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Solving Quadratic Equations

Unit Plan

Solving Quadratic Equations

Objectives

    Students will learn the characteristics and properties of quadratic equations and their graphs. They will learn how to:

  • transition between the different representations of quadratic equations (equation, table, graph, and context).

  • solve quadratic functions using factoring methods and the quadratic formula.

  • apply the solving techniques learned in the unit to real-life situations and how to decide what technique would be most appropriate in a situation.

  • interpret solutions to real-life problems and explain which solutions are reasonable for the situation.

Essential Questions

  • How can we show that algebraic properties and processes are extensions of arithmetic properties and processes and how can we use algebraic properties and processes to solve problems?

  • What functional representation would you choose to model a real-world situation and how would you explain your solution to the problem?

Related Unit and Lesson Plans

Related Materials & Resources

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Formative Assessment

  • View

    Multiple Choice Items:

    1. -5 and (9/2) are the solutions to which of the following equations?

    01a.PNG

    2. Using the Quadratic Formula, what is the solution to x² - 7x + 4 = 0?


    02a.PNG

    3. Solve the following quadratic equation by factoring: 03a.PNG

    03b.PNG

    4. Which of the following is the correct table representing y = 2x² - x - 6

    A

    x

    -2

    -1

    0

    1

    2

    y

    -8

    -7

    -6

    -5

    -4



    B

    x

    -2

    -1

    0

    1

    2

    y

    0

    -5

    -6

    -3

    4



    C

    x

    -2

    -1

    0

    1

    2

    y

    8

    2

    0

    2

    8



    D

    x

    -2

    -1

    0

    1

    2

    y

    4

    -3

    -6

    -5

    0


    5. What is the equation that most accurately represents the area of a rectangle with a length 5 cm longer than its width and with an area of 36 cm2?

    A

    (x + 9)(x + 4) = 36

    B

    (x - 9)(x - 4) = 36

    C

    x² + 5x = 36

    D

    (x + 5)² = 36

    6. The graph represents which of the following quadratic equations?

    06a.PNG



    7. The height of a baseball thrown from 4 feet above the ground at a velocity of 41 feet per second can be modeled by the equation h(t) = -16t² + 41t + 4, where h(t) represents the height of the ball above the ground after t seconds. At what time will the ball reach the ground after being thrown?

    A

    0.1 sec

    B

    4 sec

    C

    2.7 sec

    D

    1.3 sec

    8. What are the roots of 2x² - 6x + 3 = 0?

    08a.PNG

    9. Which of the following quadratic equations is in standard form?

    A     2x² + 5 = -13x
    B     x² + 6x - 9 = 0
    C     15 - 6x + x² = 0
    D     8x + 2 = -4x²


    10. What are the coordinates of the vertex and zeros for the quadratic equation y = x² + 2x - 8?

    A

    Zeros: (0, 2) & (0, -4)

    Vertex: (-9, -1)

    B

    Zeros: (0, -2) & (0, 4)

    Vertex: (0, -8)

    C

    Zeros: (-2, 0) & (4, 0)

    Vertex: (-9, -1)

    D

    Zeros: (2, 0) & (-4, 0)

    Vertex: (-1, -9)


    Multiple Choice Answer Key:

    1. D

    2. D

    3. B

    4. D

    5. C

    6. A

    7. C

    8. A

    9. B

    10. D


    Short Answer Items:

    11. Which method would be most efficient to solve the following quadratic and why? Choose the best method and then solve the equation. Show all work.


    x² - 3x - 28 = 0



    12. A cliff diver dives off a 120 foot cliff at a velocity of 4 ft per second. The height of the diver from the surface of the water below can be modeled by the equation h(t) = -16t² + 4t + 120 where h(t) is the height of the diver and t is in seconds. Find the height of the diver after 2 seconds. How long does it take the diver to reach the surface of the water below?




    13.Generate a table of five x and five y values for the equation y = x2 + 5x +19.

    x






    y










    14. Generate a graph of the quadratic equation given below. Label the coordinates of the zeros and the vertex on the graph y = x² - 9

    14a.PNG


    Short-Answer Key and Scoring Rubrics:

    11. x = 7, x = -4


    Points

    Description

    2

    • Correctly finds solutions using the method the student chose.

    • Reasoning behind the method choice effectively demonstrates understanding of the differences between methods.

    • Shows all work.

    1

    • Provides the correct answer but does not provide sufficient reasoning behind why the student chose a specific method.

    • Gaps in work or support for the problem.

