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ALGEBRA 1 OBJECTIVES Unit A Properties of Real Numbers A1. Perform operations on real numbers. (3%) (N.RN.3 *Use properties of rational and irrational numbers ) a. b. c. d. Add and subtract real numbers, including positives and negatives. Multiply real numbers, including positives and negatives. Simplify expressions using the distributive property. Divide real numbers, including positives and negatives. A2. Compare and order real numbers and find their approximate location on a number line. (3%) (N.RN.3) *Use properties of rational and irrational numbers) e. Classify whole numbers, integers, rational numbers, irrational numbers, and real numbers. f. Graph numbers on number lines. g. Compare real numbers using inequality signs (<, >, ≤, ≥) and write them in order. Unit B Expressions, Equations and Functions B1. Write an algebraic expression, which may include an exponent, and evaluate an expression using the order of operations. (3%) (A.SSE.1a *Interpret parts of an expression, such as terms, factors, and coefficients.) (F.BF.1a *Determine an explicit expression, a recursive process, or steps for calculation from a context. ) (A.SSE.2 *Use the structure of an expression to identify ways to rewrite it. ) a. Evaluate expressions (Find the value, plug in for variables and get an answer). b. Evaluate using the order of operations. c. Combine like terms. d. Translate phrases into expressions/equations e. Determine whether a given number is a solution of an equation or inequality. B2. Use unit analysis to verify that a variable expression will give a reasonable calculation based on the units in the answer. (3%) (N.Q.1 *Use units as a way to understand problems) f. Find the unit rate. g. Given an expression which models a situation, analyze the units included in the terms in the expression to verify that the expression will give a reasonable calculation based on the units in the answer. Unit 1 Solving Linear Equations/Inequalities 1. Write a linear equation/inequality to model a real-life situation and solve linear equations/inequalities in one variable. (16%) (A.CED.1 *Create equations and inequalities in one variable and use them to solve problems.) (A.REI.1 *Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step. Construct a viable argument to justify a solution method. ) (A.REI.3 *Solve linear equations in one variable, including equations with coefficients represented by letters. ) (N.Q.2 *Define appropriate quantities for the purpose of descriptive modeling.) (N.Q.3 *Choose a level of accuracy appropriate to limitations on measurement when reporting quantities ) a. Solve one-step equations/inequalities. b. Solve equations/inequalities with 2 or more steps. c. Solve equations/inequalities with a variable on both sides. d. Solve absolute value equations. e. Solve and graph compound inequalities (and/or). For example: -3 < x + 2 < 7 f. Solve/graph absolute value inequalities (and/or). g. Solve an inequality that is true for all real numbers or determine if an inequality has no solutions. h. Given a verbal description of a real-life problem, translate it into an algebraic equation/inequality. i. Use the solution of the equation/inequalities to help you solve the problem. j. Explain and justify steps in solving equations/inequalities. k. Solve an inequality that is true for all real numbers or determine if an inequality has no solutions. 2. Given an equation with two or more variables (literal equation), solve the equation for a specified variable. (5%) (A.SSE.1b *Interpret complicated expressions by viewing one or more of their parts as a single entity. ) (A.CED.4 *Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.) (A.REI.3 *Solve linear equations in one variable, including equations with coefficients represented by letters. ) l. Rewrite an equation with one variable in terms of the others. m. Rewrite an equation or formula, then evaluate the new formula. Unit 2 Graphing Linear Equations and Functions 3. Graph a linear equation or function. (16%) (A.CED.2 *Graph equations on coordinate axes with labels and scales. ) (A.REI.10 *Understand that the graph of an equation in 2 variables is the set of all its solutions plotted in the coordinate plane. ) (F.IF.2 *Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.) (F.IF.7a *Graph linear and quadratic functions and show intercepts, maxima, and minima. ) (F.IF.7b *Graph absolute value functions.) (F.IF.4 *For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. ) a. b. c. d. e. f. g. Graph the line of an equation using a table of points. Graph horizontal and vertical lines from their equations. Find the x & y intercepts of the graph of a linear equation. Graph the line of an equation using x & y intercepts. Graph the line of an equation using slope-intercept form. Solve and graph linear equations in function notation [f(x)]. Graph absolute value equations. 4. Compare linear equations by their rates of change and y-intercepts. (10%) (A.CED.2 *Create equations in 2 or more variables to represent relationships between quantities. ) (S.ID.7 *Interpret the slope and the intercept of a linear model in the context of the data.) (F.IF.9 *Compare properties of 2 functions each represented in a different way. ) (F.BF.3 *Identify the effect on the graph of replacing f(x) by f(x) + k… for specific values of k; find the value of k given the graphs) h. Using rate of change, classify lines with positive, negative, zero, or undefined rate of change, and recognize the effect rate of change has on a line. i. Find the rate of change of a line. j. Find the rate of change and y-intercept of a line using the slope-intercept form. k. Compare linear equations using their y-intercepts. 