Algebra 2, Quarter 3, Unit 3.1
Inequalities and Absolute Value Overview Number of instructional days:
5
(1 day = 45–60 minutes)
Content to be learned
Mathematical practices to be integrated
•
Given a quadratic word problem in one variable, create an inequality and solve it.
Make sense of problems and persevere in solving them.
•
State limitations/restrictions by writing equations or inequalities in the context of a quadratic problem.
•
Explain to themselves the meaning of a problem and look for entry points to its solution.
•
Determine whether solutions are viable or not in the context of the problem.
•
Analyze givens, constraints, relationships, and goals.
•
Find the approximate solutions of f(x) = g(x) using technology, including absolute value functions.
•
Consider analogous problems.
•
Explain correspondences between equations, verbal descriptions, tables, and graphs.
Model with mathematics. •
Apply known mathematics to solve problems arising in everyday life, society, and the workplace.
•
Interpret mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
•
Why are there limitations/restrictions on variables when solving real-life problems?
Essential questions •
What is the difference between the solution of an equation and the solution of an inequality?
•
What are different ways to use technology to find solutions to f(x) = g(x)?
Cumberland, Lincoln, and Woonsocket Public Schools, with process support from the Charles A. Dana Center at the University of Texas at Austin
37
Algebra 2, Quarter 3, Unit 3.1
Inequalities and Absolute Value (5 days)
Written Curriculum Common Core State Standards for Mathematical Content Creating Equations★
A-CED
Create equations that describe numbers or relationships [Equations using all available types of expressions, including simple root functions] A-CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential ★ functions. A-CED.3 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.
Reasoning with Equations and Inequalities
A-REI
Represent and solve equations and inequalities graphically [Combine polynomial, rational, radical, absolute value, and exponential functions] A-REI.11 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.
Common Core Standards for Mathematical Practice 1
Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
Cumberland, Lincoln, and Woonsocket Public Schools, with process support from the Charles A. Dana Center at the University of Texas at Austin
38
Algebra 2, Quarter 3, Unit 3.1
4
Inequalities and Absolute Value (5 days)
Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
Clarifying the Standards Prior Learning In unit 1.4, students created quadratic equations from word problems and solved them. Students also represented constraints by equations and inequalities. Also, in unit 1.4, students solved linear and polynomial systems using technology, by letting f(x) = g(x). Current Learning Students create and solve inequalities with quadratic functions. They represent constraints by systems of inequalities and find approximations using graphs, tables, and absolute value. Finding solutions to systems of quadratic and other nonlinear equations (such as absolute value functions) can be done graphically by making sure that they are in vertex form. Future Learning In unit 2.3, students will create radical equations from word problems and solve them. They will also represent constraints of radical equations. Finally, in unit 2.3, students will use technology and let f(x) = g(x).
Additional Findings Beyond Numeracy describes linear programming and how solving systems of linear inequalities relates to the real world (pp. 133–135). Science for All Americans describes how events in a system influence one another and gives applications of systems of real-world events (pp. 166–168). Benchmarks for Science Literacy describes how students learn about systems in different grade levels, developing higher-order thinking about systems in high school (p. 262).
Cumberland, Lincoln, and Woonsocket Public Schools, with process support from the Charles A. Dana Center at the University of Texas at Austin
39
Algebra 2, Quarter 3, Unit 3.1
Inequalities and Absolute Value (5 days)
Cumberland, Lincoln, and Woonsocket Public Schools, with process support from the Charles A. Dana Center at the University of Texas at Austin
40
Algebra 2, Quarter 3, Unit 3.2
Radical Functions Overview Number of instructional days:
15
(1 day = 45–60 minutes)
Content to be learned
Mathematical practices to be integrated
Remind students throughout that radical expressions can be written equivalently as rational expressions, ensuring that examples incorporate this concept.
•
•
•
Analyze, interpret, and model the characteristics and behavior of radical functions graphically, algebraically, numerically, and analytically. Include both square and cube root functions.
Make sense of problems and persevere in solving them.
