A-SSE.1.1, A-SSE.1.2-

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1 Putnam County Schools Curriculum Map Algebra Module: 4 Quadratic and Exponential Functions Instructional Window: January 9-February 17 Assessment Window: February 20 March 3 MAFS Standards Lesson A: Quadratic Functions Introduction MAFS.912.F-IF.3.8 Also Assesses A-APR.2.3, F-IF.3.7- Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. EngageNY, Module 4, Lessons 11-14/Shmoop Unit 6 & 7. Math Nation Section 5, Topic 10. MAFS.912.A-REI.2.4- Solve quadratic equations 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. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. EngageNY, Module 4, Lessons 5-7, 9/Shmoop Unit 4 & 8. Math Nation Section 5, Topic 8 (a), topics 3,4,5,6,9,10 (b). MAFS.912.A-SSE.1.2 Assessed Within A-SSE.2.3- Use the structure of an expression to identify ways to rewrite it. For example, see x 4 - y 4 as (x²)² (y²)², thus recognizing it as a difference of squares that can be factored as (x² y²)(x² + y²). EngageNY, Module 4, Lessons 1, 2/Shmoop Unit 2. Math Nation Section 5, Topics 2,5,6. MAFS.912.A-SSE.2.3 Also Assesses A-SSE.1.1, A-SSE.1.2- Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor a quadratic expression to reveal the zeros of the function it defines. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. EngageNY, Module 4, Lessons 11-14/Shmoop Unit 2, 3, 4, 7. Math Nation Section 5, Topics 3-5 (a), Topic 10 (a & b).

2 Lesson B: Quadratic Functions MAFS.912.F-BF.2.3- Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. EngageNY, Module 4, Lessons 19, 20/Shmoop Unit 7. Math Nation Section 6, Topics 7,8. MAFS.912.A-CED.1.2 Also Assesses A-REI.3.5, A-REI.3.6, A-REI Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. EngageNY, Module 4, Lessons 20-24/Shmoop Unit 6, 7, 9. Math Nation Section 6, Topic 1. MAFS.912.F-IF.2.4 Also Assesses F-IF.3.9- For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. EngageNY, Module 4, Lessons 17/Shmoop Unit 7. Math Nation Section 6, Topic 1. MAFS.912.F-IF.3.7 Assessed Within F-IF.3.8- Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. EngageNY, Module 4, Lessons 8-10/Shmoop Unit 6 & 7. Math Nation Section 6, Topics 3-6. MAFS.912.F-IF.3.8 Also Assesses A-APR.2.3, F-IF.3.7- Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. EngageNY, Module 4, Lessons 11-14/Shmoop Unit 6 & 7. Math Nation Section 6, Topics 1,4,5,6. MAFS.912.F-IF.3.9 Assessed Within F-IF.2.4- Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. EngageNY, Module 4, Lessons 19-21/Shmoop Unit 7. Math Nation Section 6, Topic 6. MAFS.912.A-REI.2.4- Solve quadratic equations 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. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. EngageNY, Module 4, Lessons 11, 12, 13, 15/Shmoop Unit 4 & 8. Math Nation Section 6, Topic 2. MAFS.912.A-REI.4.11 Also Assesses A-REI 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. Shmoop Unit 9. Math Nation Section 6, Topic 9.

3 Lesson C: Introduction to Exponential Functions MAFS.912.F-IF.1.3 Assessed Within F- LE.1.2- Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n 1. EngageNY, Module 3, Lessons 1, 2/Shmoop Unit 5. Math Nation Section 7, Topics 1,2. MAFS.912.F-LE.1.1 Also Assesses F-LE.2.5- Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. EngageNY, Module 3, Lessons 4-7/Shmoop Unit 6 & 8. Math Nation Section 7, topic 2. MAFS.912.F-LE.1.2 Also Assesses F-BF.1.1, F-IF.1.3 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). EngageNY, Module 3, Lessons 21-24/Shmoop Unit 6. Math Nation Section 7, Topics 1,3. MAFS.912.F-LE.1.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. EngageNY, Module 3, Lessons 5, 6/Shmoop Unit 6 & 8. Math Nation Section 7, Topic 2. MAFS.912.A-REI.4.10 Assessed within A-REI Understand 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. EngageNY, Module 1, Lessons 20/ Shmoop Unit 6. Math Nation Section 7, Topics 1-3.

