Topics Days in Cycle Dates Grade 8 envisionmath2.0 2017-2018 SY Benchmark Cycle 1 Benchmark Cycle 2 Benchmark Cycle 3 Cycle 4 September 5 - October 31 BM Window Nov 1-17 November 2 - January 26 BM Window Jan 29-Feb 13 January 30 - May 8 BM Window May 9-25 May 10 - June 12 Total Days: 39 Including 1 Half Day Total Days: 50 Days Including 4 Half Days Total Days Before PSSA: 50 Including 9 Half Days Total Days in Cycle: 60 Including 13 Half Days Total Days: 22 Including 1 Half Day envisionmath2.0 envisionmath2.0 envisionmath2.0 envisionmath2.0 2: Analyze and Solve Linear Equations 1-1 1-5: Real Numbers* *Note: Volume could be addressed within cubes (1-4 and 1-5). 1-6 1-10: Real Numbers (continued) 3: Use Functions to Model Relationships 4: Investigate Bivariate Data 5-1 5-2: Analyze and Solve Systems of Linear Equations 5-3 5-4: Analyze and Solve Systems of Linear Equations 6-1 6-7*: Congruence and Similarity 7: Understand and Apply the Pythagorean Theorem *Note: Lessons 6-8 to 6-10 do not contain new eligible content or PA Core content and so can be covered after the PSSA 6-7 6-10: Congruence and Similarity (cont d) 8: Solve Problems Involving Surface Area and Volume Remediate and Extend Note: A cycle is defined as the time allotted to teach the content that is on each benchmark, and assumes the benchmark is taken on the first day of the window. This means that though it is fine to give the test later in the window, you should be moving on to new content as of the above listed dates, or you will fall behind. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 1
Table of Contents Benchmark Cycle 1 Standards... 3 Benchmark Cycle 1 Scope and Sequence... 5 Benchmark Cycle 2 Standards... 7 Benchmark Cycle 2 Scope and Sequence... 11 Benchmark Cycle 3 Standards... 15 Benchmark Cycle 3 Scope and Sequence... 17 Cycle 4 Standards... 21 Cycle 4 Scope and Sequence... 22 PA Core Standards and Eligible Content by Cycle... 25 Document Information Page... 30 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 2
Benchmark Cycle 1 Standards PA Core Standard PA Eligible Content Common Core Standard 2.1.8.E.1 Distinguish between rational and irrational numbers using their properties. 2.1.8.E.4 Estimate irrational numbers by comparing them to rational numbers. 2.2.8.B.1 Apply concepts of radicals and integer exponents to generate equivalent expressions. M08.A-N.1.1.1 Determine whether a number is rational or irrational. For rational numbers, show that the decimal expansion terminates or repeats (limit repeating decimals to thousandths). M08.A-N.1.1.2 Convert a terminating or repeating decimal into a rational number (limit repeating decimals to thousandths). M08.A-N.1.1.3 Estimate the value of irrational numbers without a calculator (limit whole number radicand to less than 144). Example: 5 is between 2 and 3 but closer to 2. M08.A-N.1.1.4 Use rational approximations of irrational numbers to compare and order irrational numbers. M08.A-N.1.1.5 Locate/identify rational and irrational numbers at their approximate locations on a number line. M08.B-E.1.1.2 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 perfect squares (up to and including 12 2 ) and cube roots of perfect cubes (up to and including 5 3 ) without a calculator. Example: If x 2 = 25 then x = ± 25. 8.NS.1. 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. 8.NS.2. 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., π2). 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. 8.EE.2. 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. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 3
2.2.8.B.2 Understand the connections between proportional relationships, lines, and linear equations. 2.2.8.B.3 Analyze and solve linear equations and pairs of simultaneous linear equations M08.B-E.2.1.1 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Example: Compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. M08.B-E.2.1.2 Use similar right triangles to show and explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane. M08.B-E.2.1.3 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. M08.B-E.3.1.1 Write and identify 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]n 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). M08.B-E.3.1.2 Solve linear equations that have rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. 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. 8.EE.6. Use similar triangles to explain why the slope m is the same between any two distinct points on a nonvertical 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. 8.EE.7. 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). 8.EE.7. 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. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 4
