Algebra 1 - Unit 1 Functions - Jed's Portfolio



ALGEBRA 1 UNIT 2 MAP – Linear Functions & Solving Linear Equations – 11 Weeks | |

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|ESSENTIAL QUESTIONS |VOCABULARY |

|Interpret solutions to real-world problems through the use of multiple |Associative property |Absolute value |

|representations |Commutative property |Constant rate of change |

|How do you recognize linear change from a graph, function rule, table, or |Distributive property |Domain and Range |

|real-world situation? |Equivalent linear expressions |Independent variable |

|Given a set of bivariate data that appears to have a linear pattern in a |Systems of equations |Dependent variable |

|scatterplot, how do you find a line of best fit? What is the meaning of the |Absolute value |Explicit form |

|slope and y-intercept of that line in terms of the problem situation? |Domain and range |Recursive form |

|What are some methods to solve an absolute value equation? |Slope |Lines of best fit |

|What are some methods to solve linear inequalities? |x- and y-intercept | |

|What are the possible types of solutions for a system of two linear equations? | | |

|Give sample equations that demonstrate these types. | | |

|COMMON CORE MATHEMATICAL PRACTICES |

|Make sense of problems and persevere in solving them |Use appropriate tools strategically |

|Reason abstractly and quantitatively |Attend to precision |

|Construct viable arguments & critique the reasoning of others |Look for and make use of structure |

|Model with mathematics |Look for and express regularity in repeated reasoning |

|OVERVIEW OF UNIT |

|In looking at linear functions, patterns appear in tabular, graphic, verbal and symbolic representations.  Understanding of these patterns should be clear and connected.   In tables, looking at what happens to the |

|dependent variable as the independent variable changes, leads to a constant value.  When these table values are plotted, a straight line results.  This linear relationship can be represented recursively and |

|explicitly. Making connections between the tabular, graphic, verbal, and symbolic representations and linking these representations to real-world situations provides a conceptual framework for working further with |

|solving linear equations, inequalities, and systems. |

|Writing the equation of a line in standard form, slope-intercept form and point-slope form is part of the study of linear relationships.  Understanding the meaning of the rate of change can assist in writing equations|

|of parallel and perpendicular lines and connect the relationship of their slopes.  |

|In solving linear equations, absolute value equations, linear inequalities, and systems of equations, algebraic properties including identity and inverse elements, and the commutative, associative and distributive |

|properties will be applied.  Problem situations can be solved using tables, graphs, or symbolic manipulations. Students should justify their methods of solving and explain the meaning of the solution(s), connecting to|

|real-world situations when appropriate.  |

|COMMON CORE STATE STANDARDS |UNPACKED STANDARDS |

|Seeing Structure in Expression |Seeing Structure in Expression |

|Interpret the structure of expressions |Interpret the structure of expressions |

|A-SSE.1. Interpret expressions that represent a quantity in terms of its context. ★ |A-SSE.1. Interpret expressions that represent a quantity in terms of its context. |

|A-SSE.1a. Interpret parts of an expression, such as terms, factors, and coefficients. |A.SSE.1a. Identify the different parts of the expression and explain their meaning within the context of a |

|A-SSE.2. Use the structure of an expression to identify ways to rewrite It |problem |

| |A.SSE.2 Rewrite algebraic expressions in different equivalent forms such as factoring or combining like |

|Write expressions in equivalent forms to solve problems |terms |

|A-SSE.3. Choose and produce an equivalent form of an expression to reveal and explain properties of the | |

|quantity represented by the expression. ★ |Write expressions in equivalent forms to solve problems |

| |A.SSE.3 Use the properties of operations to create equivalent expressions. |

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|Creating Equations |Creating Equations |

|Create equations that describe numbers or relationships |Create equations that describe numbers or relationships |

|A-CED.1. Create equations and inequalities in one variable and use them to solve problems. Include equations|A.CED.1 Create linear, quadratic, rational and exponential equations and inequalities in one variable and |

|arising from linear and quadratic functions, and simple rational and exponential functions. |use them in a contextual situation to solve problems. |

|A-CED.2. Create equations in two or more variables to represent relationships between quantities; graph |A.CED.2 Create and graph equations in two or more variables to represent relationships between quantities. |

|equations on coordinate axes with labels and scales. |A.CED.3 Write and use a system of equations and/or inequalities to solve a real world problem. Recognize |

