Abspd.appstate.edu



ASE MA 3 Algebra Functions and ModelingSteve Schmidt, ABSPD, abspd.appstate.edu, schmidtsj@appstate.eduWorkshop Focus: This course will focus on building the algebra knowledge adult students need for adult secondary education, college, and career success. Workshop Goals:Understand and use the Standards for Mathematical Practice to develop mathematically proficient studentsUnderstand concepts necessary to successfully complete adult high school courses and pass high school equivalency examsIdentify course content requirements based on adult education standardsUnderstand what instructors should know and be able to do to deliver effective instructionKnow how to use technology, including the TI 30 XS calculator, to enhance learningKnow how to design and plan instruction based on assessment resultsKnow where to locate supplemental resourcesWorkshop Schedule8:30 - 10:00Welcome and introductionsAHS/Modeling/GED 2014 Overview Webb’s DOKFunctions Intro10:00 – 10:15Break10:15 – 11:45Looking at functions 4 ways Functions Matching Functions Foldables 11:45 – 12:45Lunch 12:45 – 2:00Algebra TilesModeling FunctionsWindow Washing Model2:00 – 2:15Break2:15 – 3:40Modeling Functions Polya Method 3:40 – 4:00 Resource sharingTake AwaysProfessional DevelopmentStandards for Mathematical Practice1. Makes sense of problems and perseveres in solving them? Understands the meaning of the problem and looks for entry points to solve it ? Analyzes information (givens, constraints, relationships, goals) ? Designs a plan? Monitors and evaluates progress and changes course as necessary ? Checks their answers to problems and asks, “Does this make sense?”2. Reason abstractly and quantitatively ? Makes sense of quantities and relationships? Represents a problem symbolically ? Considers the units involved? Understands and uses properties of operations3. Construct viable arguments and critique the reasoning of others? Uses definitions and previously established results in making arguments? Makes conjectures and attempts to prove or disprove through examples and counterexamples? Communicates and defends their mathematical reasoning using objects, drawings, diagrams, and actions? Listens or reads the arguments of others? Decides if the arguments of others make sense? Asks useful questions to clarify or improve the arguments4. Model with mathematics? Applies reasoning to create a plan or analyzes a real world problem? Applies formulas/equations? Makes assumptions and approximations to make a problem simpler? Checks to see if an answer makes sense and changes a model when necessary5. Use appropriate tools strategically? Identifies relevant math resources and uses them to pose or solve problems? Makes sound decisions about the use of specific tools. Examples may include:? Calculator? Concrete models? Digital Technology? Pencil/paper? Ruler, compass, protractor? Uses technological tools to explore and deepen understanding of concepts6. Attend to precision? Communicates precisely using clear definitions ? Provides carefully formulated explanations? States the meaning of symbols, calculates accurately and efficiently ? Labels accurately when measuring and graphing7. Look for and make use of structure? Looks for patterns or structure? Recognizes the significance in concepts and models and can apply strategies for solving related problems? Looks for the big picture or overview8. Look for and express regularity in repeated reasoning? Notices repeated calculations and looks for general methods and shortcuts ? Continually evaluates the reasonableness of their results while attending to details and makes generalizations based on findings? Solves problems arising in everyday lifeAdapted from Common Core Standards for Mathematics: Standards for Mathematical PracticeGED 2002 and 2014 Math Test ComparisonGED 2002GED 2014- There are 50 multiple choice questions - Each multiple choice question has 5 possible answer choices- The test is primarily multiple choice with about 46 items- Each multiple choice question has 4 possible answer choices- There are fill-in-the-blank, drop down, hot spot, and drag and drop itemsTest takers are given 90 minutes to complete the testTest takers are given 90 minutes to complete the testA formula page