Grade 6 envisionmath2.0 2017-2018 SY Benchmark Cycle 1 Benchmark Cycle 2 Benchmark Cycle 3 Cycle 4 Dates 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 Teaching Days 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; Excluding Week of Math PSSA Total Days: 22 Including 1 Half Day Topics 1 Use Positive Rational Numbers 2 Integers and Rational Numbers 3-1 3-3 Numeric and Algebraic Expressions 3-4 3-7 Numeric and Algebraic Expressions 4 Represent and Solve Equations and Inequalities 5 Understand and Use Ratio and Rate *8 Display, Describe, and Summarize Data 7 Solve Area, Surface Area, and Volume Problems 6 Understand and Use Percent Note: A Benchmark 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. *We moved Topic 6 until after the PSSA because the content is not highly tested. You may need to do a mini-lesson on basic percents before Topic 8, but you do not need to do the entire topic. We also switched the order of 7 and 8 because is the most highly tested and therefore we wanted to ensure you do it before the test. If you know you will fit in all of 7 and 8 before the PSSA, you are welcome to reverse them. 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... 6 Benchmark Cycle 2 Standards... 9 Benchmark Cycle 2 Scope and Sequence... 13 Benchmark Cycle 3 Standards... 16 Benchmark Cycle 3 Scope and Sequence... 18 Cycle 4 Standards... 22 Cycle 4 Scope and Sequence... 23 PA Core 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 Common Core Standard CC.2.1.6.E.1 Apply and extend previous understandings of multiplication and division to divide fractions by fractions. CC.2.1.6.E.2 Identify and choose appropriate processes to compute fluently with multi-digit numbers. CC.2.1.6.E.3 Develop and/or apply number theory concepts to find common factors and multiples. PA Eligible Content M06.A-N.1.1.1 Interpret and compute quotients of fractions (including mixed numbers), and solve word problems involving division of fractions by fractions. Example 1: Given a story context for (2/3) (3/4), explain that (2/3) (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) (c/d) = (a/b) (d/c) = ad/bc.) Example 2: How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi? Example 3: How many 2 1/4- foot pieces can be cut from a 15 1/2-foot board? M06.A-N.2.1.1 Solve problems involving operations (+,,, and ) with whole numbers, decimals (through thousandths), straight computation, or word problems. M06.A-N.2.2.1 Find the greatest common factor to express a sum of two whole numbers, 1 through 100, with a common factor as a multiple of a sum of two whole numbers with no common factor. Example: Express 36 + 8 as 4(9+2) M06.A-N.2.2.2 Apply the distributive property to express a sum of two whole numbers, 1 through 100, with a common factor as a multiple of a sum of two whole numbers with no common factor. Example: Express 36 + 8 as 4(9 + 2). Common Core State Standard 6.NS.A.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?. 6.NS.B.2 Fluently divide multi-digit numbers using the standard algorithm. 6.NS.B.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. 6.NS.B.4 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2). THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 3
PA Common Core Standard CC.2.1.6.E.4 Apply and extend previous understandings of numbers to the system of rational numbers. PA Eligible Content M06.A-N.3.1.1 Represent quantities in real-world contexts using positive and negative numbers, explaining the meaning of 0 in each situation (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge). M06.A-N.3.1.2 Determine the opposite of a number and recognize that the opposite of the opposite of a number is the number itself (e.g., ( 3) = 3; 0 is its own opposite). M06.A-N.3.1.3 Locate and plot integers and other rational numbers on a horizontal or vertical number line; locate and plot pairs of integers and other rational numbers on a coordinate plane. Common Core State Standard 6.NS.C.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation. 6.NS.C.6 Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., -(-3) = 3, and that 0 is its own opposite. 6.NS.C.6 Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 4
