| In what ways can relationships between variables be represented and studied?
| | In this unit, students build the foundational understandings that functions model relationships between quantities. In addition, students build an arsenal of tools to study functions throughout the course including the routine of examining functions with multiple representations. “Students should develop ways of thinking that are general and allow them to approach any type of function, work with it, and understand how it behaves, rather than see each function as a completely different animal in the bestiary” (Grade 8, High School Functions progression document, page 7). Specific features of linear, exponential, quadratic, and polynomial functions are each studied extensively in their own units in Algebra 1. Many standards that are introduced in Unit 1 will be further developed and/or utilized across these future units; look for these standards under the Unit Level Standards heading. Contexts are important for developing conceptual understanding. Drawing from contexts, students justify that one quantity depends on another and that each element of the domain corresponds to exactly one element of the range. They can qualitatively describe aspects of functions (e.g., increasing, decreasing) and reason about a function’s domain and range leading to conjectures about additional representations. In addition, students use contexts to make sense of function notation. For example, if the function h represents the height in centimeters of a bean sprout at specified days, t, then students should be able to talk about and/or identify each of the following: h(3), 5=h(t), h(6.5)=13.2, and h(t)=n. When functions are expressed symbolically, students input values into the equations to generate a table of output values and use these corresponding values to graph the function. After manually generating numeric and then graphic representations, students will use technologies such as a graphing calculator to generate both tables of values and graphs of functions. In doing so, they should pause to reflect upon the functional relationship between variables exhibited in these representations and why the representations make sense. By generating and comparing multiple cases, students recognize that functions can be organized by similar and different features like patterns of change, restrictions in the domain/range, and general shape. This organization of functions both serves as as an introduction to the families of functions that they will study throughout the course and equips students with strategies to study and represent functions. Students should not be expected to generate symbolic representations until later units. Likewise, symbolic manipulation of equations to reveal key features of functions (e.g., intercepts, maximum, horizontal asymptote) is included later in specific function units.
| Please Note: The standards listed in this section have been modified to be appropriate for this unit. Text in gray font is part of the CCSS-M standard but does not apply to this unit. Text in brackets denotes a modification that has been made to the standard. Mathematics: HS: Algebra,Creating Equations HSA-CED.A. Create equations that describe numbers or relationships. HSA-CED.A.2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Mathematics: HS: Functions,Interpreting Functions HSF-IF.B. Interpret functions that arise in applications in terms of the context. HSF-IF.B.6. Calculate and[Describe qualitatively and] interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate [Describe qualitatively] the rate of change from a graph. HSF-IF.C. Analyze functions using different representations. HSF-IF.C.9. Compare [informally] properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another[that can be represented numerically or graphically], say which has the larger maximum. Mathematics: HS: Functions, Building Functions HSF-BF.A. Build a function that models a relationship between two quantities. HSF-BF.A.1a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
| - What is a function and how can tables, graphs, algebraic rules, and verbal descriptions be used to study them?
- Given two variables, how do you decide which is the independent variable and which is the dependent variable?
- How do the shapes of graphs, the patterns in tables, the parts of algebraic rules, or verbal descriptions give clues about the ways the variables are related to one another?
- What are the advantages and disadvantages of using graphs, tables, verbal descriptions, and algebraic rules to demonstrate the relationship between two variables?
| function families of functions domain and range function notation patterns of change/recursion relationships between variables independent/dependent variable
| | Standards for Mathematical Practice Students will have opportunities to: - construct viable arguments and critique the reasoning of others when using contexts to make connections between multiple representations of functions and including function notation;
- make sense of problems and persevere in solving them developing routines for representing, examining, and interpreting functions with multiple representations; and
- model with mathematics by attending to contexts and making connections between different representations of functions.
| | |
| In what ways can equations and inequalities and systems of equations and inequalities model constraints and relationships between quantities? What processes and representations can be used to find and interpret solutions?
