Algebra II Texas Mathematics: Unpacked Content
Algebra II Texas Mathematics: Unpacked ContentWhat is the purpose of this document?To increase student achievement by ensuring educators understand specifically what the newstandards mean a student must know, understand and be able to do. This document may alsobe used to facilitate discussion among teachers and curriculum staff and to encouragecoherence in the sequence, pacing, and units of study for grade level curricula. This document,along with on going professional development, is one of many resources used to understandand teach the new math standards.What is in the document?Descriptions of what each standard means a student will know, understand, and be able to do.The “unpacking” of the standards done in this document is an effort to answer a simple question“What does this standard mean that a student must know and be able to do?” and to ensure thedescription is helpful, specific and comprehensive for educators.
At A Glance:New to Algebra II: Write equations of parabolas using specific attributes. Specifically solve systems using three linear equations and systems consisting oftwo equations, one linear and one quadratic. Use Gaussian elimination method of solving systems of equations. Addition of the cubic and cube root functions Composition of two functions Identify extraneous solutions of square root equations Formulate absolute value of linear equations Solve absolute value linear equations and inequalities Transformations with cubic, cube root and absolute value functions Add, subtract, and multiply polynomials Determine the quotient of a polynomial of degree three and of degree four whendivided by a polynomial of degree one and of degree two Determine the sum, difference, product, and quotient of rational expressions withintegral exponents of degree one and of degree two Rewrite radical expressions that contain variables to equivalent forms Solve equations involving rational exponents Use linear, quadratic, and exponential functions to model data Use regression methods to write linear, quadratic, and exponential models TEKS language no longer refers to the different types of functions as “parent”functions, they are simply referred to as “functions”Algebra II TEKS moved to Algebra I Writing equations that are parallel and perpendicular to the x and y axis (zero andundefined slope) Dividing polynomials Solving and graphing quadratics Make connections between standard and vertex form in a quadratic function Writing quadratic functions specifically using the vertex and another point Predict the effects of changes in a, h, and k using the graph of y a(x h)2 k Predict the effects of changes in a and b using the form f(x) abx Find the domain and range of exponential functions Write exponential functionsAlgebra II TEKS moved to Pre Calculus Graph and write an equation of an ellipse and a hyperbola Solving rational inequalitiesAlgebra II TEKS moved to Geometry Graph and write an equation of a circle
Instructional Implications for 2015 16: Some teachers have already been teaching many of the new Algebra II TEKS, importantto cover in depth now Remember to cover the gaps created by TEKS that have moved up or downProfessional Learning Implications for 2015 16: Important for all to know the changes in TEKS and how that affects the students’ mathprogress Collaboration among Algebra I, Algebra II, Pre Calculus, and Geometry teachers toensure no gaps occur
Algebra II Primary Focal Areas:The Primary Focal Areas are designed to bring focus to the standards at each grade bydescribing the big ideas that educators can use to build their curriculum and to guideinstruction.1. The desire to achieve educational excellence is the driving force behind the Texasessential knowledge and skills for mathematics, guided by the college and careerreadiness standards. By embedding statistics, probability, and finance, while focusing onfluency and solid understanding, Texas will lead the way in mathematics education andprepare all Texas students for the challenges they will face in the 21st century.2. The process standards describe ways in which students are expected to engage in thecontent. The placement of the process standards at the beginning of the knowledge andskills listed for each grade and course is intentional. The process standards weave theother knowledge and skills together so that students may be successful problem solversand use mathematics efficiently and effectively in daily life. The process standards areintegrated at every grade level and course. When possible, students will applymathematics to problems arising in everyday life, society, and the workplace. Studentswill use a problem solving model that incorporates analyzing given information,formulating a plan or strategy, determining a solution, justifying the solution, andevaluating the problem solving process and the reasonableness of the solution. Studentswill select appropriate tools such as real objects, manipulatives, paper and pencil, andtechnology and techniques such as mental math, estimation, and number sense to solveproblems. Students will effectively communicate mathematical ideas, reasoning, andtheir implications using multiple representations such as symbols, diagrams, graphs, andlanguage. Students will use mathematical relationships to generate solutions and makeconnections and predictions. Students will analyze mathematical relationships to connectand communicate mathematical ideas. Students will display, explain, or justifymathematical ideas and arguments using precise mathematical language in written ororal communication.3. In Algebra II, students will build on the knowledge and skills for mathematics inKindergarten Grade 8 and Algebra I. Students will broaden their knowledge of quadraticfunctions, exponential functions, and systems of equations. Students will studylogarithmic, square root, cubic, cube root, absolute value, rational functions, and theirrelated equations. Students will connect functions to their inverses and associatedequations and solutions in both mathematical and real world situations. In addition,students will extend their knowledge of data analysis and numeric and algebraicmethods.4. Statements that contain the word "including" reference content that must be mastered,while those containing the phrase "such as" are intended as possible illustrativeexamples.
