# ALEX Classroom Resource

## How Many Solutions?: Algebra 1, Episode 14: Unit 7, Lesson 5 | Illustrative Math

Classroom Resource Information

Title:

How Many Solutions?: Algebra 1, Episode 14: Unit 7, Lesson 5 | Illustrative Math

URL:

https://aptv.pbslearningmedia.org/resource/im20-math-ep14-75/how-many-solutions/

Content Source:

PBS
Type: Audio/Video

Overview:

This video lesson builds on the idea that both graphing and rewriting quadratic equations in the form of expression = 0 are useful strategies for solving equations. It also reinforces the ties between the zeros of a function and the horizontal intercepts of its graph, which students began exploring in an earlier unit.

Here, students learn that they can solve equations by rearranging them into the form expression = 0, graphing the equation y = expression, and finding the horizontal intercepts. They also notice that dividing each side of a quadratic equation by a variable is not reliable because it eliminates one of the solutions. As students explain why certain maneuvers for solving quadratic equations are acceptable and others are not, students practice constructing logical arguments (MP3).

Content Standard(s):
 Mathematics MA2015 (2016) Grade: 9-12 Algebra I 16 ) 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. Construct a viable argument to justify a solution method. [A-REI1] Mathematics MA2015 (2016) Grade: 9-12 Algebra I 17 ) Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. [A-REI3] Alabama Alternate Achievement Standards AAS Standard: M.AAS.A.HS.17- Solve an equation of the form ax + b = c where a, b, and c are positive whole numbers and the solution, x, is a positive whole number to represent a real-world problem. Mathematics MA2015 (2016) Grade: 9-12 Algebra II 4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7] Mathematics MA2015 (2016) Grade: 9-12 Algebra II 21 ) Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [A-CED2] Mathematics MA2015 (2016) Grade: 9-12 Algebra II 27 ) 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); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.* [A-REI11] Mathematics MA2015 (2016) Grade: 9-12 Algebra II with Trigonometry 4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7] Mathematics MA2015 (2016) Grade: 9-12 Algebra II with Trigonometry 13 ) Use the structure of an expression to identify ways to rewrite it. [A-SSE2] Example: See x4 - y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 - y2)(x2 + y2). Mathematics MA2015 (2016) Grade: 9-12 Algebra II with Trigonometry 21 ) Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [A-CED2] Mathematics MA2015 (2016) Grade: 9-12 Algebra II with Trigonometry 27 ) 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); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.* [A-REI11] Mathematics MA2019 (2019) Grade: 9-12 Algebra I with Probability 5. Use the structure of an expression to identify ways to rewrite it. Example: See x4 - y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 - y2)(x2 + y2). Unpacked Content Evidence Of Student Attainment:Students: Make sense of algebraic expressions by identifying structures within the expression which allow them to rewrite it in useful and more efficient ways. Teacher Vocabulary:Terms Linear expressions Equivalent expressions Difference of two squares Factor Difference of squaresKnowledge:Students know: Algebraic properties. When one form of an algebraic expression is more useful than an equivalent form of that same expression. Skills:Students are able to: Use algebraic properties to produce equivalent forms of the same expression by recognizing underlying mathematical structures.Understanding:Students understand that: Generating equivalent algebraic expressions facilitates the investigation of more complex algebraic expressions. Diverse Learning Needs: Essential Skills:Learning Objectives: ALGI.5.1: Define equivalent expressions. ALGI.5.2: Rewrite an exponential expression in an alternative way. ALGI.5.3: Rewrite a quadratic expression in an alternative way. ALGI.5.4: Rewrite a linear expression in an alternative form. ALGI.5.5: Understand that rewriting an expression in different forms in a problem context can shed light on the problem. ALGI.5.6: Recall properties of exponents. Prior Knowledge Skills:li>Give examples of the properties of operations including distributive, commutative, and associative. Recall how to find the greatest common factor. Combine like terms of a given expression. Recognize the property demonstrated in a given expression. Simplify expressions with parentheses (Ex. 5(4 + x) = 20 + 5x). Simplify an expression by dividing by the greatest common factor (Ex. 18x + 6y= 6(3x + y). Define linear expression, rational, coefficient, and rational coefficient. Example: See x4 - y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 y2)(x2 + y2). Alabama Alternate Achievement Standards AAS Standard: M.A.AAS.11.5 Solve simple algebraic equations using real-world scenarios with one variable using multiplication or division. Mathematics MA2019 (2019) Grade: 9-12 Algebra I with Probability 9. Select an appropriate method to solve a quadratic equation in one variable. a. 