| | In eighth grade, students studied how linear functions represented by equations, tables, and graphs can be used to model and solve real world situations. Particular attention was given to using the rate of change and initial value from multiple representations to model the relationship between two values with the generalizable function, y=mx+b. In Algebra 1, students continue using and making connections between representations of linear functions. A common misconception when making connections between representations is that constant rate of change and slope can be used interchangeably when “a linear function does not have slope, but the graph of a non-vertical line has a slope.” (High School Functions progression document, page 6) Unit 2 establishes a deep understanding of the characteristics of linear functions. This understanding includes exploring linear functions geometrically by analyzing the effects of transformations on the graph by replacing f(x) by f(x) + k, kf(x), f(kx) and f(x + k) for specific values of k. These understandings allow students to compare linearity to function families studied in future units (e.g., exponential, quadratic). In this unit, students will extend their focus from slope intercept form to reasoning about standard form and point slope form. Students begin to make distinctions about which of these forms are most beneficial when modeling a real world situation. Different contexts lend themselves to different forms of linear equations. Students may build a function to model a situation, using parameters from that situation (e.g., rate of change, start value, ordered pair). Other situations are more efficiently modeled with standard form (e.g., Dana purchased 3 brauts and 4 drinks for $8.50). Symbolic manipulation from one form to another can reveal new characteristics of the function or assist in solving systems of equations. In 6^{th} and 7^{th} grade students solved one and two step equations and inequalities algebraically. In 8^{th} grade students solved linear equations using graphs, tables, and algebraic manipulation. In this unit, students will apply what they know about solving equations and inequalities to solving multi-step inequalities which include variables on both sides. Students will make sense of what a solution means for an equation compared to an inequality. Students will extend their understanding of solving linear equations with two variables. First, they will manipulate equations to solve for specific variables. Second, students will justify their reasoning by supplying mathematical properties to explain each step in solving an equation. This work will help set the groundwork for mathematical proofs in tenth grade. In addition, students will extend their 8^{th} grade understanding of solving systems of linear equations to include systems of linear inequalities as representations of real world situations. Students will solve systems of linear equations exactly (e.g. with substitution principle, combination/elimination), and approximately (e.g., with graphs) with a new emphasis on the conceptual understanding and justification of why these strategies work. Students will compare and contrast the benefit of using each of these strategies in different situations. In this unit students are using what they know about linear functions to build new understandings of piecewise linear functions including absolute value functions. In addition, students will use tables and graphs to solve absolute value equations as described in HSA.REI.D.11. The Michigan State Standards no longer require students to be able to algebraically solve piecewise and absolute value functions.
| There are standards listed in this section for two reasons. - The standards have been modified to be appropriate for this unit. Text in gray font is part of the Michigan K-12 standard but does not apply to this unit. Text in brackets denotes a modification that has been made to the standard.
- The standards contain content that is developed and/or utilized across multiple units.
Modified For this Unit Creating Equations HSA-CED.A. Create equations that describe numbers or relationships. HSA-CED.A.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. Reasoning with Equations & Inequalities HSA-REI.D. Represent and solve equations and inequalities graphically. HSA-REI.D.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 HSF-IF.B. Interpret functions that arise in applications in terms of the context. HSF-IF.B.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.★ HSF-IF.C. Analyze functions using different representations. HSF-IF.C.7a. Graph linear and quadraticfunctions and show intercepts, maxima, and minima. HSF.IF.C.7.b. Graph square root, cube root,and piecewise-defined functions, including step functions and absolute value functions. HSF-IF.C.9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Building Functions HSF-BF.B. Build new functions from existing functions. HSF-BF.B.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. Linear, Quadratic, and Exponential Models HSF-LE.A. Construct and compare linear and exponential models and solve problems. HSF-LE.A.1. Distinguish between situations that can be modeled with linear functions and with [nonlinear functions] exponential functions. HSF-LE.A.1a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. HSF-LE.A.2. 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). HSF-LE.B. Interpret expressions for functions in terms of the situation they model. HSF-LE.B.5. Interpret the parameters in a linear or exponential function in terms of a context. Developed and/or utilized across multiple units Quantities HSN-Q.A. Reason quantitatively and use units to solve problems. HSN-Q.A.2. Define appropriate quantities for the purpose of descriptive modeling. HSN-Q.A.3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Creating Equations HSA-CED.A. Create equations that describe numbers or relationships. HSA-CED.A.2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Reasoning with Equations and Inequalities HSA-REI.D. Represent and solve equations and inequalities graphically. HSA-REI.D.10. 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). Interpreting Functions HSF-IF.A. Understand the concept of a function and use function notation. HSF-IF.A.1. 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). HSF-IF.B. Interpret functions that arise in applications in terms of the context. HSF-IF.B.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.★ HSF-IF.B.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.★ HSF-IF.B.6. 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. HSF-IF.C. Analyze functions using different representations. HSF-IF.C.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Building Functions HSF-BF.A. Build a function that models a relationship between two quantities. HSF-BF.A.1. Write a function that describes a relationship between two quantities. HSF-BF.A.1a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
| - How do changes in values of the parameters in a linear function change the behavior of its graph and/or table?
