8.NS.1 Supporting Content Examples of Linking Supporting Clusters to the Major Work of the Grade: Examples of Major Within-Grade Dependencies

Transcription

1 8.NS.1 Supporting Content Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. Examples of Linking Supporting Clusters to the Major Work of the Grade: Work with the number system in this grade (8.NS.1 2) is intimately related to work with radicals (8.EE.2), and both of these may be connected to the Pythagorean theorem (8.G.6 8) as well as to volume problems (8.G.9), e.g., in which a cube has known volume but unknown edge lengths. Examples of Major Within-Grade Dependencies Much of the work of grade 8 involves lines, linear equations and linear functions (8.EE.5 8; 8.F.3 4; 8.SP.2 3). Irrational numbers, radicals, the Pythagorean theorem and volume (8.NS.1 2; 8.EE.2; 8.G.6 9) are nonlinear in nature. Curriculum developers might choose to address linear and nonlinear bodies of content somewhat separately. An exception, however, might be that when addressing functions, pervasively treating linear functions as separate from nonlinear functions might obscure the concept of function per se. There should also be sufficient treatment of nonlinear functions to avoid giving students the misleading impression that all functional relationships are linear (see also 7.RP.2a). Opportunities for Connecting Mathematical Content to Mathematical Practices When students convert a fraction such as 1/7 to a decimal, they might notice that they are repeating the same calculations and conclude that the decimal repeats. Similarly, by repeatedly checking whether points are on a line through (1, 2) with slope 3, students might abstract the equation of the line in the form (y - 2)/(x - 1) = 3. In both examples, students look for and express regularity in repeated reasoning (MP.8). 8.NS.1 Supporting Content

2 8.NS.1 Supporting Content Performance Level Indicators Grade 8 Math: Sub-Claim B The student solves problems involving the Additional and Supporting Content for the grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Command Level 4: Strong Command Level 3: Moderate Command Level 2: Partial Command Rational Numbers 8.NS.1 8.NS.2 Distinguishes between rational and irrational numbers, understands that these numbers have decimal expansions and locates them approximately on a number line, and converts between terminating decimals or repeating decimals and fractional representations of rational numbers. Distinguishes between rational and irrational numbers, understands that these numbers have decimal expansions and locates them approximately on a number line, and converts between terminating decimals or repeating decimals and fractional representations of rational numbers. Distinguishes between rational and irrational numbers, understands that these numbers have decimal expansions and locates them approximately on a number line, and converts between terminating decimals or repeating decimals of the form (0.aaa ) and fractional representations of rational numbers. Distinguishes between rational and irrational numbers and understands that these numbers have decimal expansions and locates them approximately on a number line. Analyzes and generalizes patterns and structures of repeating decimals. EOY Evidence Statement Key 8.NS.1 Evidence Statement Text Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. Clarifications, limits, emphases, and other information intended to ensure appropriate variety in tasks i) Tasks do not have a context. ii) An equal number of tasks require students to write a fraction a/b as a repeating decimal, or write a repeating decimal as a fraction. iii) For tasks that involve writing a repeating decimal as a fraction the given decimal should include no more than two repeating decimals i.e , MP Calculator 7, 8 No 8.NS.1 Supporting Content

3 8.NS.1 Supporting Content EOY Sample Item (non-calculator) 8.NS.1 Supporting Content

4 8.NS.2 Supporting Content Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π 2 ). For example, by truncating the decimal expansion of 2, show that 2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. Examples of Linking Supporting Clusters to the Major Work of the Grade: Work with the number system in this grade (8.NS.1 2) is intimately related to work with radicals (8.EE.2), and both of these may be connected to the Pythagorean theorem (8.G.6 8) as well as to volume problems (8.G.9), e.g., in which a cube has known volume but unknown edge lengths. Examples of Major Within-Grade Dependencies Much of the work of grade 8 involves lines, linear equations and linear functions (8.EE.5 8; 8.F.3 4; 8.SP.2 3). Irrational numbers, radicals, the Pythagorean theorem and volume (8.NS.1 2; 8.EE.2; 8.G.6 9) are nonlinear in nature. Curriculum developers might choose to address linear and nonlinear bodies of content somewhat separately. An exception, however, might be that when addressing functions, pervasively treating linear functions as separate from nonlinear functions might obscure the concept of function per se. There should also be sufficient treatment of nonlinear functions to avoid giving students the misleading impression that all functional relationships are linear (see also 7.RP.2a). Performance Level Indicators Grade 8 Math: Sub-Claim B The student solves problems involving the Additional and Supporting Content for the grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Command Level 4: Strong Command Level 3: Moderate Command Level 2: Partial Command Rational Numbers 8.NS.1 8.NS.2 Distinguishes between rational and irrational numbers, understands that these numbers have decimal expansions and locates them approximately on a number line, and converts between terminating decimals or repeating decimals and fractional representations of rational numbers. Distinguishes between rational and irrational numbers, understands that these numbers have decimal expansions and locates them approximately on a number line, and converts between terminating decimals or repeating decimals and fractional representations of rational numbers. Distinguishes between rational and irrational numbers, understands that these numbers have decimal expansions and locates them approximately on a number line, and converts between terminating decimals or repeating decimals of the form (0.aaa ) and fractional representations of rational numbers. Distinguishes between rational and irrational numbers and understands that these numbers have decimal expansions and locates them approximately on a number line. Analyzes and generalizes patterns and structures of repeating decimals. 8.NS.2 Supporting Content