|A-CED.3. Represent constraints by equations or inequalities, and by systems of equations and/or |that the equations and inequalities represent the constraints of the problem. Use the Objective Equation and|

|inequalities, and interpret solutions as viable or nonviable options in a modeling context |the Corner Principle to determine the solution to the problem. (Linear Programming) |

|A-CED.4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving |A.CED.4 Solve multi-variable formulas or literal equations, for a specific variable. |

|equations. | |

|Reasoning with Equations and Inequalities |Reasoning with Equations and Inequalities |

|Understand solving equations as a process of reasoning and explain the reasoning |Understand solving equations as a process of reasoning and explain the reasoning |

|A-REI.1. Explain each step in solving a simple equation as following from the equality of numbers asserted |A.REI.1 Assuming an equation has a solution, construct a convincing argument that justifies each step in the|

|at the previous step, starting from the assumption that the original equation has a solution. Construct a |solution process. Justifications may include the associative, commutative, and division properties, |

|viable argument to justify a solution method. |combining like terms, multiplication by 1, etc. |

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|Solve equations and inequalities in one variable |Solve equations and inequalities in one variable |

|A-REI.3. Solve linear equations and inequalities in one variable, including equations |A.REI.3 Solve linear equations and inequalities in one variable, including coefficients represented by |

|with coefficients represented by letters. |letters. |

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|Solve systems of equations |Solve systems of equations |

|A-REI.5. Prove that, given a system of two equations in two variables, replacing one equation by the sum of |A.REI.5 Solve systems of equations using the elimination and substitution method |

|that equation and a multiple of the other produces a system with the same solutions. |A.REI.6 Solve systems of equations using graphs. |

|A-REI.6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs | |

|of linear equations in two variables. | |

|Represent and solve equations and inequalities graphically | |

|A-REI.10. Understand that the graph of an equation in two variables is the set of all its solutions plotted |Represent and solve equations and inequalities graphically |

|in the coordinate plane, often forming a curve (which could be a line). |A.REI.10 Understand that all solutions to an equation in two variables are contained on the graph of that |

|A-REI.11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = |equation. |

|g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using |A.REI.11 Explain why the intersection of y = f(x) and y = g(x) is the solution of f(x)=g(x). Find the |

|technology to graph the functions, make tables of values, or find successive approximations. Include cases |solution(s) by: |

|where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic |Using technology to graph the equations and determine their point of intersection, using tables of values, |

|functions. ★ |or using successive approximations that become closer and closer to the actual value |

|A-REI.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 |A.REI.12 Graph the solutions to a linear inequality, and a system of linear inequalities in two variables as|

|in two variables as the intersection of the corresponding half-planes. |a half-plane. |

|Interpreting Functions |Interpreting Functions |

|Understand the concept of a function and use function notation |Understand the concept of a function and use function notation |

|F-IF.1. Understand that a function from one set (called the domain) to another set (called the range) |F.IF.1 Use the definition of a function to determine whether a relationship is a function given a table, |

|assigns to each element of the domain exactly one element of the range. If f is a function and x is an |graph or words. Given the function f(x), identify x as an element of the domain, the input, and f(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|element in the range, the output. Know that the graph of the function, f, is the graph of the equation |

|graph of the equation y = f(x). |y=f(x). |

|F-IF.2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that|F.IF.2 When a relation is determined to be a function, use f(x) notation. Evaluate functions for inputs in |

|use function notation in terms of a context. |their domain. Interpret statements that use function notation in terms of the context in which they are |

| |used. |

|Interpret functions that arise in applications in terms of the context | |

|F-IF.4. For a function that models a relationship between two quantities, interpret key features of graphs |Interpret functions that arise in applications in terms of the context |

|and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of |F.IF.4 Given a function, identify key features in graphs and tables including: intercepts; intervals where |

|the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, |the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; |

|positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★ |end behavior; and periodicity. Given the key features of a function, sketch the graph. |

|F-IF.5. 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.★ |F.IF.5 Given the graph of a function, determine the practical domain of the function as it relates to the |

|F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a |numerical relationship it describes. |

|table) over a specified interval. Estimate the rate of change from a graph.★ | |

