Putnam County Schools Curriculum Map Algebra Module: 2 Instructional Window: Assessment Window: Lesson A: Introduction to Functions

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1 Putnam County Schools Curriculum Map Algebra Module: 2 Functions Instructional Window: September 29-November 2 Assessment Window: November 3 November 16 MAFS Standards Lesson A: Introduction to Functions MAFS.912.A-APR.1.1- Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. EngageNY, Module 4, Lessons 1, 2/Shmoop Unit 3 & 4. Math Nation Section 3, Topics 3,4,6. MAFS.912.F-BF.1.1 Assessed Within F-LE.1.2- Write a function that describes a relationship between two quantities. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. **Not covered in ENY. c. Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time. **Not covered in ENY. Shmoop Unit 5 & 7. Math Nation Section 3, Topic 7. 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 3, Topic 11. MAFS.912.A-CED.1.3MAFS.912.A-REI.1.1- Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Math Nation Section 3, Topic 5. MAFS.912.F-IF.1.1 Assessed Within F-IF.1.2- Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. 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. The graph of f is the graph of the equation y = f(x). EngageNY, Module 3, Lessons 1/Shmoop Unit 5 & 6. Math Nation Section 3, Topics 1,2,10. MAFS.912.F-IF.1.2 Also Assesses F-IF.1.1, F-IF.2.5- Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. EngageNY, Module 3, Lessons 1/Shmoop Unit 5. Math Nation Section 3, Topics 1,2,10. 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 3, Lessons 11-14/Shmoop Unit 7. Math Nation Section 3, Topics 8,9.

2 MAFS.912.F-IF.2.5 Assessed Within F-IF.1.1 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. EngageNY, Module 3, Lessons 8, 11/Shmoop Unit 7. Math Nation Section 3, Topic 2. 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. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. EngageNY, Module 4, Lessons 16, 17/Shmoop Unit 6 & 7. Math Nation Section 3, Topic 10. MAFS.912.A-REI.2.3 Assessed within A-CED.1.1- Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. EngageNY, Module 1, Lessons 11-15/ Shmoop Unit 4. Math Nation Section 3, Topic 5. MAFS.912.A-SSE.1.1 Assessed Within A-SSE.2.3- Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. EngageNY, Module 4, Lessons 1, 2, 6/Shmoop Unit 2 & 3. Math Nation Section 3, Topics 3,4. 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 11-14/Shmoop Unit 2. Math Nation Section 3, Topics 3,4.

3 Lesson B: Linear Functions MAFS.912.F-BF.1.1 Assessed Within F-LE.1.2- Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. **Not covered in ENY. c. Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time. **Not covered in ENY. Math Nation, Section 4, Topic 1 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 1, Lessons 25-28/ Shmoop Unit 4. Math Nation Section 4, Topics 3 and 4. MAFS.912.A-CED.1.3MAFS.912.A-REI.1.1- Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Math Nation Section 4, Topics 2-4. MAFS.912.S-ID.3.7 Assessed Within F-IF.2.6- Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. EngageNY, Module 2, Lessons 14, 15, 17, 19, 20/Shmoop Unit 10. Math Nation Section 4, Topics 2-4. 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 4, Topic 1. MAFS.912.F-IF.2.6 Also Assesses S-ID.3.7- Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. EngageNY, Module 3, Lessons 1, 4/Shmoop Unit 8. Math Nation Section 4, Topics 2-4. 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 4, Topic 1. MAFS.912.F-LE.2.5 Assessed Within F-LE.1.1 Interpret the parameters in a linear or exponential function in terms of a context. EngageNY, Module 3, Lessons 22-24/Shmoop Unit 7 & 8. Math Nation Section 4, Topics 2,3. 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 (which could be a line). EngageNY, Module 1, Lessons 20/ Shmoop Unit 6. Math Nation Section 4, Topics 2-4. Expectations to be Learned Unpacking What do these standards mean a child will know and be able to do? DOK Level