5 8.NS.2 Supporting Content EOY Evidence Statement Key 8.NS.2 Evidence Statement Text Use rational approximations of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., πr 2 ). For example, by truncating the decimal expansion of 2, show that 2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. Clarifications, limits, emphases, and other information intended to ensure appropriate variety in tasks MP Calculator i) Tasks do not have a context. 5, 7, 8 No EOY Sample Item (non-calculator) 8.NS.2 Supporting Content

6 8.EE.1 Major Content Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, = 3 3 = 1/3 3 = 1/27. Performance Level Indicators Grade 8 Math : Sub-Claim A The student solves problems involving the Major Content for grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Level 4: Strong Command Level 3: Moderate Command Level 2: Partial Command Command Expressions and Equations 8 EE.1 8 EE.2 Evaluates and generates equivalent numerical expressions applying properties of integer exponents. Solves equations of the form x 2 = p and x 3 = p, representing solutions using or symbols. Demonstrates a solid understanding of the structure of these properties within a real- world context. Evaluates and generates equivalent numerical expressions applying properties of integer exponents. Solves equations of the form x 2 = p and x 3 = p, representing solutions using or symbols. Demonstrates a general understanding of the structure of these properties within a real- world context. Evaluates and generates equivalent numerical expressions applying properties of integer exponents. Solves equations of the form x 2 = p, where p is a perfect square and solves equations of the form x 3 = p, where p is a perfect cube, by representing the positive solution of the equation. Evaluates simple numerical expressions using properties of integer exponents. Partially solves equations of the form x 2 = p, where p is a perfect square, by representing the positive solution of the equation. PBA Evidence Statement Key 8.EE.1 Evidence Statement Text Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, Clarifications, limits, emphases, and other information intended to ensure appropriate variety in tasks i) Tasks do not have a context. ii) Tasks focus on the properties of equivalence, not on simplification. iii) Half of the expressions involve one property; half of the expressions involves two or three properties. iv) Tasks should involve a single common base or potential common base, such as, a task that includes 3, 9, and 27. MP Calculator 7 No 8.EE.1 Major Content

7 8.EE.1 Major Content PBA Sample Item (non-calculator) EOY Evidence Statement Key 8.EE.1 Evidence Statement Text Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, Clarifications, limits, emphases, and other information intended to ensure appropriate variety in tasks i) Tasks do not have a context. ii) Tasks focus on the properties of equivalence, not on simplification. iii) Half of the expressions involve one property; half of the expressions involves two or three properties. iv) Tasks should involve a single common base or potential common base, such as, a task that includes 3, 9, and 27. MP Calculator 7 No 8.EE.1 Major Content

8 8.EE.1 Major Content EOY Sample Item (non-calculator) 8.EE.1 Major Content

9 8.EE.2 Major Content Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that 2 is irrational. Examples of Major Within-Grade Dependencies Much of the work of grade 8 involves lines, linear equations and linear functions (8.EE.5 8; 8.F.3 4; 8.SP.2 3). Irrational numbers, radicals, the Pythagorean theorem and volume (8.NS.1 2; 8.EE.2; 8.G.6 9) are nonlinear in nature. Curriculum developers might choose to address linear and nonlinear bodies of content somewhat separately. An exception, however, might be that when addressing functions, pervasively treating linear functions as separate from nonlinear functions might obscure the concept of function per se. There should also be sufficient treatment of nonlinear functions to avoid giving students the misleading impression that all functional relationships are linear (see also 7.RP.2a). Performance Level Indicators Grade 8 Math : Sub-Claim A The student solves problems involving the Major Content for grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Level 4: Strong Command Level 3: Moderate Command Level 2: Partial Command Command Expressions and Equations 8 EE.1 8 EE.2 Evaluates and generates equivalent numerical expressions applying properties of integer exponents. Solves equations of the form x 2 = p and x 3 = p, representing solutions using or symbols. Demonstrates a solid understanding of the structure of these properties within a real- world context. Evaluates and generates equivalent numerical expressions applying properties of integer exponents. Solves equations of the form x 2 = p and x 3 = p, representing solutions using or symbols. Demonstrates a general understanding of the structure of these properties within a real- world context. Evaluates and generates equivalent numerical expressions applying properties of integer exponents. Solves equations of the form x 2 = p, where p is a perfect square and solves equations of the form x 3 = p, where p is a perfect cube, by representing the positive solution of the equation. Evaluates simple numerical expressions using properties of integer exponents. Partially solves equations of the form x 2 = p, where p is a perfect square, by representing the positive solution of the equation. 8.EE.2 Major Content