Mathematical Process StandardsUnpackingThe student uses mathematical processes to What does this standard mean that a studentacquire and demonstrate mathematicalwill know and be able to do?understanding. The student is expected to:1(A) apply mathematics to problems arisingin everyday life, society, and the workplace;ApplyMathematical Practices1. Make sense of problems and persevere insolving them.2. Reason abstractly and quantitatively.3. Construct viable arguments and critiquethe reasoning of others.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeatedreasoning.Mathematical Process StandardsUnpackingThe student uses mathematical processes to What does this standard mean that a studentacquire and demonstrate mathematicalwill know and be able to do?understanding. The student is expected to:1(B) use a problem-solving model thatincorporates analyzing given information,formulating a plan or strategy, determining asolution, justifying the solution, andevaluating the problem-solving process andthe reasonableness of the solution;Mathematical Process StandardsUse, Formulate, Determine, Justify, EvaluateUnpackingThe student uses mathematical processes to What does this standard mean that a studentacquire and demonstrate mathematicalwill know and be able to do?understanding. The student is expected to:
1(C) select tools, including real objects,manipulatives, paper and pencil, andtechnology as appropriate, and techniques,including mental math, estimation, andnumber sense as appropriate, to solveproblems;Select, SolveMathematical Process StandardsUnpackingThe student uses mathematical processes to What does this standard mean that a studentacquire and demonstrate mathematicalwill know and be able to do?understanding. The student is expected to:1(D) communicate mathematical ideas,reasoning, and their implications usingmultiple representations, including symbols,diagrams, graphs, and language asappropriate;Communicate:Have the students:Mathematical Process StandardsUnpacking1) Keep a journal2) Participate in online forums for theclassroom like Schoology or Edmoto.3) Write a summary of their daily notes aspart of the assignments.The student uses mathematical processes to What does this standard mean that a studentacquire and demonstrate mathematicalwill know and be able to do?understanding. The student is expected to:1(E) create and use representations toorganize, record, and communicatemathematical ideas;Create, UseThe students can:1) Create and use graphic organizers forthings such as parent functions. Thisorganizer can have a column for a) the nameof the functions, b) the parent functionequation, c) the graph, d) the domain, e) therange, f) any asymptotes.This organizer can be filled out all at once, oras you go through the lessons. They couldeven use this on assessments at theteacher’s discretion.
