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# Pennsylvania Common Core Standards Ninth Grade Mathematics resources

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Algebra

A-APR. Arithmetic with Polynomials and Rational Functions

Perform arithmetic operations on polynomials.

A-APR.1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Quiz, Flash Cards, Worksheet, Game & Study Guide Polynomials and Exponents

A-CED. Creating Equations

Create equations that describe numbers or relationships.

A-CED.1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

Quiz, Flash Cards, Worksheet, Game & Study Guide Equations and inequalities

A-CED.2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Quiz, Flash Cards, Worksheet, Game & Study Guide Linear equations

Quiz, Flash Cards, Worksheet, Game & Study Guide Solving linear equations

A-CED.3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

Quiz, Flash Cards, Worksheet, Game & Study Guide Equations and inequalities

Quiz, Flash Cards, Worksheet, Game & Study Guide Solving linear equations

A-CED.4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R.

Quiz, Flash Cards, Worksheet, Game & Study Guide Solving equations and inequalities

A-REI. Reasoning with Equations and Inequalities

Represent and solve equations and inequalities graphically.

A-REI.11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

Flip Charts Introduction to Functions

Flip Charts Math Grade 8

Flip Charts Systems of Equations

Flip Charts Systems of Equations

A-REI.12. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Quiz, Flash Cards, Worksheet, Game & Study Guide Linear relationships

Solve equations and inequalities in one variable.

A-REI.3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Flip Charts Math Grade 8

Flip Charts Ratios, Rates & Proportions

Quiz, Flash Cards, Worksheet, Game & Study Guide Equations and inequalities

Quiz, Flash Cards, Worksheet, Game & Study Guide Integer operations

Quiz, Flash Cards, Worksheet, Game & Study Guide Rational numbers and operations

Quiz, Flash Cards, Worksheet, Game & Study Guide Solving equations and inequalities

Quiz, Flash Cards, Worksheet, Game & Study Guide Solving linear equations

Solve systems of equations.

A-REI.5. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Flip Charts Introduction to Functions

Flip Charts Math Grade 8

Flip Charts Systems of Equations

Flip Charts Systems of Equations

A-REI.6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

Flip Charts Introduction to Functions

Flip Charts Math Grade 8

Flip Charts Systems of Equations

Flip Charts Systems of Equations

Understand solving equations as a process of reasoning and explain the reasoning.

A-REI.1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Flip Charts Math Grade 8

Flip Charts Ratios, Rates & Proportions

Quiz, Flash Cards, Worksheet, Game & Study Guide Equations and inequalities

Quiz, Flash Cards, Worksheet, Game & Study Guide Integer operations

Quiz, Flash Cards, Worksheet, Game & Study Guide Rational numbers and operations

Quiz, Flash Cards, Worksheet, Game & Study Guide Solving equations and inequalities

Quiz, Flash Cards, Worksheet, Game & Study Guide Solving linear equations

A-SSE. Seeing Structure in Expressions

Write expressions in equivalent forms to solve problems.

A-SSE.3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

A-SSE.3(c) Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15^t can be rewritten as (1.15^(1/12))^12t approximately equals 1.012^12t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

Quiz, Flash Cards, Worksheet, Game & Study Guide Functions

Functions

F-BF. Building Functions

Build a function that models a relationship between two quantities.

F-BF.1. Write a function that describes a relationship between two quantities.

F-BF.1(a) Determine an explicit expression, a recursive process, or steps for calculation from a context.

Quiz, Flash Cards, Worksheet, Game & Study Guide Solving linear equations

F-IF. Interpreting Functions

Analyze functions using different representations.

F-IF.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

F-IF.7(a) Graph linear and quadratic functions and show intercepts, maxima, and minima.

Quiz, Flash Cards, Worksheet, Game & Study Guide Linear equations

F-IF.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

F-IF.8(b) Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)^t, y = (0.97)^t, y = (1.01)^12t, y = (1.2)^t/10, and classify them as representing exponential growth or decay.

Quiz, Flash Cards, Worksheet, Game & Study Guide Functions

Interpret functions that arise in applications in terms of the context.

F-IF.4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

Quiz, Flash Cards, Worksheet, Game & Study Guide Linear equations

F-IF.6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Quiz, Flash Cards, Worksheet, Game & Study Guide Linear equations

Understand the concept of a function and use function notation.

F-IF.1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

Quiz, Flash Cards, Worksheet, Game & Study Guide Functions

F-IF.3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n greater than or equal to 1.

Flip Charts Arithmetic & Sequences

Flip Charts Circle Around

Flip Charts Math Grade 8

Quiz, Flash Cards, Worksheet, Game & Study Guide Sequences

F-LE. Linear and Exponential Models

Construct and compare linear and exponential models and solve problems.

F-LE.1. Distinguish between situations that can be modeled with linear functions and with exponential functions.