- What are some possible strategies for solving systems of equations and inequalities? When might one strategy be more efficient than another?
- What does it mean to find solutions for equations, inequalities, and systems of equations and inequalities? How are the processes similar and different?
- How can understanding linear functions assist in investigating and representing piecewise linear functions?
| rate of change of a linear function (slope of the graph, common difference) linear patterns (recursive and explicit) models of linear functions (tables, graphs, and equations) intercepts parallel and perpendicular lines piecewise linear functions (absolute value, step) properties of operations (associative, commutative, distributive) properties of equality (addition/subtraction, multiplication/division) solutions (to equations, inequalities, systems of equations, and systems of inequalities) solution processes (algebraic, graphical, numeric) equivalent linear expressions transformations of a linear function
| | Standards for Mathematical Practice Students will have opportunities to: - construct viable arguments and critique the reasoning of others when explaining each step in solving a linear equation or inequality;
- reason abstractly and quantitatively when decontextualizing a situation in order to write a linear equation, inequality or system of linear equations or inequalities; and
- model with mathematics by creating and using multiple representations to organize, record, and communicate information about relationships between variables.
| | |
| How can exponential functions model real-world situations as seen in tables, graphs, and equations that represent these situations?
| |
In eighth grade, students used recursive thinking and various representations to analyze situations involving exponential patterns of change. This work was primarily done to distinguish nonlinear from linear functions and to give contexts for generating properties of integer exponents. Students used these properties to generate equivalent numerical expressions including those involving scientific notation. The generation of symbolic form, y=ab^x, was not necessarily required of students.
In Algebra 1, students use multiple representations to model exponential functions arising from real-world phenomena like population change, interest on investments, and radioactive decay. Similar to their work with linear functions, students use recursive thinking to generate y-values in a table for exponential functions. They notice that instead of a constant rate of change like linear functions, exponential functions have a constant growth/decay factor. Students describe recursive thinking with statements like, “to get the NEXT y-value, I multiply the y-value I have NOW by the growth factor of 1.4.” Drawing from recursive thinking, students build proficiency defining exponential functions explicitly with equations that describe the relationship between independent and dependent variables (e.g., f(x)=a*b^x, f(x)=a*b^(x-1), etc. where a and b are non-zero). Eventually, students build facility in writing explicit representations purely from contexts. In order to do so, students also distinguish between rates and factors and learn for example, that a growth rate of 2% equates to a growth factor of 1.02.
Students use and translate among representations to make sense of and problem solve within contexts. Consider the example, a dog receives a 400 mg dose of medicine. Students should see this initial dosage: in the first row of the table when the time column is 0 hours and the medicine column is 400 mg, in the graph at the point (0,400), and in the equation f(x)=400*b^x. In both the table of values and in the graph, students also see the amount of medicine approaching zero making a contextual connection to the concept of horizontal asymptotes. When students are also given the decay rate or factor, they can find the amount of medicine in the dog’s body for any given time. Graphical and tabular representations can be used to estimate the time, given an amount of medicine. (Logarithms are not used to solve exponential equations until Algebra 2.) Graphical and tabular representations can also be used to explore changing rates of change. Exponential functions provide the first useful context to find the average rate of change over a unit interval and compare rates for successive intervals.
