, Algebra I, Quarter 1


 Kerry Lindsey
 11 days ago
 Views:
Transcription
1 , Algebra I, Quarter 1 The following Practice Standards and Literacy Skills will be used throughout the course: Standards for Mathematical Practice Literacy Skills for Mathematical Proficiency 1. Make sense of problems and persevere in solving them. 1. Use multiple reading strategies. 2. Reason abstractly and quantitatively. 2. Understand and use correct mathematical vocabulary. 3. Construct viable arguments and critique the reasoning of others. 3. Discuss and articulate mathematical ideas. 4. Model with mathematics. 4. Write mathematical arguments. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Standards Student Friendly I can Statements Quantities and Modeling A1.A.REI.A.1 Explain each step in solving an equation as following from the I can solve a simple equation and justify each step using properties. equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. A1.N.Q.A.1 Use units as a way to understand problems and to guide the I can give the solution of an equation using the correct units. solution of multi step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data I can convert unit rates of measurements in multistep arithmetic problems. For displays. example, feet per second to miles per hour. A1.N.Q.A.2 Identify, Interpret, and justify appropriate quantities for the purpose of descriptive modeling. A1.N.Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. I can define appropriate quantities. I can use the scale to read the data correctly. I can determine the accuracy of values based on their limitations in the context of the situation.
2 A1.A.SSE.A.1 Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. I can define and recognize parts of an expression, such as terms, factors, and coefficients for expressions that represent a contextual quantity. I can interpret parts of an expression, such as terms, factors, and coefficients in terms of the context for expressions that represent a contextual quantity. A1.A.CED.A.1 Create equations and inequalities in one variable and use them to solve problems. I can solve multistep equations and equations with variables on both sides. I can recognize and explain when 1 solution, no solution, or infinite solutions are the result of solving an equation. I can create linear equations and inequalities in one variable and use them in a contextual situation to solve problems. A1.A.REI.B.2 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. I can solve linear equations in one variable, including those with coefficients represented by letters. I can solve and graph multistep & compound linear inequalities in one variable, including those with coefficients represented by letters. I can solve inequalities in one variable and use them in a contextual situation to solve problems. A1.A.CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. I can describe the calculations needed to model a function between two quantities. I can write equations to solve contextual problems that involve distance/time/rate, mixture, consecutive integers, and cost. I can solve multivariable formulas or literal equations for a specific variable.
3 A1.A.CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Clarification: For example, represent inequalities describing nutritional and cost constraints on combinations of different foods I can take a realworld problem and use equations and inequalities to write constraints. I can interpret solutions as viable or nonviable options in a modeling context. A1.F.IF.B.3 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. Clarification: Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Relations & Functions I can determine from context if a graph should be discrete or continuous. I can, given a linear function, identify key features in graphs and tables including: intercepts, intervals where the function is increasing, decreasing, positive or negative; relative maximums and minimums; symmetries; end behavior. I can interpret and explain features of the graph of the function in terms of the context of the problem. I can sketch a graph if given the key features of a function. A1.F.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). I can (given the function f(x)) identify x as an element of the domain, the input, and f(x) is an element in the range, the output. I can know that the graph of the function, f, is the graph of the equation y=f(x). I can write an expression in function notation. I can represent a relation in different formats (graph, table, list, and mapping). I can determine whether or not a relation is a function and use the f(x) notation.
4 A1.F.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. I can use function notation to express a contextual problem. I can evaluate functions for inputs in their domain. I can interpret statements that use function notation in terms of the context in which they are used. A1.F.IF.B.4 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Clarification: For example, if the function h(n) gives the number of personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. I can (given the graph) choose the practical domain of the function as it relates to the numerical relationship it describes. I can describe a situation that matches a graph based on its shape. I can explain when a relation is determined to be a function, use f(x) notation. I can state the appropriate domain of a function that represents a problem situation, defend my choice, and explain why other numbers might be excluded from the domain. Arithmetic Sequences A1.F.BF.A.1 Write a function that describes a relationship between two I can determine if a sequence is arithmetic or not. quantities. I can use tables, graphs and expressions to model situations. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. I can write an algebraic expression or equation to generalize the pattern in a table. I can find the common difference in an arithmetic sequence. Linear Functions I can write an arithmetic sequence using both an explicit and recursive rule. I can identify patterns and write an explicit expression using context.