is provided in the test bookletA new formula page has more algebra and surface area but fewer basic formulasThe test is designed to measure high school equivalency The test measures high school equivalency and college and career readiness There are 2 sections, one that allows the use of the Casio FX260 There is an approximate 12 minute non- calculator section (first 5 questions) and 78 minute calculator active section using the TI30XS Virtual CalculatorEach of the following content areas are about 25%: Number sense and operations, Measurement and geometry, Data statistics and probability, Algebra functions and patterns The content is 45% quantitative problem solving (number operations and geometric thinking) and 55% algebraic problem solving that are presented in academic and workforce contexts410 is the lowest passing score, students need to average 450 to pass the total test batteryScaled scores will range from 100 to 200, the high school equivalency passing score is 150 per test (600 total score), a higher college and career readiness score will be determined in late 2014Source: GED Testing Assessment GuideGED 2014 Technology Skills KeyboardingCan your students: ?Type approximately 25 words per minute, fast enough to complete a 500 word constructed response in 45 minutes? ?Use the keys: space bar, return/enter, shift, arrows, delete, backspace, tab? ?Use the numbers and punctuation keys?Using a MouseCan your students use a mouse to: ?Point to an area or word on screen? ?Left click, right click, double click? ?Highlight and select text? ?Cut, copy, and paste text? ?Select an object or window by clicking on it? ?Drag an object to a new location? ?Open/close a pop-up window? ?Select one or more radio and/or check buttons? ?Scroll horizontally and vertically?NavigationCan your students: ?Use the screen navigation commands previous/next? ?Navigate multiple windows? ?Navigate toolbars and drop-down menus? ?Navigate screen tabs? ?Use the undo and redo operations? ?Mark a section/question as a “Flag for Review” and return to section/question as needed?Free Technology ResourcesGCF Learn FreeThis free site guides students in learning how to use a mouse and computer basics as well as helping students learn Microsoft Office applications like Word and Excel. Google: gcf learn free or go to Typing WebThis free site helps students learn keyboarding. Instructors can register students to monitor their progress. Google: typing web or go to Shifts in the Mathematics Common Core Standards Prior to the common core, state standards were “an inch deep and a mile wide.” Instructors would try to move through a huge amount of material at a quick pace in order to “cover” everything by the end of the year. This led to students with a superficial grasp of a variety of concepts. When they attempted the college placement test and postsecondary math, they performed poorly. A recent study on developmental math in North Carolina showed that only one out of five students completed the developmental math sequence and took college level math classes. Studies of math instruction in countries that consistently perform well on international math tests showed three major things: focus, coherence, and rigor. Instead of trying to cover everything, these countries focused on a deeper understanding of a smaller set of skills. There is also a coherent progression, a logical pattern where similar topics are explored each year at a greater depth with the ultimate goal of algebra instruction. Developing students’ algebraic thinking skills are a major theme throughout math instruction. Finally, there is increased rigor that encourages students to think more deeply and engage in active problem solving instead of relying on teachers to just give students the answer.Standards Shift 1: Focus on…Key ideas, understandings, and skillsDeep learning of concepts stressed Fewer concepts with more depthStandards Shift 2 - Coherence…Designing learning around coherent progressions level to levelCoherence for masteryAll roads lead to algebraic thinking – abstract reasoningStandards Shift 3 - Rigor…The level of rigor is increasing Pursuing conceptual understanding, procedural skill and fluency, and applicationOther Notes:SubjectWebb’s Depth of KnowledgeLevel 