4 Expectations to be Learned Unpacking What do these standards mean a child will know and be able to do? MAFS.912.F-BF.2.3 ITEM SPECIFICATIONS Item Type Equation Editor May require creating a value or an expression. GRID May require plotting points or a transformed function. Matching Item May require matching an equation, a value of k, and an explanation of the effect on a graph. Multiple Choice May require selecting a graph or a table from a list. Open Response May require explaining the effects of a transformation. Table Item May require completing a table of values. Clarifications Students will determine the value of k when given a graph of the function and its transformation. Students will identify differences and similarities between a function and its transformation. Students will identify a graph of a function given a graph or a table of a transformation and the type of transformation that is represented. Students will graph by applying a given transformation to a function. Students will identify ordered pairs of a transformed graph. Students will complete a table for a transformed function. Assessment Limits Functions represented algebraically are limited to linear, quadratic, or exponential. Functions represented using tables or graphs are not limited to linear, quadratic, or exponential. Functions may be represented using tables or graphs. Functions may have closed domains. Functions may be discontinuous. Items should have a single transformation. Stimulus Attributes Items should be given in a mathematical context. Items may use function notation. Items may present a function using an equation, a table of values, or a graph. Response Attributes Items may require the student to explain or justify a transformation that has been applied to a function. Items may require the student to explain how a graph is affected by a value of k. Items may require the student to find the value of k. Items may require the student to complete a table of values. Calculator Neutral DOK Level MAFS.912.F-BF.2.3

5 MAFS.912.A-CED.1.2 Also Assesses A-REI.3.5, A-REI.3.6, A-REI.4.12 ITEM SPECIFICATIONS Item Type Editing Task Choice May require choosing the correct definition of a variable or completing an explanation or a proof. Equation Editor May require creating a set of equations, creating a set of inequalities, or giving an ordered pair. GRID May require graphing a representation of a set of equations, a set of inequalities, or an ordered pair; selecting a solution region; or dragging and dropping text to complete a proof. Hot Text May require selecting a solution or dragging and dropping text to complete a proof. Multiple Choice May require identifying a set of equations, a set of inequalities, a value, an ordered pair, or a graph. Multiselect May require identifying equations or inequalities. Open Response May require writing an explanation. Clarifications Students will identify the quantities in a real-world situation that should be represented by distinct variables. Students will write a system of equations given a real-world situation. Students will graph a system of equations that represents a real-world context using appropriate axis labels and scale. Students will solve systems of linear equations. Students will provide steps in an algebraic proof that shows one equation being replaced with another to find a solution for a system of equations. Students will identify systems whose solutions would be the same through examination of the coefficients. Students will identify the graph that represents a linear inequality. Students will graph a linear inequality. Students will identify the solution set to a system of inequalities. Students will identify ordered pairs that are in the solution set of a system of inequalities. Students will graph the solution set to a system of inequalities. Assessment Limits Items that require the student to write a system of equations using a real-world context are limited to a system of 2 x 2 linear equations with integral coefficients if the equations are written in the form Ax + By = C. MAFS.912.A-CED.1.2

6 Items that require the student to solve a system of equations are limited to a system of 2 x 2 linear equations with integral coefficients if the equations are written in the form Ax + By = C. Items that require the student to graph a system of equations or inequalities to find the solution are limited to a 2 x 2 system. Items that require the student to write a system of inequalities using a real-world context are limited to integer coefficients. Stimulus Attributes Items assessing A-CED.1.2 must be placed in a real-world context. Items assessing A-REI.3.5 must be placed in a mathematical context. Items assessing A-REI.3.6 and A-REI.4.12 may be set in a real-world or mathematical context. Items may result in infinitely many solutions or no solution. Response Attributes Items may require the student to choose an appropriate level of accuracy. Items may require the student to choose and interpret the scale in a graph. Items may require the student to choose and interpret units. For A-CED.1.2, items may require the student to apply the basic modeling cycle. Calculator Neutral MAFS.912.F-IF.1.3 Assessed Within F- LE.1.2 MAFS.912.F-IF.2.4 ITEM SPECIFICATIONS Item Types Equation Editor May require expressing a value, expression, or equation. MAFS.912.F-IF.1.3 Content Complexity: MAFS.912.F-IF.2.4 Content Complexity:

7 GRID May require plotting points on a coordinate plane, graphing a function, or matching and/or selecting key features as verbal descriptions to points on the graph. Hot Text May require selecting a key feature or region on a graph. Multiple Choice May require selecting a choice from a set of possible choices. Open Response May require explaining the meaning of key features or the comparison of two functions. Clarifications Students will determine and relate the key features of a function within a real-world context by examining the function s table. Students will determine and relate the key features of a function within a real-world context by examining the function s graph. Students will use a given verbal description of the relationship between two quantities to label key features of a graph of a function that model the relationship. Students will differentiate between different types of functions using a variety of descriptors (e.g., graphically, verbally, numerically, and algebraically). Students will compare and contrast properties of two functions using a variety of function representations (e.g., algebraic, graphic, numeric in tables, or verbal descriptions). Assessment Limits Functions represented algebraically are limited to linear, quadratic, or exponential. Functions may be represented using tables, graphs or verbally. Functions represented using these representations are not limited to linear, quadratic or exponential. Functions may have closed domains. Functions may be discontinuous. Items may not require the student to use or know interval notation. Key features include x-intercepts, y-intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; and end behavior. Stimulus Attributes For F-IF.2.4, items should be set in a real-world context. For F-IF.3.9, items may be set in a real-world or mathematical context. Items may use verbal descriptions of functions. Items may use function notation. Response Attributes For F-IF.2.4, items may require the student to apply the basic modeling cycle. Items may require the student to write intervals using inequalities. Items may require the student to choose an appropriate level of accuracy. Items may require the student to choose and interpret the scale in a graph. Items may require the student to choose and interpret units. Calculator No

8 MAFS.912.F-IF.3.7 Also Assesses S-ID.2.6 MAFS.912.F-IF.3.8 Also Assesses A-APR.2.3, F-IF.3.7 ITEM SPECIFICATIONS Item Type Equation Editor May require creating a value, an expression, or an equation. GRID May require plotting points, key features, or an equation on a graph; identifying key features; or selecting key features on a graph. Hot Text May require selecting key features on a graph. Multiple Choice May require selecting from a list. Multiselect May require selecting multiple responses. Open Response May require explaining and interpreting a function. Clarifications Students will identify zeros, extreme values, and symmetry of a quadratic function written symbolically. Students will classify the exponential function as exponential growth or decay by examining the base, and students will give the rate of growth or decay. Students will use the properties of exponents to interpret exponential expressions in a real-world context. Students will write an exponential function defined by an expression in different but equivalent forms to reveal and MAFS.912.F-IF.3.7 Content Complexity: MAFS.912.F-IF.3.8