10 8.EE.2 Major Content EOY Evidence Statement Key 8.EE.2 Evidence Statement Text Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that 2 is irrational. Clarifications, limits, emphases, and other information intended to ensure appropriate variety in tasks i) Tasks may or may not have a context. ii) Students are not required to simplify expressions such as 8 to 2 2. Students are required to expression the square roots of 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100; and the cube roots of 1, 8, 27, and 64. MP Calculator - No EOY Sample Item (non-calculator) 8.EE.2 Major Content

11 8.EE.3 Major Content 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 to express how many times as much one is than the other. For example, estimate the population of the United States as and the population of the world as , and determine that the world population is more than 20 times larger. Performance Level Descriptors Grade 8 Math : Sub-Claim A The student solves problems involving the Major Content for grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Command Level 4: Strong Command Level 3: Moderate Command Level 2: Partial Command Scientific Notation 8.EE.3 8.EE EE.4-2 Using scientific notation, estimates very large and very small quantities and determines how many times as large one number is in relation to another. Using scientific notation, estimates very large and very small quantities and determines how many times as large one number is in relation to another. Using scientific notation, estimates very large and very small quantities. Using scientific notation, estimates very large quantities. Performs operations with numbers expressed in scientific notation, without technology. Performs operations with numbers expressed in scientific notation, with and without technology. Performs operations with numbers expressed in scientific notation, without technology. Performs operations with numbers expressed in scientific notation, without technology. PBA Evidence Statement Key 8.EE.3 Evidence Statement Text 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 to express how many times as much one is than the other. For example estimate the population of the United States as 3 x 10 8 and the population of the world as 7 x 10 9, and determine that the world population is more than 20 times larger. Clarifications, limits, emphases, and other information intended to ensure appropriate variety in tasks MP Calculator None 4 No 8.EE.3 Major Content

12 8.EE.3 Major Content EOY Evidence Statement Key 8.EE.3 Evidence Statement Text 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 to express how many times as much one is than the other. For example estimate the population of the United States as 3 x 10 8 and the population of the world as 7 x 10 9, and determine that the world population is more than 20 times larger. Clarifications, limits, emphases, and other information intended to ensure appropriate variety in tasks MP Calculator None 4 No EOY Sample Item (non-calculator) 8.EE.3 Major Content

13 8.EE.4 Major Content 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. Opportunities for Connecting Mathematical Content to Mathematical Practices Scientific notation (8.EE.4) presents opportunities for strategically using appropriate tools (MP.5). For example, the computation (1.73 x 10-4 ) (1.73 x 10-5 ) can be done quickly with a calculator by squaring 1.73 and then using properties of exponents to determine the exponent of the product by inspection. Performance Level Descriptors Grade 8 Math : Sub-Claim A The student solves problems involving the Major Content for grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Command Level 4: Strong Command Level 3: Moderate Command Level 2: Partial Command Scientific Notation 8.EE.3 8.EE EE.4-2 Using scientific notation, estimates very large and very small quantities and determines how many times as large one number is in relation to another. Using scientific notation, estimates very large and very small quantities and determines how many times as large one number is in relation to another. Using scientific notation, estimates very large and very small quantities. Using scientific notation, estimates very large quantities. Performs operations with numbers expressed in scientific notation, without technology. Performs operations with numbers expressed in scientific notation, with and without technology. Performs operations with numbers expressed in scientific notation, without technology. Performs operations with numbers expressed in scientific notation, without technology. PBA Evidence Statement Key 8.EE.4-1 Evidence Statement Text Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Clarifications, limits, emphases, and other information intended to ensure appropriate variety in tasks i) Tasks have thin context or no context. ii) Rules or conventions for significant figures are not assessed. iii) 20% of tasks involve both decimal and scientific notation, e.g., write x 10 4 in scientific notation. MP Calculator 6, 7, 8 No 8.EE.4 Major Content

14 8.EE.4 Major Content 8.EE.4-2 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. i) Task have thin context. ii) Tasks may require students to recognize 3.7E-2 (or 3.7e-2) from technology as 3.7 x , 7, 8 Yes 8.EE.4 Major Content

