Coordinate Algebra: Unit 2 Reasoning with Equations and Inequalities PARENT RESOURCE This resource is merely a supplement to what the students are doing in their classroom. It is meant to serve as additional support for the students as they work with their parents at home. These are suggested materials and are additional help for homework and practice assignments. Unit Overview: By the end of eighth grade, your students have learned to solve linear equations in one variable and have applied graphical and algebraic methods to analyze and solve systems of linear equations in two variables. This unit builds on these earlier experiences by asking students to analyze and explain the process of solving an equation and to justify the process used in solving a system of equations. Students develop fluency writing, interpreting, and translating between various forms of linear equations and inequalities, and using them to solve problems. They master the solution of linear equations and apply related solution techniques and the laws of exponents to the creation and solution of simple exponential equations. Students explore systems of equations and inequalities, and they find and interpret their solutions. All of this work is grounded on understanding quantities and on relationships between them.
Understanding Solving Equations as a Process of Reasoning ESSENTIAL QUESTION: In what ways can problems be solved, and why should one method be chosen over another? LEARNING TARGETS: I can Understand, apply, and explain the results of using inverse operations. Justify the steps in solving equations by applying and explaining the properties of equality, inverse and identity. Use the names of the properties and common sense explanations to explain the steps in solving an equation. CONCEPT OVERVIEW: Properties of operations can be used to change expressions on either side of the equation to equivalent expressions. In addition, adding the same term to both sides of an equation or multiplying both sides by a non-zero constant produces an equation with the same solutions. Other operations, such as squaring both sides, may produce equations that have extraneous solutions. Students should focus on and master linear equations and be able to extend and apply their reasoning to other types of equations in future courses. Students will solve exponential equations with logarithms in CCGPS Advanced Algebra. Assuming an equation has a solution; students will construct a convincing argument that justifies each step in the solution process. Justifications may include the associative, commutative, and division properties, combining like terms, multiplication by 1, etc. Students will be challenged to justify each step of solving an equation. Transforming 2x 5 = 7 to 2x = 12 is possible because 5 = 5, so adding the same quantity to both sides of an equation makes the resulting equation true as well. Each step of solving an equation can be defended, much like providing evidence for steps of a geometric proof. Examples will be provided for how the same equation might be solved in a variety of ways as long as equivalent quantities are added or subtracted to both sides of the equation, the order of steps taken will not matter. 3n +2 = n 10-2 = -2 3n = n 12 -n = -n 2n = -12 2 2 n = -6 or 3n + 2 = n 10 +10 = +10 3n +12 = n -3n = -3n 12 = -2n -2-2 n = -6 or 3n +2 = n 10 -n = -n 2n +2 = -10-2 = -2 2n = -12 2 2 n = -6 The Properties of Operations Here a, b and c stand for arbitrary numbers in a given number system. The properties of operations apply to the rational number system, the real number system, and the complex number system. Associative property of addition (a + b) + c = a + (b + c) Commutative property of addition a + b = b + a Additive identity property of 0 a + 0 = 0 + a = a Existence of additive inverses For every a there exists a so that a + (-a) = (-a) + a = 0. Associative property of multiplication (a *b) * c = a *(b * c) Commutative property of multiplication a * b = b * a Multiplicative identity property of 1 a * 1 = 1 * a = a Existence of multiplicative inverses For every a 0 there exists so that Distributive property of multiplication over addition The Properties of Equality Here a, b and c stand for arbitrary numbers in the rational, real, or complex number systems. a * (b + c) = a * b + a * c Reflexive property of equality a = a Symmetric property of equality If a = b, then b = a Transitive property of equality If a = b and b = c, then a = c Addition property of equality If a = b, then a + c = b + c Subtraction property of equality If a = b, then a c = b c Multiplication property of equality If a = b, then a c = b c
Division property of equality If a = b and c 0, then a c = b c Substitution property of equality If a = b, then b may be substituted for a in any expression containing a. IMPORTANT VOCABULARY: Properties of operations and properties of equalities refer to the table above Coefficient - the number in front of the variable in a given expression Like terms terms that have the same variables and the same corresponding exponents Variable a quantity that changes or that can have different values; a symbol, usually a letter, that can stand for a variable quantity Evaluate to find the value of a mathematical expression Justify - to prove or show that the answer is valid Viable - capable of working successfully Inverse operation the operation that reverses the effect of another operation Equivalent expressions - expressions that are the same, even though they may look a little different. If you substitute in the same variable value into equivalent expressions, they will each give you the same value when you simplify. Extraneous solutions - a solution of the simplified form of an equation that does not satisfy the original equation. SAMPLE PROBLEM: 1. Which of the following equations have the same solution? Give reasons for your answer that does not depend on solving the equations. 1. x+3 = 5x 4 2. x 3 = 5x+4 3. 2x+8 = 5x 3 4. 10x+6 = 2x 8 5. 10x 8 = 2x+6 6. 0 3 x 10 1 2 x - 0 4