      OR

    • Appropriately describes the reasoning behind the method choice but has some errors in calculations.

    • Demonstrates partial understanding in solving quadratic equations.

    0

    • Incorrect solutions.

    • No explanation for the method choice.

    • Demonstrates no understanding of solving quadratics.

    12. After 2 seconds, the diver is 64 feet above the water’s surface. It takes the diver 2.9 seconds to reach the water’s surface.

    Points

    Description

    2

    • Demonstrates correct interpretation of variables.

    • Correctly answers both parts of the question.

    • Demonstrates understanding of the scenario presented.

    • Correctly interprets the relationship between situation and solving quadratic equations.

    • Shows all work.

    1

    • Correctly identifies the solution to one part of the problem.

    • Demonstrates partial understanding of the scenario presented.

    • Incorrect interpretation of variables, which leads to incorrect solutions. Yet, methods of solving are correct for their interpretation.

    0

    • Demonstrates no understanding of problem to be solved.

    • Solutions are incorrect.

    • No work presented.

    • Understanding of the relationship between problem and solving quadratics is not shown.
    13. x² + 5x + 19 = 0

    Answers will vary.

    x

    -2

    -1

    0

    1

    2

    y

    13

    15

    19

    25

    33

    Points

    Description

    2

    • Correctly generates five x- and five corresponding y-values.

    1

    • Correctly generates three or four x- and three or four corresponding y-values.

    0

    • Incorrect answer.

    • No explanation provided.

    • Does not demonstrate understanding of the relationship between the tabular and algebraic representations of quadratic equations.

    14.

    14key.PNG

    Points

    Description

    2

    • Demonstrates ability to graph quadratic equations correctly.

    • Correctly determines the vertex and zeros of the equation on the graph.

    • Clearly demonstrates understanding of graphing quadratic equations.

    1

    • Correct graph but does not label the zeros or vertex.

    • Demonstrates understanding of the zeros and vertex of a quadratic graph, but the actual graph is incorrect.

    0

    • Incorrect or no graph.

    • Does not demonstrate understanding of the zeros and vertex of a quadratic equation.

    Performance Assessment:

    The amount of money A in an account in which P dollars is invested for two years is given by equation A = P(1 + r)², where r is the interest rate compounded annually. Ben deposits the amount of $1,250 into an account and it grows to $1,428 in two years.

    1. Based on the problem, determine the values of A and P.

    2. Substitute the values of A and P into A = P(1 + r)².

    3. Write the equation from (2) in quadratic standard form. Hint: you will need to FOIL (First Outside Inside Last).

    4. Generate a graph that represents this equation.

    5. Solve the quadratic equation from the substitution of A and P into A = P(1 + r)²  ..

    6. Explain what your solution means in terms of the problem/situation.

    Performance Assessment Scoring Rubric:

    1. A = $1428 P = $1250
    2. $1428 = $1250(1 + r)2
    3. 0 = 1250r2 + 2500r – 178
    4.
    pa04.PNG

    5. a = 0.0688 = 6.88%
    6. Annual percent rate = 6.88 on $1250 on deposit for two years.

    Points

    Description

    4

    • Correctly identifies values of A and P–showing understanding of variables in the equation.

    • Correctly substitutes values of A and P into equation.

    • Correctly FOILs and arranges quadratic in standard form.

    • Work is correct, neat, and easy to follow.

    • Demonstrates an understanding of the graph of the function.

    • Is able to correctly determine the solution to the problem.

    • There is a precise and thorough explanation of the meaning of the solution.

    3

    • Correctly identifies variables.

    • Correctly uses the variables in the equation.

    • Equation is correctly derived.

    • Graph is correctly modeled.

    • Explanations and interpretations are brief but correct.

    • Mathematics is correct, but only some work is shown.

    2

    • Demonstrates understanding of the equation given and its variables but has difficulty putting the equation in standard form.

    • Has an incorrect graph, but attempts to graph a quadratic from the graph s/he has.

    • Math has some errors and gaps in the work.

    • Explanations are brief and/or incorrect.

    1

    • Work has many errors.

    • Standard form of the equation is seriously flawed.

    • Graph of the equation is seriously flawed.

    • Explanations are incorrect.

    0

    • Demonstrates no understanding of the variables and their meanings.

    • Is not able to graph the equation.

    • No work is shown or work is completely incorrect.

    • Explanations are missing.

    • Is not able to come to a final answer to the problem.
DRAFT 10/21/2010
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