PA Common Core Standard CC.2.1.6.E.4 Apply and extend previous understandings of numbers to the system of rational numbers. CC.2.2.6.B.1 Apply and extend previous understandings of arithmetic to algebraic expressions. CC.2.3.6.A.1 Apply appropriate tools to solve real-world and mathematical problems involving area, surface area, and volume. PA Eligible Content M06.A-N.3.2.1 Write, interpret, and explain statements of order for rational numbers in real-world contexts. Example: Write 3 C > 7 C to express the fact that 3 C is warmer than 7 C. M06.A-N.3.2.2 Interpret the absolute value of a rational number as its distance from 0 on the number line and as a magnitude for a positive or negative quantity in a realworld situation. Example: For an account balance of 30 dollars, write 30 = 30 to describe the size of the debt in dollars, and recognize that an account balance less than 30 dollars represents a debt greater than 30 dollars. M06.A-N.3.2.3 Solve real-world and mathematical problems by plotting points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. M06.B-E.1.1.1 Write and evaluate numerical expressions involving whole-number exponents. M06.C-G.1.1.4 Given coordinates for the vertices of a polygon in the plane, use the coordinates to find side lengths and area of the polygon (limited to triangles and special quadrilaterals). Formulas will be provided. Common Core State Standard 6.NS.C.7 Understand ordering and absolute value of rational numbers. b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write -3 o C > -7 o C to express the fact that -3 o C is warmer than -7 o C. 6.NS.C.7 Understand ordering and absolute value of rational numbers. c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of -30 dollars, write -30 = 30 to describe the size of the debt in dollars. 6.NS.C.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. 6.EE.A.1 Write and evaluate numerical expressions involving whole-number exponents. 6.G.A.3 Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 5
Benchmark Cycle 1 Scope and Sequence Suggested Dates 9/5 9/29 2 Days Per Lesson + 3 Days Topic- Lesson 1-1 1-2 Topic 1: Use Positive Rational Numbers Lesson Title Lesson Objective Eligible Content Fluently Add, Subtract, and Multiply Decimals Fluently Divide Whole Numbers and Decimals 1-3 Multiply Fractions 3-Act* 1-4 Mathematical Modeling: Stocking Up Understand Division with Fractions Add and subtract decimals with precision. Multiply decimals. Add, subtract, and multiply decimals to solve real-world problems. Use place-value structure to divide whole numbers and decimals. Divide whole numbers and decimals to solve real-world problems. Use models to multiply fractions. Multiply the numerators and then the denominators to find the product of two fractions. Multiply mixed numbers. Use mathematical modeling to represent a problem situation and to propose a solution. Test and verify the appropriateness of their math models. Explain why the results from their mathematical models may not align exactly to the problem situation. Use models to divide with fractions. Use equations to divide with fractions. M06.A-N.2.1.1 M06.A-N.2.1.1 M06.A-N.1.1.1 M06.A-N.2.1.1 M06.A-N.1.1.1 1-5 Divide Fractions by Fractions Use models to divide fractions by fractions. Use an algorithm to divide fractions by fractions. M06.A-N.1.1.1 1-6 Divide Mixed Numbers Divide with mixed numbers. Estimate the quotient of mixed numbers. M06.A-N.1.1.1 1-7 Solve Problems with Rational Numbers Solve multistep problems with fractions and decimals. M06.A-N.1.1.1 *Note: 3 Act Mathematical Modeling are not required lessons and so are not included in the scheduled days. They are optional lessons that could be done during a Flex Day. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 6
Suggested Dates 10/2-10/20 2 Days Per Lesson + 3 Days Topic- Lesson 2-1 Understand Integers 2-2 2-3 2-4 3-Act* 2-5 2-6 Topic 2: Integers and Rational Numbers Lesson Title Lesson Objectives Eligible Content Represent Rational Numbers on the Number Line Absolute Values of Rational Numbers Represent Rational Numbers on the Coordinate Plane Mathematical Modeling: The Ultimate Throw Find Distances on the Coordinate Plane Represent Polygons on the Coordinate Plane Identify opposites of integers. Compare and order integers. Use integers to represent real-world quantities and explain the meaning of 0 in each context. Plot rational numbers on a number line. Compare and order rational numbers. Use rational numbers to represent real-world quantities. Use absolute value to represent a number s distance from 0. Interpret absolute value in real-world situations. Identify and graph points with rational coordinates on the coordinate plane. Reflect points with rational coordinates across both axes. Use mathematical modeling to represent a problem situation and to propose a solution. Test and verify the appropriateness of their math models. Explain why the results from their mathematical models may not align exactly