Virtual Resources: Understanding Solving Equations as a Process of Reasoning http://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/v/why-we-do-the-same--thing-toboth-sides-multi-step-equations http://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/v/solving-equations-with-thedistributive-property-2
Solving Literal Equations and Inequalities in One Variable LESSON ESSENTIAL QUESTION: In what ways can the problem be solved, and why should one method be chosen over another? LEARNING TARGETS: I can Write equations in equivalent forms to solve problems. Analyze and solve literal equations for a specified variable. Understand and apply the properties of Inequalities. Verify that a given number or variable is a solution to the equation or inequality. Interpret the solution of an inequality in real terms. Solve linear equations and inequalities with one variable, including absolute value equations and inequalities and compound inequalities. Solve an equation for a specific variable and reason about the new equation (i.e. A=1/2 Bh: if I know that the area of a triangle is 50 units, then 100/h = B, so as h increases, B decreases.) Solve simple exponential equations using laws of exponents. Graph the solution to linear inequalities in two variables. Graph the solution to systems of linear inequalities in two variables. Identify the solutions as a region of the plane. CONCEPT OVERVIEW: In grades 6-8, students solve and graph linear equations and inequalities. There are two major reasons for discussing the topic of inequalities along with equations: First, there are analogies between solving equations and inequalities that help students understand them both. Second, the applications that lead to equations almost always lead in the same way to inequalities. Graphing experience with inequalities should be limited to graphing on a number line diagram. Despite this work, some students will still need more practice to be proficient. It may be beneficial to remind your students of the most common solving techniques, such as converting fractions from one form to another, removing parentheses in the sentences, or multiplying both sides of an equation or inequality by the common denominator of the fractions. Students must be aware of what it means to check an inequality s solution The substitution of the end points of the solution set in the original inequality should give equality regardless of the presence or the absence of an equal sign in the original sentence. The substitution of any value from the rest of the solution set should give a correct inequality. Examples and exercises will provide students with meaningful review and to introduce other important concepts, such as the use of properties and applications of solving linear equations and inequalities. The idea that the application of properties is also appropriate when working with equations or inequalities that include more than one variable, fractions and decimals is critical. Regardless of the type of numbers or variables in the equation or inequality, students have to examine the validity of each step in the solution process. Solving equations for the specified letter with coefficients represented by letters (e.g., A = ½(B + b) when solving for b) is similar to solving an equation with one variable. Students will have an opportunity to abstract from particular numbers and apply the same kind of manipulations to formulas as they did to equations. One of the purposes of doing abstraction is to learn how to evaluate the formulas in easier ways and use the techniques across mathematics and science. Students will manipulate equations containing variables with subscripts. The same variables with different subscripts (e.g., x 1 and x 2 ) should be viewed as different variables that cannot be combined as like terms. A variable with a variable subscript, such as a n, must be treated as a single variable the nth term, where variables a and n have different meaning. The graphing method can be the first step in solving systems of equations. A set of points representing solutions of each equation is found by graphing these equations. Even though the graphing method is limited in finding exact solutions and often yields approximate values, the use of it helps to discover whether solutions exist and, if so, how many are there. The next step is to turn to algebraic methods, elimination or substitution, to allow the students to find exact solutions. For any method, it is important to have a well-organized format for writing solutions. IMPORTANT VOCABULARY: Inequality any mathematical sentence that contains the symbols > (greater than), < (less than), (less than or equal to), or (greater than or equal to) Property of Inequalities- Here a, b and c stand for arbitrary numbers in the rational or real number systems.
