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## Content Area: MathematicsGrade Level Expectations: Seventh GradeStandard: 1. Number Sense, Properties, and Operations

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 2. Formulate, represent, and use algorithms with rational numbers flexibly, accurately, and efficiently Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Apply understandings of addition and subtraction to add and subtract rational numbers including integers. (CCSS: 7.NS.1)Represent addition and subtraction on a horizontal or vertical number line diagram. (CCSS: 7.NS.1)Describe situations in which opposite quantities combine to make 0.5 (CCSS: 7.NS.1a)Demonstrate p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. (CCSS: 7.NS.1b)Show that a number and its opposite have a sum of 0 (are additive inverses). (CCSS: 7.NS.1b)Interpret sums of rational numbers by describing real-world contexts. (CCSS: 7.NS.1c)Demonstrate subtraction of rational numbers as adding the additive inverse, p  q = p + (q). (CCSS: 7.NS.1c)Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. (CCSS: 7.NS.1c)Apply properties of operations as strategies to add and subtract rational numbers. (CCSS: 7.NS.1d) Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers including integers. (CCSS: 7.NS.2)Apply properties of operations to multiplication of rational numbers.6 (CCSS: 7.NS.2a)Interpret products of rational numbers by describing real-world contexts. (CCSS: 7.NS.2a)Apply properties of operations to divide integers.7 (CCSS: 7.NS.2b)Apply properties of operations as strategies to multiply and divide rational numbers. (CCSS: 7.NS.2c)Convert a rational number to a decimal using long division. (CCSS: 7.NS.2d)Show that the decimal form of a rational number terminates in 0s or eventually repeats. (CCSS: 7.NS.2d) Solve real-world and mathematical problems involving the four operations with rational numbers.8 (CCSS: 7.NS.3) Inquiry Questions: How do operations with rational numbers compare to operations with integers? How do you know if a computational strategy is sensible? Is $0.\overline9$ equal to one? How do you know whether a fraction can be represented as a repeating or terminating decimal? Relevance & Application: The use and understanding algorithms help individuals spend money wisely. For example, compare discounts to determine best buys and compute sales tax. Estimation with rational numbers enables individuals to make decisions quickly and flexibly in daily life such as estimating a total bill at a restaurant, the amount of money left on a gift card, and price markups and markdowns. People use percentages to represent quantities in real-world situations such as amount and types of taxes paid, increases or decreases in population, and changes in company profits or worker wages). Nature Of: Mathematicians see algorithms as familiar tools in a tool chest. They combine algorithms in different ways and use them flexibly to accomplish various tasks. Mathematicians make sense of problems and persevere in solving them. (MP) Mathematicians construct viable arguments and critique the reasoning of others. (MP) Mathematicians look for and make use of structure. (MP)

5 For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. (CCSS: 7.NS.1a)

6 Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. (CCSS: 7.NS.2a)

7 Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then (p/q) = (p)/q = p/(q). (CCSS: 7.NS.2b)
Interpret quotients of rational numbers by describing real-world contexts. (CCSS: 7.NS.2b)

8 Computations with rational numbers extend the rules for manipulating fractions to complex fractions. (CCSS: 7.NS.3)

## Content Area: MathematicsGrade Level Expectations: Sixth GradeStandard: 1. Number Sense, Properties, and Operations

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 2. Formulate, represent, and use algorithms with positive rational numbers with flexibility, accuracy, and efficiency Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Fluently divide multi-digit numbers using standard algorithms. (CCSS: 6.NS.2) Fluently add, subtract, multiply, and divide multi-digit decimals using standard algorithms for each operation. (CCSS: 6.NS.3) Find the greatest common factor of two whole numbers less than or equal to 100. (CCSS: 6.NS.4) Find the least common multiple of two whole numbers less than or equal to 12. (CCSS: 6.NS.4) 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.7 (CCSS: 6.NS.4) Interpret and model quotients of fractions through the creation of story contexts.8 (CCSS: 6.NS.1) Compute quotients of fractions.9 (CCSS: 6.NS.1) Solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.10 (CCSS: 6.NS.1) Inquiry Questions: Why might estimation be better than an exact answer? How do operations with fractions and decimals compare to operations with whole numbers? Relevance & Application: Rational numbers are an essential component of mathematics. Understanding fractions, decimals, and percentages is the basis for probability, proportions, measurement, money, algebra, and geometry. Nature Of: Mathematicians envision and test strategies for solving problems. Mathematicians model with mathematics. (MP) Mathematicians look for and make use of structure. (MP)

7 For example, express 36 + 8 as 4 (9 + 2). (CCSS: 6.NS.4)

8 For example, create a story context for (2/3) ๗ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ๗ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (CCSS: 6.NS.1)