Mathematical Process StandardsUnpackingThe student uses mathematical processes to What does this standard mean that a studentacquire and demonstrate mathematicalwill know and be able to do?understanding. The student is expected to:1(F) analyze mathematical relationships toconnect and communicate mathematicalideas; andAnalyzeMathematical Process StandardsUnpackingThe student uses mathematical processes to What does this standard mean that a studentacquire and demonstrate mathematicalwill know and be able to do?understanding. The student is expected to:1(G) display, explain, and justifymathematical ideas and arguments usingprecise mathematical language in written ororal communication.Display, Explain, JustifyLinear Functions, Equations, andInequalitiesUnpackingWhat does this standard mean that a studentThe student applies mathematical processes will know and be able to do?to understand that functions have distinct keyattributes and understand the relationshipbetween a function and its inverse. Thestudent is expected to:2(A) graph the functions f (x) x , f (x) 1x ,f (x) x3 , f (x) 3 x , f (x) xb , f (x) x ,and f (x) logbx where b is 2, 10, and e, and,when applicable, analyze the key attributessuch as domain, range, intercepts,symmetries, asymptotic behavior, andmaximum and minimum given an intervalGraph* This could work here as well as in theexample above. The graphs would bestressed at this standard.The students can:1) Create and use graphic organizers forthings such as parent functions. Thisorganizer can have a column for a) the nameof the functions, b) the parent functionequation, c) the graph, d) the domain, e) therange, f) any asymptotes.This organizer can be filled out all at once, or
as you go through the lessons. They couldeven use is on assessments at the teacher’sdiscretion.Linear Functions, Equations, andInequalitiesUnpackingWhat does this standard mean that a studentThe student applies mathematical processes will know and be able to do?to understand that functions have distinct keyattributes and understand the relationshipbetween a function and its inverse. Thestudent is expected to:2(B) graph and write the inverse of afunction using notation such as f -1 (x);Graph, WriteThe students should be given several graphsthat may or may not be “functions”. In teams,the students will: a) identify points on thegiven graph, b) identify the domain and rangeof the given graphs, c) switch the x and yvalues, d) graph the results using the “new”points, e) write the domain and range of theresult. The teacher can guide them to theidea that the inverse is the original graphreflected about the line y x.Linear Functions, Equations, andInequalitiesUnpackingWhat does this standard mean that a studentThe student applies mathematical processes will know and be able to do?to understand that functions have distinct keyattributes and understand the relationshipbetween a function and its inverse. Thestudent is expected to:2(C) describe and analyze the relationshipbetween a function and its inverse (quadraticand square root, logarithmic andexponential), including the restriction(s) ondomain, which will restrict its range; andDescribe, Analyze*This lesson will be similar to the one above,however this time the original functionsshould be specific functions that willdemonstrate the relationships between the
inverse functions, i.e., quadratic and squareroot, logarithmic and exponential.The students should be given several graphsthat have inverses. In teams, the studentswill: a) identify points on the given graph, b)identify the domain and range of the givengraphs, c) switch the x and y values, d)graph the results using the “new” points, e)write the domain and range of the result. Theteacher can guide them to the idea that theinverse is the original graph reflected aboutthe line y x.Next, the students should identify the type offunction the resulting graph is, write itsequation, identify its domain and range, andanalyze any restrictions on the range.Finally, have the students describe therelationship between a function and it’sinverse. This can be done as an exit ticket, asummary of the day’s work in a notebook, in aclassroom online forum, or as a warmup forthe next day’s work.Describe the relationship between a functionand its inverse for quadratic, square roots,logarithms, and exponentials.Example:Given the function f(x) x2 2x, a table ofvalues of points would includexy-20-1-1001328A table of values for the inverse, f 1, wouldreverse the values for x and y.xY
0-1038-2-1012The graphs will be reflected across theidentity function.Linear Functions, Equations, andInequalitiesUnpackingWhat does this standard mean that a studentThe student applies mathematical processes will know and be able to do?to understand that functions have distinct keyattributes and understand the relationshipbetween a function and its inverse. Thestudent is expected to:2(D) use the composition of two functions,including the necessary restrictions on thedomain, to determine if the functions areinverses of each other.Systems of Equations and InequalitiesUse, DetermineRemind the students of composition offunctions from Algebra 1. Explain whatcomposition of functions is and how to do thework. Then give them several examplesgrouped into two groups: 1) These functionsare inverses of each other, and 2) thesefunctions are not inverses of each other. Askthem to determine what the inverses have incommon that the non inverse functions do nothave in common. Lead them to theunderstanding that if two functions areinverses of each other, then f(g(x)) x andg(f(x)) x. If there is any other result, thanthe functions are not inverses of each other.UnpackingThe student applies mathematical processes What does this standard mean that a studentto formulate systems of equations andwill know and be able to do?inequalities, use a variety of methods to solve,
