| How can polynomial functions model real-world or mathematical situations as seen in tables, graphs, and equations that represent these situations?
| | In Algebra 1, students worked to make connections between representations of functions. They observed that different forms of symbolic representations of linear and quadratic functions revealed different features of the functions and vice versa. Factored form, for example, revealed rational zeroes and x-intercepts while vertex form more readily connected to translations of the curves. Students also examined recursive patterns in linear, quadratic, and exponential functions. In this unit they will make connections to these prior experiences as they extend similar reasoning and explorations to power and polynomial functions. Many standards that were introduced in Algebra 1 will be further developed and/or utilized across Algebra 2 units; look for these standards under the Unit Level Standards heading. Power functions have the general form f(x)= ax^{n}. The study of power functions includes exploring the impact of changing the values of the coefficient a and the exponent n. (In this unit, n is kept to positive whole numbers. In unit 4, the study of these models will include values of n that are negative and fractional, as well as whole numbers.) Numeric patterns in tables will be contrasted to linear and quadratic patterns of change and connected to features in the graphs. As n increases students will also note similarities and differences in shapes of the graphs and examine rotational or reflectional symmetry. Attending to the structures in both the graphs and equations will lead to generalizations about power functions with even exponents compared to power functions with odd exponents. End behavior is introduced as a way to compare and contrast the ranges. Students will use multiple representations to first explore and then generalize that the ranges for odd degree power functions are all real numbers and the ranges for even degree power functions is restricted. Polynomial functions are the sum of power functions and can be represented symbolically by: f(x) = a_{n}x^{n}+ a_{n-1}x^{n-1}+…+ a_{1}x^{1}+a_{0}x^{0}, where n is a whole number, and the coefficients, an,are real numbers. Looking for patterns in tables, graphs, and equations will help develop an understanding of characteristics of odd and even degree polynomial functions as an extension to power functions. These patterns show the relationship of the degree of the polynomial function to the possible number of zeroes and the general shape of the graphs. The key features of the graph and the relationship to the equation and table provide the basis for developing understanding of this broader class of polynomial functions. Students will also perform arithmetic operations on polynomials. They will rewrite polynomials in factored form to reveal the zeroes using clues from the tables and graphs and structures in equations of the functions. They extend their understanding of the relationship between zeroes and factors from quadratics to polynomial functions. They see that for a polynomial p(x), and a number a, p(a) = 0 if and only if (x-a) is a factor of p(x). (Division by (x-a) is postponed for unit 4 where the Remainder Theorem will be used.) This unit could also include an introduction to the Binomial Theorem and Pascal's Triangle to demonstrate expansion of (x+y)^{n}. In Algebra 1, students solved systems of linear equations both approximately, with tables and graphs, and exactly using algebraic strategies.They found approximate solutions for systems of equations consisting of both linear and exponential equations. In addition they found both approximate and exact solutions for systems that include linear, exponential and quadratic equations. In this unit, students will extend these concepts to include systems containing polynomial 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 NA for this unit Developed and/or Utilized Across Multiple Units Seeing Structures 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). 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). 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.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.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.
| - How do changes in the values of the parameters in a polynomial/power function change the behavior of the graph and/or table?
- How can the patterns in tables, graphs, and equations help determine the characteristics (e.g. shape, number of zeros, end behavior, odd/even) of polynomial functions?
- What are some similarities and differences between polynomial, quadratic, exponential, and linear functions?
- How are operations with polynomial functions similar to or different from operations with integers?
- How is solving a system consisting of a linear or a non linear equation and a polynomial equation similar/different to solving linear systems and/or quadratic equations?
| factors key features of power and polynomial graphs (symmetry, x- and y-intercepts, end behavior) models of polynomial functions (tables, graphs, equations) operations with polynomial functions polynomial function (even, odd, power) polynomial patterns (recursive, common successive difference, explicit) solutions (real or imaginary/complex roots, zeros, x-intercepts) solutions to non-linear systems
| | Standards for Mathematical Practice Students will have opportunities to: make sense of problems and persevere in solving them when exploring graph for polynomial functions of various degrees; construct arguments and critique the reasoning of others when discussing key features of odd and even polynomial functions; and - model with mathematics to represent contexts that are not well represented by linear, exponential or quadratic functions.