9/5 10/06 2 Days Per Lesson + 4 Days Benchmark Cycle 1 Scope and Sequence Suggested Dates Topic- Lesson 2-1 2-2 Lesson Title Combine Like Terms to Solve Equations Solve Equations with Variables on Both Sides 2-3 Solve Multistep Equations 2-4 2-5 Equations with No Solutions or Infinitely Many Solutions Mid-Topic Checkpoint and Performance Task Compare Proportional Relationships Topic 2: Analyze and Solve Linear Equations Lesson Objective(s) - Combine like terms - Solve equations with like terms on one side of the equation - Make sense of scenarios and represent them with equations - Solve equations with like terms on both sides of the equation - Make sense of scenarios and represent them with equations - Plan multiple solution pathways and choose one to find the solution to multistep equations Eligible Content M08.B-E.3.1.2 M08.B-E.3.1.2 M08.B-E.3.1.2 - Determine the number of solutions to an equation M08.B-E.3.1.1 - Analyze equations, linear graphs, and tables to find unit rates and compare proportional relationships M08.B-E.3.1.1 M08.B-E.3.1.2 M08.B-E.2.1.1 2-6 Connect Proportional Relationships and Slope 2-7 Analyze Linear Equations y=mx 2-8 Understand the y-intercept of a Line - Find the slope of a line using different strategies - Interpret a slope in context and relate it to steepness on a graph - Understand how the constant of proportionality and the slope relate in a linear equation - Write a linear equation in the form y = mx when the slope is given - Graph a linear equation in the form y = mx - Interpret and extend the table or graph of a linear relationship to find its y-intercept - Analyze graphs in context to determine and explain the meaning of the y-intercept M08.B-E.2.1.2 M08.B-E.2.1.3 M08.B-E.2.1.2 M08.B-E.2.1.3 M08.B-E.2.1.2 M08.B-E.2.1.3 2-9 Analyze Linear Equations y = mx + b - Graph a line from an equation in the form y = mx + b - Write an equation that represents the given graph of a line M08.B-E.2.1.2 M08.B-E.2.1.3 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 5
10/10-10/31 2 Days Per Lesson + 3.5 Days Topic 1: Real Numbers Suggested Dates Topic- Lesson Lesson Title Lesson Objective(s) Eligible Content 1-1 Rational Numbers as Decimals - Write repeating decimals as fractions M08.A-N.1.1.2 1-2 Understand Irrational Numbers - Identify an irrational number M08.A-N.1.1.1 1-3 1-4 Compare and Order Real Numbers Evaluate Square Roots and Cube Roots - Compare and order rational and irrational numbers M08.A-N.1.1.3 M08.A-N.1.1.4 M08.A-N.1.1.5 - Find square roots and cube roots of rational numbers M08.B-E.1.1.2 1-5 Solve Equations Using Square Roots and Cube Roots - Solve equations and problems, in real-world contexts involving square roots and cube roots M08.B-E.1.1.2 Mid-Topic Checkpoint and/or Performance Task Benchmark 1 Window: 11/1 11/17 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 6
Benchmark Cycle 2 Standards PA Core Standard PA Eligible Content Common Core Standard 2.2.8.B.1 Apply concepts of radicals and integer exponents to generate equivalent expressions. 2.2.8.B.3 Analyze and solve linear equations and pairs of simultaneous linear equations. M08.B-E.1.1.1 Apply one or more properties of integer exponents to generate equivalent numerical expressions without a calculator (with final answers expressed in exponential form with positive exponents). Properties will be provided. Example: 3 12 3-15 = 3 3 = 1/3 3 = 1/27. M08.B-E.1.1.2 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 perfect squares (up to and including 12 2 ) and cube roots of perfect cubes (up to and including 5 3 ) without a calculator. Example: If x 2 = 25 then x = ± 25. M08.B-E.1.1.3 Estimate very large or very small quantities by using numbers expressed in the form of a single digit times an integer power of 10, and express how many times larger or smaller one number is than another. Example: Estimate the population of the United States as 3 10 8 and the population of the world as 7 10 9, and determine that the world population is more than 20 times larger than the United States population. M08.B-E.3.1.3 Interpret solutions to a system of two linear equations in two variables as points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. M08.B-E.3.1.4 Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. 8.EE.1. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3 2 3 5 = 3 3 = 1/3 3 = 1/27. 8.EE.2. 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. 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 times 10 8 and the population of the world as 7 times 10 9, and determine that the world population is more than 20 times larger. 8.EE.8. 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. 8.EE.8. 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. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 7