and analyze reasonableness of solutions. Thestudent is expected to:3(A) formulate systems of equations,including systems consisting of three linearequations in three variables and systemsconsisting of two equations, the first linearand the second quadratic;FormulateHelp the students analyze applicationproblems with 2 or 3 variables. They shouldpractice formulating systems of equationsfrom the given real world problems. In thisitem, they are only formulating the systems,not solving them. Three examples are:Two linear equations:1) The admission fee at a small fair is 1.50for children and 4.00 for adults. On a certainday, 2200 people enter the fair and 5050 iscollected. How many children and how manyadults attended?Three linear equations:2)Billy’s Restaurant ordered 200 flowers forMother’s Day. They ordered carnations at 1.50 each, roses at 5.75 each, and daisiesat 2.60 each. They ordered mostlycarnations, and 20 fewer roses than daisies.The total order came to 589.50. How manyof each type of flower was ordered?1 Linear and 1 Quadratic3)An acorn, falling from the top of a 45 foottree is modeled by the equationh 16t² 45. Before it can hit the ground asquirrel hiding on a lower branch jumps outand intercepts it. If the squirrel's movement ismodeled by the equation h 3t 32, at whatheight did the squirrel intercept the acorn?Systems of Equations and InequalitiesUnpackingThe student applies mathematical processes What does this standard mean that a studentto formulate systems of equations andwill know and be able to do?inequalities, use a variety of methods to solve,and analyze reasonableness of solutions. Thestudent is expected to:
3(B) solve systems of three linear equationsin three variables by using Gaussianelimination, technology with matrices, andsubstitution;Solve1) This website has a good explanation ofHow to use Gaussian Elimination for a3 variable system of mination.html2) Use inverse matrices to solve a 3 variablesystem of equations with the calculator. Thisshould be done as part of the matrices unit.3) Solving by substitution is the same with3 variable systems of equations as it is with2 variable systems, there is just an extrastep.What is the x value of the solution to thesystem of equations below?x y z 8x 2y 6y z 4Systems of Equations and InequalitiesUnpackingThe student applies mathematical processes What does this standard mean that a studentto formulate systems of equations andwill know and be able to do?inequalities, use a variety of methods to solve,and analyze reasonableness of solutions. Thestudent is expected to:3(C) solve, algebraically, systems of twoequations in two variables consisting of alinear equation and a quadratic equation;SolveHere is the problem from above with a goodexplanation of the answer:Question:An acorn, falling from the top of a 45 foot treeis modeled by the equation h 16t² 45.Before it can hit the ground a squirrel hidingon a lower branch jumps out and intercepts it.If the squirrel's movement is modeled by theequation h 3t 32, at what height did thesquirrel intercept the acorn?
Answer Explanation:You want to know when the height of thesquirrel and the height of the acorn are thesame, so take your two equations and makethem equal: 16t² 45 3t 32Shuffle that around a bit (subtract 3t 32from both sides) and you get: 16t² 3t 13 0There's your quadratic equation. If you stickthis into the quadratic formula, you'll get t 1or 13/16. We can forget about negative time,so t 1 is the one that matters. That's thetime at which the squirrel intercepts theacorn.To convert that back into height, plug it intoeither of your original equations. (A good wayto check your work is to put it into both andmake sure they match.)h 3t 32 3 32 29They intercept at 29 feet, in one second.Systems of Equations and InequalitiesUnpackingThe student applies mathematical processes What does this standard mean that a studentto formulate systems of equations andwill know and be able to do?inequalities, use a variety of methods to solve,and analyze reasonableness of solutions. Thestudent is expected to:3(D) determine the reasonableness ofsolutions to systems of a linear equation anda quadratic equation in two variables;DetermineWhen solving quadratic equations, 2solutions are expected. However, both
solutions will not always make sense. Thestudents need to understand ideas likenegative time doesn’t exist, and negativedistance is invalid. Students should have theopportunity to determine whether asolution is reasonable. This would be a greatforum topic that would give the kids a chanceto discuss t
Algebra II Texas Mathematics: Unpacked Content . essential knowledge and skills for mathematics, guided by the college and career . skills listed for each grade and course is intentional. The process standards weave t
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5th Grade Mathematics Unpacked Content Updated September, 2015 5thGrade Mathematics Unpacked Content This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers.
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