1Level 2Level 3MathematicsRequires students to recall or observe facts, definitions, or terms. Involves simple one-step procedures. Involves computing simple algorithms (e.g. sum, quotient).Examples:Recall or recognize a fact, term or propertyRepresent in words, pictures or symbols in a math object or relationshipPerform routine procedure like measuringRequires students to make decisions of how to approach a problem. Requires students to compare, classify, organize, estimate, or order data. Typically involves two-step procedures.Examples:Specify and explain relationships between facts, terms, properties or operationsSelect procedure according to criteria and perform itSolve routine multiple-step problemsRequires reasoning, planning, or use of evidence to solve problem or algorithm. May involve activity with more than one possible answer. Requires conjecture or restructuring of problems. Involves drawing conclusions from observations, citing evidence, and developing logical arguments for concepts. Uses concepts to solve non-routine problems.Examples:Analyze similarities and differences between proceduresFormulate original problem given situationFormulate mathematical model for complex situationSource GED 2014 Assessment Guide Solve each problem and identify the Depth of Knowledge for each question. Algebra 1 EOC: Webb’s Depth of Knowledge Level _______Katie and Jennifer are playing a game. Katie and Jennifer each started with 100 points.At the end of each turn, Katie’s points doubled.At the end of each turn, Jennifer’s points increased by 200.At the start of which turn will Katie first have more points than Jennifer?GED 2014: Webb’s Depth of Knowledge Level _______Hartley opened a food truck business to sell food on the street. On day 2, the business earned $112. On day 5, the business earned $367. Hartley assumes that the earnings will continue to increase at the same rate. How much will the business earn on day 10?$ GED 2014 Algebra Assessment TargetsCommon Core State Standards ReferencesAlgebraic Problem Solving Assessment Targets Content IndicatorsRange of Depth of Knowledge (DOK)A.1Write, evaluate, and compute with expressions and polynomials7.EE.1A.1.aAdd, subtract, factor, multiply and expand linear expressions with rational coefficients.1-26.EE.2A.1.bEvaluate linear expressions by substituting integers for unknown quantities.1-26.EE.2; 6.EE.6A.1.cWrite linear expressions as a part of word-to-symbol translations or to represent common settings.1-2A-APR.1A.1.dAdd, subtract, multiply polynomials, including multiplying two binomials, or divide factorable polynomials.1-26.EE.2A.1.eEvaluate polynomial expressions by substitiuting integers for unknown qualities.1-2A-SSE.2;A-SSE.3;A-SSE.4A.1.fFactor polynomial expressions1-26.EE.2; 6.EE.6A.1.gWrite polynomial expressions as part of word-to-symbol translations or to represent common settings.1-26.EE.3A.1.hAdd, subtract, multiply and divide rational expressions.1-26.EE.2A.1.iEvaluate rational expressions by substituting integers for unknown quantities.1-26.EE.2; 6.EE.6A.1.jWrite rational expressions as part of word-to-symbol translations or to represent common settings.1-2A.2Write, manipulate, and solve linear equations7.EE.4; 8.EE.7; A-REI.3A.2.aSolve one-variable linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms or equations with coefficients represented by letters.1-27.EE.4; A-CED.1; A-CED.2A.2.bSolve real-world problems involving linear equations.1-26.EE.6; A-CED.1;A-CED.2A.2.cWrite one-variable and multi-variable linear equations to represent context.1-28.EE.6;A-REI.6A.2.dSolve a system of two simultaneous linear equations by graphing, substitution, or linear combination. Solve real-world problems leading to a system of linear equations.1-2A.3Write, manipulate, solve, and graph linear inequalitiesA-REI.3A.3.aSolve linear inequalities in one variable with rational number coefficients.1-26.EE.8; 7.EE.4A.3.bIdentify or graph the solution to a one variable linear inequality on a number line.1-27.EE.4;A-CED.1;A-CED.2A.3.cSolve real-world problems involving inequalities.1-26.EE.2;A-CED.1;A-CED.2A.3.dWrite linear inequalities in one variable to represent context.1-2A.4Write, manipulate, and solve quadratic equationsA.REI.4A.4.aSolve quadratic equations in one variable with rational coefficients and real solutions, using appropriate methods. (e.g., quadratic formula, completing the square, factoring, inspection)1-2A-CED.1A.4.bWrite one-variable quadratic equations to represent context.1-2A.5Connect and interpret graphs and functions6.NS.6A.5.aLocate points in the coordinate plane.18.F.4A.5.bDetermine the slope of a line from a graph, equation, or table.1-28.EE.5A.5.cInterpret unit rate as the slope in a proportional relationship. 