2.2.8.C.1 Define, evaluate, and compare functions. 2.2.8.C.2 Use concepts of functions to model relationships between quantities. M08.B-F.2.1.1 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models and in terms of its graph or a table of values. M08.B-F.1.1.2 Compare properties of two functions each represented in a different way (i.e., algebraically, graphically, numerically in tables, or by verbal descriptions). 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. M08.B-F.1.1.3 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. M08.B-F.2.1.1 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models and in terms of its graph or a table of values. M08.B-F.2.1.2 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch or determine a graph that exhibits the qualitative features of a function that has been described verbally. 8.F.4. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. 8.F.2. 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. 8.F.3. 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 is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. 8.F.4. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. 8.F.5. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 8
2.4.8.B.1 Analyze and/or interpret bivariate data displayed in multiple representations. 2.4.8.B.2 Understand that patterns of association can be seen in bivariate data utilizing frequencies. M08.D-S.1.1.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative correlation, linear association, and nonlinear association. M08.D-S.1.1.2 For scatter plots that suggest a linear association, identify a line of best fit by judging the closeness of the data points to the line. M08.D-S.1.1.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. Example: In a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. M08.D-S.1.2.1 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 associations between the two variables. Example: Given data on whether students have a curfew on school nights and whether they have assigned chores at home, is there evidence that those who have a curfew also tend to have chores? 8.SP.1. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. 8.SP.2. 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. 8.SP.3. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. 8.SP.4. 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. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores? THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 9
2.2.8.C.1 Define, evaluate, and compare functions. 2.2.8.C.2 Use concepts of functions to model relationships between quantities. M08.B-F.1.1.1 Determine whether a relation is a function. M08.B-F.2.1.2 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch or determine a graph that exhibits the qualitative features of a function that has been described verbally. 8.F.1. 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. 8.F.5. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 10
11/02-11/28 2 Days Per Lesson + 3.5 Days Benchmark Cycle 2 Scope and Sequence Topic 1: Real Numbers (continued) Suggested Dates Topic- Lesson Lesson Title Lesson Objective(s) Eligible Content 1-6 Use Properties of Integer Exponents - Understand the properties of exponents - Generate equivalent expressions with exponents M08.B-E.1.1.1 1-7 More Properties of Integer Exponents - Simplify expressions with negative and zero exponents - Evaluate expressions with negative and zero exponents M08.B-E.1.1.1 1-8 Use Powers of 10 to Estimate Quantities - Estimate very large and very small quantities by rounding and then writing that number as a single digit times a power of 10 M08.B-E.1.1.3 1-9 Understand Scientific Notation - Use scientific notation to write very large or very small quantities - Convert numbers written in scientific notation to standard form Preparing for: M08.B-E.1.1.4 1-10 Operations with Numbers in Scientific Notation - Apply number properties to calculations with numbers in scientific notation M08.B-E.1.1.4 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 11