Exponent rules facilitate the development of explicit representations from recursive thinking. In addition, they allow students to make efficient comparisons between populations and manipulate algebraic representations to reveal new properties of the relationship being described. (SMP 2 and SMP 7 and MI: Math HSA-SSE.B.3c ) http://www.corestandards.org/Math/Practice/ In this work, students must move beyond the Grade 8 rules, generating equivalent numeric expressions with integer exponents, and extend their understanding to applications of rational exponents including rewriting expressions involving radicals and rational exponents using the properties of exponents.
| There are standards listed in this section for two reasons. - The standards have been modified to be appropriate for this unit. Text in gray font is part of the Michigan K-12 standard but does not apply to this unit. Text in brackets denotes a modification that has been made to the standard.
- The standards contain content that is developed and/or utilized across multiple units.
Modified For this Unit Creating Equations HSA-CED.A. Create equations that describe numbers or relationships. HSA-CED.A.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. Reasoning with Equations and Inequalities HSA-REI.D. Represent and solve equations and inequalities graphically. HSA-REI.D.10. 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). HSA-REI.D.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 HSF-IF.C. Analyze functions using different representations. HSF-IF.C.7e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Building Functions HSF-BF.B. Build new functions from existing functions. HSF-BF.B.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. Linear, Quadratic, and Exponential Models HSF-LE.A. Construct and compare linear and exponential models and solve problems. HSF-LE.A.1a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. HSF-LE.A.2. 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). HSF-LE.A.3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. HSF-LE.B. Interpret expressions for functions in terms of the situation they model. HSF-LE.B.5. Interpret the parameters in a linear or exponential function in terms of a context. Developed and/or Utilized Across Multiple Units Quantities HSN-Q.A. Reason quantitatively and use units to solve problems. HSN-Q.A.2. Define appropriate quantities for the purpose of descriptive modeling. HSN-Q.A.3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Seeing Structure in Expressions HSA-SSE.A. Interpret the structure of expressions. HSA-SSE.A.1. Interpret expressions that represent a quantity in terms of its context. HSA-SSE.A.1a. Interpret parts of an expression, such as terms, factors, and coefficients. HSA-SSE.A.1b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)ⁿ as the product of P and a factor not depending on P. HSA-SSE.A.2. Use the structure of an expression to identify ways to rewrite it. For example, see x⁴ – y⁴ as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Creating Equations HSA-CED.A. Create equations that describe numbers or relationships. HSA-CED.A.2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Interpreting Functions HSF-IF.A. Understand the concept of a function and use function notation. HSF-IF.A.1. 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). HSF-IF.A.2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. HSF-IF.B. Interpret functions that arise in applications in terms of the context. HSF-IF.B.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.★ HSF-IF.B.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.★ HSF-IF.B.6. 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. HSF-IF.C. Analyze functions using different representations. HSF-IF.C.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. HSF-IF.C.9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Building Functions HSF-BF.A. Build a function that models a relationship between two quantities. HSF-BF.A.1. Write a function that describes a relationship between two quantities. HSF-BF.A.1a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
| - What are some similarities and differences between exponential and linear functions?
- What similarities do all tables of exponential functions share?
- How do changes in values of the parameters in an exponential function change the behavior of the graph and/or table?
- How do you recognize exponential growth or decay from a graph, equation, table, or real-world situation?
- How can understanding rules of exponents assist in representing and interpreting exponential functions from specific contexts?
| asymptotic behavior exponential growth and decay exponential patterns (recursive and explicit) growth/decay factor models of exponential functions (tables, graphs, equations) properties of exponents
| | Standards for Mathematical Practice Students will have opportunities to: - look for and express regularity in repeated reasoning as seen in tables to express mathematical ideas regarding exponential patterns of change precisely;
- attend to precision when giving answers that represent very large or very small values; and
- use appropriate tools strategically when selecting, applying, and translating among mathematical representations to solve exponential equations.
| | |
| How can quadratic functions model real-world situations as seen in tables, graphs, and equations that represent these situations?