5 A1.F.LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Recognize that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. I can describe whether a given situation in question has a linear pattern of change or a nonlinear pattern of change. I can show that linear functions change at the same rate over time. I can describe situations where one quantity changes at a constant rate per unit interval as compared to another. c. Recognize situations in which a quantity grows or decays by a constant factor per unit interval relative to another. A1.F.LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two inputoutput pairs (include reading these from a table.) I can write an arithmetic sequence, given either the first few terms of a sequence, a graph of the sequence, two terms in the sequence, or a table of values of the sequence. I can graph an arithmetic sequence and identify it as a discrete, linear function. A1.F.LE.B.4 Interpret the parameters in a linear or exponential function in terms of a context. Clarification: For example, The total cost of a plumber who charges 50 dollars for a house call and 85 dollars per hour would be expressed as the function y= 85x +50. If the rate were raised to 90 dollars per hour, how would the function change? A1.F.IF.C.6 Graph functions expressed symbolically and show key features of the graph, by hand and using technology. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. I can explain the meaning of the coefficients, factors, exponents, and/or intercepts in a linear function based on the context of a situation. I can determine the parent function for the line f(x) = x. I can identify and compare the different forms of linear functions. (pointslope, standard, slopeintercept) I can identify the intercepts of a function. I can compare the key features of two linear functions represented in different ways. I can describe a line as a translation of the parent function y = x.
6 A1.F.BF.B.2 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. Clarification: i) Identifying the effect on the graph of replacing f(x) by f(x) + k, k f(x), and f(x+k) for specific values of k (both positive and negative) is limited to linear, quadratic, and absolute value functions. ii) f(kx) will not be included in Algebra 1. It is addressed in Algebra 2. iii) Experimenting with cases and illustrating an explanation of the effects on the graph using technology is limited to linear functions, quadratic functions, absolute value, and exponential functions with domains in the integers. iv) Tasks do not involve recognizing even and odd functions. A1.F.IF.B.5 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. I can transform linear functions. I can find the value of k, given the graphs of a parent function, f(x), and the transformed function: f(x) + k, k f(x), or f(x + k). I can use technology to experiment with the graphs of various functions when transforming the equations using different values of k and form conjectures based on those experiments. I can identify the effect on the graph given a single transformation on a function (symbolic or graphic). I can explain the connection between average rate of change and the slope formula. I can estimate the average rate of change over a specified interval of a function presented either algebraically, in a table, or from the function s graph. I can interpret the meaning of the average rate of change (using units) as it relates to a realworld problem. I can compare the rates of change of two or more functions when represented with function notation, with a graph, or with a table.
7 A1.S.ID.C.5 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. I can explain the meaning of the slope in terms of the units stated in the data. I can explain the meaning of the yintercept in terms of the units stated in the data. I can show that a linear function has a constant rate of change. I can describe and explain the difference between having no slope and having zero slope. A1.F.IF.B.4 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Clarification: For example, if the function h(n) gives the number of personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. A1.A.CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations with two variables on coordinate axes with labels and scales. I can predict and articulate what happens when the slope changes and/or y intercept changes. I can (given a linear graph) choose the practical domain of the function as it relates to the numerical relationship it describes. I can state the appropriate domain of a linear function that represents a problem situation, defend my choice, and explain why other numbers might be excluded from the domain. I can create equations in two or more variables to represent relationships between quantities. I can graph equations in two variables on a coordinate plane and label the axes and scales. I can decide which functions are relatively easy to sketch accurately by hand and which should be graphed using technology.
8 *Note: Classification of function families is approached by introducing 1 function at a time in the standard class; comparing graphs and equations occurs as each function family is introduced. Honors classes can introduce these graphs simultaneously and compare key features. Honors Addendum: Note for Teachers of Honors: Do not teach this Honors Addendum at the end of the quarter. Embed the Honors Addendum within the regular Scope & Sequence. Describe and solve problems using the relationship between parallel and perpendicular lines. Classify graphs and equations into the appropriate function families, limited to: linear, quadratic, absolute value, square root, cube root, exponential, and cubic. I can determine when lines are parallel, perpendicular, or simply intersecting. I can write the equation of a line parallel or perpendicular to a given line. I can identify a function as being linear, quadratic, absolute value, square root, cube root, exponential, or cubic based on the shape of its graph. I can identify a function as being linear, quadratic, absolute value, square root, cube root, exponential, or cubic based on the key features of its equation. I can identify the domain and range of the parent function of linear, quadratic, absolute value, square root, cube root, exponential, and cubic functions.