11/29 12/22 2 Days per Lesson + 5.5 Days Topic 3: Use Functions to Model Relationships Suggested Dates Topic- Lesson Lesson Title Lesson Objective(s) Eligible Content 3-1 Understand Relations and Functions - Identify whether a relation is a function - Interpret a Function M08.B-F.1.1.1 3-2 Connect Representations of Functions - Identify functions by their equations, tables, and graphs - Represent linear and nonlinear functions with graphs M08.B-F.1.1.1 3-3 Compare Linear and Nonlinear Functions - Use different representations to compare linear and nonlinear functions M08.B-F.1.1.2 M08.B-F.1.1.3 Mid-Topic Checkpoint and/or 3-Act Mathematical Modeling M08.B-F.1.1.1 M08.B-F.1.1.2 M08.B-F.1.1.3 3-4 Construct Functions to Model Linear Relationships - Write an equation in the form y = mx + b to describe a linear function M08.B-F.1.1.2 M08.B-F.2.1.1 3-5 Intervals of Increase and Decrease - Describe the behavior of a function in different intervals M08.B-F.2.1.2 3-6 Sketch Functions from Verbal Descriptions - Draw a sketch of a graph for a function that has been described verbally - Analyze and interpret the sketch of a graph of a function M08.B-F.2.1.2 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 12
01/03 1/19 2 Days per Lesson + 2 Days Topic 4: Investigate Bivariate Data Suggested Dates Topic- Lesson Lesson Title Lesson Objective(s) Eligible Content 4-1 Construct and Interpret Scatter Plots - Construct a scatter plot to model paired data - Utilize a scatter plot to identify and interpret the relationship between paired data M08.D-S.1.1.1 4-2 Analyze Linear Associations - Recognize whether the paired data have a linear association, a nonlinear association, or no association M08.D-S.1.1.2 M08.B-F.1.1.3 M08.B-F.2.1.1 4-3 Use Linear Models to Make Predictions - Use the slope and y-intercept of a trend line to make a prediction - Make a prediction when no equation is given by drawing trend lines and writing the equation of the linear model M08.D-S.1.1.3 M08.B-F.1.1.3 M08.B-F.2.1.1 Mid-Topic Checkpoint and/or Performance Task M08.D-S.1.1.1 M08.D-S.1.1.2 M08.D-S.1.1.3 M08.B-F.1.1.3 M08.B-F.2.1.1 4-4 Interpret Two-Way Frequency Tables - Organize paired categorical data into a two-way frequency table - Compare and make conjectures about data displayed in a twoway frequency table M08.D-S.1.2.1 4-5 Interpret Two-Way Relative Frequency Tables - Construct two-way relative frequency tables - Compare and make conjectures about data displayed in a twoway relative frequency table M08.D-S.1.2.1 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 13
01/22-01/26 2 Days Per Lesson + 1 Day Topic 5: Analyze and Solve Systems of Linear Equations Suggested Dates Topic. Lesson Lesson Title 5-1 Estimate Solutions by Inspection Lesson Objective(s) - Examine the graphs of a linear system of equations to determine the number of solutions to the system - Compare the equations in a linear system to determine the number of solutions Eligible Content M08.B-E.3.1.4 M08.B-E.3.1.5 5-2 Solve Systems by Graphing - Create and examine graphs of linear systems of equations to determine the solution M08.B-E.3.1.3 M08.B-E.3.1.5 Mid-Topic Checkpoint and/or Performance Task M08.B-E.3.1.3 M08.B-E.3.1.4 M08.B-E.3.1.5 Benchmark 2 Window 01/29-2/13 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 14
Benchmark Cycle 3 Standards PA Core Standard PA Eligible Content Common Core Standard 2.2.8.B.3 Analyze and solve linear equations and pairs of simultaneous linear equations. 2.2.8.B.3 Analyze and solve linear equations and pairs of simultaneous linear equations. 2.3.8.A.2 Understand and apply congruence, similarity, and geometric transformations using various tools. M08.B-E.3.1.3 Interpret solutions to a system of two linear equations in two variables as points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. M08.B-E.3.1.4 Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. M08.B-E.3.1.5 Solve real-world and mathematical problems leading to two linear equations in two variables. M08.B-E.3.1.3 Interpret solutions to a system of two linear equations in two variables as points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. M08.C-G.1.1.1 Identify and apply properties of rotations, reflections, and translations. Example: Angle measures are preserved in rotations, reflections, and translations. M08.C-G.1.1.2 Given two congruent figures, describe a sequence of transformations that exhibits the congruence between them. 8.EE.8. 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. 8.EE.8. 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. 8.EE.8. 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. 8.G.1. Verify experimentally the properties of rotations, reflections, and translations: a. Lines are taken to lines, and line segments to line segments of the same length. b. Angles are taken to angles of the same measure. c. Parallel lines are taken to parallel lines. 8.G.2. Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 15