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| How can visual displays and statistical calculations be used to organize, analyze, and evaluate data sets?
| | In this unit, students will build on the knowledge and experience developed in the 6^{th} grade and 7^{th} grade univariate statistics units. (Bivariate data was the statistical focus in 8^{th} grade and Algebra 1.) “They will develop a more formal and precise understanding of statistical inference, which requires a deeper understanding of probability. Students learn that formal inference procedures are designed for studies in which the sampling or assignment of treatments was random, and these procedures may not be informative when analyzing non-randomized studies, often called observational studies.” (Progressions for the CCSS-M High School Statistics and Probability, p. 2)“Students now move beyond analyzing data to making sound statistical decisions based on probability models.” (Progressions for the CCSS-M High School Statistics and Probability, p.8). In addition, students will deepen their understanding of summarizing, representing and interpreting data on a single measurement variable by applying more complex measurement tools. As in past data units, students: (1) formulate questions, (2) design a plan to collect data, (3) analyze data, and (4) interpret results/draw conclusions. This unit will build upon students’ understanding of statistical variability (quartiles, range, mean absolute deviation*(MAD), outliers) and their capacity to summarize and describe distributions (measure of center, shape) that were developed in middle school. They will now learn to give more precise answers to questions like deciding on which is the more appropriate measure of center, mean or median, and show their deeper understanding by justifying this choice using statistical reasoning. Two major shifts in focus and complexity from middle school to Algebra 2 are the ability to distinguish among distributions that were skewed or approximately symmetric to whether or not they are approximately normal as well as building on their understanding of MAD and applying it to the concept of standard deviation as a measure of variation. Students will study data sets with generally normal distributions. In a normal distribution, the relationship between the mean, median and the percentages within one, two and three standard deviations from the mean are key to calculating and interpreting data. For a normal data set, students should be able to sketch a graph of the data distribution given information about the measure of center and the distribution and the reverse, describe the measures of center and distribution given a graph. With respect to random sampling, students move beyond analyzing data, as they did in middle school, to making sound statistical decisions based on probability models. For example, students will apply their understanding of the mean and standard deviation of a data set to fit it to a normal distribution in order to estimate population percentages (S.ID.4). They will use their understanding of calculating margin of error in estimating a population quantity and extend it to the random assignment of treatments to available units in an experiment. Students will recognize the purposes of and differences among sample surveys, randomized experiments and observational studies as they make inferences and justify conclusions from real world contexts. *NOTE: The mean absolute deviation (MAD), also called the average deviation of a data set {x_{1}, x_{2}, ..., x_{n}}, is the average of the absolute deviations and is an abstract statistic of statistical distribution or set of data.
| 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.
| - How is data collected (types of samples)?
- What are some possible sources of bias that occur in collecting data? What are some methods of reducing bias?
- What is the difference between observational studies, randomized experiments, and sample surveys? What conclusions can be drawn from each?
- Given a frequency distribution graph, what is the relationship between the median and the mean in a distribution that is skewed to the left? Skewed to the right? Explain why this relationship exists.
- How can you use mean and standard deviation of a normal distribution to compare two pieces of data?
| bias census experimental study interpret dot plots, relative frequency histograms, bar graphs, and box plots margin of error measures of center (mean, median, mode) measures of variation (percentiles, quartiles, range, IQR, variance, standard deviation) normal distribution observational study outlier sample sampling methods simulation skewed distribution symmetric distribution
| | Standards for Mathematical Practice Students have opportunities to: - construct viable arguments and critique the reasoning of others by using and communicating information about data distributions coherently and clearly to support arguments;
- Reason abstractly and quantitatively to build new statistical knowledge through problem solving; and
- Model with mathematics to identify and apply statistical techniques to data and data distributions to solve problems with contexts that are outside of mathematics.