15 8.EE.4 Major Content PBA Sample Item (non-calculator) PBA Sample Item (calculator) EOY Evidence Statement Key 8.EE.4-1 Evidence Statement Text Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Clarifications, limits, emphases, and other information intended to ensure appropriate variety in tasks i) Tasks have thin context or no context. ii) Rules or conventions for significant figures are not assessed. iii) 20% of tasks involve both decimal and scientific notation, e.g., write x 10 4 in scientific notation. MP Calculator 6, 7, 8 No 8.EE.4-2 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. i) Task have thin context. ii) Tasks may require students to recognize 3.7E-2 (or 3.7e-2) from technology as 3.7 x , 7, 8 Yes 8.EE.4 Major Content

16 8.EE.4 Major Content EOY Sample Item (calculator) 8.EE.4 Major Content

17 8.EE.5 Major Content 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. Examples of Key Advances from Grade 7 to Grade 8 Students build on previous work with proportional relationships, unit rates, and graphing to connect these ideas and understand that the points (x, y) on a nonvertical line are the solutions of the equation y = mx + b, where m is the slope of the line as well as the unit rate of a proportional relationship (in the case b = 0). Students also formalize their previous work with linear relationships by working with functions rules that assign to each input exactly one output. Examples of Opportunities for In-Depth Focus: When students work toward meeting this standard, they build on grades 6 7 work with proportions and position themselves for grade 8 work with functions and the equation of a line. Examples of Major Within-Grade Dependencies An important development takes place in grade 8 when students make connections between proportional relationships, lines and linear equations (8.EE, second cluster). Making these connections depends on prior grades work, including 7.RP.2 and 6.EE.9. There is also a major dependency within grade 8 itself: The angle-angle criterion for triangle similarity underlies the fact that a non-vertical line in the coordinate plane has equation y = mx + b. Therefore, students must do work with congruence and similarity (8.G.1 5) before they are able to justify the connections among proportional relationships, lines and linear equations. Hence the indicated geometry work should likely begin at or near the very start of the year. Much of the work of grade 8 involves lines, linear equations and linear functions (8.EE.5 8; 8.F.3 4; 8.SP.2 3). Irrational numbers, radicals, the Pythagorean theorem and volume (8.NS.1 2; 8.EE.2; 8.G.6 9) are nonlinear in nature. Curriculum developers might choose to address linear and nonlinear bodies of content somewhat separately. An exception, however, might be that when addressing functions, pervasively treating linear functions as separate from nonlinear functions might obscure the concept of function per se. There should also be sufficient treatment of nonlinear functions to avoid giving students the misleading impression that all functional relationships are linear (see also 7.RP.2a). Opportunities for Connecting Mathematical Content to Mathematical Practices When students convert a fraction such as 1/7 to a decimal, they might notice that they are repeating the same calculations and conclude that the decimal repeats. Similarly, by repeatedly checking whether points are on a line through (1, 2) with slope 3, students might abstract the equation of the line in the form (y - 2)/(x - 1) = 3. In both examples, students look for and express regularity in repeated reasoning (MP.8). 8.EE.5 Major Content

18 8.EE.5 Major Content Performance Level Descriptors Grade 8 Math : Sub-Claim A The student solves problems involving the Major Content for grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Command Level 4: Strong Command Level 3: Moderate Command Level 2: Partial Command Proportional Relationships and Linear Equations Interprets scientific notation that has been generated by technology. Interprets scientific notation that has been generated by technology. Graphs linear relationships, in the form y=mx+b, including proportional relationships. Graphs linear relationships, in the form y=mx+b, including proportional relationships. 8.EE EE EE F.3-1 Graphs linear relationships in the form y=mx+b, including proportional relationships. Interprets the unit rate as the slope of the graph of a proportional relationship and applies these concepts to solve real-world problems. Graphs linear relationships in the form y=mx+b, including proportional relationships. Interprets the unit rate as the slope of the graph of a proportional relationship and applies these concepts to solve real-world problems. Interprets the unit rate as the slope of the graph of a proportional relationship and applies these concepts to solve real-world problems. Compares two different proportional relationships represented in different ways. Interprets the unit rate as the slope of the graph of a proportional relationship. Makes some comparisons between two different proportional relationships represented in different ways. Compares two different proportional relationships represented in different ways. Compares two different proportional relationships represented in different ways. Interprets y=mx+b as defining a linear function. Interprets y=mx+b as defining a linear function. Uses similar triangles to show that the slope is the same between any two distinct points on a non-vertical line in the coordinate plane. 8.EE.5 Major Content