2.3.8.A.3 Understand and apply the Pythagorean Theorem to solve problems. M08.C-G.1.1.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures, using coordinates. M08.C-G.1.1.4 Given two similar two-dimensional figures, describe a sequence of transformations that exhibits the similarity between them. M08.C-G.2.1.1 Apply the converse of the Pythagorean theorem to show a triangle is a right triangle. M08.C-G.2.1.2 Apply the Pythagorean theorem to determine unknown side lengths in right triangles in realworld and mathematical problems in two and three dimensions. (Figures provided for problems in three dimensions will be consistent with Eligible Content in grade 8 and below.) M08.C-G.2.1.3 Apply the Pythagorean theorem to find the distance between two points in a coordinate system. 8.G.3. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. 8.G.4. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar twodimensional figures, describe a sequence that exhibits the similarity between them. 8.G.6. Explain a proof of the Pythagorean Theorem and its converse. 8.G.7. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. 8.G.8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 16
01/30 02/07 2 Days per Lesson + 3 Days Benchmark Cycle 3 Scope and Sequence Topic 5: Analyze and Solve Systems of Linear Equations (continued) Suggested Date Topic- Lesson Lesson Title Lesson Objective(s) Eligible Content 5-3 Solve Systems by Substitution - Understand how substitution can be used to solve a linear system of equations - Apply this understanding to interpret the results with one solution, no solution or infinitely many solutions M08.B-E.3.1.4 M08.B-E.3.1.5 5-4 Solve Systems by Elimination - Understand how the process of elimination can be used to solve a system of linear equations with no solution, one solution, or infinitely many solutions - Apply this understanding to solve mathematical and realworld problems M08.B-E.3.1.4 M08.B-E.3.1.5 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 17
2/08 3/13 2 Days Per Lesson + 6.5 Days Topic 6: Congruence and Similarity Suggested Date Topic- Lesson Lesson Title 6-1 Analyze Translations 6-2 Analyze Reflections 6-3 Analyze Rotations Lesson Objective(s) - Use coordinates to describe the rules of a translation - Translate a two-dimensional figure on a coordinate plane by mapping each of its vertices - Understand reflections as a type of transformation and how they differ from translations - Use coordinates to describe the image created by a reflection - Reflect a two-dimensional figure on a coordinate plane - Identify and perform a rotation - Describe a rotation - Determine how a rotation affects a two-dimensional figure Eligible Content M08.C-G.1.1.1 M08.C-G.1.1.3 M08.C-G.1.1.1 M08.C-G.1.1.3 M08.C-G.1.1.1 M08.C-G.1.1.3 6-4 Compose Transformations - Describe and perform a sequence of transformations - Apply their knowledge of transformations to solve problems M08.C-G.1.1.1 M08.C-G.1.1.3 6-5 Understand Congruent Figures Mid-Topic Checkpoint and/or Performance Assessment 6-6 Describe Dilations - Use a sequence of transformations to justify congruence of figures - Understand that reflections, rotations, and translations are actions that produce congruent geometric figures - Verify the properties of a dilation - Graph the image of a dilation given a fixed center and a common scale factor M08.C-G.1.1.2 M08.C-G.1.1.3 M08.C-G.1.1.3 M08.C-G.1.1.4 6-7 Understand Similar Figures - Perform a sequence of transformations to identify similar figures M08.C-G.1.1.3 M08.C-G.1.1.4 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 18
3/14 4/13 2 Days Per Lesson + 6 Days and ELA PSSA Week Topic 7: Understand and Apply the Pythagorean Theorem Suggested Dates Topic. Lesson Lesson Title Lesson Objective Eligible Content 7-1 Understand the Pythagorean Theorem - Understand a proof of the Pythagorean Theorem - Use the Pythagorean Theorem to find the length of the hypotenuse or a leg of a right triangle Preparing for M08.C-G.2.1.1 M08.C-G.2.1.2 7-2 Understand the Converse of the Pythagorean Theorem - Understand and apply the converse of the Pythagorean Theorem to identify right triangles. - Use the converse of the Pythagorean Theorem to analyze two-dimensional shapes M08.C-G.2.1.1 7-3 Apply the Pythagorean Theorem to Solve Problems - Use the Pythagorean Theorem and its converse to solve problems M08.C-G.2.1.2 7-4 Find Distance in the Coordinate Plane - Apply the Pythagorean Theorem to find distance between two points in the coordinate plane - Use the Pythagorean Theorem to find the perimeter of a figure and to solve problems on the coordinate plane M08.C-G.2.1.3 4/16 4/20 MATH PSSA THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 19