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| What is the connection between exponential and logarithmic functions? What patterns of change are modeled by logarithmic functions as seen in real-world situations, and the tables, graphs, and functions rules that represent these situations?
| | The study of exponential functions begins in grade 8 and is a key component of the Algebra I curriculum. In Algebra II, students use their understanding of exponential functions with the definition of inverse functions to explore logarithms. The concept of function inverses is linked to composition of functions. Introducing and using composition of functions in this unit provides the opportunity to verify whether one function is the inverse of another. Recognizing the inverse relationship between exponential and logarithmic functions will help students to understand the definition of a logarithm; to know and be able to use the properties of logarithms; to make graphical connections between the two functions; and to solve exponential equations not only by using graphs and tables, but also the properties of logarithms. Connections to real-world situations are found in looking at situations that use a logarithmic scale to report values such as the Richter scale, the pH scale and the measurement of sound intensity using a decibel scale.
| 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 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. Linear, Quadratic, and Exponential Models HSF-LE.A. Construct and compare linear and exponential models and solve problems. 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). 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. Interpreting Functions HSF-IF.C. Analyze functions using different representations. 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.
| - How can the properties of logarithms be used to write algebraic expressions in equivalent forms?
- What types of real world relationships are best described using a logarithmic scale? Why?
- What relationships - graphical, algebraic, numeric - exist between a function and its inverse?
- Why can't a logarithm have an argument of zero or a negative number?
- What are the similarities and differences between exponential and logarithmic functions?
| asymptote base of a logarithm base ten logarithms (common logarithms) composition of functions domain e end behavior exponential function exponential models (compound interest, populations, radioactivity) f(x) - e^{x} f(x) = ab^{x} inverse function logarithmic function logarithmic scales (Richter scale for earthquakes, decibel for acoustic power, entropy, pH for acidity, stellar magnitude scale for brightness of stars) log_{b}x = y natural logarithms properties of exponents properties of logarithms range transformation of functions
| | Standards for Mathematical Practice Students will have opportunities to: - make sense of problems and persevere in solving them: apply and adapt a variety of appropriate strategies to solve exponential and logarithmic functions;
- model with mathematics: recognize and apply mathematics in contexts outside of mathematics; and
- use appropriate tools strategically: select, apply, and translate among mathematical representations of exponential and logarithmic functions to solve problems
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| How does understanding polynomial functions (and other function families) aid in making sense of rational functions?
| | This unit introduces students to the family of rational functions by building on knowledge of linear, quadratic, and polynomial functions studied extensively in previous units. These familiar function families are used as expressions and combined using the four arithmetic operations in a way that is similar to how integers are combined to form rational numbers. This leads to a general form: f(x)=g(x)/h(x), where g(x) and h(x) are polynomial functions. The study of rational functions also creates opportunities for students to engage with beginning concepts of limits, that will be studied in future courses, as they examine the end behavior and behavior near asymptotes. Because of the content and algebraic complexity in this unit, technology plays a vital role in allowing students to investigate key features of rational functions and the relationships between representations. Technology opens access to concepts early in the learning trajectory and facilitates problem solving allowing more students to make connections and reason about this new function family. For example, technology allows students to efficiently create and move between various representations in a fraction of the time it would take to generate the representations with paper and pencil. The development of a deep understanding of function classes and their characteristics as called for in the CCSS-M's function domain requires the effective use of technology in teaching and learning. In a functions-based approach to algebra, each family begins with a parent function, in this case f(x)=1/x, that can be transformed into other members of the function family by "replacing f(x) by f(x)+k, kf(x), f(kx), and f(x+k) for specific values of k (both positive and negative)" (CCSS-M, HSF-BF.B.3) In the case of rational functions, k may also take on the form of a polynomial function. Students should examine and discuss the effect of these changes (e.g. multiplying by a factor of 1/(1x+2) would, in effect, create another asymptote and change the end behavior in a similar way to that of a polynomial function moving from odd to even degree). These two algebraic forms of rational functions (i.e., the general form, f(x)=g(x)/h(x), and the parent function, f(x)=1/x) are useful in different situations. The general form is useful for many applications (e.g. forming a "rate" with two functions, one representing profit based on number of attendees for an event and one representing ticket price based on number of attendees, with the rate function representing profit per ticket) while the parent function is useful when examining ideas of transformation in the plane, and algebraic relationships between different rational functions. Students should experience both during this unit and be provided with multiple opportunities to use and make connections among graphic, tabular, and symbolic representations of rational functions. In this unit, students also learn to operate on rational expressions and relate to experiences with rational operations. The analogous set of properties with rational numbers provides the rationale for the use of synthetic division and polynomial long division. Operations on rational expressions are presented as tools to better understand features and behaviors of rational functions (e.g. zeros, asymptotes, and end behaviors). Students should have experiences that drive home these ideas and foster both a need for and understandings of procedures.