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| |F.IF.6 Calculate the average rate of change over a specified interval of a function presented symbolically |

|Analyze functions using different representations |or in a table. Estimate the average rate of change over a specified interval of a function from the |

|F-IF.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases |function’s graph. Interpret, in context, the average rate of change of a function over a specified interval.|

|and using technology for more complicated cases.★ | |

| |Analyze functions using different representations |

|F-IF.7a. Graph linear and quadratic functions and show intercepts, maxima, and minima. |F.IF.7 Graph functions expressed symbolically and show key features of the graph. Graph simple cases by |

|F-IF.9. Compare properties of two functions each represented in a different way (algebraically, graphically,|hand, and use technology to show more complicated cases including: |

|numerically in tables, or by verbal descriptions) |F-IF.7a. Linear and quadratic functions showing intercepts, quadratic functions showing intercepts, maxima, |

| |or minima. |

| |F.IF.9 Compare the key features of two functions represented in different ways. For example, compare the end|

| |behavior of two functions; one of which is represented graphically and the other is represented |

| |symbolically. |

|Building Functions |Building Functions |

|Build a function that models a relationship between two quantities |Build a function that models a relationship between two quantities |

|F-BF.1. Write a function that describes a relationship between two quantities.★ |F.BF.1 Write a function that describes a relationship between two quantities |

|F-BF.1a. Determine an explicit expression, a recursive process, or steps for calculation from a context. |F.BF.1a From context, write an explicit expression, define a recursive process, or describe the calculations|

| |needed to model a function between two quantities. |

|F-BF.1b. Combine standard function types using arithmetic operations |F.BF.1b. Combine standard function types, such as linear and exponential, using arithmetic operations. |

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|Linear, Quadratic, and Exponential Models |Linear, Quadratic, and Exponential Models |

|Construct and compare linear, quadratic, and exponential models and solve problems |Construct and compare linear, quadratic, and exponential models and solve problems |

|F-LE.1. Distinguish between situations that can be modeled with linear functions and with exponential |F.LE.1 Given a contextual situation, describe whether the situation in question has a linear pattern of |

|functions. |change or an exponential pattern of change. |

|F-LE.1a. Prove that linear functions grow by equal differences over equal intervals, and that exponential |F.LE.1a Show that linear functions change at the same rate over time and that exponential functions change |

|functions grow by equal factors over equal intervals. |by equal factors over time. |

|F-LE.1b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to| |

|another. |F.LE.1b Describe situations where one quantity changes at a constant rate per unit interval as compared to |

| |another |

|Interpret expressions for functions in terms of the situation they model | |

|F-LE.5 Interpret the parameters in a linear or exponential function in terms of a context. |Interpret expressions for functions in terms of the situation they model |

| |F.LE.5 Based on the context of a situation, explain the meaning of the coefficients, factors, exponents, |

|★-Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. |and/or intercepts in a linear or exponential function. |

|Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear | |

|throughout the high school standards indicated by a star symbol. | |

|SAMPLE TASKS |

|ESSENTIAL QUESTION 1 – Multiple representations |

|Filling a swimming pool |

|Investigating costs |

|Miles to your destination |

|ESSENTIAL QUESTION 2 – Recognizing linear change |

|Linear or not |

|Rates of Change |

|ESSENTIAL QUESTION 3 – Line of best fit |

|Bike weights |

|Height of corn |

|College affordability |

|ESSENTIAL QUESTION 4 – Absolute value equations |

|Medical equipment |

|Setting thermostat |

|ESSENTIAL QUESTION 5 – Inequalities |

|Envelope construction |

|ESSENTIAL QUESTION 6 – Systems of equations |

|Candles |

|MARS – this folder contains a PDF file which contains the entire lesson (directions, all activities, and sample student |

|work). A DOC file has been created that quickly gives you the outline of the lesson. A PPT presentation has been created that can be used to project sample |

|student work. |

|ASSESSMENT RESOURCES |

|Mathematics Assessment Project Tasks: |

|​A Golden Crown?:  Archimedes famously solved a problem for a King who thought his crown might be a fake.  In this task, you must work out whether the crown is |

|pure gold. |

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|Best Buy Tickets:  Susie is organizing the printing of tickets for a show. She has collected prices from several printers.  Your task is to use graphs and |

|algebra to advise Susie on how to choose the best printer. |

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|OTHER RESOURCES |

|NCTM Illuminations ([pic] ) |

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|Exploring Linear Data: Students model linear data in a variety of settings that range from car repair costs to sports to medicine. Students work to construct |

|scatterplots, interpret data points and trends, and investigate the notion of line of best fit. |

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|Growth Rate: Given growth charts for the heights of girls and boys, students will use slope to approximate rates of change in the height of boys and girls at |

|different ages. Students will use these approximations to plot graphs of the rate of change of height vs. age for boys and girls.  |