Determine domain, range, intercepts, increasing and decreasing intervals, end behavior, and maximum and/or minimum values of radical functions. Solve equations involving radical expressions, including solving problems where an extraneous solution exists.
•
Analyze givens, constraints, and goals.
•
Consider analogous problems and try special cases in simpler forms of the original problem in order to gain insight into its solution.
•
Transform algebraic expressions.
•
Explain tables, graphs, and the important features and relationships.
Model with mathematics
•
Determine the inverse of a radical function.
•
•
Use technology to create a table of values and graph and approximate solutions to radical equations.
Use a function to describe how one quantity depends on another.
•
Make assumptions and approximations to simplify a complicated situation.
•
Map relationships using two-way tables and graphs.
Essential questions •
What are the characteristics of radical functions?
•
How can radical functions be described using multiple representations?
•
What are some real-world applications of radical functions?
•
How are a radical function and its inverse function related?
Cumberland, Lincoln, and Woonsocket Public Schools, with process support from the Charles A. Dana Center at the University of Texas at Austin
41
Algebra 2, Quarter 3, Unit 3.2
Radical Functions (15 days)
Written Curriculum Common Core State Standards for Mathematical Content Creating Equations★
A-CED
Create equations that describe numbers or relationships [Equations using all available types of expressions, including simple root functions] A-CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential ★ functions. A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph ★ equations on coordinate axes with labels and scales. A-CED.3 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. A-CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving ★ equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.
Reasoning with Equations and Inequalities
A-REI
Understand solving equations as a process of reasoning and explain the reasoning [Simple radical and rational] A-REI.2
Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
Interpreting Functions
F-IF
Interpret functions that arise in applications in terms of the context [Emphasize selection of appropriate models] F-IF.4
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.
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.
Cumberland, Lincoln, and Woonsocket Public Schools, with process support from the Charles A. Dana Center at the University of Texas at Austin
42
Algebra 2, Quarter 3, Unit 3.2
Radical Functions (15 days)
Reasoning with Equations and Inequalities
A-REI
Represent and solve equations and inequalities graphically [Combine polynomial, rational, radical, absolute value, and exponential functions] A-REI.11 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.
Interpreting Functions
F-IF
Analyze functions using different representations [Focus on using key features to guide selection of appropriate type of model function] F-IF.7
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.
Building Functions
F-BF
Build new functions from existing functions [Include simple radical, rational, and exponential functions; emphasize common effect of each transformation across function types] F-BF.4
Find inverse functions. a.
Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠ 1.
Common Core Standards for Mathematical Practice 1
Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. Cumberland, Lincoln, and Woonsocket Public Schools, with process support from the Charles A. Dana Center at the University of Texas at Austin
43
Algebra 2, Quarter 3, Unit 3.2
4
Radical Functions (15 days)
Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
Clarifying the Standards Prior Learning In units 1.4, 2.1, and 2.2, students created linear, polynomial, and absolute value equations and inequalities to solve problems. Students represented constraints of linear, polynomial, and absolute value functions, and interpreted results. Students have solved equations in one variable since grade 6. Current Learning Students analyze, interpret, and model the characteristics of radical functions graphically, algebraically, and analytically, and they solve radical equations. Characteristics of radical functions, including domain, range, intercepts, increasing and decreasing intervals, end behavior, and maximum values. Students continue to develop working with inverses. Future Learning In precalculus and calculus, students will use radical functions when studying composition of continuous functions and getting domain restrictions. In unit 4.2, algebra 2 students will analyze the characteristics of radical function graphs.
Additional Findings Principles and Standards for School Mathematics discusses how to represent and analyze mathematical situations and structures using algebraic symbols. According to this source, fluency with algebraic symbolism helps students represent and solve problems in the curriculum. Students need to be fluent in generating algebraic expressions, combining them, and re-expressing them in alternative forms (p. 300).