| 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 Real Number System HSN-RN.B. Use properties of rational and irrational numbers and that the product of a nonzero rational number and an irrational number is irrational. HSN-RN.B.3. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. 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.A. Understand solving equations as a process of reasoning and explain the reasoning. HSA-REI.A.2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. HSA-REI.D. Represent and solve [rational] 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. Developed and/or Utilized Across Multiple Units Creating Equations HSA-CED.A. Create equations that describe numbers or relationships. HSA-CED.A.2. Create [rational] equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Interpreting Functions 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.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. Interpreting Functions HSF-IF.B. Interpret functions that arise in applications in terms of the context. 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.★
| How can equations and tables of values for rational functions help reveal key features their graphs? - How can the key features of graphs of rational functions be used to create an algebraic model?
How do different forms of rational functions highlight structures where polynomial functions or transformations can aid in analyzing rational functions? (+) How does understanding operations with rational numbers inform operations with rational expressions?
| asymptote (horizontal, vertical, slant) continuity (continuous, discontinuous, holes/undefined points) domain and range end behavior rational function solutions to rational equations (extraneous solutions and solutions) intercepts (x-intercept, y-intercept)
| | Standards for Mathematical Practice Students will have opportunities to: - construct viable arguments and critique the reasoning of others by analyzing and using multiple representations of rational functions;
look for and make use of structure within rational functions (represented in multiple ways) by making connections to knowledge of other families of functions (such as polynomial functions); reason abstractly and quantitatively within mathematical and real world contexts involving rational functions, connecting operations with rational numbers to operations with rational functions, and transferring meaning between multiple representations of rational functions; and - use appropriate tools strategically while working with rational functions by making thoughtful decisions about when to use technology and taking into account its limitations in representation.
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| What connection exists between arithmetic and geometric sequences and linear and exponential functions?
| | Students have already had experience with arithmetic and geometric sequences. As they studied linear and exponential functions these sequences appeared in tabular representations; this unit will allow students to formalize this knowledge. Students will take NOW-NEXT recursive equations and write them using subscripted notation. Students will learn to model problems involving sequential change, represent them using sigma notation, and analyze their model to solve problems. Understanding the differences and characteristics of arithmetic and geometric sequences will help students develop strategies to find sums of arithmetic and geometric series and develop formulas for these sums. Technology in the form of sequential mode of graphing calculators, CAS systems, and computer spreadsheets will provide additional resources for students to explore these topics.
| 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 Linear, Quadratic, and Exponential Models HSF-LE.A. Construct and compare linear and exponential models and solve problems. HSF-LE.A.1b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. HSF-LE.A.1c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
| - Given a sequence of numbers, how can you determine if it is arithmetic, geometric, or neither?
- Find the recursive and explicit formulas for a given arithmetic or geometric sequence.
- Translate between sigma notation and expanded form.
- What are the strategies for finding an arithmetic or geometric series?