9 , Algebra I, Quarter 2 The following Practice Standards and Literacy Skills will be used throughout the course: Standards for Mathematical Practice Literacy Skills for Mathematical Proficiency 1. Make sense of problems and persevere in solving them. 1. Use multiple reading strategies. 2. Reason abstractly and quantitatively. 2. Understand and use correct mathematical vocabulary. 3. Construct viable arguments and critique the reasoning of others. 3. Discuss and articulate mathematical ideas. 4. Model with mathematics. 4. Write mathematical arguments. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Standards A1.S.ID.A.1 Represent data with plots on the real number line (dot plots, histograms, and box plots). Statistical Models Student Friendly I can Statements I can construct dot plots, histograms, and box plots for data on a real number line. I can make predictions based on graphs and data sets. A1.S.ID.A.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. I can construct and interpret various forms of data representations, (including line graphs, bar graphs, circle graphs, histograms, scatterplots, boxandwhiskers, stemandleaf, and frequency tables). I can describe a situation using center and spread. I can use the correct measure of center and spread to describe a distribution that is symmetric or skewed. I can identify outliers (extreme data points) and their effects on data sets. I can compare two or more different data sets using the center and spread of each.
10 A1.S.ID.A.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). I can interpret differences in data sets in context. I can interpret differences due to possible effects of outliers. I can identify a set of data as being skewed left, skewed right, or having a normal distribution. I can find and interpret the population standard deviation of a set of data. I can use the bell curve to analyze data distributions. A1.S.ID.B.4 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Scatterplots & Regression Models I can determine when linear, quadratic, and exponential models should be used to represent a data set. I can determine whether linear and exponential models are increasing or decreasing. b. Fit a linear function for a scatter plot that suggests a linear association. I can determine the relationship shown in a scatterplot. I can sketch the function of best fit on the scatter plot. I can use technology to find the function of best fit for a scatter plot. I can find the equation for the line of best fit and use it to make predictions. I can sketch a line of best fit on a scatter plot that appears linear. I can write the equation of the line of best fit (y=mx+b) using technology or by using two points on the best fit line.
11 A1.S.ID.C.6 Use technology to compute and interpret the correlation coefficient of a linear fit. I can use a calculator or computer to find the correlation coefficient for a linear association. I can interpret the meaning of the correlation coefficient in the context of the data. A1.S.ID.C.7 Distinguish between correlation and causation. I can distinguish between correlation and causation. A1.A.CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Clarification: For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Unit 7 Systems of Equations and Inequalities I can write and use a system of equations and/or inequalities to solve a realworld problem. I can represent constraints by equations or inequalities and by systems of equations and/or inequalities. I can interpret solutions as viable or nonviable options in a modeling context. I can solve systems of equations and inequalities with and without technology. A1.A.REI.D.5 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). A1.A.REI.D.6 Explain why the xcoordinates 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 approximate solutions using technology. Clarification: Include cases where f(x) and/or g(x) are linear, quadratic, absolute value, and exponential functions. For example, f(x) = 3x + 5 and g(x) = x^ Exponential functions are limited to domains in the integers. I can explain that all solutions to an equation in two variables are contained on the graph of that equation. I can explain why the intersection of y=f(x) and y=g(x) for any linear function. (I can find the solutions by using technology to graph the equations and determine their point of intersection, using tables of values, or using successive approximations that become closer and closer to the actual value.)
12 A1.A.REI.D.7 Graph the solutions to a linear inequality in two variables as a halfplane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding halfplanes. I can graph linear inequalities on the coordinate plane and describe the meaning of the solution set. I can graph the solutions to a linear inequality in two variables as a halfplane, excluding the boundary for noninclusive inequalities I can graph the solution set to a system of linear inequalities in two variables as the intersection of their corresponding halfplanes. A1.A.REI.C.4 Write and solve a system of linear equations in context. I can graph a system of equations and determine its solution. I can use technology to solve a linear system graphically. A1.F.IF.B.4 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Clarification: For example, if the function h(n) gives the number of personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Absolute Value I can solve a system of equations algebraically, by using substitution and elimination. I can solve a system of linear equations in two variables graphically. I can determine if a linear system has no solution, 1 solution of infinite solutions. I can choose the best method for solving a linear system, and I can justify my method. I can state the appropriate domain of a function that represents a problem situation, defend my choice, and explain why other numbers might be excluded from the domain.
