Transitional Algebra. Semester 1 & 2. Length of Unit. Standards: Functions

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1 Semester 1 & 2 MP.1 MP.2 Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Length of Unit Progress Monitoring Short cycle Weekly or bi-weekly formative assessment opportunities MP.3 Construct viable arguments and critique the reasoning of others. Standards of Mathematical Practices MP.4 MP.6 Model with mathematics. Attend to precision. MP.7 Look for and make use of structure. MP.8 Look for and express regularity in repeated reasoning. Standards: Functions Graph all functions: (6 weeks) Algebra 2 Standards: A.SSE.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). IF.1 Priority Standards 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).

2 IF.6 IF.7b If.8a IF.9 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. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. 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. 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. Algebra 1 standards: SSE.3b REI.10 IF.7a Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. 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). Graph linear and quadratic functions and show intercepts, maxima, and minima. Linear, Quadratic, Cubic, Exponential, Absolute Value, Square Root (Radical) Identify Parent Functions & graphs Use graphing calculator to identify domain, range, maximum, minimum, intercepts, end behavior, concavity, and locating point. Depth on domain and range Transfer graphs to own paper using x and y intercept and locating point. Locating Point characteristic point that determines the position of the graph based on translations Linear (0, 0) Quadratic vertex Cubic point of inflection

3 Square Root (0, 0) where it transitions from imaginary to real Exponential y-intercept Absolute Value vertex Operations of functions Transformations Translate, flip, stretch graphically & algebraically *Graphing is a large focus of this unit Graphing activity from Jordan, Kristie s functions activity PROBLEM SOLVING OPPORTUNITIES IMBEDDED WITHIN ALL AREAS Standards: Exponents Properties of Exponents (6 weeks at the most) Standards: F-IF.3 F-IF7e F-IF8b N-RN-1 Priority Standards Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n 1. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay. 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. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.

4 N-RN-2 A-SSE-1a A-SSE-3b F-LE1c F-LE2 F-LE4 A-CED2 F-BF5 Rewrite expressions involving radicals and rational exponents using the properties of exponents. Interpret parts of an expression, such as terms, factors, and coefficients. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. 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). For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. Review Exponential Function tie to property of exponents* Look at tables of exponent function Look for pattern showing repeated multiplication Explore negative exponents with relationship to graph and table of values Explore zero and one as exponents Explore horizontal asymptote as x goes to negative infinity Explore as x goes to infinity end behavior Determine locating point with function Properties of Exponents* Use prime factorization to reinforce properties (2xy 3 ) 2 = 2xy 3 2xy 3 = 2 2 x x y y y y y y = 4x 2 y 6 Do not state the properties discover the properties Discover the following properties: multiplying bases adding exponents, dividing bases subtracting exponents, exponent to an exponent, negative exponents Connect to factoring

5 Logarithms Expand and condense logarithms Rewriting logarithms and exponentials (progress monitor) *Reviewing Algebra 1 and prior to algebra 1, these are the areas when talking about mastery that have to be there (This maybe an area that takes a little longer depending on your students needs) PROBLEM SOLVING OPPORTUNITIES IMBEDDED WITHIN ALL AREAS- Exponential growth and decay, mortgages, car loans, compound daily monthly, bacteria growth, population What are they? How do we use them Connect to factoring Using what we know about rules of exponents to make a problem easier Fractional exponents Quizzes and exit tickets frequently Standards: Radicals Simplifying Radicals (3 to 4 weeks) Standards: A-REI.2 A-REI.10 F.IF.1 F.IF7b N.RN.1 Priority Standards Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. 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). 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). Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer

6 N.RN.2 N.RN.3 N.CN1 N.CN2 N.CN3 exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5. Rewrite expressions involving radicals and rational exponents using the properties of exponents. 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. Know there is a complex number i such that i2 = 1, and every complex number has the form a + bi with a and b real. Use the relation i2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. A.CED.1 Review Square Root Function Graph with graphing calculator Explore table of values looking for patterns Explore domain and range Discuss end behavior The meaning of exponent of ½ Dividing by two is the same as multiplying by ½ Review properties of exponents using rational exponents Number Sense activity Find all square roots of perfect squares from 1 to 100 Estimate the square root of one number between each perfect square to the nearest tenth and place on a number line ?? Talk about irrational and rational numbers with number line Radicals expressions 32 = 16 2 = x 2 = 16 2 x 2 = 4x 2

7 Complex numbers and solving rational equations Find inverse of (if time permits) END FIRST SEMESTER Instructional Notes: Semester 1

8 Standards: Factoring Factoring (6 weeks) Standards: SSE2 Priority Standards 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). Why do we factor? Use words problems to establish relevance of factoring GCF (how far into) constant or variable Leading coefficient of 1 and -1 Verifying solution for correctness by multiplying answers back out Complete square (if time permits within the 6 weeks of the unit) Making the connection of how it leads into algebra 2 Quadratic problem solving Simplify ratios Find least common multiples of polynomials This must be at mastery Standards: System of Equations and Inequalities System of Equations & Inequalities (6 weeks) CED.1 REI.11 Priority Standards 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

9 REI.2 REI.4a REI.4b REI.7 A.SSE3b IF4a (only 4F I could not find IF4a) IF7a IF8a REI12 f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Use the method of completing the square to transform 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. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = 3x and the circle x2 + y2 = 3. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. 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 Revised 08/14/14 Page 4 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. Graph linear and quadratic functions and show intercepts, maxima, and minima. 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. Graph the solutions to a linear inequality in two variables as a half plane (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 half-planes.

10 REI6 REI1 IF9 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Explain each step in solving a simple equation as following from the 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. 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. Clarifying expression and equations Solving linear equations & inequalities Simplifying polynomials Solve quadratics Solve polynomials Kristie has some resources Standards: System of Equations and Inequalities TBD System of Equations & Inequalities (6 weeks) Utilize situation (word problems) Solve logically, tables, graph, equations (graphically, substitution and elimination) Discuss which method is the best Always check solution and interpret meaning Tie in linear, absolute value and quadratic graphs Use graphing calculator to identify domain, range, maximum, minimum, intercepts, end behavior, concavity, and locating point. Connect to systems with quadratics, absolute values and linear equations Number of solutions Standards: Project Based Assessment

11 An end of the year culminating project that synthesizes the content learned within the course. Authenticity of the mathematics that can transfer knowledge to Algebra 2 (these projects could be used the following years as examples of the concepts) Rubric (needed) Project example: Create your own logo, rollercoaster Standards: Problem Solving (embedded throughout the course) ******problem solving - throughout course Skill level Critical thinking Number sense Logic and reasoning Solving linear Solving expressions Number talks (fractions)

12 Instructional Notes: Semester 2 Transitional Algebra