19 8.EE.5 Major Content PBA Evidence Statement Key 8.EE EE.5-2 Evidence Statement Text 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 a greater speed. Clarifications, limits, emphases, and other information intended to ensure appropriate variety in tasks MP Calculator i) Tasks may or may not contain context. 1, 5 Yes i) Tasks may or may not contain context. 7 Yes PBA Sample Item (calculator) 8.EE.5 Major Content

20 8.EE.5 Major Content PBA Sample Item (calculator) 8.EE.5 Major Content

21 8.EE.5 Major Content PBA Sample Item (calculator) 8.EE.5 Major Content

22 8.EE.5 Major Content EOY Evidence Statement Key 8.EE EE.5-2 Evidence Statement Text 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 a greater speed. Clarifications, limits, emphases, and other information intended to ensure appropriate variety in tasks MP Calculator i) Tasks may or may not contain context. 1, 5 Yes i) Tasks may or may not contain context. 7 Yes EOY Sample Item (calculator) 8.EE.5 Major Content

23 8.EE.5 Major Content EOY Sample Item (calculator) 8.EE.5 Major Content

24 8.EE.6 Major Content 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. Examples of Key Advances from Grade 7 to Grade 8 Students build on previous work with proportional relationships, unit rates, and graphing to connect these ideas and understand that the points (x, y) on a nonvertical line are the solutions of the equation y = mx + b, where m is the slope of the line as well as the unit rate of a proportional relationship (in the case b = 0). Students also formalize their previous work with linear relationships by working with functions rules that assign to each input exactly one output. Examples of Major Within-Grade Dependencies An important development takes place in grade 8 when students make connections between proportional relationships, lines and linear equations (8.EE, second cluster). Making these connections depends on prior grades work, including 7.RP.2 and 6.EE.9. There is also a major dependency within grade 8 itself: The angle-angle criterion for triangle similarity underlies the fact that a non-vertical line in the coordinate plane has equation y = mx + b. Therefore, students must do work with congruence and similarity (8.G.1 5) before they are able to justify the connections among proportional relationships, lines and linear equations. Hence the indicated geometry work should likely begin at or near the very start of the year. Much of the work of grade 8 involves lines, linear equations and linear functions (8.EE.5 8; 8.F.3 4; 8.SP.2 3). Irrational numbers, radicals, the Pythagorean theorem and volume (8.NS.1 2; 8.EE.2; 8.G.6 9) are nonlinear in nature. Curriculum developers might choose to address linear and nonlinear bodies of content somewhat separately. An exception, however, might be that when addressing functions, pervasively treating linear functions as separate from nonlinear functions might obscure the concept of function per se. There should also be sufficient treatment of nonlinear functions to avoid giving students the misleading impression that all functional relationships are linear (see also 7.RP.2a). 8.EE.6 Major Content

25 8.EE.6 Major Content Performance Level Descriptors Grade 8 Math : Sub-Claim A The student solves problems involving the Major Content for grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Command Level 4: Strong Command Level 3: Moderate Command Level 2: Partial Command Proportional Relationships and Linear Equations Interprets scientific notation that has been generated by technology. Interprets scientific notation that has been generated by technology. Graphs linear relationships, in the form y=mx+b, including proportional relationships. Graphs linear relationships, in the form y=mx+b, including proportional relationships. 8.EE EE EE F.3-1 Graphs linear relationships in the form y=mx+b, including proportional relationships. Interprets the unit rate as the slope of the graph of a proportional relationship and applies these concepts to solve real-world problems. Graphs linear relationships in the form y=mx+b, including proportional relationships. Interprets the unit rate as the slope of the graph of a proportional relationship and applies these concepts to solve real-world problems. Interprets the unit rate as the slope of the graph of a proportional relationship and applies these concepts to solve real-world problems. Compares two different proportional relationships represented in different ways. Interprets the unit rate as the slope of the graph of a proportional relationship. Makes some comparisons between two different proportional relationships represented in different ways. Compares two different proportional relationships represented in different ways. Compares two different proportional relationships represented in different ways. Interprets y=mx+b as defining a linear function. Interprets y=mx+b as defining a linear function. Uses similar triangles to show that the slope is the same between any two distinct points on a non-vertical line in the coordinate plane. 8.EE.6 Major Content