with the problem situation. Use absolute value to find the distance between two points that lie on the same horizontal or vertical line on a coordinate plane. Solve real-world and mathematical problems involving distances on the coordinate plane. Find side lengths of polygons on the coordinate plane. Find the perimeter of polygons on the coordinate plane. M06.A-N.3.1.1 M06.A-N.3.1.2 M06.A-N.3.2.1 M06.A-N.3.1.3 M06.A-N.3.2.1 M06.A-N.3.2.2 M06.A-N.3.1.3 M06.A-N.3.1.1 M06.A-N.3.1.1 M06.A-N.3.2.2 M06.A-N.3.1.3 M06.A-N.3.2.3 M06.C-G.1.1.4 *Note: 3 Act Mathematical Modeling are not required lessons and so are not included in the scheduled days. They are optional lessons that could be done during a Flex Day. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 7
Suggested Dates 10/23-10/31 1 Day Per Lesson + ½ Day Topic- Lesson 3-1 3-2 3-3 Topic 3: Numeric and Algebraic Expressions Lesson Title Lesson Objective Eligible Content Understand and Represent Exponents Find Greatest Common Factor and Least Common Multiple Write and Evaluate Numerical Expressions Write expressions using whole-number exponents to represent real-world and mathematical problems. Evaluate expressions with whole-number exponents. Find the prime factorization of a whole number. Find the greatest common factor (GCF) and the least common multiple (LCM) of two whole numbers. Use the GCF and the Distributive Property to add. Use the GCF and the LCM to solve problems. Evaluate expressions using the order of operations. Insert grouping symbols in a numerical expression to affect the value of the expression. Benchmark 1 Window: 11/1 11/17 M06.B-E.1.1.1 M06.A-N.2.2.1 M06.A-N.2.2.2 M06.B-E.1.1.1 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 8
Benchmark Cycle 2 Standards PA Core Standard PA Eligible Content Common Core State Standard CC.2.2.6.B.1 Apply and extend previous understandings of arithmetic to algebraic expressions. M06.B-E.1.1.2 Write algebraic expressions from verbal descriptions. Example: Express the description five less than twice a number as 2y 5. 6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers. a. Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation "Subtract y from 5" M06.B-E.1.1.3 Identify parts of an expression using mathematical terms (e.g., sum, term, product, factor, quotient, coefficient, quantity). Example: Describe the expression 2(8 + 7) as a product of two factors. M06.B-E.1.1.4 Evaluate expressions at specific values of their variables, including expressions that arise from formulas used in real-world problems. Example: Evaluate the expression b2 5 when b = 4. as 5 - y. 6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers. b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms. 6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers. c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s 3 and A = 6 s 2 to find the volume and surface area of a cube with sides of length s = 1/2. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 9
PA Core Standard PA Eligible Content Common Core State Standard M06.B-E.1.1.5 Apply the properties of operations to generate equivalent expressions. Example 1: Apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x. Example 2: Apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6(4x + 3y). Example 3: Apply properties of operations to y + y + y to produce the equivalent expression 3y. CC.2.2.6.B.2 Understand the process of solving a one variable equation or inequality and apply to real world and mathematical problems. M06.B-E.2.1.1 Use substitution to determine whether a given number in a specified set makes an equation or inequality true. M06.B-E.2.1.2 Write algebraic expressions to represent real-world or mathematical problems. 6.EE.A.3 Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y. 6.EE.B.5 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. 6.EE.B.6 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. M06.B-E.2.1.3 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q, and x are all nonnegative rational numbers. M06.B-E.2.1.4 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem and/or represent solutions of such inequalities on number lines. 6.EE.B.7 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. 6.EE.B.8 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 10