13 A1.F.IF.C.6 Graph functions expressed symbolically and show key features of the graph, by hand and using technology. b. Graph square root, cube root, and piecewisedefined functions, including step functions and absolute value functions. A1.F.IF.B.4 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Clarification: For example, if the function h(n) gives the number of personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. A1.F.BF.B.2 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. Clarification: i) Identifying the effect on the graph of replacing f(x) by f(x) + k, k f(x), and f(x+k) for specific values of k (both positive and negative) is limited to linear, quadratic, and absolute value functions. I can identify key features of an absolute value function, by looking at the graph, a table of values, or the equation. I can graph absolute value functions. I can identify the parent function of an absolute value graph as f(x)= x. I can identify transformations of an absolute value function from the parent function by looking at the graph or the equation. I can state the appropriate domain of a function that represents a problem situation, defend my choice, and explain why other numbers might be excluded from the domain. I can transform absolute value functions. I can use technology to experiment with the graphs of various functions when transforming the equations using different values of k. I can form conjectures based on my experiments with substituting different values into the general (parent) functions. ii) f(kx) will not be included in Algebra 1. It is addressed in Algebra 2. iii) Experimenting with cases and illustrating an explanation of the effects on the graph using technology is limited to linear functions, quadratic functions, absolute value, and exponential functions with domains in the integers. iv) Tasks do not involve recognizing even and odd functions.
14 *Note: Classification of function families is approached by introducing 1 function at a time in the standard class; comparing graphs and equations occurs as each function family is introduced. Honors classes can introduce these graphs simultaneously and compare key features. Honors Addendum: Note for Teachers of Honors: Do not teach this Honors Addendum at the end of the quarter. Embed the Honors Addendum within the regular Scope & Sequence. Informally assess the fit of a function by plotting and analyzing residuals. This standard can be embedded with A1.S.ID.B.4 I can compute the residuals (observed value minus predicted value) for the set of data and the function of best fit. I can construct a scatter plot of the residuals. I can analyze the residual plot to determine whether the function is an appropriate fit.
15 , Algebra I, Quarter 3 The following Practice Standards and Literacy Skills will be used throughout the course: Standards for Mathematical Practice Literacy Skills for Mathematical Proficiency 1. Make sense of problems and persevere in solving them. 1. Use multiple reading strategies. 2. Reason abstractly and quantitatively. 2. Understand and use correct mathematical vocabulary. 3. Construct viable arguments and critique the reasoning of others. 3. Discuss and articulate mathematical ideas. 4. Model with mathematics. 4. Write mathematical arguments. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Standards Student Friendly I can Statements Geometric Sequences & Exponential Functions A1.F.LE.A.2 Construct linear and exponential functions, including arithmetic I can identify a sequence as being geometric. and geometric sequences, given a graph, a description of a relationship, or two inputoutput pairs (include reading these from a table). I can identify the common ratio of a geometric sequence. I can write the explicit and recursive rule for a geometric sequence, given either the sequence, a graph of the sequence, a table of values, or two consecutive terms in the sequence. I can differentiate between an arithmetic and geometric sequence, based on if the sequence has a common difference or common ratio between successive terms. I can create linear and exponential functions, from a simple context, given the following situations: arithmetic and geometric sequences, a graph, a description of a relationship, and two points which can be read from a table.