26 8.EE.6 Major Content PBA Evidence Statement Key 8.C C.5.1 Evidence Statement Text Base reasoning on the principle that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane. Content Scope: Knowledge and skills articulated in 8.EE.6 Apply geometric reasoning in a coordinate setting, and/or use coordinates to draw geometric conclusions. Content Scope: Knowledge and skills articulated in 8.EE.6 Clarifications, limits, emphases, and other information intended to ensure appropriate variety in tasks i) Tasks require students to derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intersecting the vertical axis at b. i) Note especially the portion of 8.EE.6 before the semicolon. (this was omitted in the December, 2014 update) MP 2, 3, 7, 8 Calculator Yes 2, 3, 5 Yes EOY Evidence Statement Key 8.EE.6-1 Evidence Statement Text 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. Clarifications, limits, emphases, and other information intended to ensure appropriate variety in tasks i) Tasks do not have a context. ii) Given a non-vertical line in the coordinate plane, tasks might for example require students to choose two pairs of points and record the rise, run, and slope relative to each pair and verity that they are the same. (omitted in December, 2014 update) iii) The testing interface can provide students with a calculation aid of the specified kind for these tasks. (omitted in December, 2014 update) MP Calculator 2, 7 Yes 8.EE.6 Major Content

27 8.EE.6 Major Content EOY Sample Item (calculator) 8.EE.6 Major Content

28 8.EE.7a Major Content Solve linear equations in one variable. a. 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). Fluency Expectations or Examples of Culminating Standards: Students have been working informally with one-variable linear equations since as early as kindergarten. This important line of development culminates in grade 8 with the solution of general one-variable linear equations, including cases with infinitely many solutions or no solutions as well as cases requiring algebraic manipulation using properties of operations. Coefficients and constants in these equations may be any rational numbers. Examples of Major Within-Grade Dependencies Much of the work of grade 8 involves lines, linear equations and linear functions (8.EE.5 8; 8.F.3 4; 8.SP.2 3). Irrational numbers, radicals, the Pythagorean theorem and volume (8.NS.1 2; 8.EE.2; 8.G.6 9) are nonlinear in nature. Curriculum developers might choose to address linear and nonlinear bodies of content somewhat separately. An exception, however, might be that when addressing functions, pervasively treating linear functions as separate from nonlinear functions might obscure the concept of function per se. There should also be sufficient treatment of nonlinear functions to avoid giving students the misleading impression that all functional relationships are linear (see also 7.RP.2a). Examples of Opportunities for In-Depth Focus: This is a culminating standard for solving one-variable linear equations. Opportunities for Connecting Mathematical Content with Mathematical Practices Solving an equation such as 3(x 1/2) = x + 2 requires students to see and make use of structure (MP.7). Performance Level Descriptors Grade 8 Math : Sub-Claim A The student solves problems involving the Major Content for grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Level 4: Strong Command Level 3: Moderate Command Level 2: Partial Command Command Solving Linear Equations 8.EE.7 8.EE.C.Int. 1 Solves complex mathematical and real- world linear equations in one variable, with rational number coefficients, including those that require use of the distributive property and combining of like terms. Solves mathematical and realworld linear equations in one variable, with rational number coefficients, including those that require use of the distributive property or combining of like terms. Solves linear equations in one variable, with rational number coefficients, including those that require use of the distributive property and combining like terms. Solves linear equations in one variable, with rational number coefficients, including those that require use of the distributive property or combining like terms. 8.EE.7a Major Content

29 8.EE.7a Major Content PBA Evidence Statement Key 8.C.2 Evidence Statement Text Given an equation or system of equations, present the solution steps as a logical argument that concludes with the set of solutions (if any). Content Scope: Knowledge and skills articulated in 8.EE.7a, 8.EE.7b, 8.EE.8b Clarifications, limits, emphases, and other information intended to ensure appropriate variety in tasks i) Tasks may have three equations, but students are only required to analyze two equations at a time. MP Calculator 3, 6 Yes EOY Evidence Statement Key 8.EE.C.Int.1 Evidence Statement Text Solve word problems leading to linear equations in one variable whose solutions require expanding expressions using the distributive property and collecting like terms. Clarifications, limits, emphases, and other information intended to ensure appropriate variety in tasks i) For an example of an illustrative task, see 2009 CCRS: If a bar of soap balances ¾ of a bar of soap and ¾ of a pound, how much does the bar of soap weigh? At least 80% of tasks should involve contextual word problems (a noncontextual word problem could be the sum of two times a number and 8 is 16 ). MP Calculator 4, 6, 7 Yes 8.EE.7a Major Content