| | In eighth grade students began exploring different function families by studying linear functions and how they compare to nonlinear functions. In Algebra 1, they continue this exploration by developing a deeper understanding of linear functions and formalizing their understanding of exponential functions; they studied the key characteristics found in tables, graphs, and equations for linear and exponential functions. In this unit, students will use knowledge of these characteristics to build a new understanding of quadratic functions (e.g., end behavior, symmetry, constant second difference, maximum) which will include making comparisons among the three function families. Both exponential functions and quadratic functions have changing rates of change as seen in tables and graphs. As such, quadratics provide another useful context to find the average rate of change over a unit interval and compare rates for successive intervals. In this unit students will study quadratic functions represented algebraically in several forms: standard/polynomial form, f(x) = ax^{2} + bx + c; factored form, f(x) = a (x - p)(x - q); and vertex form, f(x) = a(x - h) ^{2} + k. They will use applets and/or graphing utilities to analyze the graphs and tables for each of these forms and identify what each form reveals about the function. Once, they understand the usefulness of different forms, students will employ algebraic skills to generate equivalent forms. Students will also explore how changing the parameters in algebraic forms affect the graphs and/or tables of these functions. In this unit, students use multiple representations to build a conceptual understanding of quadratic functions. In addition, they create tables and graphs to find solutions to quadratic equations which lays the foundation for the more abstract work of using algebraic strategies to solve quadratic equations in unit 5. Some of the more sophisticated work in unit 5 includes finding roots that may be imaginary/complex. (A more formal study of operations with imaginary/complex numbers will be done in Algebra 2.) Understanding the connections between x-intercepts and the roots of quadratic equations along with transformations of the graphs of quadratics helps students conceptualize the need for imaginary numbers.
| There are standards listed in this section for two reasons. - The standards have been modified to be appropriate for this unit. Text in gray font is part of the Michigan K-12 standard but does not apply to this unit. Text in brackets denotes a modification that has been made to the standard.
- The standards contain content that is developed and/or utilized across multiple units.
Modified For this Unit The Complex Number System HSN-CN.A. Perform arithmetic operations with complex numbers. HSN-CN.A.1. Know there is a complex number i such that i² = –1, and every complex number has the form a + bi with a and b real. Creating Equations HSA-CED.A. Create equations that describe numbers or relationships. HSA-CED.A.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. HSF-IF.C. Analyze functions using different representations. HSF-IF.C.7a. Graph linear and quadratic functions and show intercepts, maxima, and minima. Building Functions HSF-BF.B. Build new functions from existing functions. HSF-BF.B.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. Developed and/or Utilized Across Multiple Units Quantities HSN-Q.A. Reason quantitatively and use units to solve problems. HSN-Q.A.2. Define appropriate quantities for the purpose of descriptive modeling. HSN-Q.A.3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Seeing Structure in Expressions HSA-SSE.A. Interpret the structure of expressions. HSA-SSE.A.1. Interpret expressions that represent a quantity in terms of its context. HSA-SSE.A.1a. Interpret parts of an expression, such as terms, factors, and coefficients. HSA-SSE.A.1b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)ⁿ as the product of P and a factor not depending on P. HSA-SSE.A.2. Use the structure of an expression to identify ways to rewrite it. For example, see x⁴ – y⁴ as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Creating Equations HSA-CED.A. Create equations that describe numbers or relationships. HSA-CED.A.2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Reasoning with Equations and Inequalities HSA-REI.D. Represent and solve equations and inequalities graphically. HSA-REI.D.10. 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). Interpreting Functions HSF-IF.A. Understand the concept of a function and use function notation. HSF-IF.A.1. 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). HSF-IF.A.2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. HSF-IF.B. Interpret functions that arise in applications in terms of the context. HSF-IF.B.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.★ HSF-IF.B.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.★ HSF-IF.B.6. 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. HSF-IF.C. Analyze functions using different representations. HSF-IF.C.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. HSF-IF.C.9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Building Functions HSF-BF.A. Build a function that models a relationship between two quantities. HSF-BF.A.1. Write a function that describes a relationship between two quantities. HSF-BF.A.1a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
| - What are the characteristics of quadratic functions when represented in a graph, equation, table, and/or real-world situation?