5. Write a rule for a function or make a graph given the inputs and outputs, or write the pairing given the rule or a graph. (5%) (F.IF.1 *Understand that a function from one set (domain) to another set (range) assigns to each element of the domain exactly one element of the range.) (F.IF.5 *Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. ) l. List domain and range of a function m. Write the rule of a function given the input and the output. n. Write a rule for a function when given the graph of the function. Note to teacher: Include domain constraints. For example, you can’t have half a person. Unit 3 Writing Linear Equations 6. Write a linear equation to model a real-life situation and use the equation to help solve a problem. (16%) (F.IF.6 *Calculate and interpret the average rate of change of a function over a specified interval. Estimate the rate of change from a graph.) (F.LE.2 *Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or 2 input-output pairs.) (F.IF.3 *Recognize that sequences are functions, sometimes defined, recursively, whose domain is a subset of the integers ) (F.BF.1b *Combine standard function types using arithmetic operations.) (F.BF.2 *Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the 2 forms.) (F.BF.4a *Find inverse function of linear equation.) a. b. c. d. e. f. g. h. i. Identify the slope/rate of change and a point from point-slope form. Write linear equations in slope-intercept form. Write linear equations in point-slope form. Write linear equations in standard form. Write the equation of a line taken from a graph. Write the equations of lines that are parallel and/or perpendicular to a given line. Use a linear equation to solve a problem. Relate arithmetic sequences to linear functions (F.IF.3) Find the inverse of a linear function. 7. Construct a scatter plot, determine an equation for a line of best fit, make predictions based on a set of data, make conjectures about possible relationships between two characteristics of a sample based on a set of data. (10%) (F.IF.6 *Calculate and interpret the average rate of change of a function over a specified interval. Estimate the rate of change from a graph.) (S.ID.3 *Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).) (S.ID.6a,b,c *Represent data on 2 quantitative variables on a scatter plot, and describe how the variables are related.) (S.ID.8/9 *Compute (using technology) and intercept the correlation coefficient of a linear fit./Distinguish between correlation and causation.) j. Find a line of best fit for a set of data. k. Make a scatter plot. l. Write an equation for line of best fit, include the rate of change. m. Make a prediction based on a scatter plot. n. Determine the correlation between two characteristics. (positive/negative) _______________________________Semester____________________________________ 2 Unit 4 Systems of Equations and Inequalities 8. Use and solve systems of linear equations with 2 variables. (9%) (A.REI.5,6,7 *Solve systems of equations, including a linear equation and a quadratic equation) (A.REI.11 *Explain why the x-coordinates of the points where the graphs of the equations y=f(x) and y=g(x) intersect are the solutions of the equation f(x)=g(x). Find solutions using technology to graph the functions, make tables of values, or find successive approximations.) a. b. c. d. e. for sections a-c Solve system of equations by graphing. Solve system of equations by substitution. Solve system of equations by elimination. Determine the number of solutions of a system of linear equations. Determine whether or not an ordered pair is a solution of a system. 9. Use and solve systems of linear inequalities with 2 variables. (8%) (A.CED.3 *Represent constraints by equations of inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. ) (A.REI.12 *Graph the solutions to a linear inequality in 2 variables as a half-plane and graph the solution set to a system of linear inequalities in 2 variables as the intersection of the corresponding half-planes.) f. Graph a linear inequality on a coordinate plane. g. Solve of system of linear inequalities by graphing. h. Use linear programming to find maximums and minimums. Unit 5 Exponents and Exponential Functions 10. Use properties and definitions of exponents to simplify expressions. (8%) (N.RN.1 *Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.) (N.RN.2 *Rewrite expressions involving radicals and rational exponents using the properties of exponents. ) (A.SSE.2 *Use the structure of an expression to identify ways to rewrite it. ) a. Simplify expressions using multiplication & division properties of exponents. b. Simplify expressions using zero & negative properties of exponents. c. Simplify expressions with fractional exponents. 11. Use scientific notation to represent very large and very small numbers. (4%) (N.Q.3 *Choose a level of accuracy appropriate to limitations on measurement when reporting quantities ) d. Convert numbers between scientific notation and standard form. e. Multiply and divide using scientific notation. f. Judge reasonableness of scientific notation computations. 