9 explain different properties of the function, and students will determine which form of the function is the most appropriate for interpretation for a real-world context. Students will find the zeros of a polynomial function when the polynomial is in factored form. Students will create a rough graph of a polynomial function in factored form by examining the zeros of the function. Students will use the x-intercepts of a polynomial function and end behavior to graph the function. Students will identify the x- and y-intercepts and the slope of the graph of a linear function. Students will identify zeros, extreme values, and symmetry of the graph of a quadratic function. Students will identify intercepts and end behavior for an exponential function. Students will graph a linear function using key features. Students will graph a quadratic function using key features. Students will graph an exponential function using key features. Students will identify and interpret key features of a graph within the real-world context that the function represents. Assessment Limits For A-APR.2.3, the leading coefficient should be an integer and the polynomial s degree is restricted to 3 or 4. The polynomial function should not have a zero with multiplicity. The polynomial should be given in factored form. For F-IF.3.8a, items that require the student to transform a quadratic equation to vertex form, the coefficient of the linear term must be an even factor of the coefficient of the quadratic form. For F-IF.3.7e and F-IF.3.8b, exponential functions are limited to simple exponential growth and decay functions and to exponential functions with one translation. Base e should not be used. For F-IF.3.8, items may specify a required form using an equation or using common terminology such as standard form. In items that require the student to interpret the vertex or a zero of a quadratic function within a real-world context, the student should interpret both the x-value and the y-value. For F-IF.3.7a, quadratic functions that are given in the form y = ax2 + bx + c, a, b, and c must be integers. Quadratic functions given in vertex form y = a(x h)2 + k, a, h, and k must be integers. Quadratic functions given in other forms should be able to be rewritten and adhere to one of the two previous forms. Stimulus Attributes Items may require the student to identify a correct graph. Items may be set in a mathematical or real-world context. Items may use function notation. Items should not require the student to complete a sign chart for a polynomial. Response Attributes For F-IF.3.7, items may require the student to apply the basic modeling cycle. Items may require the student to choose an appropriate level of accuracy. Items may require the student to choose and interpret the scale in a graph. Items may require the student to choose and interpret units. Responses with square roots should require the student to rewrite the square root so that the radicand has no square factors. Calculator Neutral

10 MAFS.912.F-IF.3.9 Assessed Within F-IF.2.4 MAFS.912.F-LE.1.1 Also Assesses F-LE.2.5 ITEM SPECIFICATIONS Item Type Editing Task Choice May require choosing a model, a parameter, and/or an interpretation. Equation Editor May require creating a value or an expression. GRID May require dragging and dropping expressions or statements to a graph. Hot Text May require dragging and dropping justifications or interpretations. Matching Item May require matching parameters with interpretations. Multiple Choice May require selecting an interpretation from a list. Multiselect May require selecting multiple values. Open Response May require analyzing the growth of a function or explaining parameters of a function. Clarifications Students will determine whether the real-world context may be represented by a linear function or an exponential MAFS.912.F-IF.3.9 MAFS.912.F-LE.1.1 Content Complexity: Level 3: Strategic Thinking & Complex Reasoning

11 function and give the constant rate or the rate of growth or decay. Students will choose an explanation as to why a context may be modeled by a linear function or an exponential function. Students will interpret the rate of change and intercepts of a linear function when given an equation that models a realworld context. Students will interpret the x-intercept, y-intercept, and/or rate of growth or decay of an exponential function given in a real-world context. Assessment Limits Exponential functions should be in the form a(b)x + k. Stimulus Attributes Items should be set in a real-world context. Items may use function notation. Response Attributes Items may require the student to apply the basic modeling cycle. Items may require the student to choose a parameter that is described within the real-world context. Items may require the student to choose an appropriate level of accuracy. Items may require the student to choose and interpret the scale in a graph. Items may require the student to choose and interpret units. Calculator No MAFS.912.F-LE.1.2 Also Assesses F-BF.1.1, F-IF.1.3 ITEM SPECIFICATIONS MAFS.912.F-LE.1.2 Content Complexity:

12 Item Types Editing Task Choice May require choosing an expression, function, or definition of a variable. Equation Editor May require creating a value, creating an expression, creating a function, or showing steps for a calculation. GRID May require ordering of steps for a calculation from a context. Hot Text May require dragging and dropping values or expressions to construct a function. Multiple Choice May require selecting a choice from a set of possible choices. Multiselect May require choosing equivalent functions. Open Response May require explaining and interpreting a resulting function. Table Item May require completing missing cells in a table. Clarifications Students will write a linear function, an arithmetic sequence, an exponential function, or a geometric sequence when given a graph that models a real-world context. Students will write a linear function, an arithmetic sequence, an exponential function, or a geometric sequence when given a verbal description of a real-world context. Students will write a linear function, an arithmetic sequence, an exponential function, or a geometric sequence when given a table of values or a set of ordered pairs that model a real-world context. Students will write an explicit function, define a recursive process, or complete a table of calculations that can be used to mathematically define a real-world context. Students will write a function that combines functions using arithmetic operations and relate the result to the context of the problem. Students will write a function to model a real-world context by composing functions and the information within the context. Students will write a recursive definition for a sequence that is presented as a sequence, a graph, or a table. Assessment Limits In items where the student must write a function using arithmetic operations or by composing functions, the student should have to generate the new function only. In items where the student constructs an exponential function, a geometric sequence, or a recursive definition from input-output pairs, at least two sets of pairs must have consecutive inputs. In items that require the student to construct arithmetic or geometric sequences, the real-world context should be discrete. In items that require the student to construct a linear or exponential function, the real-world context should be continuous. Stimulus Attributes Items should be set in a real-world context. Items may use function notation. In items where the student builds a function using arithmetic operations or by composition, the functions may be given using verbal descriptions, function notation or as equations. Response Attributes For F-BF.1.1b and c, the student may be asked to find a value. For F-LE.1.2 and F-BF.1.1, items may require the student to apply the basic modeling cycle. In items where the student writes a recursive formula, the student may be expected to give both parts of the formula.

13 The student may be required to determine equivalent recursive formulas or functions. Items may require the student to choose an appropriate level of accuracy. Items may require the student to choose and interpret the scale in a graph. Items may require the student to choose and interpret units. Calculator Neutral MAFS.912.F-LE.1.3 ITEM SPECIFICATIONS Item Types Editing Task Choice May require choosing a function and/or a justification. Equation Editor May require creating a value or an expression. GRID May require selecting a part of a graph or table. Hot Text May require rearranging equations. Multiple Choice May require selecting a value or an expression from a list. Multiselect May require selecting multiple values. Open Response May require explaining what happens to a function for large values of x or explaining a comparison. Clarifications Students will compare a linear function and an exponential function given in real-world context by interpreting the functions graphs. Students will compare a linear function and an exponential function given in a real-world context through tables. Students will compare a quadratic function and an exponential function given in real-world context by interpreting the functions graphs. Students will compare a quadratic function and an exponential function given in a real-world context through tables. Assessment Limits Exponential functions represented in graphs or tables should be able to be written in the form a(b)x + k. For exponential relationships, tables or graphs must contain at least one pair of consecutive values. Stimulus Attributes Items should give a graph or a table. MAFS.912.F-LE.1.3 Content Complexity:

14 Items should be given in a real-world context. Items may use function notation. Response Attributes Items may require the student to apply the basic modeling cycle. Items may require the student to choose an appropriate level of accuracy. Items may require the student to choose and interpret the scale in a graph. Items may require the student to choose and interpret units. Calculator No MAFS.912.A-REI.2.4 ITEM SPECIFICATIONS Item Type Editing Task Choice May require choosing steps in a derivation of the quadratic formula. Equation Editor May require creating a value or an expression. GRID May require dragging and dropping text to complete the derivation of the quadratic formula, or to drag and drop text to complete steps for solving a quadratic equation. Hot Text May require rearranging equations. Matching Item May require matching quadratic equations with the type of solution (complex or real). Multiple Choice May require selecting a value or an expression from a list. Multiselect May require selecting multiple values. Open Response May require writing an explanation of a step in a solution. Clarifications Students will rewrite a quadratic equation in vertex form by completing the square. Students will use the vertex form of a quadratic equation to complete steps in the derivation of the quadratic formula. MAFS.912.A-REI.2.4

