Algebra II Unit 1 Snap Shot Unit Title Cluster Statements Standards in this Unit Unit 1 Polynomial, Rational and Radical Relationships Extend the properties of exponents to rational numbers Reason quantitatively and use units to solve problems Perform arithmetic operations with complex numbers Use complex numbers in polynomial identities and equations Interpret the structure of expressions Write expressions in equivalent forms to solve problems Understand the relationship between zeros and factors of polynomials Use polynomial identities to solve problems Rewrite rational expressions Understand solving equations as a process of reasoning and explain the reasoning Solve equations and inequalities in one variable Solve systems of equations Represent and solve equations and inequalities graphically Analyze functions using different representations Translate between the geometric description and the equation for a conic section N.RN.1 (major) N.RN.2 (major)(cross-cutting MD only) N.Q.2★ (supporting)(cross-cutting) N.CN.1(additional) N.CN.2 (additional) N.CN.7 (additional) A.SSE.2 (major)(fluency)(cross-cutting) A.SSE.3c ★ A.SSE.4 (major) A.APR.2 (major) A.APR.3 (major)(cross-cutting) A.APR.4 (additional) A.APR.6 (additional)(fluency) A.REI.1(major)(cross-cutting) A.REI.2(major) A.REI.4b(supporting)(cross-cutting) A.REI.6(additional)(cross-cutting) A.REI.7(additional) A.REI.11★ (major)(cross-cutting) F.IF.7c★ (supporting) G.GPE.2(additional)
numbers A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. ★ c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. (supporting) Ability to connect experience with properties of exponents from Unit 4 of Algebra I to more complex expressions A.SSE.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. ★ (major) Note: Consider extending this standard to infinite geometric series in curricular implementations of this course description. Knowledge of the difference between an infinite and a finite series Ability to apply the formula for the sum of a finite geometric series : Sum of Finite Geometric Progression The sum in geometric progression (also called geometric series) is given by Equation (1) Multiply both sides of Equation (1) by will have Equation (2) Subtract Equation (2) from Equation (1)
A.APR.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x). (major) Ability to make connections between factors, roots and evaluating functions Ability to use both long division and synthetic division Students should be able to identify integer roots of a polynomial equation by graphing. Using these integer roots a student should then use division to factor until reaching quadratic form allowing them to identify all roots of the polynomial equation. A.APR.3 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. (major)(cross-cutting) Knowledge of the differences in the end behavior of the graphs as dictated by the leading coefficient and whether the function is even or odd Ability to capture the graphical behavior of polynomial functions which have roots with multiplicity greater than one Students should be able to factor higher degree polynomials that have rational roots and are therefore easier to factor. Students should be able to produce a rough graph of a polynomial that is in factored form. Cluster Note: This cluster has many possibilities for optional enrichment, such as relating the example in A.APR.4 to the solution of the system u2+v2=1, v = t(u+1), relating the Pascal triangle property of binomial coefficients to (x+y)n+1 = (x+y)(x+y)n, deriving explicit formulas for the coefficients, or proving the binomial theorem by induction. A.APR.4 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples. (additional) Knowledge of the process for proving identities Ability to see, use and manipulate the structure in an expression as needed to prove an identity To prove an identity, logical steps should be used to manipulate one side of the equation to match the other side of the equation. When proving polynomial identities one may not use properties of equality and must restrict their work to using techniques such as factoring, combining like terms, multiplying polynomials etc.
A.APR.6 Rewrite simple rational expressions in different forms; write in the form where are polynomials with the degree of less than the degree of, using inspection, long division, or, for the more complicated examples, a computer algebra system. (additional)(fluency) Ability to make connections to the Remainder Theorem Examples by long division by inspection A.REI.1 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. Ability to identify the mathematical property (addition property of equality, distributive property, etc.) used at each step in the solution process as a means of justifying a step Example: A student could justify each step of solving equations as shown below.
. A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. (major) Note: The limitations on rational functions apply to the rational expressions in this standard. Limitations: In this course rational functions are limited to those whose numerators are of degree at most one and denominators of degree at most 2. Ability to connect prior experience with solving simple equations in one variable to solving equations which require new strategies and additional steps Ability to make connections between the domain of a function and extraneous solutions Ability to identify extraneous solutions Students should be asked to make a connection between solving linear equations in one variable and solving rational equations in one variable whose numerators are of degree at most one. A.REI.4b Solve quadratic equations by inspection ( e.g. for ), taking square roots, completing the square, the quadratic formula and factoring as appropriate to the initial form of the equations. (supporting)(cross-cutting) Recognize when the quadratic formula gives Ability to solve quadratic equations using various methods and recognize the most efficient method Ability to use the value of the discriminant to determine if a quadratic equation has one double solution, two unique solutions or no real
Example: geometric series A.SSE.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. ★ A geometric series is a series where there is a constant ratio between successive terms. Examples 2+20+200+2000+… (Common Ratio =10) 100 + 10 + 1 + +… (Common Ratio =) (Common Ratio = )
Note: Consider extending this standard to infinite geometric series in curricular implementations of this course description. focus G.GPE.2 Derive the equation of a parabola given the focus and directrix. (additional) The parabola is defined as "the set of all points P in a plane equidistant from a fixed line and a fixed point in the plane." The fixed line is called the directrix, and the fixed point is called the focus.