4/23 5/08 2 Days Per Lesson + Science PSSA Week and 1.5 Days Topic 8: Surface Area and Volume Suggested Dates Topic- Lesson Lesson Title Lesson Objective Eligible Content 8-1 Find Surface Area of Three- Dimensional Figures - Calculate the surface area of cylinders, cones and spheres Preparing for M08.C-G.3.1.1 8-2 Find Volume of Cylinders - Identify and use the correct formula to find the volume of cylinders - Recognize the relationship between the formulas for volume of a rectangular prism and the volume of a cylinder M08.C-G.3.1.1 Benchmark 3 Window 5/9 5/25 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 20
Cycle 4 Standards PA Core Standard PA Eligible Content Common Core Standard 2.3.8.A.1 Apply the concepts of volume of cylinders, cones, and spheres to solve realworld and mathematical problems. M08.C-G.3.1.1 Apply formulas for the volumes of cones, cylinders, and spheres to solve real-world and mathematical problems. Formulas will be provided. 8.G.9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve realworld and mathematical problems. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 21
4/23 5/25 2 Days Per Lesson + 4.5 Days Cycle 4 Scope and Sequence Topic 8: Surface Area and Volume Suggested Dates Topic- Lesson Lesson Title Lesson Objective Eligible Content 8-1 Find Surface Area of Three- Dimensional Figures - Calculate the surface area of cylinders, cones and spheres Preparing for M08.C-G.3.1.1 8-2 Find Volume of Cylinders - Identify and use the correct formula to find the volume of cylinders - Recognize the relationship between the formulas for volume of a rectangular prism and the volume of a cylinder M08.C-G.3.1.1 8-3 Find Volume of Cones - Find the volume of a cone - Recognize the relationship between volume of a cylinder and volume of a cone M08.C-G.3.1.1 8-4 Find Volume of Spheres - Calculate the volume of a sphere - Recognize the relationship between the formula for the volume of a cone and the volume of a sphere M08.C-G.3.1.1 6-7 Understand Similar Figures - Perform a sequence of transformations to identify similar figures M08.C-G.1.1.3 M08.C-G.1.1.4 6-8 Angles, Lines, and Transversals - Identify relationships between angles formed by parallel lines and a transversal - Determine the measures of angles formed by parallel lines and a transversal - Reason about parallel lines M07.C-G.2.1.2 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 22
6-9 Interior and Exterior Angles of Triangles - Determine unknown measures of interior and exterior angles in triangles - Write and solve algebraic equations to find angle measures Not assessed in PA 6-10 Angle-Angle Triangle Similarity - Determine triangle similarity by comparing the angle measures of triangles - Solve algebraic problems involving similar triangles M07.C-G.1.1.2 Remediate and Extend 5/29 6/12 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 23
Remediation and Extension Options 5/29-6/12 11 Days Option 1: Revisit 3-Act Math Lessons (6-8 env) Each Topic featured a 3-Act Mathematical Modeling lesson that you may or may not have had time to do with your students. During the remainder of the year, you can revisit key ideas from 6 th grade and provide students with opportunities to engage authentically in the Standards for Mathematical Practice through a series of 3-Act Lessons. You can also find more lessons at this link: https://whenmathhappens.com/3-act-math/. Option 2: Reteach Based on Common Core-identified focal areas. Option 3: Miscellaneous (Financial Literacy, Quantitative Reasoning, Statistics, etc.) You taught many topics this year. The topics below are identified as the most critical for 8 th grade. If you want to revisit something, one or more of these topics would be a good idea. 8.EE.A Work with radicals and integer exponents. 8.EE.B Understand the connections between proportional relationships, lines, and linear equations. 8.EE.C Analyze and solve linear equations and pairs of simultaneous linear equations. 8.F.A Define, evaluate, and compare functions. 8.F.B Use functions to model relationships between quantities. 8.G.A Understand congruence and similarity using physical models, transparencies, or geometry software. 8.G.B Understand and apply the Pythagorean Theorem. The Federal Reserve Bank of Philadelphia has posted free lesson plans for elementary teachers (including K) on financial literacy at: https://philadelphiafed.org/education/teachers/lesson-plans. Many of these lessons are tied to early children s literature, as well. In addition, Census.gov has a number of data analysis activities, using real U.S. Census data, at: https://www.census.gov/schools/activities/math.html. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 24