PA Core Standard PA Eligible Content Common Core State Standard CC.2.2.6.B.3 Represent and analyze quantitative relationships between dependent and independent variables. M06.B-E.3.1.1 Write an equation to express the relationship between the dependent and independent variables. Example: In a problem involving motion at a constant speed of 65 units, write the equation d = 65t to represent the relationship between distance and time. M06.B-E.3.1.2 Analyze the relationship between the dependent and independent variables using graphs and tables and/or relate these to an equation. 6.EE.C.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. CC.2.1.6.D.1 Understand ratio concepts and use ratio reasoning to solve problems. M06.A-R.1.1.1 Use ratio language and notation (such as 3 to 4, 3:4, 3/4) to describe a ratio relationship between two quantities. Example 1: The ratio of girls to boys in a math class is 2:3 because for every 2 girls there are 3 boys. Example 2: For every five votes candidate A received, candidate B received four votes. M06.A-R.1.1.2 Find the unit rate a/b associated with a ratio a:b (with b 0) and use rate language in the context of a ratio relationship. Example 1: This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar. Example 2: We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger. 6.RP.A.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, "The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak." "For every vote candidate A received, candidate C received nearly three votes." 6.RP.A.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b 0, and use rate language in the context of a ratio relationship. For example, "This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar." "We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger." THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 11
PA Core Standard PA Eligible Content Common Core State Standard M06.A-R.1.1.3 Construct tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and/or plot the pairs of values on the coordinate plane. Use tables to compare ratios. 6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. a. Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 12
Benchmark Cycle 2 Scope and Sequence Suggested Dates 11/2-11/22 2 Days Per Lesson + 2 Whole And 3 Half Days Topic- Lesson 3-4 Write Algebraic Expressions Topic 3 Continued: Numeric and Algebraic Expressions Lesson Title Lesson Objective Eligible Content 3-5 Evaluate Algebraic Expressions 3-Act* Mathematical Modeling: The Field Trip 3-6 Generate Equivalent Expressions 3-7 Simplify Algebraic Expressions Write an algebraic expression to model a pattern. Write an algebraic expression from a word phrase. Use precise mathematical language when identifying parts of an expression. Evaluate algebraic expressions, including those with whole numbers, decimals, and fractions. Use mathematical modeling to represent a problem situation and to propose a solution. Test & verify the appropriateness of mathematical models. Explain why the results from their mathematical models may not align exactly to the problem situation. Write equivalent algebraic expressions. Identify equivalent algebraic expressions. Justify whether two expressions are equivalent. Use properties of operations to simplify algebraic expressions by combining like terms. M06.B-E.1.1.2 M06.B-E.1.1.3 M06.B-E.1.1.4 M06.B-E.2.1.2 M06.B-E.1.1.5 M06.B-E.1.1.5 *Note: 3 Act Mathematical Modeling are not required lessons and so are not included in the scheduled days. They are optional lessons that could be done during a Flex Day. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 13
Suggested Date 11/20-1/9 2 Days Per Lesson + 4 Whole And 4 Half Days Topic- Lesson 4-1 Topic 4: Represent and Solve Equations and Inequalities Lesson Title Lesson Objective Eligible Content Understand Equations and Solutions 4-2 Apply Properties of Equality 4-3 4-4 4-5 4-6 Write and Solve Addition and Subtraction Equations Write and Solve Multiplication and Division Equations Write and Solve Equations with Rational Numbers Understand and Write Inequalities 4-7 Solve Inequalities 3-Act* 4-8 4-9 4-10 Mathematical Modeling: Checking a Bag Using Dependent and Independent Variables Use Patterns to Write and Solve Equations Relate Tables, Graphs, and Equations Identify equations and variables. Use substitution to find solutions to equations. Use the properties of equality to keep both sides of an equation equal. Identify which properties of equality are used to write equivalent expressions. Write one-variable addition and subtraction equations. Use inverse relationships and properties of equality to solve one-step addition and subtraction equations. Write one-variable multiplication and division equations. Use inverse relationships and properties of equality to solve one-step multiplication and division equations. Write and solve equations