- Given an arithmetic or geometric sequence, what information is needed to find the nth term?
| arithmetic sequence arithmetic series convergence divergence explicit formulas finite series geometric sequence geometric series infinite series nth term recursive formulas subscripted notation sum of a series summation/sigma notation
| | Standards for Mathematical Practice Students will have opportunities to: - look for and make use of structure: understand how mathematical ideas interconnect and build on one another to produce a coherent whole;
- model with mathematics: use representations to model and interpret physical, social, and mathematical phenomena; and
- reason abstractly and quantitatively: build new mathematical knowledge of sequential change through problem solving.
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| How can algebraic and geometric ideas be used to explore and connect representations of quadratic relations from the conic sections?
| | This unit is an extension of the Quadratic Units from Algebra 1. Students will build on prior experiences with quadratic functions (parabolas) graphing, factoring, completing the square, solving and transforming parabolas graphically and algebraically this time encountering equations with imaginary solutions (no real solutions) since not all solutions to quadratic equations are real numbers. Therefore, the study of complex numbers will be included in this unit. They will also explore quadratic relations, more commonly known as conic sections, by making connections between algebraic and geometric relationships. Conic sections are all considered quadratic curves because the highest power of any variable is two. Slicing a double napped cone (two cones with a shared apex) with planes at various angles will yield a parabola, circle, ellipse, or hyperbola. Students explore the locus of points that determine the four conic sections and use the distance formula along with geometric descriptions to generate equations for each of the conic sections. Observing the equations (both standard and transformation form) and the resulting graphs should include identifying type of conic section shown and all of its special properties.
| 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. Geometric Measurement & Dimensions HSG-GMD.B. Visualize the relation between two-dimensional and three-dimensional objects HSG-GMD.B.4. Identify cross-sectional shapes of slices of three-dimensional objects [cones], and identify three-dimensional objects generated by rotations of two-dimensional objects. Developed and/or utilized across multiple units Seeing Structure in Expressions HSA-SSE.B. Write expressions in equivalent forms to solve problems. HSA-SSE.B.3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. 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 & 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. HSF-IF.C.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. 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. HSF-BF.A.1b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. 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.
| - What shapesresultfrompassing a plane through a cone?
- How are the equation and the properties of a circle and ellipse similar and different?
- Given the equation of a conic section, how can you identify whether the graph will be a circle, ellipse, parabola, or hyperbola?
- How can you use a line for the directrix and a point for the focus to sketch a resulting parabola?
- What methods are available to convert an equationin the general conic form into a form more suitable for graphing?
| relationship of circles, ellipses, and hyperbolas to cones circle ellipse parabola hyperbola locus of points completing the square discriminant symmetry, lines and axes of graphs of conic sections major axis andminor axis transverse axis and conjugate axis asymptotes focus, foci vertex, vertices distance formula directrix eccentricity i =√-1 complex number (a + bi, where a and b are Real Numbers) conjugate of a complex number Standard form Ax^{2} + Bxy + Cy^{2} + Dx+Ey + F = 0 Transformation form (x-h)^{2}+ (y-k)^{2}= r^{2 } (x-h)^{}^{2}+ (y-k)^{2} = 1 a^{2 }^{}b^{2} (x-h)^{2}_ (y-k)^{2} = 1 a^{2} b^{2} x = a(y-k)^{2}+ h or y = a(x-h)^{2}+k
| | Standards for Mathematical Practice Students have opportunities to: - make sense of problems and persevere in solving them by building new knowledge of conic sections through problem solving;
- model with mathematics to develop an understanding of how mathematical ideas interconnect and build on one another to produce a coherent whole; and
- use appropriate tools strategically particularlytechnology based tools to recognize and use connections among mathematical ideas.