30 8.EE.7a Major Content EOY Sample Item (calculator) 8.EE.7a Major Content

31 8.EE.7b Major Content Solve linear equations in one variable. b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Fluency Expectations or Examples of Culminating Standards: Students have been working informally with one-variable linear equations since as early as kindergarten. This important line of development culminates in grade 8 with the solution of general one-variable linear equations, including cases with infinitely many solutions or no solutions as well as cases requiring algebraic manipulation using properties of operations. Coefficients and constants in these equations may be any rational numbers. Examples of Major Within-Grade Dependencies Much of the work of grade 8 involves lines, linear equations and linear functions (8.EE.5 8; 8.F.3 4; 8.SP.2 3). Irrational numbers, radicals, the Pythagorean theorem and volume (8.NS.1 2; 8.EE.2; 8.G.6 9) are nonlinear in nature. Curriculum developers might choose to address linear and nonlinear bodies of content somewhat separately. An exception, however, might be that when addressing functions, pervasively treating linear functions as separate from nonlinear functions might obscure the concept of function per se. There should also be sufficient treatment of nonlinear functions to avoid giving students the misleading impression that all functional relationships are linear (see also 7.RP.2a). Examples of Opportunities for In-Depth Focus: This is a culminating standard for solving one-variable linear equations. Opportunities for Connecting Mathematical Content with Mathematical Practices Solving an equation such as 3(x 1/2) = x + 2 requires students to see and make use of structure (MP.7). Performance Level Descriptors Grade 8 Math : Sub-Claim A The student solves problems involving the Major Content for grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Level 4: Strong Command Level 3: Moderate Command Level 2: Partial Command Command Solving Linear Equations 8.EE.7 8.EE.C.Int. 1 Solves complex mathematical and real- world linear equations in one variable, with rational number coefficients, including those that require use of the distributive property and combining of like terms. Solves mathematical and realworld linear equations in one variable, with rational number coefficients, including those that require use of the distributive property or combining of like terms. Solves linear equations in one variable, with rational number coefficients, including those that require use of the distributive property and combining like terms. Solves linear equations in one variable, with rational number coefficients, including those that require use of the distributive property or combining like terms. 8.EE.7b Major Content

32 8.EE.7b Major Content PBA Evidence Statement Key 8.EE.7.b 8.C.2 Evidence Statement Text Solve linear equations in one variable. b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Given an equation or system of equations, present the solution steps as a logical argument that concludes with the set of solutions (if any). Content Scope: Knowledge and skills articulated in 8.EE.7a, 8.EE.7b, 8.EE.8b Clarifications, limits, emphases, and other information intended to ensure appropriate variety in tasks MP Calculator i) Tasks do not have a context. 6, 7 No i) Tasks may have three equations, but students are only required to analyze two equations at a time. 3, 6 Yes PBA Sample Item (non-calculator) EOY Evidence Statement Key 8.EE.7.b 8.EE.C.Int.1 Evidence Statement Text Solve linear equations in one variable. b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Solve word problems leading to linear equations in one variable whose solutions require expanding expressions using the distributive property and collecting like terms. Clarifications, limits, emphases, and other information intended to ensure appropriate variety in tasks MP Calculator i) Tasks do not have a context. 6, 7 No i) For an example of an illustrative task, see 2009 CCRS: If a bar of soap balances ¾ of a bar of soap and ¾ of a pound, how much does the bar of soap weigh? At least 80% of tasks should involve contextual word problems (a noncontextual word problem could be the sum of two times a number and 8 is 16 ). 8.EE.7b Major Content 4, 6, 7 Yes

33 8.EE.7b Major Content EOY Sample Item (non-calculator) EOY Sample Item (non-calculator) 8.EE.7b Major Content

34 8.EE.7b Major Content EOY Sample Item (calculator) 8.EE.7b Major Content

35 8.EE.8a Major Content Analyze and solve pairs of simultaneous linear equations. a. 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. Examples of Major Within-Grade Dependencies Much of the work of grade 8 involves lines, linear equations and linear functions (8.EE.5 8; 8.F.3 4; 8.SP.2 3). Irrational numbers, radicals, the Pythagorean theorem and volume (8.NS.1 2; 8.EE.2; 8.G.6 9) are nonlinear in nature. Curriculum developers might choose to address linear and nonlinear bodies of content somewhat separately. An exception, however, might be that when addressing functions, pervasively treating linear functions as separate from nonlinear functions might obscure the concept of function per se. There should also be sufficient treatment of nonlinear functions to avoid giving students the misleading impression that all functional relationships are linear (see also 7.RP.2a). Examples of Opportunities for In-Depth Focus: When students work toward meeting this standard, they build on what they know about twovariable linear equations, and they enlarge the varieties of real-world and mathematical problems they can solve. Performance Level Descriptors Grade 8 Math : Sub-Claim A The student solves problems involving the Major Content for grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Level 4: Strong Command Level 3: Moderate Command Level 2: Partial Command Command Simultaneous Linear Equations 8.EE.8a 8.EE.8b-1 8.EE.8b-2 8.EE.8b-3 Analyzes and solves mathematical and real-world problems leading to pairs of simultaneous linear equations graphically, algebraically and by inspection. Understands the relationship between the graphic representation and the algebraic solution to the system. Verifies a solution utilizing multiple methods to prove accuracy. Analyzes and solves mathematical and real- world problems leading to pairs of simultaneous linear equations graphically, algebraically and by inspection. Understands the relationship between the graphic representation and the algebraic solution to the system. Analyzes and solves mathematical problems leading to pairs of simultaneous linear equations graphically and algebraically. Solves mathematical problems leading to pairs of simultaneous linear equations graphically or by inspection. 8.EE.8a Major Content