15 Students will solve a simple quadratic equation by inspection or by taking square roots. Students will solve a quadratic equation by choosing an appropriate method (i.e., completing the square, the quadratic formula, or factoring). Students will validate why taking the square root of both sides when solving a quadratic equation will yield two solutions. Students will recognize that the quadratic formula can be used to find complex solutions. Assessment Limits In items that require the student to transform a quadratic equation to vertex form, the coefficient of the linear term must be an even factor of the coefficient of the quadratic term. In items that require the student to solve a simple quadratic equation by inspection or by taking square roots, equations should be in the form ax2 = c or ax2 + d = c, where a, c, and d are rational numbers and where c is not an integer that is a perfect square and c d is not an integer that is a perfect square. In items that allow the student to choose the method for solving a quadratic equation, equations should be in the form ax2 + bx + c = d, where a, b, c, and d are integers. Items may require the student to recognize that a solution is nonreal but should not require the student to find a nonreal solution. Stimulus Attributes The formula must be given in the item for items that can only be solved using the quadratic formula. Items should be set in a mathematical context. Items may use function notation. Response Attributes Items may require the student to complete a missing step in the derivation of the quadratic formula. Items may require the student to recognize equivalent solutions to the quadratic equation. Responses with square roots should require the student to rewrite the square root so that the radicand has no square factors. Calculator Neutral

16 MAFS.912.A.REI.4.10 Assessed within A-REI.4.11 STANDARD DECONSTRUCTION: Students should understand that equations with two variables can be represented graphically. The shape that results on the coordinate plane is a visual representation of all the solutions to that equation. What does that mean? It means we re not just pulling rabbits out of hats! The equations actually mean something visually. An equation with two variables can be anything from y = x to x2 + y2 = 4 to 19x13 = y. Some are simpler than others, of course, but they all have an x and a y. That means instead of having an equation with one variable (and therefore one solution), we can have many different solutions. Graphically, we can represent these solutions by drawing a curve or line through all the pairs of solutions (one for x and one for y) that work for that particular equation. MAFS.912.A.REI.4.10 Complexity: Level 1: Recall Let s take the equation 7x 18 = y and see how we can represent this graphically. How do we prove to a student that indeed, a line of a two-variable equation, when graphed, shows all of the solutions? Let s show them how to pull that rabbit out of the hat themselves. Since any two points define a line, all we need to do is input two values for x and see what the output y values are. We ll pick three points just to be sure our graph is a line and not some weird curve. Let s pick the numbers -1, 0, and 3 for x. Plugging in the numbers for x into our equation 7x 18 = y gives us -25, -18, and 3 for the y values. So the points in our graph become (-1, -25), (0, -18), and (3, 3). If we graph these on the x-y coordinate plane, we ll have this:

17 If your students don t believe you, prove to them that the equation and graph correspond to one another. Take a point on the line that is easily identifiable, say (2, -4), and plug the values into the equation. If we do that, we ll have -4 = 7(2) 18, which simplifies to -4 = -4. That way, students will be sure that points on the line or curve are valid solutions to the equation, and vice versa. But don t stop there. It s also important to prove the opposite. For example, the coordinate (4, 1), which is not on the line, is also not a solution to our equation. If we plug in the coordinates, we can confirm this: 1 = 7(4) 18 is false because This means (4, 1) isn t a solution to our equation and not a point on the line. Now you can pat yourself on the back and prove to students that teachers aren t just up to some magic tricks. Everything in math pretty much works as it s supposed to. This method can be applied to two-variable equations of higher orders. The generic shapes of these equations (such as quadratics making a parabola) should be known and associated with each other already. Otherwise, students will need to graph several points before verifying the graph that corresponds with the particular equation. MAFS.912.A-REI.4.11 Also Assesses A-REI.4.10 ITEM SPECIFICATIONS Item Types Equation Editor May require creating a value, an equation, or an expression. GRID May require identifying points where f(x) = g(x). Hot Text May require dragging labels to a graph or dragging and dropping numbers and symbols to complete a solution. Matching Item May require choosing ordered pairs that are solutions of a function. Multiple Choice May require selecting a value or an expression from a list. Multiselect May require selecting multiple values. Open Response May require creating a written response. Table Item May require completing missing cells in a table. Clarifications Students will find a solution or an approximate solution for f(x) = g(x) using a graph. Students will find a solution or an approximate solution for f(x) = g(x) using a table of values. Students will find a solution or an approximate solution for f(x) = g(x) using successive approximations that give the solution to a given place value. Students will justify why the intersection of two functions is a solution to f(x) = g(x). Students will verify if a set of ordered pairs is a solution of a function. Assessment Limits MAFS.912.A- REI.4.11