Exactly one of the following is true: a < b, a = b, a > b. If a > b and b > c then a > c. If a > b, then b < a. If a > b, then a < b. If a > b, then a ± c > b ± c. If a > b and c > 0, then a c > b c. If a > b and c < 0, then a c < b c. If a > b and c > 0, then a c > b c. If a > b and c < 0, then a c < b c. Solution any value for a variable that makes an equation true Literal equations- an equation where variables represent known values Equivalent equations- equivalent equations are the equations that have the same solution SAMPLE PROBLEM: 1. The perimeter of a rectangle is given by P = 2W + 2L. Solve for W and restate in words the meaning of this new formula in terms of the meaning of the other variables.
Virtual Resources: Solving Literal Equations and Inequalities in One Variable http://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/v/one-step-inequalities http://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/v/one-step-inequalities-2 http://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/v/absolute-value-inequalities http://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/v/absolute-value-equations http://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/v/equations-with-variables-onboth-sides
Verifying Solutions to Systems of Equations LESSON ESSENTIAL QUESTION: How can a solution for a given System of Equations be verified? LEARNING TARGETS: I can Explain the use of the multiplication property of equality to solve a system of equations. Explain why the sum of two equations is justifiable in the solving of a system of equations (property of equality). Relate the process of linear combinations with the process of substitution for solving a system of linear equations. Explain how to solve a system of two equations and justify the method used to find the solution. Explain why replacing one equation by the sum of that equation with a multiple of the other produces a system with the same solution. Create a system of equations to represent relationships between quantities. CONCEPT OVERVIEW: The focus of this standard is to provide mathematics justification for the addition (elimination) and substitution methods of solving systems of equations that transform a given system of two equations into a simpler equivalent system that has the same solutions as the original. The Addition and Multiplication Properties of Equality allow finding solutions to certain systems of equations. In general, any linear combination, m(ax + By) + n(cx + Dy) = me +nf, of two linear equations Ax + By = E and Cx + Dy = F intersecting in a single point contains that point. The multipliers m and n can be chosen so that the resulting combination has only an x-term or only a y-term in it. That is, the combination will be a horizontal or vertical line containing the point of intersection. In the proof of a system of two equations in two variables, where one equation is replaced by the sum of that equation and a multiple of the other equation, produces a system that has the same solutions, let point (x 1, y 1 ) be a solution of both equations: Ax 1 + By 1 = E (true) Cx 1 + Dy 1 = F (true) Replace the equation Ax + By = E with Ax + By + k(cx + Dy) on its left side and with E + kf on its right side. The new equation is Ax + By + k(cx + Dy) = E + kf. The students will show that the ordered pair of numbers (x 1, y 1 ) is a solution of this equation by replacing (x 1, y 1 ) in the left side of this equation and verifying that the right side really equals E + kf: Ax 1 + By 1 + k(cx 1 + Dy 1 ) = E + kf (true) Systems of equations are classified into two groups, consistent or inconsistent, depending on whether or not solutions exist. The solution set of a system of equations is the intersection of the solution sets for the individual equations. Students will understand the benefit of making the appropriate selection of a method for solving systems (graphing vs. addition vs. substitution). This depends on the type of equations and combination of coefficients for corresponding variables, without giving a preference to either method. Initially simple, real-world examples will help students recognize a graph as a set of solutions to an equation. For example, if the equation y = 6x + 5 represents the amount of money paid to a babysitter (i.e., $5 for gas to drive to the job and $6/hour to do the work), then every point on the line represents an amount of money paid, given the amount of time worked. The students will explore visual ways to solve an equation such as 2x + 3 = x 7 by graphing the functions y = 2x + 3 and y = x 7. Students should recognize that the intersection point of the lines is at (-10, -17). They should be able to verbalize that the intersection point means that when x = -10 is substituted into both sides of the equation, each side simplifies to a value of -17. Therefore, -10 is the solution to the equation. This same approach can be used whether the functions in the original equation are linear, nonlinear or both. Using technology, students will graph a function and use the trace function to move the cursor along the curve. Discuss the meaning of the ordered pairs that appear at the bottom of the calculator, emphasizing that every point on the curve represents a solution to the equation. They will begin with simple linear equations and how to solve them using the graphs and tables on a graphing calculator. Then, they will advance to nonlinear situations so they can see that even complex equations that might involve quadratics, absolute value, or