2A-CED.2;F-IF.7A.5.dGraph two-variable linear equations1-28.F.3; 8.F.5; F-IF.5A.5.eFor a function that models a linear or nonlinear relationship between two quantities, interpret key features of graphs and tables in terms of quantities, and sketch graphs showing key features of graphs and tables in terms of quantities, and sketch graphs showing key features given a verbal 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.1-2A.6Connect coordinates, lines, and equationsA-CED.2A.6.aWrite the equation of a line with a given slope through a given point.1-2A-CED.2A.6.bWrite the equation of a line passing through two given distinct points.2G-GPE.5A.6.cUse slope to identify parallel and perpendicular lines and to solve geometric problems. 1-2A.7Compare, represent, and evaluate functions8.EE.5A.7.aCompare two different proportional relationships represented in different ways. Examples include but are not limited to: compare a distance-time graph to a distance-time equation to determine which of the two moving objects has a greater speed.28.F.1; F-IF.1A.7.bRepresent or identify a function in a table or graph as having exactly one output (one element in the range) for each input (each element in the domain)1-2F-IF.2A.7.cEvaluate linear and quadratic functions for values in their domain when represented using function notation.1-28.F.2; F-IF.9A.7.dCompare properties of two linear or quadratic functions each represented in a different way (algebraically, numerically in tables, graphically or by verbal descriptions). Examples include but are not limited to: given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.2Teaching Functions Using a One to One CorrespondenceVacation Winners Toaster/Function MachineWhat is a function?A relationship where each input (x) has only one matching output (y). Function Examples in 4 Forms: Table/Equation/Words/GraphTableThis table shows a function since each input (x) has only one output (y):XY0- 1122538EquationThis equation shows a function since there can be only one output for every input: y = 3x - 1As we plug in different input values for x, we will always get only one output value for y. Putting in 0 for x will give us - 1 for y, putting in 1 for x will give us 2 for y, and this pattern will continue. WordsWe can also describe a function in words: y is equal to three times x minus one. A real life example could be: Keisha sells programs at football games. For each program she sells, she earns $3. Her cost to produce each program is $1. Her profit is described by the function, profit (y) is equal to three times the number of programs she sells (x) minus the one dollar production cost. GraphThe graph for the function y = 3x -1 is shown here: Function NotationName of the function 43574318986500 The function9890497014600 f (x) = 3x + 4119240544486Tells what number to plug in the function176713110294973x + 403x + 42001047199705910010820833359646 2 2f (x) is the same as y: y = 2x + 1 is the same as f(x) = 2(x) + 1. We use f(x) since it makes it easier to see what happens as we put in different inputs for x:f(x) = 2(x) + 1f(0) = 2(0) + 1 f(0) = 1f(1) = 2(1) + 1 f(1) = 3f(2) = 2(2) + 1 f(2) = 5f(3) = 2(3) + 1 f(3) = 7Linear, Quadratic, and Exponential Function GraphsLinearQuadraticExponentialDescription:Straight lineU Shape (parabola) that opens up or downGrows/Shrinks fast and levels off on 1 side Graph looks like:Equation y = 2xy = 3x2 - 8y = 2xHow to tell from the equation what the graph looks like:The x term is to the first power (x)The x term is to the second power (x2)The x term is the exponent (2x)Graph specific information:In the equation y = mx + b, if m is negative, the line decreases (goes down) as we look from left to right: y = - x + 5If m is positive, the line increases (goes up) as we look from left to right: y = .5x + 2If the number with the x2 is negative, the U shape opens down. If the number with the x2 is positive, the U shape opens up:In the equation y = ax , if a is bigger than 1 the curve will grow fast. If a is between 0 and 1, the curve will shrink fast. Finding the x and y Intercepts in a Linear FunctionTo find the x intercepts in a linear function like y = 2x + 6, we are going to set y = 0 and solve for x. To find the y intercept, we’ll set x = 0 and solve for y:x InterceptSet y = 0 and solve for x:y = 2x + 60 = 2x + 6 - 6 = 2x -3 = xWhen y = 0, x is -3 so we have the ordered pair: (-3,0)y InterceptSet x = 0 and solve for y:y = 2x + 6y = 2(0) + 6 y = 6When x = 0, y is 6 so we have the ordered pair: (0,6)Quadratic Functions - Minimums and MaximumsGraphs of quadratic functions either have minimums and maximums. The maximum is the highest point on the upside down U shape (or parabola) where we have the largest y value. It is also called the vertex. The minimum is the lowest point on the upward facing U shape (or parabola) where we have the smallest y value. It is also called the vertex. Quadratic Functions - SymmetryThe axis of symmetry is the vertical (straight up and down) line through the vertex on the graph of a quadratic function. It cuts the U shape in half. Quadratic Functions – InterceptsThe x intercept(s) in a quadratic function are the places where the graph crosses the x axis (where y = 0). The y intercept is the place where the graph crosses the y axis (where x = 0).Increasing/Decreasing/ConstantWe always read graphs like we read a book from left to right. Look to see where the graph is going up, down, or staying the same:Where is this graph increasing, decreasing, and remaining constant?PeriodicityFunctions that have periodicity repeat their values in regular intervals or periods. End BehaviorYoutube: How to Describe End Behavior of Functions How Can We Tell Which of Two Functions Has a Greater Rate of Change?California offers five cents for every aluminum can brought in to recycle. Michigan offers 10 cents for every aluminum can turned in for recycling. If we represent California by the equation y = .05x and Michigan by the following table, which function has the greater rate of change?Cans (x)Money (y)1.102.203.304.405.50101.00In the table, the rate of change is found by taking the change in y divided by the change in x:.20 - .10 = .10 = .10 2 – 1 1In the equation, to find the rate of change, look at the number with the x term (sometimes called the independent variable). In y = .05x , .05 is the rate of change. So, the function in the table has the greater rate of change (.10 is bigger than .05). Adapted from Understanding and Comparing Functions Lesson 5 of 6 (Youtube)Let’s PracticeJamal can walk a certain number of miles per hour as described in this equation: y = 4x. His friend Marquisha’s speed is shown on the chart below. Who can walk faster?Time (x)In HoursMiles (y).53161.59212530Your TurnCreate 2 functions and show them by a graph, in words, in an equation, or in a table on a piece of flip chart paper (or on a whiteboard). Have another group guess which has the bigger rate of change. Activities with FunctionsFunction MatchingThis activity gives students practice in noticing four representations of a function: words, graphs, equations, and tables. You will be given 13 functions and must match the cards representing the function in words, graphs, equations, and tables. Be careful with the graphs as not all the scales are the same! The master for this activity is on the abspd.appstate.edu website. Adapted from Foldables This activity takes a minimum of prep time and all you need are a few sheets of card stock. Cut the card stock in half and have students fold it in fourths. Give students a function equation and have them show it in words, graphs, and as a table. You could also vary the activity and have students start with either a function in words/graphs/tables and show the other forms. Adapted from Math Activities for Migrating to the Common Core Classroom (Donna Ware)Real Life FunctionsAnother way to develop algebraic thinking in adult learners and answer the, “When are we ever gonna use this?” question is to have students