that involve fractions, decimals, and mixed numbers. Understand the symbols required to write an inequality. Write inequalities to describe mathematical or real-world situations. Describe solutions to an inequality. Represent solutions to an inequality on a number line. Use mathematical modeling to represent a problem situation and to propose a solution. Test and verify the appropriateness of their math models. Explain why the results from their mathematical models may not align exactly to the problem situation. Identify dependent variables. Identify independent variables. Analyze relationships between variables by using tables. Write equations to represent the relationships between variables. Analyze the relationships between dependent and independent variables using tables, graphs, and equations. M06.B-E.2.1.1 M06.B-E.2.1.3 M06.B-E.2.1.2 M06.B-E.2.1.3 M06.B-E.2.1.2 M06.B-E.2.1.3 M06.B-E.2.1.2 M06.B-E.2.1.3 M06.B-E.2.1.4 M06.B-E.2.1.4 M06.B-E.2.1.1 M06.B-E.2.1.2 M06.B-E.2.1.4 M06.B-E.3.1.1 M06.B-E.3.1.1 M06.B-E.3.1.2 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 14
Suggested Date 1/10-1/28 2 Days Per Lesson + 2 Days Topic- Lesson 5-1 Understand Ratios 5-2 Generate Equivalent Ratios 5-3 Compare Ratios 5-4 Represent and Graph Ratios Topic 5: Understand and Use Ratio and Rate Lesson Title Lesson Objective Eligible Content 5-5 Understand Rates and Unit Rates Use ratios to describe relationship between 2 quantities. Use bar diagrams and double number line diagrams to model ratio relationships. Use multiplication and division to find equivalent ratios. Solve problems by finding equivalent ratios. Use ratio tables to compare ratios. Compare ratios to solve problems. Represent equivalent ratios on graphs. Use ratio tables and graphs to solve problems. Use rates to describe ratios in which the terms have different units. Use rates and unit rates to solve problems. M06.A-R.1.1.1 M06.A-R.1.1.3 M06.A-R.1.1.3 M06.A-R.1.1.1 M06.A-R.1.1.2 M06.A-R.1.1.1 Benchmark 2 Window: 1/29 2/13 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 15
Benchmark Cycle 3 Standards PA Core Standard PA Eligible Content Common Core State Standard CC.2.1.6.D.1 Understand ratio concepts and use ratio reasoning to solve problems. M06.A-R.1.1.3 Construct tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and/or plot the pairs of values on the coordinate plane. Use tables to compare ratios. 6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. a. Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the CC.2.4.6.B.1 Demonstrate an understanding of statistical variability by displaying, analyzing, and summarizing distributions. M06.A-R.1.1.4 Solve unit rate problems including those involving unit pricing and constant speed. Example: If it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed? M06.A-R.1.1.5 Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percentage. M06.D-S.1.1.1 Display numerical data in plots on a number line, including line plots, histograms, and boxand-whisker plots. M06.D-S.1.1.2 Determine quantitative measures of center (e.g., median, mean, mode) and variability (e.g., range, interquartile range, mean absolute deviation). coordinate plane. Use tables to compare ratios. 6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed? 6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. 6.SP.B.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots. 6.SP.B.5 Summarize numerical data sets in relation to their context, such as by: THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 16
CC.2.4.6.B.1 Demonstrate an understanding of statistical variability by displaying, analyzing, and summarizing distributions. CC.2.3.6.A.1 Apply appropriate tools to solve real-world and mathematical problems involving area, surface area, and volume. M06.D-S.1.1.3 Describe any overall pattern and any deviations from the overall pattern with reference to the context in which the data were gathered. M06.D-S.1.1.4 Relate the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. M06.C-G.1.1.1 Determine the area of triangles and special quadrilaterals (square, rectangle, parallelogram, rhombus, and trapezoid). Formulas will be provided. M06.C-G.1.1.2 Determine the area of irregular or compound polygons. Example: Find area of a room in the shape of irregular polygon by composing & decomposing. M06.C-G.1.1.3 Determine the volume of right rectangular prisms with fractional edge lengths. Formulas will be provided. M06.C-G.1.1.4 Given coordinates for the vertices of a polygon in the plane, use the coordinates to find side lengths and area of the polygon (limited to triangles and special quadrilaterals). Formulas will be provided. M06.C-G.1.1.5 Represent three-dimensional figures using nets made of rectangles and triangles. M06.C-G.1.1.6 Determine the surface area of triangular and rectangular prisms (including cubes). Formulas will be provided. c. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. 6.SP.B.5 Summarize numerical data sets in relation to their context, such as by: d. Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. 6.G.A.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. 6.G.A.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world & mathematical problems. 6.G.A.3 Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. 6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 17
Benchmark Cycle 3 Scope and Sequence Suggested Date 2/1-2/7 2 Days per Lesson + 3 Days (Italicized lessons are supplemental and not included in count) Topic 5 Continued: Understand and Use Ratio and Rate Topic- Lesson Lesson Title Lesson Objective Eligible Content 5-6 Compare Unit Rates Use ratio reasoning to compare unit rates and solve problems. M06.A-R.1.1.3 Use unit rates to solve problems involving constant speed. 5-7 Solve Unit Rate Problems Use unit rates to solve problems involving unit price. M06.A-R.1.1.4 Solve unit rate problems using an equation. Use mathematical modeling to represent a problem 3-Act* situation and to propose a solution. Mathematical Modeling: Get in Test and verify the appropriateness of their math modes. Line Explain why the results from their mathematical models M06.A-R.1.1.4 may not align exactly to the problem situation. 5-8 Ratio Reasoning: Convert Use ratio reasoning and conversion factors to convert Customary Units customary units of measure. M06.A-R.1.1.3* 5-9 Ratio Reasoning: Convert Metric Use ratio reasoning and conversion factors to convert metric Units units of measure. M06.A-R.1.1.3 5-10 Relate Customary and Metric Use ratio reasoning and conversion factors to convert between Units customary and metric units of measure. M06.A-R.1.1.3* *Note: 3 Act Mathematical Modeling are not required lessons and so are not included in the scheduled days. They are optional lessons that could be done during a Flex Day. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 18
Suggested Date 2/8-3/8 2 Days Per Lesson + 3 Whole And 3 Half Days Topic- Lesson Topic 8: Display, Describe, and Summarize Data Lesson Title Lesson Objective Eligible Content 8-1 Recognize Statistical Questions 8-2 Summarize Data Using Mean, Median, Mode, and Range 8-3 Display Data in Box Plots 8-4 8-5 8-6 Display Data in Frequency Tables and Histograms Summarize Data Using Measures of Variability Choose Appropriate Statistical Measures 8-7 Summarize Data Distributions 3-Act* Mathematical Modeling: Vocal Range Identify statistical questions. Write statistical questions and display the collected data. Determine the mean, median, mode, and range of a data set. Display data in a box plot. Interpret and analyze a box plot. Organize data into equal intervals and display data in a frequency table or histogram. Interpret and analyze a histogram. Calculate the mean absolute deviation (MAD) and interquartile range (IQR) of a data set. Select the most appropriate measure of center and variability for a data set. Use measures to describe data sets. Describe the center, spread, and overall shape of a data set. Summarize numerical data sets using measures of center and related measures of variability. Use mathematical modeling to represent a problem situation and to propose a solution. Test and verify the appropriateness of their math models. Explain why the results from their mathematical models may not align exactly to the problem situation. M06.D-S.1.1.2* M06.D-S.1.1.2 M06.D-S.1.1.1 M06.D-S.1.1.1 M06.D-S.1.1.2 M06.D-S.1.1.4 M06.D-S.1.1.3 M06.D-S.1.1.2 M06.D-S.1.1.3 *Note: 3 Act Mathematical Modeling are not required lessons and so are not included in the scheduled days. They are optional lessons that could be done during a Flex Day. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 19