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|Supply and Demand: This activity focuses on having students create and solve a system of linear equations in a real-world setting. By solving the system, |

|students will find the equilibrium point for supply and demand. Students should be familiar with finding linear equations from two points or slope |

|and y-intercept. |

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|Line of Best Fit: This activity allows the user to enter a set of data, plot the data on a coordinate grid, and determine the equation for a line of best fit. |

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|Absolutely! (TI-84+):  Students solve linear absolute value equations in a single variable and use their graphing calculator to verify solutions. |

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| FCAT Real World Solving Linear Equations/Inequalities (TI-84+):  Activity explores the vocabulary and methods of solving systems of equations and includes |

|classroom Activity "Catch Me If You Can" with response sheet and assessment focused on real world problem solving of linear equations and systems (MA.D.2.4.2).|

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| Inequality Graphing App (TI-84+):  Students explore inequalities by entering inequalities using symbols, plot their graphs (including union and intersection |

|shades), store (x, y) coordinate pairs as lists, enter inequalities with vertical lines in an X= editor, and trace points of interest (such as intersections) |

|between functions. |

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|​An Introduction to Solving Literal Equations​(TI-Nspire):  Students will look at literal equations and investigate how to solve them for different variables. |

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| Slopes of Perpendicular Lines​(TI-Nspire):  Students will construct perpendicular lines and then examine slopes in a table and as a scatterplot to write a |

|function that describes the relationship. |

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| ​Solving Linear Inequalities by Graphing - Calculating Mileage​(TI-Nspire):  The mathematics goal of this activity is to deepen students' understanding of |

|solving linear inequalities by graphing. At the end of a 90 minutes class, the students should be able to:  Solve linear inequalities by using the TI-Nspire |

|calculator; Solve real world problems by applying the concept of solving linear inequalities; Complete the group activity collaboratively; and Make a |

|reflection about the concept learned for the day. |

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|Linear Functions and Inequalities: In the first of these two lessons, students explore how to write an equation of a linear function when given a set of data, |

|interpret the meaning of the slope and y-intercept, and then use the equation to find other values of x and y. The second lesson provides students with an |

|introduction to solving equations and inequalities numerically (using a table), graphically, and algebraically. |

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| The Yo-Yo Problem: The lesson starts with the presentation of the yo-yo problem. Students then complete a hands-on activity involving a design created with |

|pennies that allows them to explore a linear pattern and express that pattern in symbolic form. Algebra tiles are introduced as the students practice solving |

|linear equations. Working from the concrete to the abstract is especially important for students who have difficulty with mathematics, and algebra tiles help |

|students make this transition. In addition to using algebra tiles, students also use symbolic manipulation and the graphing calculator. Finally, the students |

|return to solve the yo-yo problem. A very special feature of this lesson is the effective use of peer tutors in this inclusion classroom.  |

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| The Algebra Lab: High School:  ​This resource provides a comprehensive program using manipulatives for teaching and learning algebra.  The entire book can be |

|downloaded as pdf files.  Some of the activities can be done without the manipulatives and the goal is to have students move from the use of manipulatives to |

|symbol manipulation alone. |

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|SUGGESTED SEQUENCE |

|Number of Days |Task |Corresponding Text Sections |

| |Rate of Change – EQ 2 |5.1, 5.3*, 2.5* |

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| | |*The task Rate of Change can be used to teach 5.3 and 2.5 |

| | |if you revisit and reengage the task. |

| |Miles to Your Destination – EQ 1 |5.4 |

| |TEXT - Standard Form |5.5 |

| |Bike Weights – Question 3 |5.7 |

| |Candles – EQ 6 |6.1 |

| |TEXT - Systems by Substitution, Elimination |6.2-6.3 |

| |Graphing Inequalities – EQ 5 |6.5 |

| |Systems of Inequalities |6.6 |

| |Medical Equipment – EQ 4 |3.7 |

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Linear Functions and Solving Linear Equations

Solving one-variable equations and inequalities can be implemented in the unit at your discretion when you find it to be appropriate.

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1CJOJQJU[pic]^JaJmHnHtH u[pic]8There is not a suggested task for standard form, solving systems by substitution and elimination, and solving inequalities algebraically.

Tasks may exist for these sections on the Cloud but they were not recommended by the Algebra 1 committee over the summer. This does not imply that they are not quality tasks.

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