PA Core Standards and Eligible Content by Cycle PA Core Standard CC.2.1.8.E.1 Distinguish between rational and irrational numbers using their properties. PA Eligible Content M08.A-N.1.1.1 Determine whether a number is rational or irrational. For rational numbers, show that the decimal expansion terminates or repeats (limit repeating decimals to thousandths). M08.A-N.1.1.2 Convert a terminating or repeating decimal to a rational number (limit repeating decimals to thousandths). Cycle 1 Cycle 2 Cycle 3 Cycle 4 CC.2.1.8.E.4 Estimate irrational numbers by comparing them to rational numbers M08.A-N.1.1.3 Estimate the value of irrational numbers without a calculator (limit whole number radicand to less than 144). M08.A-N.1.1.4 Use rational approximations of irrational numbers to compare and order irrational numbers. M08.A-N.1.1.5 Locate/identify rational and irrational numbers at their approximate locations on a number line. CC.2.2.8.B.1 Apply concepts of radicals and integer exponents to generate equivalent expressions. M08.B-E.1.1.1 Apply one or more properties of integer exponents to generate equivalent numerical expressions without a calculator (with final answers expressed in exponential form with positive exponents). Properties will be provided. M08.B-E.1.1.2 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 perfect squares (up to and including 12 2 ) and cube roots of perfect cubes (up to and including 5 3 ) without a calculator. M08.B-E.1.1.3 Estimate very large or very small quantities by using numbers expressed in the form of a single digit times an integer power of 10 and express how many times larger or smaller one number is than another. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 25
M08.B-E.1.1.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Express answers in 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 (e.g., interpret 4.7EE9 displayed on a calculator as 4.7 10 9 ). CC.2.2.8.B.2 Understand the connections between proportional relationships, lines, and linear equations. CC.2.2.8.B.3 Analyze and solve linear equations and pairs of simultaneous linear equations. M08.B-E.2.1.1 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. M08.B-E.2.1.2 Use similar right triangles to show and explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane. M08.B-E.2.1.3 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. M08.B-E.3.1.1 Write and identify 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). M08.B-E.3.1.2 Solve linear equations that have rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. M08.B-E.3.1.3 Interpret solutions to a system of two linear equations in two variables as points of intersection of their graphs because points of intersection satisfy both equations simultaneously. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 26
M08.B-E.3.1.4 Solve systems of two linear equations in two variables algebraically and estimate solutions by graphing the equations. Solve simple cases by inspection. M08.B-E.3.1.5 Solve real-world and mathematical problems leading to two linear equations in two variables. CC.2.2.8.C.1 Define, evaluate, and compare functions. CC.2.2.8.C.2 Use concepts of functions to model relationships between quantities. M08.B-F.1.1.1 Determine whether a relation is a function. M08.B-F.1.1.2 Compare properties of two functions, each represented in a different way (i.e., algebraically, graphically, numerically in tables, or by verbal descriptions). M08.B-F.1.1.3 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. M08.B-F.2.1.1 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models and in terms of its graph or a table of values. M08.B-F.2.1.2 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch or determine a graph that exhibits the qualitative features of a function that has been described verbally. CC.2.3.8.A.1 Apply the concepts of volume of cylinders, cones, and spheres to solve real world and mathematical problems. M08.C-G.3.1.1 Apply formulas for the volumes of cones, cylinders, and spheres to solve real-world and mathematical problems. Formulas will be provided. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 27