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| How can the unit circle be used to develop a circular definition of trigonometric functions?
| | Previous studies of trigonometry have included right triangle trigonometric relationships and the Laws of Sines and Cosines. In this unit students develop a circular definition of trigonometric functions using both degree and radian measure. The unit circle is the basis for these definitions and leads to graphs of the sine and cosine curves. Students can then define the tangent and inverse trigonometric functions based on the sine and cosine. Making connections between the other trigonometric functions and the graphs and tables of the sine and cosine functions provides a context for students to develop the graphs and tables of the tangent, cotangent, secant, and cosecant functions. Building on their previous knowledge of transformations (both algebraic and geometric), students explore and graph the effects of transformations (amplitude, period, and phase shift) on the sine and cosine curves.
| There are standards listed in this section for two reasons.or a function that models a relationship between two quantities, interpret key features of graphs a - 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 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.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. Developed and/or utilized across multiple units Quantity 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. 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. 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.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.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.1b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. 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.
| - What is the relationship between degree and radian measures of angles? Why or when would you use degree or radian?
- How can the effect of transformations on the sine and cosine curves be seen in the graphs and tables of these functions?
- How can the unit circle be used to generate the sine and cosine graphs?
- Why are the trigonometric functions periodic?
- How do you know if a function is periodic?
| amplitude asymptote cosecant cosine cotangent degree and radian relationship and conversion domain maxima minima period phase shift range relate graphs of trigonometric functions to their inverses secant sine tangent transformations of trig function from the parent functions unit circle
| | Standards for Mathematical Practice Students will have opportunities to: - look for and express regularity in repeated reasoning observing the relationships between the sine and cosine and using the unit circle to generate the periodic graphs of these functions;
- model with mathematics using trigonometric functions and transformations of these functions to model periodic behavior; and
- use appropriate tools strategically such as graphing calculators or computer applets to analyze the sine and cosine functions and changes in their characteristics as parameter changes are made and to find values of trigonometric functions for specific angle measures.
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| How can the ideas of independence and conditional probability, along with expected value, be used to evaluate the outcomes of decisions in a variety of contexts?
| | Although students have had informal experiences with probability in elementary school, their first formal introduction to the study of probability begins in seventh grade. Students conducted experiments or simulations and learned about calculating empirical (experimental) probabilities from their results. They made comparisons of empirical and theoretical probabilities and used theoretical probabilities to predict relative frequencies for a simple chance event. Students learned to set up probability models from a theoretical perspective (such as rolling two dice) or modeling a situation with or without technology and using the empirical data as an estimate of the theoretical outcomes. With compound events, students used tree diagrams, organized lists, or tables to generate the sample space for the event. They use the sample space to calculate the probability of a particular event occurring. At the high school level, students will extend their knowledge of probability and use their previous experience with simulations as a basis for more complicated ideas. Students will use their experience in making sample spaces in middle school and extend the use of these sample spaces to determine the probability of an event occurring to including the probability of event A and event B occurring, of event A or event B occurring; and the probability of an event not occurring. Students will study a variety of problems in order to develop an understanding of independent and dependent events. They learn that the second event has not been influenced by what occurred in the first event when the two events are independent. The idea of conditional probability, i.e. finding the probability of one event occurring given that another event has already occurred, is studied in this unit and can be used as another way to view independence. In terms of probability notation, conditional probability can be expressed as P (A/B) = P (A and B)/P (B). In determining independence of events A and B, when P (A/B) = P (A) and P (B/A) = P (B), then A and B are independent. Another way to determine independence is through the construction of a two-way frequency table, used when two categories are associated with each object being classified. After all of these calculations, it is important to be able to recognize and understand conditional probability and independence and be able to explain theses concepts in everyday language. Following exploring and understanding the ideas of conditional probability and independence, students are introduced to the rules of probability. These include the Addition Rule P (A or B) = P (A) + P (B) – P (A and B), and the Multiplication Rule P (A and B) = P (A) P (B/A). They should be able to use and interpret these rules. Additionally the use of permutations and combinations will help in determining probabilities and solving problems. Use of Pascal’s Triangle is particularly helpful in determining combinations. When using applications of probability to solve problems, often a numerical quantity is more useful than a description of possible outcomes. Graphing a probability distribution gives a different perspective. This is where the expected value is introduced and interpreted as the mean of the probability distribution. Students will develop a probability distribution from a sample space in which the probabilities were assigned either theoretically or empirically. They will then calculate the expected value. The use of expected values and probabilities are widely used to solve problems and evaluate decisions. Probabilities can be used to weigh decisions depending on the probability of outcomes. For example how much a company charges for an extended warranty depends on the cost of repairing or replacing an item and the probability that that item will fail. Games of chance depend on the ideas of expected value. Analyzing decisions and strategies involve probability concepts. Students should recognize the wide impact that probability can have on decisions they make.