locus See definition of directrix The set of all points whose coordinates satisfy a single equation or one or more algebraic conditions. Remainder Theorem A.APR.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x). When you divide a polynomial by the remainder will be successive approximations 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 the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where As it appears in this standard, successive approximations would refer to the process of using guess and check to determine when . After substituting a value into each expression one would use clues from the results as to what would be a better guess. This process would continue over and over again until the output values from the two expressions come closer to being equal to one another until a specified degree of precision is reached. Example: To find where by using successive approximations you might try
F.IF.7 A.APR.3 Quadratic Expressions and Equations This activity provides a means for having students to work collaboratively and provides a legitimate reason for movement while providing practice that should help to strengthen student’s proficiency with graphing quadratics and identifying key characteristics. Practice This activity is best used as a review or for skill maintenance. Guiding questions will show how this activity can be used to introduce transformation of graphs. This activity is called “Pass the Problem.” Pass the Problem works well as a means of reviewing multiple concepts and can be used to provide distributed practice over time. A.REI.11 Systems of Equations The activity in this lesson seed requires students to solve systems of equations which are comprised of one linear equation and one quadratic equation. It could easily be adapted to solve systems comprised of one linear and any other type of function covered in this unit or other units. Students also explore systems which have two, one or no solutions. Practice This lesson seed describes a cooperative learning activity where students work collaboratively where students will build their proficiency with solving systems of equations graphically. A.APR.4 Pythagorean Triples This is a cooperative learning activity that will help build understanding of how is used to describe a numerical relationship. Motivation This lesson seed could be used as motivation for a lesson that targets standard A.APR.4
A.REI.2 A.REI.11 Solving Radical Equations and Extraneous Solutions This lesson seed describes an investigation that students could use to help them recognize extraneous solutions that are often produced when solving a radical equation algebraically. Investigation This lesson seed describes an investigation that could be used after teaching the algebraic methods for solving radical equations. This activity will provide additional opportunities for students to solve radical equations algebraically and graphically. This activity will also help students to understand the extraneous solutions. A.SSE.2 A.SSE.3 A.SSE.3a A.SSE.3b A.APR.6 F.IF.1 F.IF.7d Rational Functions This lesson seed begins with an activity to be completed in groups that is designed to build conceptual understanding about the characteristics of rational functions that lead to vertical asymptotes or holes in the graph. This leads into a discussion of the domain of a rational function and the restrictions caused by these characteristics. Enrichment The activities in this lesson seed could be used to extend the study of rational functions. The lesson seed helps students to discover the relationship between the function rule and the graph of a rational function and the restrictions on the domain and how they can be determined from each. A.REI.2 Solving Rational Equations-Octahedron This lesson seed provides practice in solving radical equations. Practice This lesson seed describes a novel activity. Students solve rational equations and then match problems and solutions on provided puzzle pieces that are then cut out a glued together to form an octahedron. Sample Assessment Items The items included in this component will be aligned to the standards in the unit. Topic Standards Addressed Link Notes Complex Numbers N.CN.2 A.SSE.2 This task asks students to perform computations involving complex numbers using the fact that
Understand solving equations as a process of reasoning and explain the reasoning A-REI.1 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. i) Tasks are limited to quadratic equations. i) Tasks are limited to simple rational or radical equations. Solve equations and inequalities in one variable A-REI.4b Solve quadratic equations in one variable. b) Solve quadratic equations by inspection (e.g., for), 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. i) Tasks do not require students to write solutions for quadratic equations that have roots with nonzero imaginary parts. However, tasks can require the student to recognize cases in which a quadratic equation has no real solutions. Note, solving a quadratic equation by factoring relies on the connection between zeros and factors of polynomials (cluster Cluster A-APR.B is formally assessed in Algebra II. i) In the case of equations that have roots with nonzero imaginary parts, students write the solutions as a ± bi for real numbers a and b. Solve Systems of Equations A.REI.6 Solve systems of equations exactly and approximately (e.g. with graphs), focusing on pairs of linear equations in two variables. Tasks have real-world context. Tasks have hallmarks of modeling as a mathematical practice (less defined tasks, more of the modeling cycle, Tasks are limited to 3x3 systems