18 In items where a function is represented by an equation, the function may be an exponential function with no more than one translation, a linear function, or a quadratic function. In items where a function is represented by a graph or table, the function may be any continuous function. Stimulus Attributes Items may be set in a mathematical or real-world context. Items may use function notation. Items must designate the place value accuracy necessary for approximate solutions. Response Attributes Items may require the student to complete a missing step in an algebraic justification of the solution of f(x) = g(x). Items may require the student to explain the role of the x-coordinate and the y-coordinate in the intersection of f(x) = g(x). Items may require the student to explain a process. Items may require the student to record successive approximations used to find the solution of f(x) = g(x). Calculator Neutral MAFS.912.A-SSE.1.2 Assessed Within A-SSE.2.3 MAFS.912.A-SSE.1.2

19 MAFS.912.A-SSE.2.3 Also Assesses A-SSE.1.1, A-SSE.1.2 ITEM SPECIFICATIONS Item Type Editing Task Choice May require choosing equivalent forms of an expression or an interpretation of a parameter. Equation Editor May require creating an equivalent expression or numerical response. GRID May require dragging and dropping steps in completing the square of a quadratic expression, or in rewriting an expression using algebraic structure. Hot Text May require dragging terms, factors, coefficients, or expressions to complete an equivalent expression or to complete an interpretation. Matching Item May require matching equivalent expressions. Multiple Choice May require selecting an expression or a value from a set of options. Multiselect May require selecting expressions or values from a set of options. Open Response May require constructing a written response. Clarifications Students will use equivalent forms of a quadratic expression to interpret the expression s terms, factors, zeros, maximum, minimum, coefficients, or parts in terms of the real-world situation the expression represents. Students will use equivalent forms of an exponential expression to interpret the expression s terms, factors, coefficients, or parts in terms of the real-world situation the expression represents. Students will rewrite algebraic expressions in different equivalent forms by recognizing the expression s structure. Students will rewrite algebraic expressions in different equivalent forms using factoring techniques (e.g., common factors, grouping, the difference of two squares, the sum or difference of two cubes, or a combination of methods to factor completely) or simplifying expressions (e.g., combining like terms, using the distributive property, and other operations with polynomials). Assessment Limits In items that require the student to transform a quadratic equation to vertex form, the coefficient of the linear term must be an even factor of the coefficient of the quadratic term. For A-SSE.1.1, items should not ask the student to interpret zeros, the vertex, or axis of symmetry when the quadratic expression is in the form ax2 + bx + c (see F-IF.3.8). For A-SSE.2.3b and A-SSE.1.1, exponential expressions are limited to simple growth and decay. If the number e is used then its approximate value should be given in the stem. For A-SSE.2.3a and A-SSE.1.1, quadratic expressions should be univariate. For A-SSE.2.3b, items should only ask the student to interpret the y-value of the vertex within a real-world context. For A-SSE.2.3, items should require the student to choose how to rewrite the expression. In items that require the student to write equivalent expressions by factoring, the given expression may have integral common factors, be a difference of two squares up to a degree of 4, be a quadratic, ax2 + bx + c, where a > 0 and a, b, and c are integers, or be a polynomial of four terms with a leading coefficient of 1 and highest degree of 3. Stimulus Attributes Items assessing A-SSE.2.3 and A-SSE.1.1 must be set in a real-world context. Items that require an equivalent expression found by factoring may be in a real-world or mathematical context. Items should contain expressions only. MAFS.912.A-SSE.2.3

20 Response Attributes Items may require the student to choose an appropriate level of accuracy. Items may require the student to choose and interpret units. For A-SSE.1.1 and A-SSE.2.3, items may require the student to apply the basic modeling cycle. Calculator Neutral