CC.2.3.8.A.2 Understand and apply congruence, similarity, and geometric transformations using various tools. M08.C-G.1.1.1 Identify and apply properties of rotations, reflections, and translations. M08.C-G.1.1.2 Given two congruent figures, describe a sequence of transformations that exhibits the congruence between them. M08.C-G.1.1.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. M08.C-G.1.1.4 Given two similar two-dimensional figures, describe a sequence of transformations that exhibits the similarity between them. CC.2.3.8.A.3 Understand and apply the Pythagorean Theorem to solve problems. M08.C-G.2.1.1 Apply the converse of the Pythagorean theorem to show a triangle is a right triangle. M08.C-G.2.1.2 Apply the Pythagorean theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. (Figures provided for problems in three dimensions will be consistent with Eligible Content in grade 8 and below.) M08.C-G.2.1.3 Apply the Pythagorean theorem to find the distance between two points in a coordinate system. CC.2.4.8.B.1 Analyze and/or interpret bivariate data displayed in multiple representations. M08.D-S.1.1.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative correlation, linear association, and nonlinear association. M08.D-S.1.1.2 For scatter plots that suggest a linear association, identify a line of best fit by judging the closeness of the data points to the line. M08.D-S.1.1.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 28
CC.2.4.8.B.2 Understand that patterns of association can be seen in bivariate data utilizing frequencies. M08.D-S.1.2.1 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 associations between the two variables. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 29
Document Information Page Overview of Contents of Document The Cover Page lays out the Topics taught within each Cycle as well as the corresponding dates. The Benchmark Cycle Standards pages that precede each cycle outline all of the standards that are taught within that cycle. These are, therefore, all of the standards that may be on that Cycle s Benchmark. The Benchmark Cycle Scope and Sequence pages provide suggested pacing that allows for 1 day per lesson plus some flex days. We recommend proactively using the flex days for: reviewing pre-requisite content, splitting lessons over multiple days, assessing, reteaching, and doing projects. Of course, some of these will also be taken by field trips and other school activities. The PA Core Standards and Eligible Content by Cycle page lists all of the standards and indicates in which cycle(s) they are taught. What is a Cycle? We want to offer clarity on what appears on each benchmark. Additionally, there should be sufficient time to teach that content before it is tested. Because each school administers the benchmark on a different day, not necessarily corresponding with the last day of the Term, we have created Cycles. Each Cycle contains the content that is to be taught and tested on a given benchmark. Please refer to the dates on the Cover Page to ensure you are aware of the beginning and ending dates for each Cycle. What If I Fall Behind? We trust you to make decisions about what is best for your students. This pacing will prepare you for the Benchmarks and PSSA, but it is a suggested, not mandated, pacing. You may also wish to move at a faster pace. Do not feel you should slow down to match this guide. If you are concerned about content that you may not reach before the PSSA, consider implementing number talks and other short routines and games. For example, a lot of Geometry vocabulary and concepts could be taught through Which Once Doesn t Belong. Rather than pushing to cover content, or using test prep resources, content can be infused through short but meaningful structures. The table below is purely for informational purposes. We used the PSSA Mathematics Blueprint to calculate what percent of tested content you will cover, based on how much of the book you teach before the test. Again, we recommend depth and understanding over coverage, but we wanted to provide this information for your planning purposes. Percent if Topics 1-8 Are Taught Percent if Topics 1-7 Are Taught Percent if Topics 1-6 Are Taught Percent if Topics 1-5 Are Taught 100% 96% 88.2% 79.5% THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 30