Suggested Date 3/9-4/6 2 Days Per Lesson + 3.5 Days Can Skip italicized lesson beyond the standard Topic- Lesson 7-1 Topic 7: Solve Area, Surface Area, and Volume Problems Lesson Title Lesson Objective Eligible Content Find Areas of Parallelograms and Rhombuses 7-2 Solve Triangle Area Problems 7-3 Find Areas of Trapezoids and Kites 7-4 Find Areas of Polygons 7-5 Represent Solid Figures Using Nets Use a formula to find the areas of parallelograms and rhombuses. Find the base or height of a parallelogram or rhombus when the area and the height or base are known. Find the areas of triangles, including right triangles. Find the corresponding base or height of a triangle. Find the areas of trapezoids. Find the area of kites. Find the areas of polygons by composing and decomposing shapes, including polygons on the coordinate plane. Classify solid figures. Identify solid figures from nets. Draw nets of solid figures. M06.C-G.1.1.1 M06.C-G.1.1.1 M06.C-G.1.1.1 M06.C-G.1.1.2 M06.C-G.1.1.4 M06.C-G.1.1.5 Use mathematical modeling to represent a problem 3-Act* situation and to propose a solution. Mathematical Modeling: That s a Test and verify the appropriateness of their math models. Wrap Explain why the results from their mathematical models may not align exactly to the problem situation. CC.2.3.6.A.1 Find the surface area of rectangular prisms, including 7-6 Find Surface Areas of Prisms cubes. M06.C-G.1.1.6 Find the surface area of triangular prisms. 7-7 Find Surface Areas of Pyramids Find the surface areas of square and triangular pyramids. M06.C-G.1.1.6* 7-8 Find Volume with Fractional Edge Use cubes and a formula to find the volume of a Lengths rectangular prism or a cube with fractional edge lengths. M06.C-G.1.1.3 This leaves the week of the ELA PSSA (4/9 4/13) for remediation or as flex time in case you are behind. 4/16 4/20 MATH PSSA THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 20
Topic 6 will not be on the benchmark, so its standards are included in Cycle 4. To ensure you have time in Cycle 4 to finish Topic 6 and to review and extend content before the end of the year, it is recommended that you start Topic 6 now. Topic 6: Understand and Use Percent Suggested Date 4/23-5/8 2 Days Per Lesson + 2 Whole and 4 Half Days Topic- Lesson Lesson Title Lesson Objective Eligible Content 6-1 Understand Percent 6-2 6-3 Relate Fractions, Decimals, and Percents Represent Percents Greater Than 100 and Less Than 1 Represent the percent of a whole. Find the percent of a whole. Write equivalent values as fractions, decimals, and percents. Write fractions as decimals and percents when the denominator of the fraction is not 100. Write percents that are greater than 100. Write percents that are less than 1. Benchmark 3 Window: 5/9 5/25 M06.A-R.1.1.5 M06.A-R.1.1.5 M06.A-R.1.1.5 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 21
PA Common Core Standard CC.2.1.6.D.1 Understand ratio concepts and use ratio reasoning to solve problems. CC.2.1.6.D.1 Understand ratio concepts and use ratio reasoning to solve problems. PA Eligible Content Cycle 4 Standards M06.A-R.1.1.1 Use ratio language and notation (such as 3 to 4, 3:4, 3/4) to describe a ratio relationship between two quantities. Example 1: The ratio of girls to boys in a math class is 2:3 because for every 2 girls there are 3 boys. Example 2: For every five votes candidate A received, candidate B received four votes. M06.A-R.1.1.5 Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percentage. Common Core State Standard 6.RP.A.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, "The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak." "For every vote candidate A received, candidate C received nearly three votes." 6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 22
5/10-5/23 2 Days Per Lesson + 3.5 Days Cycle 4 Scope and Sequence Topic 6 Continued: Understand and Use Percent Suggested Topic- Date Lesson Lesson Title Lesson Objective Eligible Content 6-4 Estimate to Find Percent Estimate the percent of a number. M06.A-R.1.1.5 Use the decimal form of a percent to find the percent of a 6-5 Find the Percent of a Number number. M06.A-R.1.1.5 Write an equation to solve a percent problem. 6-6 Find the Whole Given a Part and the Percent Find the whole amount when given a part and the percent. M06.A-R.1.1.5 Use mathematical modeling to represent a problem 3-Act* situation and to propose a solution. Mathematical Modeling: Ace the M06.A-R.1.1.1 Test and verify the appropriateness of their math models. Test M06.A-R.1.1.3 Explain why the results from their mathematical models may not align exactly to the problem situation. *Note: 3 Act Mathematical Modeling are not required lessons and so are not included in the scheduled days. They are optional lessons that could be done during a Flex Day. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 23