- What are some similarities and differences between quadratic, exponential, and linear functions?
- How do changes in the values of the parameters in a quadratic function change the behavior of the graph and/or table?
- What do different algebraic forms of quadratic equations reveal about graphical features of quadratic functions?
| forms of quadratic functions (factored, standard/polynomial, vertex) key features of quadratic graphs (vertex, axis of symmetry, minimum, maximum, x-intercept, y-intercept, end behavior) models of quadratic functions (tables, graphs, equations,algebra tiles) quadratic patterns (recursive, common second difference, explicit) solutions to quadratic equations (zeros, real and imaginary roots, x-intercepts) transformations of quadratic functions
| | Standards for Mathematical Practice Students will have opportunities to: - look for and make use of structure in representations of quadratic functions (e.g.,reflecting a given point over the line of symmetry to find another point);
- make sense of problems and persevere in solving them by transforming algebraic forms of quadratic functions to reveal features of the function; and
- use appropriate tools strategically to examine how changes in the values of the parameters in a quadratic function change the behavior of its graph and/or table.
| | |
| What processes and representations can be used to find and interpret solutions of quadratic equations?
| |
In the last unit, students used multiple representations to make sense of quadratic functions and solve problems with tables and graphs. In this unit, students solve quadratic functions using what they learned from the last unit and begin to use algebraic strategies to solve quadratic equations. Students should identify which of these strategies is most efficient for any given situation. Additionally, when using algebraic strategies, students will produce real and imaginary/complex solutions. However, a more formal study of operations with imaginary/complex numbers will be done in Algebra 2.
As students use algebraic strategies to find real or imaginary/complex solutions to quadratic equations, they can use what they know about transforming quadratic functions to see if their solutions makes sense. Similarly, students can use what they know about key features of quadratics (e.g., symmetry) to analyze or justify their answers.
The use of algebra tiles or area models provide opportunities for students to generate and make sense of algebraic strategies like completing the square and factoring. For example, as students repeatedly convert quadratics in standard form (represented with tiles) to vertex form, they see a pattern from the concrete model that leads to the procedure for completing the square. Once students understand the concept of completing the square, they can use it to make sense of the quadratic formula. (See an example of this process by using the following link.
https://www.youtube.com/watch?v=tHhO1_Snpsw)
In unit 2, students solved systems of linear equations both approximately, with tables and graphs, and exactly using algebraic strategies. In unit 3, students found approximate solutions for systems of equations consisting of both linear and exponential equations. In this unit, students extend those experiences to again find both approximate and exact solutions for systems that include linear, exponential and quadratic equations.
| There are standards listed in this section for two reasons. - The standards have been modified to be appropriate for this unit. Text in gray font is part of the Michigan K-12 standard but does not apply to this unit. Text in brackets denotes a modification that has been made to the standard.
- The standards contain content that is developed and/or utilized across multiple units.
Modified For this Unit The Complex Number System HSN-CN.A. Perform arithmetic operations with complex numbers. HSN-CN.A.1. Know there is a complex number i such that i² = –1, and every complex number has the form a + bi with a and b real. Creating Equations HSA-CED.A. Create equations that describe numbers or relationships. HSA-CED.A.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. Reasoning with Equations and Inequalities HSA-REI.D. Represent and solve equations and inequalities graphically. HSA-REI.D.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 [quadratic] logarithmic functions. Developed and/or Utilized Across Multiple Units Seeing Structure in Expressions HSA-SSE.A. Interpret the structure of expressions. HSA-SSE.A.2. Use the structure of an expression to identify ways to rewrite it. For example, see x⁴ – y⁴ as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Reasoning with Equations & Inequalities HSA-REI.D. Represent and solve equations and inequalities graphically. HSA-REI.D.10. 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). Building Functions HSF-BF.A. Build a function that models a relationship between two quantities. HSF-BF.A.1. Write a function that describes a relationship between two quantities. HSF-BF.A.1a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
| - What is the relationship between the number of real roots and the graph of a quadratic equation? Why does this relationship exist?