9 In general, (a/b) ๗ (c/d) = ad/bc.). (CCSS: 6.NS.1)

10 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? (CCSS: 6.NS.1)

## Content Area: MathematicsGrade Level Expectations: Fifth GradeStandard: 1. Number Sense, Properties, and Operations

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 2. Formulate, represent, and use algorithms with multi-digit whole numbers and decimals with flexibility, accuracy, and efficiency Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Fluently multiply multi-digit whole numbers using standard algorithms. (CCSS: 5.NBT.5) Find whole-number quotients of whole numbers.3 (CCSS: 5.NBT.6)Use strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. (CCSS: 5.NBT.6)Illustrate and explain calculations by using equations, rectangular arrays, and/or area models. (CCSS: 5.NBT.6) Add, subtract, multiply, and divide decimals to hundredths. (CCSS: 5.NBT.7)Use concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. (CCSS: 5.NBT.7)Relate strategies to a written method and explain the reasoning used. (CCSS: 5.NBT.7) Write and interpret numerical expressions. (CCSS: 5.OA)Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. (CCSS: 5.OA.1)Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.4 (CCSS: 5.OA.2) Inquiry Questions: How are mathematical operations related? What makes one strategy or algorithm better than another? Relevance & Application: Multiplication is an essential component of mathematics. Knowledge of multiplication is the basis for understanding division, fractions, geometry, and algebra. There are many models of multiplication and division such as the area model for tiling a floor and the repeated addition to group people for games. Nature Of: Mathematicians envision and test strategies for solving problems. Mathematicians develop simple procedures to express complex mathematical concepts. Mathematicians construct viable arguments and critique the reasoning of others. (MP) Mathematicians model with mathematics. (MP)

3 with up to four-digit dividends and two-digit divisors. (CCSS: 5.NBT.6)

4 For example, express the calculation "add 8 and 7, then multiply by 2" as $2 \times (8 + 7)$. Recognize that 3 ื (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product. (CCSS: 5.OA.2)

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 3. Formulate, represent, and use algorithms to add and subtract fractions with flexibility, accuracy, and efficiency Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Use equivalent fractions as a strategy to add and subtract fractions. (CCSS: 5.NF)Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.5 (CCSS: 5.NF.2)Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions6 with like denominators. (CCSS: 5.NF.1)Solve word problems involving addition and subtraction of fractions referring to the same whole.7 (CCSS: 5.NF.2) Inquiry Questions: How do operations with fractions compare to operations with whole numbers? Why are there more fractions than whole numbers? Is there a smallest fraction? Relevance & Application: Computational fluency with fractions is necessary for activities in daily life such as cooking and measuring for household projects and crafts. Estimation with fractions enables quick and flexible decision-making in daily life. For example, determining how many batches of a recipe can be made with given ingredients, the amount of carpeting needed for a room, or fencing required for a backyard. Nature Of: Mathematicians envision and test strategies for solving problems. Mathematicians make sense of problems and persevere in solving them. (MP) Mathematicians reason abstractly and quantitatively. (MP) Mathematicians look for and make use of structure. (MP)

5 For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2. (CCSS: 5.NF.2)

6 in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.). (CCSS: 5.NF.1)

7 including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. (CCSS: 5.NF.2)

## Content Area: MathematicsGrade Level Expectations: Fourth GradeStandard: 1. Number Sense, Properties, and Operations

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 3. Formulate, represent, and use algorithms to compute with flexibility, accuracy, and efficiency Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Use place value understanding and properties of operations to perform multi-digit arithmetic. (CCSS: 4.NBT)Fluently add and subtract multi-digit whole numbers using standard algorithms. (CCSS: 4.NBT.4)Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. (CCSS: 4.NBT.5)Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. (CCSS: 4.NBT.6)Illustrate and explain multiplication and division calculation by using equations, rectangular arrays, and/or area models. (CCSS: 4.NBT.6) Use the four operations with whole numbers to solve problems. (CCSS: 4.OA)Interpret a multiplication equation as a comparison.13 (CCSS: 4.OA.1)Represent verbal statements of multiplicative comparisons as multiplication equations. (CCSS: 4.OA.1)Multiply or divide to solve word problems involving multiplicative comparison.14 (CCSS: 4.OA.2)Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. (CCSS: 4.OA.3)Represent multistep word problems with equations using a variable to represent the unknown quantity. (CCSS: 4.OA.3)Assess the reasonableness of answers using mental computation and estimation strategies including rounding. (CCSS: 4.OA.3)Using the four operations analyze the relationship between choice and opportunity cost (PFL) Inquiry Questions: Is it possible to make multiplication and division of large numbers easy? What do remainders mean and how are they used? When is the correct answer not the most useful answer? Relevance & Application: Multiplication is an essential component of mathematics. Knowledge of multiplication is the basis for understanding division, fractions, geometry, and algebra. Nature Of: Mathematicians envision and test strategies for solving problems. Mathematicians develop simple procedures to express complex mathematical concepts. Mathematicians make sense of problems and persevere in solving them. (MP) Mathematicians construct viable arguments and critique the reasoning of others. (MP) Mathematicians look for and express regularity in repeated reasoning. (MP)

