonableness of their results. This informs and strengthens their understanding of the structure of mathematics which leads to fluency. AZ.M2.8.MP.8