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. Explain how the quadratic formula is derived from this form. b. Solve quadratic equations by inspection (such as x2 = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation, and recognize that some solutions may not be real. Unpacked Content Evidence Of Student Attainment:Students: Solve quadratic equations where both sides of the equation have evident square roots by inspection. Transform quadratic equations to a form where the square root of each side of the equation may be taken, including completing the square. Use the method of completing the square on the equation in standard form ax2+bx+c=0 to derive the quadratic formula. Identify quadratic equations which may be solved efficiently by factoring, and then use factoring to solve the equation. Use the quadratic formula to solve quadratic equations. Explain when the roots are real or complex for a given quadratic equation, and when complex write them as a ± bi. Demonstrate that a proposed solution to a quadratic equation is truly a solution by making the original true.Teacher Vocabulary:Completing the square Quadratic equations Quadratic formula Inspection Imaginary numbers Binomials TrinomialsKnowledge:Students know: Any real number has two square roots, that is, if a is the square root of a real number then so is -a. The method for completing the square. Notational methods for expressing complex numbers. A quadratic equation in standard form (ax2+bx+c=0) has real roots when b2-4ac is greater than or equal to zero and complex roots when b2-4ac is less than zero.Skills:Students are able to: Accurately use properties of equality and other algebraic manipulations including taking square roots of both sides of an equation. Accurately complete the square on a quadratic polynomial as a strategy for finding solutions to quadratic equations. Factor quadratic polynomials as a strategy for finding solutions to quadratic equations. Rewrite solutions to quadratic equations in useful forms including a ± bi and simplified radical expressions. Make strategic choices about which procedures (inspection, completing the square, factoring, and quadratic formula) to use to reach a solution to a quadratic equation.Understanding:Students understand that: Solutions to a quadratic equation must make the original equation true and this should be verified. When the quadratic equation is derived from a contextual situation, proposed solutions to the quadratic equation should be verified within the context given, as well as mathematically. Different procedures for solving quadratic equations are necessary under different conditions. If ab=0, then at least one of a or b must be zero (a=0 or b=0) and this is then used to produce the two solutions to the quadratic equation. Whether the roots of a quadratic equation are real or complex is determined by the coefficients of the quadratic equation in standard form (ax2+bx+c=0).Diverse Learning Needs: Essential Skills:Learning Objectives: ALGI.9.1: Define quadratic equation and zero product property. ALGI.9.2: Solve one-step equations using addition and subtraction that are set equal to zero. ALGI.9.3: Solve two-step equations using addition and subtraction that are set equal to zero. a. ALGI.9.4: Define completing the square. ALGI.9.5: 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. ALGI.9.6: Derive the quadratic formula from the form (x - p)= q. b. ALGI.9.7: Define quadratic formula, factoring, square root, complex number, and real number. ALGI.9.8: Solve quadratic equations by completing the square. ALGI.9.9: Solve quadratic equations by the quadratic formula. ALGI.9.10: Solve quadratic equations by factoring. ALGI.9.11: Solve quadratic equations by taking square roots. ALGI.9.12: Recognize when the quadratic formula gives complex solutions. ALGI.9.13: Write complex solutions as a ±bi for real numbers a and b. Prior Knowledge Skills:Identify perfect squares and square roots. Define square root, expressions, and approximations. Explain the distributive property. Calculate an expression in the correct order (Ex. exponents, mult./div. from left to right, and add/sub. from left to right). Recalving one-step equations. List given information from the problem. Identify the unknown, in a given situation, as the variable. Test the found number for accuracy by substitution. Example: Is 5 an accurate solution of 2(x + 5)=12? Calculate a solution to an equation by combining like terms, isolating the variable, and/or using inverse operations. Define equation and variable. Set up an equation to represent the given situation, using correct mathematical operations and variables. Recognize the correct order to solve expressions with more than one operation. Calculate a numerical expression (Ex. V=4x4x4). Choose the correct value to replace each variable in the algebraic expression (Substitution). Alabama Alternate Achievement Standards AAS Standard: M.A.AAS.11.9 Identify equivalent expressions given a linear expression using arithmetic operations. Mathematics MA2019 (2019) Grade: 9-12 Algebra I with Probability 10. Select an appropriate method to solve a system of two linear equations in two variables. a. Solve a system of two equations in two variables by using linear combinations; contrast situations in which use of linear combinations is more efficient with those in which substitution is more efficient. b. Contrast solutions to a system of two linear equations in two variables produced by algebraic methods with graphical and tabular methods. Unpacked Content Evidence Of Student Attainment:Students: Choose an appropriate method