4 MAFS.912.A-APR.1.1: ITEM SPECIFICATIONS: Item Types Editing Task Choice May require completing an informal argument on closure. Equation Editor May require creating a value or an expression. GRID May require dragging and dropping expressions/statements to complete an informal argument. Hot Text May require dragging and dropping values/expressions to complete a polynomial. Matching Item May require matching equivalent polynomials. Multiple Choice May require selecting a value or an expression from a list. Multiselect May require selecting all equivalent expressions. Open Response May require creating a written explanation. MAFS.912.A- APR.1.1 Content Complexity: Level 1: Recall Clarifications Students will relate the addition, subtraction, and multiplication of integers to the addition, subtraction, and multiplication of polynomials with integral coefficients through application of the distributive property. Students will apply their understanding of closure to adding, subtracting, and multiplying polynomials with integral coefficients. Students will add, subtract, and multiply polynomials with integral coefficients. Assessment Limits Items set in a real-world context should not result in a nonreal answer if the polynomial is used to solve for the unknown. In items that require addition and subtraction, polynomials are limited to monomials, binomials, and trinomials. The simplified polynomial should contain no more than six terms. Items requiring multiplication of polynomials are limited to a product of: two monomials, a monomial and a binomial, a monomial and a trinomial, two binomials, and a binomial and a trinomial. Stimulus Attributes Items may be set in a mathematical or real-world context. Items may use function notation. Response Attributes Items may require the student to write the answer in standard form. Items may require the student to recognize equivalent expressions. Items may require the student to rewrite expressions with negative exponents, but items must not require the student to rewrite rational expression as seen in the standard MAFS.912.A-APR.4.7. Calculator No

5 MAFS.912.F-BF.1.1 Assessed Within F-LE.1.2 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. MAFS.912.F-BF.1.1 Content Complexity: Level 3: Strategic Thinking & Complex Reasoning MAFS.912.F-BF.2.3 Level 2: Basic Application of Skills & Concepts 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

6 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 MAFS.912.A-CED.1.2 Also assesses MAFS.912.A-REI.3.5 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. MAFS.912.A-REI.3.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. MAFS.912.A-REI.4.12 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. MAFS.912.A- CED.1.2 Content Complexity: Level 2: Basic Application of Skills & Concepts ITEMS SPECIFICATIONS: Item Types 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.

7 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. 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

8 STANDARD DECONSTRUCTION: This standard has two significant components. The first is translating word problems into equations with two or more variables. The more the merrier. Well, maybe not in this case. Translating word problems to create simple equations with two or more variables is not that different conceptually from creating equations with one variable. The main difference is that more complicated mathematical relationships such as systems of equations, functions, and proportions may develop (along with nausea, headaches, and spontaneous yodeling). In any case, this aspect of this standard should be taught with the previous one. The second component is creating graphs of equations on coordinate axes, which incorporates multiple skills such as visual perception, interpreting data, and synthesizing information. Such graphs relate to equations with multiple equations by relating one variable to another. Take lines, for example. In the form y = mx + b, we can look at either x or y and any defined value for x will give us a defined value for y, and vice versa. Graphs can help visualize these relationships between variables and facilitate the connection of equations to the graphs that represent them. Yearnin for more graphin? Don t worry. There ll be more down the line. MAFS.912.A-CED.1.3 ITEMS SPECIFICATIONS: Item Types Editing Task Choice May require choosing a definition for a variable or a correct interpretation of a solution. Equation Editor May require creating a set of equations, inequalities, or values. GRID May require graphing a representation. Hot Text May require selecting a representation or dragging and dropping text to interpret solutions. Multiple Choice May require identifying an equation, an inequality, or a value. Multiselect May require selecting constraints, variable definitions, or equations that would model a context. Open Response May require writing an explanation. MAFS.912.A- CED.1.3 Content Complexity: Level 3: Strategic Thinking & Complex Reasoning Clarifications Students will write constraints for a real-world context using equations, inequalities, a system of equations, or a system of

9 inequalities. Students will interpret the solution of a real-world context as viable or not viable. Assessment Limits In items that require the student to write an equation as a constraint, the equation may be a linear function. In items that require the student to write a system of equations to represent a constraint, the system is limited to a 2 x 2 with integral coefficients. In items that require the student to write a system of inequalities to represent a constraint, the system is limited to a 2 x 2 with integral coefficients. Stimulus Attributes Items must be set in a real-world context. Items may use function notation. 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. Items may require the student to apply the basic modeling cycle. Calculator Neutral

