Keira Godwin Time Allotment: 3 das Unit Objectives: Upon completion of this unit, students will be able to: o Simplif comple rational fractions. o Solve comple rational fractional equations. o Solve quadratic equations b factoring, completing the square, and the quadratic formula. o Determine the discriminant and thus the nature of the roots for a quadratic equation. o Graph quadratic equations b hand (using a table and graph paper). o Determine the maimum/minimum, verte, ais of smmetr and the domain and range of a given quadratic equation. o Graph quadratic equations using a calculator. o Transformations of quadratic equations. (Vertical and horizontal shifts). o Put a quadratic equation into verte form using completing the square. h k o Graph circles given an equation of a circle. o Determine the center and radius of a circle given an equation of a circle. o Change equation of a circle into center-radius form using completing the square. o Solve sstems of equations (linear and quadratic) to find points of intersection. New York State learning standards A.A.3 Add, subtract, and multipl monomials and polnomials A.A.5 Find values of a variable for which an algebraic fraction is undefined A.A.6 Simplif fractions with polnomials in the numerator and denominator b factoring both and renaming them to lowest terms A.A.7 Add or subtract fractional epressions with monomial or like binomial denominators A.A. Solve all tpes of linear equations in one variable A.A.5 Solve equations involving fractional epressions A.G.4 Identif and graph linear, quadratic (parabolic), absolute value, and eponential functions A.G.5 Investigate and generalize how changing the coefficients of a function affects its graph A.G.0 Determine the verte and ais of smmetr of a parabola, given its graph (See A.A.4) Note: The verte will have an ordered pair of integers and the ais of smmetr will have an integral value. A.A.4 Determine the verte and ais of smmetr of a parabola, given its equation (See A.G.0 ) G.G.7 Write the equation of a circle, given its center and radius or given the endpoints of a diameter G.G.73 Find the center and radius of a circle, given the equation of the circle in center-radius form G.G.74 Graph circles of the form ( h) + ( k) = r A.A.47 Determine the center-radius form for the equation of a circle in standard form G.G.70 Solve sstems of equations involving one linear equation and one quadratic equation graphicall Assesment Plan- Throughout this unit students will be given numerous problems in class to work on independentl. While the are working I will be able to monitor and check for their understanding and give individual help to those who do not understand. There will be homework assigned ever night and the students will be given an opportunit to ask questions the following da. B spending this class time reviewing homework it will not onl reinforce the material but it allows for the teacher to asses which questions are not clear to the students and which topics might need further instruction and review. This combined with the classwork will allow the teacher to decide if the pace and content should be modified.
There will be two tests throughout the unit. The first will cover simplifing comple rational fractions, and solving quadratics. This will allow a focus on the methods necessar for the rest of the unit (especiall completing the square since it will be used to change equations into the forms the students will need later on). The second test will be a combination of all topics learned and will also contain word problems to test the students application of the concepts learned. Da Lesson Objectives Warm-up/Development Duration Mied # s, Simplifing Comple Rational Epressions -Change numbers from mied numbers into improper fractions. - Simplif fractions b factoring and canceling out terms. -Do-now changing mied numbers into improper fractions. -discuss and define what a comple rational epression is. -method to simplif. -several eamples for in class practice. -homework-worksheet Solving fractional equations 3 Solving quadratic equations 4 Quadratic Formula 5 Graphing Quadratics b hand -Factor denominators to find a LCD(Least Common Denominator) -Solve Fractional Equations for missing variable(s). -Check for etraneous roots and determine which roots to reject. -Solve b Factoring. -Solve b Completing the square. - Solve using quadratic formula. - Determine the nature of the roots using the discriminant. -Construct a table to determine coordinates of the quadratic equation. - determine domain and range of the given equation. -determine the verte, ma/min. - determine ais of smmetr (using formula). -go over homework - discuss the steps to solve fractional equations. - several in class eamples. - homework-worksheet on solving fractional equations. -do now- chart reviewing the terms coefficient of, and constant. - review solving b factoring. - discuss the steps to solve using completing the square. - several eamples for Classwork. - homework completing the square worksheet - do now- review completing the square steps and do an eample. -derivation of quadratic formula using completing the square.(give as eample and then derive for them). -discuss the nature of the roots based off the value of the discriminant. - Eamples using quadratic formula. Homework- worksheet asking for discriminant, nature of the roots, and solve b quadratic formula. -do-now- solve equation b completing the square, and solve equation b quadratic formula.( eamples) - introduce topic and review making a table (remind them it is the same as graphing a line). -define verte, ma/min, domain/range, ais of smm. And show how to find each. - give several eamples for them to tr. -Homework- one eample and review packet. 6 Graphing quadratics using calculator - Graph a quadratic using calculator. - find table using calculator to help them sketch graph. - Do now- graph b hand and determine verte, ma/min, domain and range, and ais of smm of a quadratic equation. - Show students how to use calculator to graph and how to obtain table from
calculator. -Homework- review packet and eample using the calculator. 7 Review Review concepts for the test Target Game 8 Test Formative Assessment of student s knowledge of content. 9 Transformations -Understand the vertical transformations of the graph of =^. -Understand the horizontal shifts of the graph of =^. -Change quadratic into verte form b completing the square. 0 Circle equation/ graphing circles Solving sstems of equations(linea r and quadratic) -Find the radius and center of a circle based on the equation of a circle. -Change a quadratic equation into h k r form. -determine intersection of two equations (linear and quadratic) b substitution and solving for a variable. -do-now-graph =^, =^+, = ^-3 and describe what happens to each graph. -discuss what makes the quadratic graph move up and down, and left and right. -show an eample and tell steps to put equation into verte form and find verte easil from it. - do- now find the radius of a circle given the center and a point on the line. - discuss the equation of a circle and how to determine the radius and center. -eplain how completing the square can manipulate circle equation into center-radius form. - homework tet book pg 490-33 odd - do-now find the equation of a circle given a diameter. -Give steps to solve sstems of equations. -give several eamples -homework-tet boo pg 497-498 -77 odd Review Review concepts for the test Target Game 3 Test Formative assessment of a student s knowledge of content.
Name Test: Rational Epressions, Completing the square, Quadratic formula Simplif:.) 8 9 4 7 4 z z.) 5 5 3 3.) 3 4.) a a 5.) Perform the indicated operation: 6.) 8 6 6 6 7.) 8
Solve: 4 8.) 5 5 Solve b Completing the Square: 9.) 6 6 0 Solve b using the Quadratic Formula: 0.) 6 7 5 0 What is the Nature of the Roots?.) 6 3
Name: Test on Quadratics and Circles.) Given : 8 6 a. Put the equation into verte form: b. State the coordinates of the verte: c. State the equation of the ais of smmetr: d. Find the roots:.) Solve the following sstem algebraicall: (ou do not need to check): 5 3 5 3.) Find the equation of the circle: 8 4 4.) Find the equation of the circle centered at (,-) and passing through (5,6).
5.) The amount s (in pounds per acre) of sugar produced from sugarbeets can be modeled b the function: s 0.0655n 7.855n 556 Where n is the amount (in pounds per acre) of nitrogen fertilizer used. How much fertilizer should ou use to maimize sugar production? What is the maimum amount of sugar ou produce? (Source: sugarbeet research and education board of Minnesota and North Dakota) 6.) A truck is traveling on a one wa street approaching a tunnel that is 8 feet high and the tunnel opening is the shape of a parabola with the equation 8. The truck is 9 feet high and 8 feet wide. Decide if the truck can make it through the tunnel and justif our answer. Hint: verte = (0,8).