for solving a system of two linear equations (e.g., substitution, addition, tables, graphing). Solve and justify solutions. Contrast solutions to a system of two linear equations to determine which method is more efficient. Understand that tables and graphs of systems of equations my produce estimates rather than exact solutions. Provide reasonable approximations when appropriate in a graph or table.Teacher Vocabulary:Solution of a system of linear equations Substitution method Elimination method Graphically solve System of linear equations Solving systems by addition Tabular methodsKnowledge:Students know: Appropriate use of properties of addition, multiplication and equality. Techniques for producing and interpreting graphs of linear equations. Techniques for producing and interpreting tables of linear equations. The conditions under which a system of linear equations has 0, 1, or infinitely many solutions.Skills:Students are able to: Accurately perform the operations of multiplication and addition, and techniques for manipulating equations. Graph linear equations precisely. Create tables and locate solutions from the tables for systems of linear equations. Use estimation to find approximate solutions on a graph. Contrast solution methods and determine efficiency of a method for a given problem situation.Understanding:Students understand that: The solution of a linear system is the set of all ordered pairs that satisfy both equations. Solving a system by graphing or with tables can sometimes lead to approximate solutions. A system of linear equations will have 0, 1, or infinitely many solutions.Diverse Learning Needs: Essential Skills:Learning Objectives: ALGI.10.1: Solve a system of equations using three methods (Substitution, Elimination, and Graphing. ALGI.10.2: Distinguish the similarities and differences between the three methods of solving systems of equations. Prior Knowledge Skills:Solve a system of equation by graphing. Solve a system of equation by elimination. Solve a system of equation by substitution. Understand the meaning of the solution to a system of equations. Graph a linear equation. Alabama Alternate Achievement Standards AAS Standard: M.A.AAS.11.9 Identify equivalent expressions given a linear expression using arithmetic operations. Mathematics MA2019 (2019) Grade: 9-12 Algebra I with Probability 12. Create equations in two or more variables to represent relationships between quantities in context; graph equations on coordinate axes with labels and scales and use them to make predictions. Limit to contexts arising from linear, quadratic, exponential, absolute value, and linear piecewise functions. Unpacked Content Evidence Of Student Attainment:Students: Given a contextual situation expressing a relationship between quantities with two or more variables, Model the relationship with equations and graph the relationship on coordinate axes with labels and scales. Make predictions about the contextual situation using the graphs of the equations.Teacher Vocabulary:Piecewise functionsKnowledge:Students know: When a particular two variable equation accurately models the situation presented in a contextual problem.Skills:Students are able to: Write equations in two variables that accurately model contextual situations. Graph equations involving two variables on coordinate axes with appropriate scales and labels. Make predictions about the context using the graph.Understanding:Students understand that: There are relationships among features of a contextual problem, a created mathematical model for that problem, and a graph of that relationship. Diverse Learning Needs: Essential Skills:Learning Objectives: ALGI.12.1: Solve the equations represented by real-world situations. ALGI.12.2: Set up an equation to represent the given situation, using correct mathematical operations and variables. ALGI.12.3: Given a contextual situation, interpret and defend the solution in the context of the original problem. ALGI.12.4: Explain how to draw informal inferences from data distributions.ALGI.12.5: Define ordered pair and coordinate plane. ALGI.12.6: Create equations with two variables (exponential, quadratic and linear). ALGI.12.7: Graph equations on coordinate axes with labels and scales (exponential, quadratic, and linear). ALGI.12.8: Identify an ordered pair and plot it on the coordinate plane. Prior Knowledge Skills:Demonstrate how to plot points on a coordinate plane using ordered pairs from a table. Plot independent (input) and dependent (output) values on a coordinate plane. Draw and label a coordinate plane. Define dependent variable, independent variable, ordered pairs, input, output, and coordinate plane. Alabama Alternate Achievement Standards AAS Standard: M.A.AAS.11.11 Select an equation or inequality involving one operation (limit to addition or subtraction) with one variable that represents a real-world problem. Solve the equation. Mathematics MA2019 (2019) Grade: 9-12 Algebra I with Probability 14. Given a relation defined by an equation in two variables, identify the graph of the relation as the set of all its solutions plotted in the coordinate plane. Note: The graph of a relation often forms a curve (which could be a line). Unpacked Content Evidence Of Student Attainment:Students: Given a relation defined by an equation in two variables, Verify that any ordered pair in the relation that makes the equation