10 STANDARD DECONSTRUCTION: Students have already translated words into algebraic equations (sometimes with more than one variable!) and have actually taken the time to solve the problem. We hate to make you the villain here, but you have to tell them their work isn t over. Students now have to interpret the results. This standard is about one thing: analysis. Well actually it s about three things: 1. Creating equations/inequalities or systems of equations/inequalities. 2. Solving these equations/inequalities or system of equations. 3. Interpreting the answer properly. To analyze problems in which multiple relationships affect multiple variables, students must be able to create systems of equations, solve them, and interpret the results appropriately. Creating Systems of Equations In order to create systems of equations from word problems or other contexts, students need to be able to differentiate the relations and create equations for each. To support this, should already be able to create equations from word problems. Creating equations from a word problem or similar context is a three-step translation process: 1. Translate the equality or inequality. (=, <, >,, or ) 2. Translate the operations. (+,,,,x!,n! ) 3. Translate the numbers and variables. Systems of equations are identified during step one of this process. Students need to be able to read a problem and identify how many equality and inequality relations are described. Then, they should write each down separately. Once these are written down, students perform steps two and three (translating the operations, numbers, and variables) independently for each equation. They should also simplify each equation individually before working to solve the system of equations. Solving Systems of Equations Solving systems of equations can be done through substitution or adding the two equations together to cancel out one of the variables. The goal is to eliminate one variable so that we can find the solution for the other and then substitute that answer back in to find the value of the second variable. Hopefully, students already know how to solve systems of equations. After all, it s necessary in order to interpret results from a set of equations. Interpreting Results from Systems of Equations In many situations, students struggle to understand what an algebraic result means in the context of a word problem. This is especially true when systems of equations are involved and when they arrive at solutions that are correct algebraically, but incorrect in context. For instance, a question about how many tops hats a giraffe can wear might produce the number 6.25 as the answer. This might make sense algebraically, but in the context of giraffes wearing top hats how can a giraffe wear one fourth of a hat? The logical answer would then have to be 6 (although logic might not be our biggest concern if we re talking about giraffes in top hats). Such algebraic solutions also present a challenge when multiple roots are encountered or when cancelling rational expressions. Describing what values of a variable are allowed when recording the variable information is one strategy for dealing with this problem. (For instance, we could write that giraffes only wear top hats in whole numbers.)

11 A common error is to report an answer based on a different variable in the problem. Recording the variable information helps to prevent such errors. If this is a common issue, students should try highlighting the quantity of interest in the problem, the matching variable name, and the eventual result. They can then check all highlighted items to ensure that they match. Multiple variables present even more of a challenge, as students often need to find information for more than one variable in a problem. Highlighting the different information requested in different colors is one strategy. Another possibility is to have students solve for all variables before interpreting a solution; however, this becomes troublesome in more complex problems. Interpreting results should be performed throughout the algebra curriculum. With enough practice, students will perform whatever strategies work best for them naturally. MAFS.912.S-ID.3.7 Assessed Within F-IF.2.6 MAFS.912.F-IF.1.1 Assessed Within F-IF.1.2 MAFS.912.S-ID.3.7 Content Complexity: Level 2: Basic Application of Skills & Concepts MAFS.912.F-IF.1.1 Content Complexity: Level 1: Recall MAFS.912.F-IF.1.2 Also Assesses F-IF.1.1, F-IF.2.5 ITEM SPECIFICATIONS Item Types Equation Editor May require expressing a value, an inequality, an expression, or a function. GRID May require mapping a relation, or choosing ordered pairs. Hot Text May require dragging and dropping values or a set of values. Matching Item May require selecting cells in a table that associate a function to its domain, values for inputs, or to choose elements of the domain of a relation. Multiple Choice May require selecting a choice from a set of possible domains. Multiselect May require selecting functions from a set of relations. Open Response May require explaining the relationship of related values, or to interpret within a context. Table Item May require completing a table of values. MAFS.912.F-IF.1.2 Content Complexity: Level 2: Basic Application of Skills & Concepts Clarifications Students will evaluate functions that model a real-world context for inputs in the domain. Students will interpret the domain of a function within the real-world context given. Students will interpret statements that use function notation within the real-world context given. Students will use the definition of a function to determine if a relationship is a function, given tables, graphs, mapping diagrams, or sets of ordered pairs.