Option 1: Revisit 3-Act Math Lessons Option 2: Reteach Based on the Major Work of the Grade Option 3: Pre-teach Based on 7 th Grade Needs Option 4: Financial Literacy Remediation and Extension Options 5/24-6/12 13 Days 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/. You taught many topics this year. The four below are identified as the most critical for 6 th grade. If you want to revisit something, one or more of these topics would be a good idea. 6.RP.A Understand ratio concepts and use ratio reasoning to solve problems. 6.NS.A Apply and extend previous understandings of multiplication and division to divide fractions by fractions. 6.NS.C Apply and extend previous understandings of numbers to the system of rational numbers. 6.EE.A Apply and extend previous understandings of arithmetic to algebraic expressions. 6.EE.B Reason about and solve one-variable equations and inequalities. 6.EE.C Represent and analyze quantitative relationships between dependent and independent variables. You can check in with your 7 th grade teacher to ask what content you could pre-teach or what content they would like you to revisit. The first two Topics in 7 th grade are: 1) Integers and Rational Numbers 2) Analyze and Use Proportional Relationships. You can access the 7 th grade Teacher s Guide and materials online for reference. 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
CC.2.1.6.D.1 Understand ratio concepts and use ratio reasoning to solve problems. PA Core Eligible Content by Cycle PA Core Standards PA Eligible Content Cycle 1 Cycle 2 Cycle 3 Cycle 4 M06.A-R.1.1.1 Use ratio language and notation (such as 3 to 4, 3:4, 3/4) to describe a ratio relationship between two quantities. Example 1: The ratio of girls to boys in a math class is 2:3 because for every 2 girls there are 3 boys. Example 2: For every five votes candidate A received, candidate B received four votes. M06.A-R.1.1.2 Find the unit rate a/b associated with a ratio a:b (with b 0) and use rate language in the context of a ratio relationship. Example 1: This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar. Example 2: We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger. CC.2.1.6.E.1 Apply and extend previous understandings of multiplication and division to divide fractions by fractions. M06.A-R.1.1.3 Construct tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and/or plot the pairs of values on the coordinate plane. Use tables to compare ratios. M06.A-R.1.1.4 Solve unit rate problems including those involving unit pricing and constant speed. Example: If it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed? M06.A-R.1.1.5 Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percentage. M06.A-N.1.1.1 Interpret and compute quotients of fractions (including mixed numbers), and solve word problems involving division of fractions by fractions. Example 1: Given a story context for (2/3) (3/4), explain that (2/3) (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) (c/d) = (a/b) (d/c) = ad/bc.) Example 2: How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi? Example 3: How many 2 1/4-foot pieces can be cut from a 15 1/2- foot board? THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 25
PA Core Standards PA Eligible Content Cycle 1 Cycle 2 Cycle 3 Cycle 4 CC.2.1.6.E.2 Identify and choose appropriate M06.A-N.2.1.1 Solve problems involving operations (+,,, and ) with whole numbers, decimals (through thousandths), straight computation, or word problems. processes to compute fluently with multi-digit numbers. CC.2.1.6.E.3 Develop and/or apply number theory concepts to find common factors and multiples. M06.A-N.2.2.1 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. M06.A-N.2.2.2 Apply the distributive property to express a sum of two whole numbers, 1 through 100, with a common factor as a multiple of a sum of two whole numbers with no common factor. Example: Express 36 + 8 as 4(9 + 2). CC.2.1.6.E.4 Apply and extend previous understandings of numbers to the system of rational numbers. M06.A-N.3.1.1 Represent quantities in real-world contexts using positive and negative numbers, explaining the meaning of 0 in each situation (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge). M06.A-N.3.1.2 Determine the opposite of a number and recognize that the opposite of the opposite of a number is the number itself (e.g., ( 3) = 3; 0 is its own opposite). M06.A-N.3.1.3 Locate and plot integers and other rational numbers on a horizontal or vertical number line; locate and plot pairs of integers and other rational numbers on a coordinate plane. M06.A-N.3.2.1 Write, interpret, and explain statements of order for rational numbers in real-world contexts. Example: Write 3 C > 7 C to express the fact that 3 C is warmer than 7 C. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 26