3 e.g., interpret 35 = 5 ื 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. (CCSS: 4.OA.1)

4 e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. (CCSS: 4.OA.2)

## Content Area: MathematicsGrade Level Expectations: Third GradeStandard: 1. Number Sense, Properties, and Operations

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 3. Multiplication and division are inverse operations and can be modeled in a variety of ways Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Represent and solve problems involving multiplication and division. (CCSS: 3.OA)Interpret products of whole numbers.7 (CCSS: 3.OA.1)Interpret whole-number quotients of whole numbers.8 (CCSS: 3.OA.2)Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities.9 (CCSS: 3.OA.3)Determine the unknown whole number in a multiplication or division equation relating three whole numbers.10 (CCSS: 3.OA.4)Model strategies to achieve a personal financial goal using arithmetic operations (PFL) Apply properties of multiplication and the relationship between multiplication and division. (CCSS: 3.OA)Apply properties of operations as strategies to multiply and divide.11 (CCSS: 3.OA.5)Interpret division as an unknown-factor problem.12 (CCSS: 3.OA.6) Multiply and divide within 100. (CCSS: 3.OA)Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division13 or properties of operations. (CCSS: 3.OA.7)Recall from memory all products of two one-digit numbers. (CCSS: 3.OA.7) Solve problems involving the four operations, and identify and explain patterns in arithmetic. (CCSS: 3.OA)Solve two-step word problems using the four operations. (CCSS: 3.OA.8)Represent two-step word problems using equations with a letter standing for the unknown quantity. (CCSS: 3.OA.8)Assess the reasonableness of answers using mental computation and estimation strategies including rounding. (CCSS: 3.OA.8)Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations.14 (CCSS: 3.OA.9) Inquiry Questions: How are multiplication and division related? How can you use a multiplication or division fact to find a related fact? Why was multiplication invented? Why not just add? Why was division invented? Why not just subtract?... Relevance & Application: Many situations in daily life can be modeled with multiplication and division such as how many tables to set up for a party, how much food to purchase for the family, or how many teams can be created. Use of multiplication and division helps to make decisions about spending allowance or gifts of money such as how many weeks of saving an allowance of $5 per week to buy a soccer ball that costs$32?. Nature Of: Mathematicians often learn concepts on a smaller scale before applying them to a larger situation. Mathematicians construct viable arguments and critique the reasoning of others. (MP) Mathematicians model with mathematics. (MP) Mathematicians look for and make use of structure. (MP)

7 e.g., interpret 5 ื 7 as the total number of objects in 5 groups of 7 objects each. (CCSS: 3.OA.1)
For example, describe a context in which a total number of objects can be expressed as 5 ื 7. (CCSS: 3.OA.1)

8 e.g., interpret 56 ๗ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. (CCSS: 3.OA.2)
For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ๗ 8. (CCSS: 3.OA.2)

9 e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (CCSS: 3.OA.3)

10 For example, determine the unknown number that makes the equation true in each of the equations 8 ื ? = 48, 5 = ? ๗ 3, 6 ื 6 = ?. (CCSS: 3.OA.4)

11 Examples: If 6 ื 4 = 24 is known, then 4 ื 6 = 24 is also known. (Commutative property of multiplication.) 3 ื 5 ื 2 can be found by 3 ื 5 = 15, then 15 ื 2 = 30, or by 5 ื 2 = 10, then 3 ื 10 = 30. (Associative property of multiplication.) Knowing that 8 ื 5 = 40 and 8 ื 2 = 16, one can find 8 ื 7 as 8 ื (5 + 2) = (8 ื 5) + (8 ื 2) = 40 + 16 = 56. (Distributive property.) (CCSS: 3.OA.5)

12 For example, find 32 ๗ 8 by finding the number that makes 32 when multiplied by 8. (CCSS: 3.OA.6)

13 e.g., knowing that 8 ื 5 = 40, one knows 40 ๗ 5 = 8. (CCSS: 3.OA.7)

14 For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends. (CCSS: 3.OA.9)

## Content Area: MathematicsGrade Level Expectations: Second GradeStandard: 1. Number Sense, Properties, and Operations

3 e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (CCSS: 2.OA.1)

4 e.g., by pairing objects or counting them by 2s. (CCSS: 2.OA.3)

## Content Area: MathematicsGrade Level Expectations: High SchoolStandard: 2. Patterns, Functions, and Algebraic Structures