rational functions can be solved fairly easily using this same strategy. While a standard graphing calculator does not simply solve an equation for the user, it can be used as a tool to approximate solutions. The students will use the table function on a graphing calculator to solve equations. For example, to solve the equation x 2 = x + 12, students can examine the equations y = x 2 and y = x + 12 and determine that they intersect when x = 4 and when x = -3 by examining the table to find where the y-values are the same. IMPORTANT VOCABULARY: System of equations- is a collection of one or more linear equations, with the same set of unknowns. Solution set - the intersection of the solution sets for the individual equations Elimination method- is the process of eliminating one of the variables in a system of equations using addition or subtraction in conjunction with multiplication or division and solving the system of equations Elimination by multiplication multiplying and combining equations in a system in order to eliminate a variable. Elimination by addition adding and combining equations in a system in order to eliminate a variable Elimination by substitution solving one of the equations for one variable in terms of the other, and then substitute that into the other equation. SAMPLE PROBLEM: 1. Create a new system using both the addition and multiplication properties of equality. Then verify that the new system has the same solution as the original. { }
Virtual Resources: Verifying Solutions to Systems of Equations http://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/v/solving-systems-by-elimination http://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/v/solving-systems-by-substitution-1 http://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/v/practice-thinking-about-number-of-solutions-tosystems http://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/v/solving-systems-by-graphing
Verifying Solutions to Systems of Inequalities LESSON ESSENTIAL QUESTION: How can a solution for a given System of Equations be verified? LEARNING TARGETS: I can Explain why replacing one equation by the sum of that equation with a multiple of the other produces a system with the same solution. Create a system of equations to represent relationships between quantities. Graph a system of linear inequalities on a coordinate plane. Explain that the solution set for a system of linear inequalities is the intersection of the shaded regions (half-planes) of both inequalities. Check points in the intersection of the half-planes to verify that they represent a solution to the system. CONCEPT OVERVIEW: This part of the unit will investigate real-world examples of two-dimensional inequalities. For example, students might explore what the graph would look like for money earned when a person earns at least $6 per hour. (The graph for a person earning exactly $6/hour would be a linear function, while the graph for a person earning at least $6/hour would be a half-plane including the line and all points above it.) Applications such as linear programming will help your students to recognize how businesses use constraints to maximize profit while minimizing the use of resources. These situations often involve the use of systems of two variable inequalities. IMPORTANT VOCABULARY: Half-plane the result of graphing an inequality on the coordinate plane, which creates a boundary that cuts the coordinate plane in half. System of linear inequalities- set of two or more linear inequalities containing two or more variables Solution of system of linear inequalities- all the ordered pairs that satisfy all the linear inequalities in the system. Solution region all points (a, b) such that when x = a, and y = b, all inequalities are true. SAMPLE PROBLEMS: 1. A publishing company publishes a total of no more than 100 magazines every year At least 30 of these are women s magazines, but the company always publishes at least as many women s magazines as men s magazines Find a system of inequalities that describes the possible number of men s and women s magazines that the company can produce each year consistent with these policies. Graph the solution set.
2. Create a context that represents the shaded area. Write the system of inequalities that models the meaning of the context. Describe the connections between the contexts, inequalities, and graphs.
Virtual Resources: Verifying Solutions to Systems of Inequalities http://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/v/graphing-systems-of-inequalities http://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/v/graphing-systems-of-inequalities-2 http://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/v/graphing-systems-of-inequalities-3