explore situations where linear functions are found. Once students have brainstormed and identified areas where functions are found in real life, they should represent the function in words, as a graph, in an equation, and as a table. One idea to get you started:- If gas costs $3.25 a gallon and I buy none, it costs $0. If I buy 1 gallon it costs $3.25, 2 gallons cost $6.50, 3 gallons cost $9.75 etc. The follow up activity to this is a gallery walk where students look at other students work posted around the classroom. Adapted from Life Functions by Donna Parrish in The Math Practitioner, Spring 2013. Algebra Tiles Practice 1. Show x – 42. Show 5x2- 2x3. Show 4x2- 2 x + 64. Show x2 - 7x + 45. Simplify -2x2 + 6x + 3 - 4x + 46. Simplify 3x2 - 3x + x2 + 2x - 47. Simplify x2 - 2x + 2 + 3x2 – 2x + 48. Simplify 2x2 - 2x + 1 + 7x - 4 9. (2x2 – 3x + 2) + (-3x2 – x)10. (x2 - 3) + (-3x2 – 3x + 1)11. (-5x2 – x – 3) + (4x – 2)12. (x2 + 4x + 4) + (-2x2 – 6x + 3)13. (x2 – 6x – 1) – (-3x2 + 2x + 2)14. (4x2 + 4x + 2) – (-x2 + 3x – 2)15. (x2 + 2x + 6) – (-4x2 + 6x – 2)16. (3x2 – 6x – 1) – (2x2 – x – 4)17. x – 2 = 518. 2x – 1 = 319. 3x + 2 = 520. 2x + 3 = 921. (x – 2) (2x + 1)22. (3x + 1) (x – 2)23. (x + 3)(x – 2)24. (2x - 2) (x + 2)Polya Problem Solving MethodUnderstand the problemWhat are you asked to do?Will a picture or diagram help you understand the problem? Can you rewrite the problem in your own words?Create a planUse a problem solving strategy:Guess and check Solve an easier problemMake a list ExperimentDraw a picture or diagram Act it outLook for a pattern Work backwardsMake a table Change your viewpointUse a variable Carry out the planBe patient Be persistent Try different strategiesLook back and reflectDoes your answer make sense? Are all the questions answered?What other ways are there to solve this problem?What did you learn from solving this problem? Understand the problemCreate a planCarry out the planLook back and reflectProblems from NC Algebra 1 End of Course Test 1. What scenario could be modeled by the graph below?The number of pounds of apples, y, minus 2 times the number of pounds of oranges, x, is at most 5.The number of pounds of apples, y, minus half the number of pounds of oranges, x, is at most 5.The number of pounds of apples, y, plus two times the number of pounds of oranges, x, is at most 5. The number of pounds of apples, y, plus half the number of pounds of oranges, x, is at most 5.2. The function of f(t) = -5t2 + 20t + 60 models the approximate height of an object t seconds after it is launched. How many seconds does it take the object to hit the ground?Suppose that the function f(x) = 2x +12 represents the cost to rent x movies a month from an internet movie club. Makayla now has $10. How many more dollars does Makayla need to rent 7 movies next month?The table below shows the cost of a pizza based on the number of toppings.Number of Toppings (n)Cost (C)1$122$13.503$154$16.50Which function represents the cost of a pizza with n toppings?C(n) = 12 +1.5(n - 1)C(n) = 1.5n + 12C(n) = 12 + nC(n) = 12n Source: ResourcesProfessional DevelopmentGoogle: nc basic skills training calendar The 2013-14 training calendar shows all the professional development opportunities available across the state. Professional development will be held in each region. Adult High School Adult High School Algebra Standards Google: nc dpi algebra 1 unpacked These are North Carolina’s standards for high school Algebra 1AHS Algebra 1 Course for AHS Algebra 1 under continuing education courses This is a complete Algebra 1 course that can be used for Adult High School2003 NC Standard Course of Study and Algebra 1 CrosswalkGoogle: nc dpi algebra 1 crosswalk This site compares the new Common Core Algebra 1 standards with the 2003 NC Standard Course of StudyAlgebra ResourcesGoogle: get the math the Math show how algebra is used in the real world in music, sports, fashion, videogames, restaurants, and special effects. ManipulativesGoogle: national library of virtual manipulatives site has algebra manipulatives for instruction across grade levels Algebra Tiles Google: algebra 4 all learnportThis site provides an easy to use set of virtual algebra tiles. ................
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