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 4. Solutions to equations, inequalities and systems of equations are found using a variety of tools Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Create equations that describe numbers or relationships. (CCSS: A-CED)Create equations and inequalities20 in one variable and use them to solve problems. (CCSS: A-CED.1)Create equations in two or more variables to represent relationships between quantities and graph equations on coordinate axes with labels and scales. (CCSS: A-CED.2)Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.21 (CCSS: A-CED.3)Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.22 (CCSS: A-CED.4) Understand solving equations as a process of reasoning and explain the reasoning. (CCSS: A-REI)Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. (CCSS: A-REI.1)Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. (CCSS: A-REI.2) Solve equations and inequalities in one variable. (CCSS: A-REI)Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. (CCSS: A-REI.3)Solve quadratic equations in one variable. (CCSS: A-REI.4)Use the method of completing the square to transform any quadratic equation in x into an equation of the form $(x  p)^2 = q$ that has the same solutions. Derive the quadratic formula from this form. (CCSS: A-REI.4a)Solve quadratic equations23 by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. (CCSS: A-REI.4b)Recognize when the quadratic formula gives complex solutions and write them as a ฑ bi for real numbers a and b. (CCSS: A-REI.4b) Solve systems of equations. (CCSS: A-REI)Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. (CCSS: A-REI.5)Solve systems of linear equations exactly and approximately,24 focusing on pairs of linear equations in two variables. (CCSS: A-REI.6)Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.25 (CCSS: A-REI.7) Represent and solve equations and inequalities graphically. (CCSS: A-REI)Explain that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve.26 (CCSS: A-REI.10)Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x);27 find the solutions approximately.28 * (CCSS: A-REI.11)Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. (CCSS: A-REI.12) Inquiry Questions: What are some similarities in solving all types of equations? Why do different types of equations require different types of solution processes? Can computers solve algebraic problems that people cannot solve? Why? How are order of operations and operational relationships important when solving multivariable equations? Relevance & Application: Linear programming allows representation of the constraints in a real-world situation identification of a feasible region and determination of the maximum or minimum value such as to optimize profit, or to minimize expense. Effective use of graphing technology helps to find solutions to equations or systems of equations. Nature Of: Mathematics involves visualization. Mathematicians use tools to create visual representations of problems and ideas that reveal relationships and meaning. Mathematicians construct viable arguments and critique the reasoning of others. (MP) Mathematicians use appropriate tools strategically. (MP)

20 Include equations arising from linear and quadratic functions, and simple rational and exponential functions. (CCSS: A-CED.1)

21 For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. (CCSS: A-CED.3)

22 For example, rearrange Ohm's law V = IR to highlight resistance R. (CCSS: A-CED.4)

23 e.g., for $x^2 = 49$. (CCSS: A-REI.4b)

24 e.g., with graphs. (CCSS: A-REI.6)

25 For example, find the points of intersection between the line y = 3x and the circle $x^2 + y^2 = 3$. (CCSS: A-REI.7)

26 which could be a line. (CCSS: A-REI.10)

27 Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. (CCSS: A-REI.11)

28 e.g., using technology to graph the functions, make tables of values, or find successive approximations. (CCSS: A-REI.11)

## Content Area: MathematicsGrade Level Expectations: Eighth GradeStandard: 2. Patterns, Functions, and Algebraic Structures

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 2. Properties of algebra and equality are used to solve linear equations and systems of equations Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Solve linear equations in one variable. (CCSS: 8.EE.7)Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions.2 (CCSS: 8.EE.7a)Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. (CCSS: 8.EE.7b) Analyze and solve pairs of simultaneous linear equations. (CCSS: 8.EE.8)Explain 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. (CCSS: 8.EE.8a)Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.3 (CCSS: 8.EE.8b)Solve real-world and mathematical problems leading to two linear equations in two variables.4 (CCSS: 8.EE.8c) Inquiry Questions: What makes a solution strategy both efficient and effective? How is it determined if multiple solutions to an equation are valid? How does the context of the problem affect the reasonableness of a solution? Why can two equations be added together to get another true equation? Relevance & Application: The understanding and use of equations, inequalities, and systems of equations allows for situational analysis and decision-making. For example, it helps people choose cell phone plans, calculate credit card interest and payments, and determine health insurance costs. Recognition of the significance of the point of intersection for two linear equations helps to solve problems involving two linear rates such as determining when two vehicles traveling at constant speeds will be in the same place, when two calling plans cost the same, or the point when profits begin to exceed costs. Nature Of: Mathematics involves visualization. Mathematicians use tools to create visual representations of problems and ideas that reveal relationships and meaning. Mathematicians make sense of problems and persevere in solving them. (MP) Mathematicians use appropriate tools strategically. (MP)

2 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). (CCSS: 8.EE.6a)

3 For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. (CCSS: 8.EE.8b)

4 For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. (CCSS: 8.EE.8c)