16 A1.A.SSE.B.3c Use the properties of exponents to rewrite exponential expressions. Clarification: For example, the growth of bacteria can be modeled by either f(t) = 3 (t+2) or g(t) = 9(3 t ) because the expression 3 (t+2) can be rewritten as (3 t )(3 2 ) = 9(3 t ). A1.F.IF.B.4 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Clarification: For example, if the function h(n) gives the number of personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. A1.F.LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions. a) Recognize that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b) Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. I can define an exponential function as f(x)=ab x. I can rewrite exponential functions using the properties of exponents. I can use properties of exponents, including rational exponents (such as power of a power, product of powers, power of a product, and rational exponents, etc ) to write an equivalent form of an exponential function to reveal and explain specific information about its approximate rate of growth or decay (including negative and zero exponents). I can write exponential functions to represent growth and decay. I can (given an exponential graph) choose the practical domain of the function as it relates to the numerical relationship it describes. I can describe a situation that matches a graph based on its shape. I can state the appropriate domain of a function that represents a problem situation, defend my choice, and explain why other numbers might be excluded from the domain. I can describe whether a given situation in question has a linear pattern of change or an exponential pattern of change. I can show that linear functions change at the same rate over time and that exponential functions change by equal factors over time. I can describe situations where one quantity changes at a constant rate per unit interval as compared to another. c) Recognize situations in which a quantity grows or decays by a constant I can describe situations where a quantity grows or decays at a constant factor per unit interval relative to another. percent rate per unit interval as compared to another. Polynomial Operations
17 A1.A.SSE.A.1 Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. I can decompose polynomial expressions and make sense of multiple factors and terms by explaining the meaning of the individual parts focusing on quadratic and exponential expressions. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. A1.A.APR.A.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. A1.A.SSE.A.2 Use the structure of an expression to identify ways to rewrite it. I can add, subtract, and multiply polynomials. I can classify a polynomial based on its number of terms, as well as identify the degree, the leading coefficient, and the constant term. I can rewrite algebraic expressions in different equivalent forms such as combining like terms, using the distributive property, and factoring. Graphing & Solving Quadratic Functions I can factor using greatest common factors and grouping. I can factor a trinomial with a leading coefficient of 1. I can factor a trinomial with a leading coefficient other than 1. I can factor using the difference of two squares. I can factor perfect square trinomials. I can choose the appropriate methods for factoring a polynomial.
18 A1.F.IF.C.6 Graph functions expressed symbolically and show key features of the graph, by hand and using technology. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. I can graph quadratic functions and show intercepts, maxima, and minima. I can identify the axis of symmetry, vertex, and intercepts of a quadratic from the graph of the quadratic. I can algebraically find the axis of symmetry, vertex, and intercepts of a quadratic when given the equation in standard form. I can identify the parent function of a quadratic as f(x)=x 2 I can identify the transformations of a quadratic function when compared to the parent function. A1.A.APR.B.2 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. I can find the zeros of a polynomial when the polynomial is factored. I can use the zeros of a function to sketch the graph of the function.
19 A1.A.REI.B.3 Solve quadratic equations and inequalities in one variable. a. Use the method of completing the square to rewrite any quadratic equation in x into an equation of the form (x p) 2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection (e.g., for x 2 = 49), taking square roots, completing the square, knowing and applying the quadratic formula, and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions. I can transform a quadratic equation written in standard form to an equation in vertex form (x p) 2 = q by completing the square (only with a leading coefficient of 1.) I can derive the quadratic formula by completing the square on the standard form of a quadratic equation. I can recognize that a quadratic will have nonreal solutions when there is a negative under the radical. I can solve quadratic equations by taking square roots. I can understand why taking the square root of both sides of an equation yields two solutions. I can solve quadratic equations by completing the square. I can solve quadratic equations by factoring. I can use the quadratic formula to solve any quadratic equation, recognizing the formula produces all complex solutions. I can compute the discriminants and interpret the results as having no real solutions, 1 real solution, or 2 real solutions. A1.A.SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor a quadratic expression to reveal the zeros of the function it defines. b. Complete the square in a quadratic expression in the form A 2 + Bx + C where A = 1 to reveal the maximum or minimum value of the function it defines. I can recognize the connection among factors, solutions (roots), zeros of related functions, and xintercepts in quadratic functions. I can complete the square to rewrite a quadratic expression (ax 2 +bx+c) with the form a(xh) 2 +k. I can predict whether a quadratic will have a minimum or a maximum based on the value of a.