true is a point on the graph. Show that there are an infinite number of ordered pairs that satisfy the equation.Teacher Vocabulary:Relation Curve (which could be a line) Graphically Finite solutions Infinite solutionsKnowledge:Students know: Appropriate methods to find ordered pairs that satisfy an equation. Techniques to graph the collection of ordered pairs to form a curve. Skills:Students are able to: Accurately find ordered pairs that satisfy an equation. Accurately graph the ordered pairs and form a curve.Understanding:Students understand that: An equation in two variables has an infinite number of solutions (ordered pairs that make the equation true), and those solutions can be represented by a curve in the coordinate plane.Diverse Learning Needs: Essential Skills:Learning Objectives: ALGI.14.1: Understand that the graph of an equation is the solution of an equation. ALGI.14.2: Graph a linear equation and use the graph to determine the solution set. ALGI.14.3: Use a given graph to determine the solution set. ALGI.14.4: Plot given points from a table. Prior Knowledge Skills:Identify the slope-intercept form (y=mx+b) of an equation where m is the slope and y is the y-intercept. Graph a function given the slope-intercept form of an equation. Demonstrate how to plot points on a coordinate plane using ordered pairs from table. Recall how to plot ordered pairs on a coordinate plane. Name the pairs of integers and/or rational numbers of a point on a coordinate plane. Alabama Alternate Achievement Standards AAS Standard: M.A.AAS.12.14 When given a relation in table form, identify the graph that represents the relation. (Ex: The points (5,5); (6,4); (3,7) are given to the student along with three graphs, and the student chooses the graph that represents the relation.) Mathematics MA2019 (2019) Grade: 9-12 Algebra I with Probability 15. Define a function as a mapping from one set (called the domain) to another set (called the range) that assigns to each element of the domain exactly one element of the range. a. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Note: If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. b. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Limit to linear, quadratic, exponential, and absolute value functions. Unpacked Content Evidence Of Student Attainment:Students: Given input/output relations between two variables in graphical form, verbal description, set of ordered pairs, or algebraic model, distinguish between those that are functions and non-functions. Using functional notation, Evaluate functions for inputs. Interpret statements in terms of context. Given a contextual relationship that may be represented as a function, Determine that exactly one element of the range (output) is assigned to each element of the domain (input) by the function. Relate the domain to its graph and to the quantitative relationship it describes.Teacher Vocabulary:Domain Range Function Relation Function notation Set notationKnowledge:Students know: Distinguishing characteristics of functions. Conventions of function notation. In graphing functions the ordered pairs are (x,f(x)) and the graph is y = f(x).Skills:Students are able to: Evaluate functions for inputs in their domains. Interpret statements that use function notation in terms of context. Accurately graph functions when given function notation. Accurately determine domain and range values from function notation. Understanding:Students understand that: A function is a mapping of the domain to the rangeFunction notation is useful in contextual situations to see the relationship between two variables when the unique output for each input relation is satisfied.Diverse Learning Needs: Essential Skills:Learning Objectives: ALGI.15.1: Define domain, range, relation, function, table of values, input, and output. ALGI.15.2: Understand the graph of a function is the set of ordered pairs consisting of an input and the corresponding output. ALGI.15.3: Understand that a function is a rule that assigns to each input exactly one output. ALGI.15.4: Identify the equation of a function, given its graph. ALGI.15.5: Find the range of a function given its domain. ALGI.15.6: Recognize that f(x) and y are the same. ALGI.15.7: Recall how to complete input/output tables. ALGI.15.8: Recall how to read/interpret information from a table. ALGI.15.9: Define function notation. ALGI.15.10: Translate a simple word problem into function notation. ALGI.15.11: Evaluate function when given x-value. Prior Knowledge Skills:Analyze the graph to determine the rate of change. Generate the slope of a line using given ordered pairs. Define linear functions, nonlinear functions, slope, and y-intercept Identify ordered pairs. Plot points on a coordinate plane., then connect points for the vertices to sketch a polygon. Alabama Alternate Achievement Standards AAS Standard: M.A.AAS.12.15 Use the vertical line test to determine if a given relation is a function. Mathematics MA2019 (2019) Grade: 9-12 Algebra I with Probability 19. 