- How does paying attention to the parameters of a quadratic function and its graph assist in finding solutions?
- What are strategies for solving quadratic equations? When might one strategy be more efficient than another?
- How is solving a system consisting of a linear or a non linear equation and a quadratic equation similar/different to solving linear systems and/or quadratic equations?
| algebraic strategies for solving (completing the square, factoring, quadratic formula) discriminant forms of quadratic functions (factored, standard, vertex) imaginary/complex numbers key features of quadratics graphs (vertex, axis of symmetry, minimum, maximum, x and y intercepts, end behavior) models of quadratic functions (tables, graphs, equations, algebra tiles) quadratic patterns (recursive, common second difference, explicit) solutions (real or imaginary/complex roots, zeroes, x-intercepts) solutions to non-linear systems
| | Standards for Mathematical Practice Students will have opportunities to: - look for and make use of structure through the parameters and the features of the graph to find solutions to quadratic equations;
- look for and express regularity in repeated reasoning when using the quadratic formula to identify real or imaginary/complex roots of a quadratic equation; and
- model with mathematics by using algebra tiles to make sense of the procedures like completing the square and factoring.
| | |
| In analyzing bivariate sets of data, how do scatter plots and two-way tables help make sense of the data and assist in making predictions?
| | In this unit students will create and analyze data displays for both categorical and quantitative data. In eighth grade students constructed and interpreted two-way tables summarizing data on two categorical variables collected from the same subjects. They used relative frequencies calculated for rows or columns to describe possible association between the two variables. In this unit, students will deepen their knowledge of relative frequencies to find and use conditional, marginal and joint relative frequencies. In Algebra 2, students will use their understanding of two-way tables to analyze a sample space, decide if events are independent, and to approximate conditional probabilities. In sixth, seventh and eighth grade students represented and analyzed both univariate and bivariate quantitative data. In eighth grade students constructed and interpreted scatter plots focusing on estimating lines of best fit and informally analyzing the closeness of the fit. They also described patterns including clustering, outliers, positive or negative association and linear or nonlinear association. In this unit students formalize their analysis of the line of best fit by using residuals to analyze the variance in a bivariate data set. This is an extension of 6th and 7th grade standards where students find the average distance from the mean to analyze the variance in an univariate data set. In addition, students will analyze the relation between the two variables in a linear model by using technology to compute and interpret the correlation coefficient. However, a cause and effect relationship is not necessarily related to the strength of the correlation, and students should recognize instances where causation is unrelated to this strength. Students will also use what they know about exponential and quadratic functions from previous units to fit a function to the data to solve problems in the context of the data. Solving problem might include describing patterns, like they did in eighth grade, or making predictions inside and outside of the data set.
| There are standards listed in this section for two reasons. - The standards have been modified to be appropriate for this unit. Text in gray font is part of the Michigan K-12 standard but does not apply to this unit. Text in brackets denotes a modification that has been made to the standard.
- The standards contain content that is developed and/or utilized across multiple units.
Modified For this Unit n/a Developed and/or Utilized Across Multiple Units Quantities HSN-Q.A. Reason quantitatively and use units to solve problems. HSN-Q.A.2. Define appropriate quantities for the purpose of descriptive modeling. HSN-Q.A.3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
| - When looking at both categorical and quantitative data how can the strength of the association between two variables be analyzed?
- What impact can an outlier have on the correlation coefficient for a scatter plot?
- What is the difference between correlation and causation?
- How are the parameters from different functions modeling data interpreted in real- world contexts?
- How are joint, marginal, and conditional relative frequencies similar? Different?
| causation correlation (positive, negative,strong, weak, moderate, linear, nonlinear) correlation coefficient line of best fit outlier relative frequencies (joint, marginal, conditional) residual scatter plot two-way table
| | Standards for Mathematical Practice Students will have opportunities to: model with mathematics to fit a function to the data and solve problems in the context of the data; use appropriate tools strategically to create scatter plots, measuring and analyzing the strength of the correlation coefficient; and - reason abstractly and quantitatively to create a coherent representation of the data presented in problems.
| | |