20 A1.F.IF.C.7 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a) Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. A1.F.BF.B.2 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. Clarification: i) Identifying the effect on the graph of replacing f(x) by f(x) + k, k f(x), and f(x+k) for specific values of k (both positive and negative) is limited to linear, quadratic, and absolute value functions. ii) f(kx) will not be included in Algebra 1. It is addressed in Algebra 2. iii) Experimenting with cases and illustrating an explanation of the effects on the graph using technology is limited to linear functions, quadratic functions, absolute value, and exponential functions with domains in the integers. iv) Tasks do not involve recognizing even and odd functions. A1.F.IF.C.8 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). I can use the process of factoring and completing the square in a quadratic function to determine the vertex, axis of symmetry, direction of opening, and zeros/roots from the graph of a quadratic function. I can graph a quadratic function from standard and vertex form. I can describe all of the properties of a quadratic function. I can transform a variety of functions including linear, quadratic, and absolute value. I can use technology to experiment with the graphs of various functions when transforming the equations using different values of k. I can form conjectures based on my experiments with substituting different values into the general (parent) functions. I can use a variety of function representations (algebraically, graphically, numerically in tables, or by verbal descriptions) to compare and contrast properties of two functions. *Note: Classification of function families is approached by introducing 1 function at a time in the standard class; comparing graphs and equations occurs as each function family is introduced. Honors classes can introduce these graphs simultaneously and compare key features.
21 Honors Addendum: Note for Teachers of Honors: Do not teach this Honors Addendum at the end of the quarter. Embed the Honors Addendum within the regular Scope & Sequence. Graph exponential functions showing intercepts and end behavior. I can graph and translate exponential functions. *Complete the square to solve quadratic equations. *This extends learning objectives from A1.A.REI.B.3. I can complete the square with leading coefficients greater than 1.
22 , Algebra I, Quarter 4 The following Practice Standards and Literacy Skills will be used throughout the course: Standards for Mathematical Practice Literacy Skills for Mathematical Proficiency 1. Make sense of problems and persevere in solving them. 1. Use multiple reading strategies. 2. Reason abstractly and quantitatively. 2. Understand and use correct mathematical vocabulary. 3. Construct viable arguments and critique the reasoning of others. 3. Discuss and articulate mathematical ideas. 4. Model with mathematics. 4. Write mathematical arguments. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Standards Student Friendly I can Statements Modeling with Quadratics A1.F.IF.B.4 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. I can (given the graph) choose the practical domain of the function as it relates to the numerical relationship it describes. Clarification: For example, if the function h(n) gives the number of personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. I can describe a situation that matches a graph based on its shape. I can explain when a relation is determined to be a function, use f(x) notation. I can state the appropriate domain of a function that represents a problem situation, defend my choice, and explain why other numbers might be excluded from the domain. Comparing Linear, Exponential, and Quadratic Models
23 A1.F.LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions. a) Recognize that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b) Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c) Recognize situations in which a quantity grows or decays by a constant factor per unit interval relative to another. A1.F.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. I can describe whether a given situation in question has a linear, quadratic, or exponential pattern of change. I can show that linear functions change at the same rate over time and that exponential functions change by equal factors over time. I can describe situations where one quantity changes at a constant rate per unit interval as compared to another. I can determine if a table of values is linear, exponential, or quadratic. I can write an equation in the appropriate form given a table of values or a graph of the equation. I can make the connection using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or any other polynomial function. A1.F.IF.C.6 Graph functions expressed symbolically and show key features of the graph, by hand and using technology. Piecewise Functions I can graph square root, cube root, and piecewise functions including step and absolute value. b. Graph square root, cube root, and piecewise I can evaluate a piecewise function for specific inputs from looking at the defined functions, including step functions and graph or by using the equation. absolute value functions. I can graph piecewise functions given the equation. Operations with Radicals & Rationalizing the Denominator
24 Note: Classification of function families is approached by introducing 1 function at a time in the standard class; comparing graphs and equations occurs as each function family is introduced. Honors classes can introduce these graphs simultaneously and compare key features. Honors Addendum: Note for Teachers of Honors: Do not teach this Honors Addendum at the end of the quarter. Embed the Honors Addendum within the regular Scope & Sequence. Rewrite simple rational expressions in different forms; write a(x) / b(x) in the form q(x)+r(x) / b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. Write the equation of piecewise functions given the graph. I can compare rational expressions by writing them in different but equivalent forms. I can multiply and divide rational expressions and simplify using equivalent forms. I can perform addition and subtraction of rational expressions. I can write the equation of piecewise functions given the graph.