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Common Core Learning Standards GRADE 8 Mathematics EXPRESSIONS & EQUATONS Common Core Learning Standards Work with radicals and integer exponents. Concepts Embedded Skills Properties of exponents Divide the numerical expressions with integer exponents with like bases by subtracting the exponents. Evaluate numerical expressions with integer exponents. Write a numerical expression with a negative exponent as an equivalent numerical expression with a positive exponent (write the base as a fraction). Multiply numerical expressions with integer exponents with like bases by adding the exponents. 8.EE.1. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, = Vocabulary Base Exponent Integer Expression Monomial Coefficient Numerical expression Add and subtract and divide numerical expressions with integer exponents. Rigorous Sample Tasks Scaffolded Sample Tasks Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only. Common Core Learning Standards Work with radicals and integer exponents. Concepts Embedded Skills Square roots and cube roots Calculate the solution to x2 = p, where p is a positive rational number by taking the square root of both sides. Calculate the solution to x3 = p, where p is a positive rational number by taking the cube root of both sides. Evaluate a square root of a perfect square. Name numbers that are perfect squares and nonperfect squares. Evaluate a cube root of a perfect cube. 8.EE.2. Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that is irrational. Name numbers that are perfect cubes and nonperfect cubes. Vocabulary Square root Cube root Squared Cubed Solution Perfect square Perfect cube Exponent Inverse operation Index Rational Irrational Identify that non-perfect squares and non-perfect cubes are irrational numbers (focus on ). Rigorous Sample Tasks Scaffolded Sample Tasks Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only. Common Core Learning Standards Work with radicals and integer exponents. Concepts Scientific notation 8.EE.3. Use numbers expressed in the form of a single digit times a whole-number power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 X and the population of the world as 7 X , and determine that the world population is more than 20 times larger. Rigorous Sample Tasks Embedded Skills Vocabulary Compare the magnitude (size) of 2 or more numbers written in scientific notation. Rewrite numbers in standard form in scientific notation. Expand numbers written in scientific notation into standard form. Divide numbers in scientific notation to compare their sizes. Scientific Notation Magnitude Standard Form Estimate Expand Scaffolded Sample Tasks Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only. Common Core Learning Standards Work with radicals and integer exponents. Concepts Embedded Skills Add, subtract, multiply, and divide numbers written Operations with scientific in scientific notation, applying laws of exponents. Find appropriate units for very large and small notation 8.EE.4. 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 (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. Rigorous Sample Tasks quantities. Demonstrate on the calculator and identify that E on the calculator means x 10a. Vocabulary Scientific notation Decimal notation Powers of 10 Standard form Exponents Scaffolded Sample Tasks Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only. Common Core Learning Standards Understand the connections between proportional relationships, lines, and linear equations. Concepts Graphing relationships 8.EE.5. 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. Rigorous Sample Tasks Embedded Skills Identify the slope of a linear relationship from equations, tables, and graphs. Create and label an appropriate quadrant I graph. Describe unit rate as the slope of a graph. Vocabulary Proportions Unit rate Slope Direct variation Compare two different proportional relationships represented in different ways (graph vs. table vs. equation vs. verbal description). Scaffolded Sample Tasks Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only. Common Core Learning Standards Understand the connections between proportional relationships, lines, and linear equations. Concepts Embedded Skills Linear equations Determine the slope of a line by counting the rise over the run of the given line. Explain why the slope of a line is the same for any two points on the graph using rise over run. Explain slope as a constant rate of change (rise over run). Given a line that passes through the origin, write the equation for the line in the form y = mx, where the slope is found using rise over run. Given a line that passes the vertical axis at point other than the origin, write the equation for the line in the form y = mx + b, where the slope is found using rise over run and b is where the line intercepts the vertical axis. 8.EE.6. Use similar triangles to explain why the slope m is 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 b. Rigorous Sample Tasks Vocabulary Slope Y-intercept Slope intercept form Similar triangles Non-vertical line Origin Constant rate of change Scaffolded Sample Tasks Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only. Common Core Learning Standards Analyze and solve linear equations and pairs of simultaneous linear equations. Concepts Solving linear equations 8.EE.7a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. 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). Rigorous Sample Tasks Embedded Skills Solve a linear equation with one solution. Solve a linear equation with infinitely many solutions. Solve a linear equation with no solution. Check the solution to an equation. Explain the differences between one solution, no solution, and infinitely many. Vocabulary Equation Variable Infinite solution Linear No solution Inverse operation Distributive property Combine like terms Scaffolded Sample Tasks Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only. Common Core Learning Standards Analyze and solve linear equations and pairs of simultaneous linear equations. Concepts Multi-step equations 8.EE.7b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Rigorous Sample Tasks Embedded Skills Simplify equations using the distributive property, combining like terms, and inverse operations. Solve multi-step linear equations with rational number coefficients. Add, subtract, multiply, and divide rational numbers. Vocabulary Distributive property Combine like terms Coefficient Inverse operations Equations Rational numbers Scaffolded Sample Tasks Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only. Common Core Learning Standards Analyze and solve linear equations and pairs of simultaneous linear equations. Concepts Systems of linear equations 8.EE.8a. 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. Rigorous Sample Tasks Embedded Skills Define the solution to a linear system of equations as the intersection point on a graph. Graph a system of linear equations. Identify the point of intersection to a system of linear equations. Vocabulary System of equations Solution Point of intersection Scaffolded Sample Tasks Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only. Common Core Learning Standards Concepts Embedded Skills Analyze and solve linear equations and pairs of simultaneous linear equations. Systems of equations algebraically Solve a system of linear equations algebraically with one solution. Solve a system of linear equations algebraically with no solution. Solve a system of linear equations algebraically with infinitely many solutions. Estimate the solution of a system of linear equations by graphing. Solve simple systems of linear equations by inspection. 8.EE.8b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. Rigorous Sample Tasks Vocabulary Elimination Substitution Algebraically Graphically One solution No solution Infinitely many solutions Systems of equations Scaffolded Sample Tasks Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only. Common Core Learning Standards Analyze and solve linear equations and pairs of simultaneous linear equations. Concepts Real world situations 8.EE.8c. Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Rigorous Sample Tasks Embedded Skills Write a system of linear equations from a word problem. Solve a system of linear equations created from a word problem. Graph points on a coordinate plane. Calculate the slope of a line using the slope formula. Create an appropriate coordinate plane based on the constraints of a word problem. Vocabulary Intersect Linear equation Solution Variable Slope/rate of change Y-intercept Constraints Coordinate plane Scaffolded Sample Tasks Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.