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Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 5. Capstone Problem —
Curves Ahead; Topic Notes
STRAND 3: QUADRATIC AND NONLINEAR FUNCTIONS
TOPIC 3.5: CURVES AHEAD (CAPSTONE)
Topic Notes
Mathematical focus
Participants will examine the general quadratic equation y = Ax 2 + Bx + C to
determine the role of the constants A, B, and C. Participants will also design
scenarios of real world problems that require quadratics around specific quadratic
equations.
Topic overview
This topic contains 3 tasks. Two topics are for the participants and a third task that is
a modification of these for the high school Algebra 1 student. The tasks are:
Task 3.5.1: Curves Ahead — Teacher Version
Task 3.5.2: Discussion and Discovery
Task 3.5.3: Curves Ahead — Student Version
TExES Standards focus
TExES Standard II.006 Patterns and algebra. The teacher understands linear and
quadratic functions, analyzes their algebraic and graphical properties, and uses them to
model and solve problems. The beginning teacher:
(E)
Makes connections between the y = ax2 + bx +c and the y = a(x –
h)2 + k representations of a quadratic function and its graph.
(F)
Solves problems involving quadratic functions using a variety of
methods (e.g., factoring, completing the square, using the quadratic
formula, using a graphing calculator).
(G)
Models and solves problems involving linear and quadratic
equations and inequalities using a variety of methods, including
technology.
TEKS/TAKS focus
TEKS A.9 Quadratic and other nonlinear functions. The student understands that the
graphs of quadratic functions are affected by the parameters of the function and
can interpret and describe the effects of changes in the parameters of quadratic
functions. The student is expected to:
(B)
investigate, describe, and predict the effects of changes in a on the
graph of y = ax2 + c;
(C)
investigate, describe, and predict the effects of changes in c on the
graph of y = ax2 + c; and
(D)
analyze graphs of quadratic functions and draw conclusions.
November 24, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
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Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 5. Capstone Problem —
Curves Ahead; Topic Notes
TEKS A.10 Quadratic and other nonlinear functions. The student understands there
is more than one way to solve a quadratic equation and solves them using appropriate
methods. The student is expected to:
(A)
solve quadratic equations using concrete models, tables, graphs,
and algebraic methods; and
(B)
make connections among the solutions (roots) of quadratic
equations, the zeros of their related functions, and the horizontal
intercepts (x-intercepts) of the graph of the function.
High School TAKS Objective 5: The student will demonstrate an understanding of
quadratics and other nonlinear functions.
Materials
Task
Copies of Task
Graphing Calculator
Grid Paper
Colored Pencils
Chart paper
3.5.1
X
X
X
X
3.5.2
X
X
X
X
X
3.5.3
X
X
X
X
Procedure
1. Task 3.5.1. This activity can either be assigned to groups or to individuals. Give
each participant a copy of the task. At the end of questions 6, 10, and 14, bring
the participants together as a whole to discuss their results.
2. Task 3.5.2. This activity is designed to be assigned to groups of 3 or 4. Assign
each group problem 1. Have the group record their findings on chart paper. After
the groups have completed the problem, have each group report their findings. A
class discussion should follow. Repeat this process for problems 2 and 3.
3. Task 3.5.3. This activity need not be completed by the participants but is an
example of how Task 3.5.1 and 3.5.2 might be modified for their high school
Algebra 1 students. Discussion should include ways in which task 1 has been
modified and why, as well as, what parts of task 1 and 2 are not appropriate for
the Algebra 1 students and why.
Summary
These tasks emphasize the importance of the quadratic in areas of real life.
Participants will be more confident in their creation of story problems that are
connected with a quadratic equation. In addition, participants will have a better
understanding of the role of A, B, and C in the function y = Ax 2 + Bx + C .
Participants will also observe the different forms of the quadratics understand the
difference between them.
November 24, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
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Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 5. Capstone Problem —
Curves Ahead; Topic Notes
Extensions
Similar exercises can be created using different quadratic equations than the ones
provided.
Assessment/Transition to the Classroom
Participants should complete the Teacher’s Journal, recording their responses, and
making any modifications necessary to make the tasks completely classroom
ready.
November 24, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.