12 Students will determine the feasible domain of a function that models a real-world context. Assessment Limits Items that require the student to determine the domain using equations within a context are limited to exponential functions with one translation, linear functions, or quadratic functions. For F-IF.1.2, in items that require the student to find a value given a function, the following function types are allowed: quadratic, polynomials whose degrees are no higher than 6, square root, cube root, absolute value, exponential except for base e, and simple rational. Items may present relations in a variety of formats, including sets of ordered pairs, mapping diagrams, graphs, and input/output models. In items requiring the student to find the domain from graphs, relationships may be on a closed or open interval. In items requiring the student to find domain from graphs, relationships may be discontinuous. Items may not require the student to use or know interval notation. Stimulus Attributes For F-IF.1.1, items may be set in a real-world or mathematical context. For F-IF.1.2, items that require the student to evaluate may be written in a mathematical or real-world context. Items that require the student to interpret must be set in a real-world context. For F-IF.2.5, items must be set in a real-world context. Items must use function notation. Response Attributes For F-IF.2.5, 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. Items may require the student to write domains using inequalities. Calculator Neutral

13 MAFS.912.F-IF.1.3 Assessed Within F- LE.1.2 MAFS.912.F-IF.1.3 Content Complexity: Level 2: Basic Application of Skills & Concepts MAFS.912.F-IF.2.4 Also Assesses F-IF.3.9 ITEM SPECIFICATIONS Item Type Equation Editor May require expressing a value, expression, or equation. 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. MAFS.912.F-IF.2.4 Level 2: Basic Application of Skills & Concepts Clarifications Students will determine and relate the key features of a function within a real-world context by examining the function s table.

14 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

15 MAFS.912.F-IF.2.5 Assessed Within F-IF.1.1 MAFS.912.F-IF.2.6 Also Assesses S-ID.3.7 ITEM SPECIFICATIONS Item Type Equation Editor May require creating rate of change as a numeric value. Hot Text May require dragging and dropping phrases or values. Matching Item May require matching a value with an interpretation. Multiple Choice May require selecting a statement about the rate of a data display, an interpretation, or context. Multiselect May require selecting multiple statements about the rate of change and/or the constant term in a given context. Open Response May require explaining the rate of change or y-intercept in context. MAFS.912.F-IF.2.5 Level 2: Basic Application of Skills & Concepts MAFS.912.F-IF.2.6 Level 2: Basic Application of Skills & Concepts Clarifications Students will calculate the average rate of change of a continuous function that is represented algebraically, in a table of values, on a graph, or as a set of data. Students will interpret the average rate of change of a continuous function that is represented algebraically, in a table of values, on a graph, or as a set of data with a real-world context.

16 Students will interpret the y-intercept of a linear model that represents a set of data with a real-world context. Assessment Limits Items requiring the student to calculate the rate of change will give a specified interval that is both continuous and differentiable. Items should not require the student to find an equation of a line. Items assessing S-ID.3.7 should include data sets. Data sets must contain at least six data pairs. The linear function given in the item should be the regression equation. For items assessing S-ID.3.7, the rate of change and the y-intercept should have a value with at least a hundredths place value. Stimulus Attributes Items may require the student to apply the basic modeling cycle. Items should be set in a real-world context. Items may use function notation. Items may require the student to choose and interpret variables. 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. Calculator Neutral

17 MAFS.912.F-IF.3.7 Assessed Within F-IF.3.8 MAFS.912.F-IF.3.7 Level 2: Basic Application of Skills & Concepts MAFS.912.F-LE.1.2 Also Assesses F-BF.1.1, F-IF.1.3 ITEM SPECIFICATIONS Item Type 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. MAFS.912.F-LE.1.2 Content Complexity: Level 2: Basic Application of Skills & Concepts 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

18 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. 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.2.5 Assessed Within F-LE.1.1 MAFS.912.F-LE.2.5 Content Complexity: Level 2: Basic Application of Skills & Concepts

19 MAFS.912.A.REI.2.3 Assessed within A-CED.1.1 STANDARD DECONSTRUCTION: With linear equations, students should be able to find the solution or solutions that make the equation true. We usually get one or several specific answers with equations, but inequalities sing a slightly different tune. As different as Under Pressure and Ice Ice Baby. MAFS.912.A.REI.2. 3 Complexity: Level 2: Basic Application of Skills & Concepts Students should first understand the difference between an equation and inequality. An equation uses the = sign while an equality may use <, >,, or. If we find that x 2, we know that x can be 2 or anything less than 2. If we know that x < 2, x cannot be 2, but it can be anything less than 2. With inequalities, students should find the set of numbers that make the inequality true. Inequalities won t tell us exactly which number x will equal. Instead, it ll give us a range of possible x values, all of which will work for the inequality. Students should also know how to work with inequalities. Algebraically, they aren t that different from an equal sign. Still, multiplying and dividing by negative numbers switches the direction of the sign (1 > -2 but multiplying both sides by -1 gives us -1 < 2). If students are unsure, it might be helpful for them to visualize inequalities on a number line. Sometimes, letters may represent constants and coefficients in equations. Students should know how to treat these as numbers. For instance, the answer to the equation x + 4m = 2x + m would be written as x = 3m. It s okay for our solution to be in terms of m because m is treated as a constant. 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. MAFS.912.A.REI Complexity: Level 1: Recall 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. 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