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). a. Find the approximate solutions of an equation graphically, using tables of values, or finding successive approximations, using technology where appropriate. Note: Include cases where f(x) is a linear, quadratic, exponential, or absolute value function and g(x) is constant or linear. Unpacked Content Evidence Of Student Attainment:Students: Given two functions (where f(x) is linear, quadratic ,absolute value, or exponential and g(x) is constant or linear) that intersect, Graph each function and identify the intersection point(s). Explain solutions for f(x) = g(x) as the x-coordinate of the points of intersection of the graphs, and explain solution paths. Use technology, tables, and successive approximations to produce the graphs, as well as to determine the approximation of solutions.Teacher Vocabulary:Functions Linear functions Absolute value functions Exponential functions IntersectionKnowledge:Students know: Defining characteristics of linear, polynomial, absolute value, and exponential graphs. Methods to use technology and tables to produce graphs and tables for two functions.Skills:Students are able to: Determine a solution or solutions of a system of two functions. Accurately use technology to produce graphs and tables for linear, quadratic, absolute value, and exponential functions. Accurately use technology to approximate solutions on graphs.Understanding:Students understand that: By graphing y=f(x) and y=g(x) on the same coordinate plane, the x-coordinate of the intersections of the two equations is the solution to the equation f(x) = g(x)Diverse Learning Needs: Essential Skills:Learning Objectives: ALGI.19.1: Define function, function notation, linear, polynomial, rational, absolute value, exponential, and logarithmic functions, and transitive property. ALGI.19.2: Explain, using the transitive property, why the x-coordinates of the points of the graphs are solutions to the equations. ALGI.19.3: Find solutions to the equations y = f(x) and y = g(x) using the graphing calculator. ALGI.19.4: Solve equations for y. ALGI.19.5: Demonstrate use of a graphing calculator, including using a table, making a graph, and finding successive approximations. Prior Knowledge Skills:Test the formula V= lwh and V=Bh with the experimental findings. Apply area formulas to solve real-world mathematical problems. Define algebraic expression and variable. Alabama Alternate Achievement Standards AAS Standard: M.A.AAS.12.18 Interpret the meaning of a point on the graph of a line. (Ex.: On a graph of football ticket purchases, trace the graph to a point and tell the number of tickets purchased and the total cost.) Mathematics MA2019 (2019) Grade: 9-12 Algebra I with Probability 20. 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, using technology where appropriate. Unpacked Content Evidence Of Student Attainment:Students: Given a linear inequality in two variables or a system of linear inequalities, graph solutions and solution sets using the appropriate notation (dotted or solid line). Teacher Vocabulary:half-planes System of linear inequalities. Boundaries Closed half-plane Open half-planeKnowledge:Students know: When to include and exclude the boundary of linear inequalities. Techniques to graph the boundaries of linear inequalities. Methods to find solution regions of a linear inequality and systems of linear inequalities.Skills:Students are able to: Accurately graph a linear inequality and identify values that make the inequality true (solutions). Find the intersection of multiple linear inequalities to solve a system.Understanding:Students understand that: Solutions to a linear inequality result in the graph of a half-plane. Solutions to a system of linear inequalities are the intersection of the solutions of each inequality in the system.Diverse Learning Needs: Essential Skills:Learning Objectives: ALGI.20.1: Define the half-plane as the shaded region. ALGI.20.2: Determine the intersecting shaded region is the solution to the system. ALGI.20.3: Graph the lines of the systems and shade the appropriate region. ALGI.20.4: Determine the shaded region is the solution to the inequality. ALGI.20.5: Graph an inequality and shade the appropriate region. ALGI.20.6: Determine whether a line should be solid or dotted, depending on the inequality symbol. ALGI.20.7: Recognize inequality symbols >, < . Prior Knowledge Skills:Define function, ordered pairs, input, output. Identify the slope-intercept form (y=mx+b) of an equation where m is the slope and y is the y-intercept. Demonstrate how to plot points on a coordinate plane using ordered pairs from table. Generate the slope of a line using given ordered pairs. Recall how to graph inequalities on a number line. Show how to graph on Cartesian plane. Show how to plot points on a Cartesian plane. Define ordered pairs. Graph the solution set on a number line for the inequality used to represent the situation. Recall how to plot ordered pairs on a coordinate plane. Identify which signs indicate the location of a point in a coordinate plane. Demonstrate when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. Name the pairs of integers and/or rational numbers of a point on a coordinate plane. Define ordered pairs. Alabama Alternate Achievement Standards AAS Standard: M.A.AAS.12.18 Interpret the meaning of a point on the graph of a line. (Ex.: On a graph of football ticket purchases, trace the graph to a point and tell the number of tickets purchased and the total cost.)
Tags: coefficients, complex solutions, coordinate axes, domain, equation, expression, function, graph, inequalities, labels, linear equations, quadratic equations, range, scales, solution, systems of linear equations, variables, xcoordinate