F-LE.1(a) Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

Quiz, Flash Cards, Worksheet, Game & Study Guide Functions

Quiz, Flash Cards, Worksheet, Game & Study Guide Linear equations

Geometry

G-CO. Congruence

Understand congruence in terms of rigid motions

G-CO.6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a rigid motion on a figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

Quiz, Flash Cards, Worksheet, Game & Study Guide Patterns in geometry

Experiment with transformations in the plane

G-CO.1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

Flip Charts Dot to Dot

Flip Charts Math Grade 8

Flip Charts Perimeter & Area

Flip Charts Plane Geometry

Flip Charts Ratios, Rates & Proportions

Quiz, Flash Cards, Worksheet, Game & Study Guide Patterns in geometry

Quiz, Flash Cards, Worksheet, Game & Study Guide Perimeter and area

Quiz, Flash Cards, Worksheet, Game & Study Guide Plane figures

Quiz, Flash Cards, Worksheet, Game & Study Guide Similarity and scale

Quiz, Flash Cards, Worksheet, Game & Study Guide Three dimensional geometry/Measurement

G-CO.2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

Quiz, Flash Cards, Worksheet, Game & Study Guide Patterns in geometry

Quiz, Flash Cards, Worksheet, Game & Study Guide Similarity and scale

G-CO.3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Quiz, Flash Cards, Worksheet, Game & Study Guide Patterns in geometry

G-CO.4. Develop definitions of rotations, reflections and translations in terms of angles, circles, perpendicular lines, parallel lines and line segments.

Quiz, Flash Cards, Worksheet, Game & Study Guide Patterns in geometry

G-CO.5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Quiz, Flash Cards, Worksheet, Game & Study Guide Patterns in geometry

Prove geometric theorems

G-CO.9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

Flip Charts Math Grade 8

Flip Charts Plane Geometry

G-GMD. Geometric Measurement and Dimension

Explain volume formulas and use them to solve problems

G-GMD.3. Use volume formulas for cylinders, pyramids, cones and spheres to solve problems.

Quiz, Flash Cards, Worksheet, Game & Study Guide Three dimensional geometry/Measurement

G-GPE. Expressing Geometric Properties with Equations

Use coordinates to prove simple geometric theorems algebraically

G-GPE.5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

Quiz, Flash Cards, Worksheet, Game & Study Guide Plane figures

G-GPE.7. Use coordinates to compute perimeters of polygons and areas for triangles and rectangles, e.g. using the distance formula.

Quiz, Flash Cards, Worksheet, Game & Study Guide Plane figures

G-SRT. Similarity, Right Triangles, and Trigonometry

Prove theorems involving similarity

G-SRT.5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Flip Charts Math Grade 8

Flip Charts Ratios, Rates & Proportions

Understand similarity in terms of similarity transformations

G-SRT.1. Verify experimentally the properties of dilations:

G-SRT.1(a) A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

Quiz, Flash Cards, Worksheet, Game & Study Guide Similarity and scale

G-SRT.1(b) The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Quiz, Flash Cards, Worksheet, Game & Study Guide Similarity and scale

G-SRT.2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

Flip Charts Math Grade 8

Flip Charts Ratios, Rates & Proportions

Quiz, Flash Cards, Worksheet, Game & Study Guide Ratios, proportions and percents

Quiz, Flash Cards, Worksheet, Game & Study Guide Similarity and scale

Mathematical Practices

MP.1. Make sense of problems and persevere in solving them.

Quiz, Flash Cards, Worksheet, Game & Study Guide Mathematical processes

MP.2. Reason abstractly and quantitatively.

Quiz, Flash Cards, Worksheet, Game & Study Guide Mathematical processes

Number and Quantity

N-RN. The Real Number System

Use properties of rational and irrational numbers.

N-RN.3. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

Flip Charts Math Grade 8

Flip Charts The Real Numbers

Extend the properties of exponents to rational exponents.

N-RN.2. Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Quiz, Flash Cards, Worksheet, Game & Study Guide Polynomials and Exponents

Statistics and Probability

S-CP. Conditional Probability and the Rules of Probability

Understand independence and conditional probability and use them to interpret data

S-CP.2. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

Quiz, Flash Cards, Worksheet, Game & Study Guide Theoretical probability and counting

S-ID. Interpreting Categorical and Quantitative Data

Summarize, represent, and interpret data on two categorical and quantitative variables

S-ID.5. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal and conditional relative frequencies). Recognize possible associations and trends in the data.

Quiz, Flash Cards, Worksheet, Game & Study Guide Using graphs to analyze data

S-ID.6. Represent data on two quantitative variables on a scatter plot and describe how the variables are related.

S-ID.6(b) Informally assess the fit of a model function by plotting and analyzing residuals.

Flip Charts Math Grade 8

Flip Charts Organizing & Displaying Data

S-ID.6(c) Fit a linear function for scatter plots that suggest a linear association.

Quiz, Flash Cards, Worksheet, Game & Study Guide Linear relationships