20 x-y coordinate plane, we ll have this: 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-SSE.1.1 Assessed within A-SSE.2.3 STANDARD DECONSTRUCTION: Students should understand the vocabulary for the parts that make up the whole expression and be able to identify those parts and interpret their meaning in terms of a context. MAFS.912.A- SSE.1.1 Complexity: Level 2: Basic Application of Skills & Concepts The only way that could be more general is if it said, Do things to things in terms of other things. Luckily, we have just a smidgeon more to work with. At its core, this standard wants students to start thinking of math as a language, not a pile of numbers. Just like any other language, math can help us communicate thoughts and ideas with each other, but students need to know the basics before they can really understand it. Your students probably already have some idea of what an expression is in a general sense. Start from this point. At its simplest, an expression is a thought or idea communicated by language. In the same way, a mathematical expression can be considered a mathematical thought or idea communicated by the language of mathematics. Emphasize that mathematics is a language, just as English, French, German, and Pig Latin are languages. Students should use the ocabulary-vay of athematics-may correctly to become fluent in it. After all, the best way to learn a new language is to immerse yourself in it. (a.) Example. Just like English has nouns, verbs, and adjectives, mathematics has terms, factors, and coefficients. Well, sort of.

21 Students should know that terms are the pieces of the expression that are separated by plus or minus signs, except when those signs are within grouping symbols like parentheses, brackets, curly braces, or absolute value bars. Every mathematical expression has at least one term. For instance, the expression 3x + 2 has two terms: 3x and 2. A term that has no variables is often called a constant because it never changes. Within each term, there can be two or more factors, the numbers and/or variables multiplied together. The term 3x has two factors: 3 and x. There are always at least two factors, though one of them may be the number 1, which isn t usually written. But that 1 is always there...watching us. Finally, a coefficient is a factor (usually numeric) that is multiplying a variable. Using the example, the 3 in the first term is the coefficient of the variable x. The order or degree of a mathematical expression is the largest sum of the exponents of the variables when the expression is written as a sum of terms. For the example 3x + 2, the order is 1, since the variable x in the first term has an exponent of 1 and there are no other terms with variables. The expression 5x2 3x + 2 has order 2, whereas the expression 3xy + 5x2y3 7x + 32y4 has order 5, because the exponents of x and y in the second term are 2 and 3, respectively, and = 5. No other term has a higher exponent sum. Now that we have our words, we can start putting them together and make expressions. A good way to see if students really understand an expression like 3x + 2 is to have them translate mathematical expressions into English and vice versa. For instance, the expression 3x + 2 could also be written as, the sum of 3 times a number and 2, or, 2 more than three times a number. Clearly, it s much easier to write the mathematical expression than to write it in English (not to mention Pig Latin). The two are directly related to each other, however, and students should be able to translate back and forth. At first, students might want to make use of a dictionary like the table below to help them go from one language to the other.

22 b). Example. Let s consider a more complex expression: 5x (2 4y). In English, this could be stated as the difference between 5 times a number and the quantity 4 times another number less than 2. That s a mouthful, and this expression isn t even that complex! It should be obvious why we do math in symbol notation now: it s much easier to write. Notice how the English expression mentioned a number and another number. This is a clue that two different variables must be used in the mathematical expression. These two variables might represent two different physical quantities in some situation, and the expression shows how each quantity contributes to the overall behavior. How many terms are in that expression? Your students will probably say three, but there are only two the way the expression is written. The two terms inside the parentheses are treated as a single thing, so the first term is 5x and the second term is -(2 4y). Since there is no number immediately following the minus sign in the second term, we assume the number is actually 1. So the second term could be written as -1(2 4y). This should be interpreted as a -1 multiplying everything inside the parentheses. Of course, we can have expressions, which have even more variables, if there are more changing or unknown quantities involved. For example, the compound interest expression P(1 + r)n has three variables: P, r, and n, each representing a different physical quantity. As written, this expression has only one term, consisting of two factors, P and (1 + r)n. The first factor depends only on P, while the second depends on r and n. MAFS.912.A-SSE.1.2 Assessed Within A-SSE.2.3 MAFS.912.A- SSE.1.2 Level 2: Basic Application of Skills & Concepts