36 8.EE.8a Major Content PBA Evidence Statement Key 8.EE.8a 8.C.1.2 Evidence Statement Text Analyze and solve pairs of simultaneous linear equations. a. 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. Base reasoning on the principle that the graph of an equation is two variables is the set of all its solutions plotted in the coordinate plane. Content Scope: Knowledge and skills articulated in 8.EE.8a PBA Sample Item (non-calculator) Clarifications, limits, emphases, and other information intended to ensure appropriate variety MP in tasks i) Tasks do not have a context. 2, 5, 6, 7 None 2, 3, 5, 6, 7 Calculator No Yes 8.EE.8a Major Content

37 8.EE.8a Major Content PBA Sample Item (calculator) 8.EE.8a Major Content

38 8.EE.8a Major Content EOY Evidence Statement Key 8.EE.8a 8.EE.C.Int.1 Evidence Statement Text Analyze and solve pairs of simultaneous linear equations. a. 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 word problems leading to linear equations in one variable whose solutions require expanding expressions using the distributive property and collecting like terms. Clarifications, limits, emphases, and other information intended to ensure appropriate variety MP in tasks i) Tasks do not have a context. 2, 5, 6, 7 i) For an example of an illustrative task, see 2009 CCRS: If a bar of soap balances ¾ of a bar of soap and ¾ of a pound, how much does the bar of soap weigh? At least 80% of tasks should involve contextual word problems (a noncontextual word problem could be the sum of two times a number and 8 is 16 ). Calculator No 4, 6, 7 Yes 8.EE.8a Major Content

39 8.EE.8a Major Content EOY Sample Item (calculator) 8.EE.8a Major Content

40 8.EE.8a Major Content EOY Sample Item (non-calculator) 8.EE.8a Major Content

41 8.EE.8b Major Content Analyze and solve pairs of simultaneous linear equations. b. 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. Examples of Major Within-Grade Dependencies Much of the work of grade 8 involves lines, linear equations and linear functions (8.EE.5 8; 8.F.3 4; 8.SP.2 3). Irrational numbers, radicals, the Pythagorean theorem and volume (8.NS.1 2; 8.EE.2; 8.G.6 9) are nonlinear in nature. Curriculum developers might choose to address linear and nonlinear bodies of content somewhat separately. An exception, however, might be that when addressing functions, pervasively treating linear functions as separate from nonlinear functions might obscure the concept of function per se. There should also be sufficient treatment of nonlinear functions to avoid giving students the misleading impression that all functional relationships are linear (see also 7.RP.2a). Examples of Opportunities for In-Depth Focus: When students work toward meeting this standard, they build on what they know about twovariable linear equations, and they enlarge the varieties of real-world and mathematical problems they can solve. Performance Level Descriptors Grade 8 Math : Sub-Claim A The student solves problems involving the Major Content for grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Level 4: Strong Command Level 3: Moderate Command Level 2: Partial Command Command Simultaneous Linear Equations 8.EE.8a 8.EE.8b-1 8.EE.8b-2 8.EE.8b-3 Analyzes and solves mathematical and real-world problems leading to pairs of simultaneous linear equations graphically, algebraically and by inspection. Understands the relationship between the graphic representation and the algebraic solution to the system. Verifies a solution utilizing multiple methods to prove accuracy. Analyzes and solves mathematical and real- world problems leading to pairs of simultaneous linear equations graphically, algebraically and by inspection. Understands the relationship between the graphic representation and the algebraic solution to the system. Analyzes and solves mathematical problems leading to pairs of simultaneous linear equations graphically and algebraically. Solves mathematical problems leading to pairs of simultaneous linear equations graphically or by inspection. 8.EE.8b Major Content

42 8.EE.8b Major Content PBA Evidence Statement Key 8.C.2 Evidence Statement Text Given an equation or system of equations, present the solution steps as a logical argument that concludes with the set of solutions (if any). Content Scope: Knowledge and skills articulated in 8.EE.7a, 8.EE.7b, 8.EE.8b Clarifications, limits, emphases, and other information intended to ensure appropriate variety in tasks i) Tasks may have three equations, but students are only required to analyze two equations at a time. MP Calculator 3, 6 Yes 8.EE.8b Major Content