Cumberland, Lincoln, and Woonsocket Public Schools, with process support from the Charles A. Dana Center at the University of Texas at Austin
44
Algebra 2, Quarter 3, Unit 3.3
Exponential and Logarithmic Functions Overview Number of instructional days:
15
(1 day = 45–60 minutes)
Content to be learned • •
•
•
Mathematical practices to be integrated
Convert abct = d into logarithmic form. (Focus on bases 2, 10, and e.)
Model with mathematics.
⎛ d⎞
Convert log b ⎜ ⎟ = ct . ⎝ a⎠ (Focus on bases 2, 10, and e and situations where a and c are 1.) Evaluate logarithms using technology. (The LOG key, the ln key, and using the change of base formula for other bases.
•
Solve problems arising in everyday life.
•
Analyze relationships mathematically to draw conclusions.
•
Interpret mathematical results in the context of the situation and reflect on whether or not the results make sense.
Use appropriate tools strategically.
Solve logarithmic equations using the properties of logs
•
Analyze graphs of functions and solutions generated using a graphing calculator.
(e.g., 3log 2 x + log 2 7 = log 2 56 or 3log 2 x + log 2 7 = 100 ).
•
Solve real-life application problems (e.g., compound interest, growth and decay, and science applications such as earthquake problems.)
•
Graph simple logarithmic functions by hand showing intercepts and end behavior.
•
Use technology to graph more complicated logarithmic functions and find intercepts and end behavior.
Essential questions •
What are the advantages of understanding the properties of logarithms?
•
What are three real-life examples where logarithms or exponentials are used?
•
What is the relationship between logarithmic and exponential expressions?
•
When either graphing or solving, when is it necessary to use a calculator?
•
What is the relationship between common logarithms and natural logarithms?
•
Why do all logarithmic graphs have a vertical asymptote?
Cumberland, Lincoln, and Woonsocket Public Schools, with process support from the Charles A. Dana Center at the University of Texas at Austin
45
Algebra 2, Quarter 1, Unit 3.3
Exponential and Logarithmic Functions (15 days)
Written Curriculum Common Core State Standards for Mathematical Content Linear, Quadratic, and Exponential Models★
F-LE
Construct and compare linear, quadratic, and exponential models and solve problems [Logarithms as solutions for exponentials] F-LE.4
For exponential models, express as a logarithm the solution to abct = d where a, c, and d are ★ numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
Interpreting Functions
F-IF
Analyze functions using different representations [Focus on using key features to guide selection of appropriate type of model function] F-IF.7
Graph functions expressed symbolically and show key features of the graph, by hand in simple ★ cases and using technology for more complicated cases. e.
Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
Common Core Standards for Mathematical Practice 4
Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
Cumberland, Lincoln, and Woonsocket Public Schools, with process support from the Charles A. Dana Center at the University of Texas at Austin
46
Algebra 2, Quarter 1, Unit 3.3
5
Exponential and Logarithmic Functions (15 days)
Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
Clarifying the Standards Prior Learning In grade 8, students initially learned about functions, and in algebra 1, students were introduced to and graphed exponential functions. Current Learning In algebra 2, students convert between exponential and logarithmic forms and apply properties of logarithms. The main bases focused on are bases 2, 10, and e. Students solve problems and real-life application problems. Students graph logarithmic functions by hand and with technology. Future Learning In unit 3.2 of algebra 2, students will be introduced to graphing transformations of all functions including exponential and logarithmic. In the next course, students will be introduced to inverses.
Additional Findings Science for All Americans discusses models that can be used to relate to equations (pp. 168–172). Principles and Standards for School Mathematics, Algebra Standards, grades 9–12 (p. 395). Beyond Numeracy compares and contrasts linear and exponential growth and gives real-life examples. In addition, models are discussed that can be used to relate to equations (pp. 62–63; 71–72).
Cumberland, Lincoln, and Woonsocket Public Schools, with process support from the Charles A. Dana Center at the University of Texas at Austin
47
Algebra 2, Quarter 1, Unit 3.3
Exponential and Logarithmic Functions (15 days)
Cumberland, Lincoln, and Woonsocket Public Schools, with process support from the Charles A. Dana Center at the University of Texas at Austin
48