| 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 n/a
| - How can you generate the numerical values of Pascal’s Triangle?
- How do you recognize when to use conditional probability rules?
- What is the difference between permutations and combinations? Give an example of a situation where each would be used.
- If the probability of an event occurring is p, what is the probability of that event not occurring? Explain why your answer makes sense.
- What is the meaning of expected value?
| Pascal’s triangle, and its connections to combinations Permutation P(n,k)=n!/(n-k)! Combination C(n,k)=n!/[(n-k)!k!] Fundamental Principle of Counting Tree Diagram Sample space Probability Distribution Independent events Multiplication Rule P(A and B)=P(A) • P(B) Area Model Dependent events Mutually exclusive events Addition Rules for Mutually Exclusive events Compound events Complementary events Conditional probability P(A|B)=P(A and B)/P(B) Applications of probability to real-world situations Simulation Law of Large Numbers Expected Value Two-way frequency table
| | Standards of Mathematical Practice Students will have opportunities to: - make sense of problems and persevere in solving them using probability that arise in mathematics and in other contents; and
- use appropriate technology tools strategically to explore and deepen understanding of probability concepts.
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| How can matrices and vectors be used to solve problems in mathematics and other related fields?
| | In this unit, students first use matrices as a way to organize and display data. After constructing matrices, students can then use the properties of matrices to analyze data and solve a variety of problems from different contexts. Students, without the use of technology, can attain understanding of basic matrix operations. However, one important use of matrices is to solve systems of equations. In doing this, the use of technology to find matrix inverses significantly simplifies the process. This can be accomplished by the use of a graphing calculator or computer software. Using matrices to solve systems of equations provides students tools to solve linear programming problem situations that can be very complex. Another important application for matrices is the representation of geometric shapes in matrix form. These matrix representations can be used in combination with matrices representing various transformations in the plane to show movement and animation of objects mathematically. Matrices are often used to solve problems involving vectors. Because students have studied matrices in this unit, it makes sense to include the study of vectors and using matrices to help solve these problems. Students should understand the components of vectors, how to add and subtract vectors, and how to multiply a vector by a scalar. Problems related to vectors are commonly used in solving physics problems. Although vectors are designated as STEM standards, it would be worthwhile for exposure and interdisciplinary cooperation to include vector problems in instruction.
| 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 n/a
| - How can you tell whether two matrices can be multiplied together?
- What process is used to solve a system of linear equations using matrices?
- How can matrices be used to represent a polygon?
- Given a polygon represented in matrix form, how can you use a matrix to rotate the polygon 180^{0}?
- How can two vectors be added together?
- How do you represent a vector by using a matrix?
| associative property commutative property Cramer's Rule determinant dimension element identity matrix inverse matrix linear programming matrix operations multiplication by a scalar solving systems of equations transformation matrices vector components magnitude direction initial point terminal point add and subtract vectors resultant parallelogram rule scalar scalar multiplication
| | Standards for Mathematical Practice Students will have opportunities to: - attend to precision: create and use matrix representations to organize, record, and communicate mathematical ideas;
- look for and make use of structure: understand how mathematical ideas interconnect and build on one another to produce a coherent whole; and
- reason abstractly and quantitatively: apply and adapt a variety of appropriate strategies to solve problems using matrices.
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