12. Write and graph exponential growth and exponential decay functions. (8%) (A.SSE.3c *Use the properties of exponents to transform expressions for exponential functions. ) (F.IF.8b *Use the properties of exponents to interpret expressions for exponential functions.) (F.BF.3 *Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or as a polynomial function.) (N.Q.3 *Choose a level of accuracy appropriate to limitations on measurement when reporting quantities ) (F.IF.7e *Graph exponential functions.) (F.BF.1b *Combine standard function types using arithmetic operations.) (F.BF.3 *Identify the effect on the graph of replacing f(x) by f(x) + k… for specific values of k; find the value of k given the graphs) (F.LE.1c *Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. ) (F.LE.5 *Interpret the parameters in a linear or exponential function in terms of a context) (A.CED.1 *Create equations and inequalities in one variable and use them to solve problems. ) (F.IF.4 *For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. ) (F.IF.5 *Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. ) g. Write a rule for an exponential growth/decay function when given a table of ordered pairs. h. Graph exponential growth & decay functions. i. Solve exponential growth & decay problems. j. Write a geometric sequence. Unit 6 Data Analysis 13. Calculate statistical measures of a set of data. (4%) (S.ID.2 *Use statistics appropriate to the shape of data distribution to compare center (median, mean) of 2 or more different data sets.) a. b. c. d. Find mean. Find median. Find mode. Determine the best measure of central tendency for a given situation. 14. Represent and analyze data distributions. (9%) (S.ID.1 *Represent data with plots on the real number line (dot plots, histograms, and box plots. ) (S.ID.2 *Use statistics appropriate to the shape of data distribution to compare spread (interquartile range, standard deviation) of 2 or more different data sets.) (S.ID.3 *Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).) (S.ID.5 *Summarize categorical data for 2 categories in 2 way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.) e. Find range and inter-quartile ranges. f. Analyze surveys and samples, identify bias in a survey. g. Construct stem-and-leaf plots. h. Construct box-and-whisker graphs. i. Construct histograms. j. Construct a line plot. Unit 7 Polynomials and Factoring 15. Perform operations on polynomials. (8%) (A.APR.1 *Perform arithmetic operations on polynomials.) (A.SSE.1 *Interpret expressions that represent a quantity in terms of its context. Interpret parts of an expression, such as terms, factors, and coefficients. e.g. Name the number of terms…, Name the degree of the polynomial. ) a. Identify the coefficients, including leading, of a polynomial. b. State the degree of polynomial. c. Add & Subtract polynomials. d. Multiply by a monomial. e. Multiply 2 binomials. f. Multiply polynomials with many terms. g. Multiply special product polynomials. 16. Factor polynomials. (6%) (A.SSE.2 *Use the structure of an expression to identify ways to rewrite it. ) h. Factor out a common factor. i. Factor trinomials into two binomials. j. Factor the difference of two perfect squares. k. Factor by grouping. Unit 8 Quadratic Equations and Functions 17. Graph quadratics. (8%) (F.BF.3 *Identify the effect on the graph of replacing f(x) by f(x) + k… for specific values of k; find the value of k given the graphs) (F.LE.1a *Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.) (F.LE.1b *Recognize situations in which 1 quantity changes at a constant rate per unit interval relative to another ) - is the equation linear, quadratic, or exponential? (F.LE.5 *Interpret the parameters in a linear or exponential function in terms of a context (F.IF.4 *For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. ) (F.IF.7a *Graph linear and quadratic functions and show intercepts, maxima, and minima. ) a. Graph simple parabolas in the form of (ax2+b). b. Graph parabolas with a “b“ term (ax2+bx+c). c. Identify/analyze functions (linear, quadratic, exponential, absolute value, piecewise, and step) 18. Solve quadratic equations and functions. (6%) (A.SSE.3a *Factor a quadratic expression to reveal the zeros of the function it defines ) (A.SSE.3b *Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines) (A.REI.4a,b *Solve quadratic equations in one variable.) (F.IF.8a *Use the process of factoring and completing the square in a quadratic function to show zeros. ) (A.CED.1 *Create equations and inequalities in one variable and use them to solve problems.) (A.REI.7 *Solve a simple system consisting of a linear equation and a quadratic equation in a two variables algebraically and graphically.) (F.IF.5 *Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. ) d. Solve simple quadratic equations. e. Solve quadratic equations by graphing. f. Solve quadratic equations by factoring. g. Solve quadratic equations by completing the square. h. Solve quadratic equations by using the quadratic formula. i. Use the discriminant to find the number of solutions to a quadratic. Unit 9 Radicals and Geometry 19. Simplify radicals. (2%) a. b. c. d. Simplify radicals (break them down--factor trees). Rationalize denominators (don‘t get caught with your radical down). Add & Subtract radicals. Multiply radicals. 20. Solve and graph radical equations. (2%) (F.IF.7b *Graph square root functions.) (F.BF.3 *Identify the effect on the graph of replacing f(x) by f(x) + k… for specific values of k; find the value of k given the graphs) (F.LE.5 *Interpret the parameters in a linear or exponential function in terms of a context ) e. f. g. h. Graph radical equations. Solve simple radical equations. Identify extraneous solutions. Solve equations with two radicals.