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Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Topic Notes
STRAND 3: QUADRATIC AND NONLINEAR FUNCTIONS
TOPIC 3.2: QUADRATICS AND SOLUTIONS
Topic Notes
Mathematical focus
The purpose of these tasks is to examine more closely the relationships among the
different solution methods for quadratic equations and to analyze quadratic graphs
from a application perspective.
Topic overview
Five different tasks are included in this topic:
Task 3.2.1: Applications—Nets and Changing Height
Task 3.2.2: Solutions Chart—Quadratic solutions
Task 3.2.3: Parameter Changes from Tables—Look at the Moves
Task 3.2.4: Patterns of Projectiles—What Goes Up
Task 3.2.5: Parameter Changes from Graphs—The Autobiography of a Cat
The tasks involve quadratic applications and graphing surface area; finding solutions of
to quadratic equations using graphs, tables, the quadratic formula, and concrete
representations; recognizing and applying parameter changes from tables; analyzing
quadratic graphs; and matching parameter changes in a story to a graphical
representation.
TExES Standards focus
TExES Standard II.004 Patterns and algebra. The teacher uses patterns to model and
solve problems and formulate conjectures. The beginning teacher:
(A) Recognizes and extends patterns and relationships in data presented in tables,
sequences, or graphs.
TExES Standard II.005 Patterns and algebra. The teacher understands attributes of
functions, relations, and their graphs. The beginning teacher:
(B) Identifies the mathematical domain and range of functions and relations and
determines reasonable domains for given situations.
(C) Understands that a function represents a dependence of one quantity on
another and can be represented in a variety of ways (e.g., concrete models,
tables, graphs, diagrams, verbal descriptions, symbols).
(F) Performs operations (e.g., sum, difference, composition) on functions, finds
inverse relations, and describes results symbolically and graphically.
(G) Uses graphs of functions to formulate conjectures of identities [e.g., y = x2 –
1 and y = (x – 1)(x + 1), y = log x3 and y = 3 log x, y = sin(x +2) and y = cos x].
November 23, 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 2. Quadratics and Solutions;
Topic Notes
TExES Standard III.011 Geometry and measurement. The teacher understands
measurement as a process. The beginning teacher:
(B) Applies formulas for perimeter, area, surface area, and volume of geometric
figures and shapes (e.g., polygons, pyramids, prisms, cylinders, cones,
spheres) to solve problems.
(C) Recognizes the effects on length, area, or volume when the linear dimensions
of plane figures or solids are changed.
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:
(A) determine the domain and range for quadratic functions in given situations;
(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.
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.
Materials
Task
Copy of task
Graphing calculator
Grid paper for nets
#1#1
x
x
x
#2
#3
#4
#5
x
x
x
x
x
x
x
x
Procedure
Task 3.2.1. Applications—Nets and Changing Height Read and discuss the directions
for Nets and Changing Height with the class before beginning the activity. Participants
work in groups of 2 to 4; assign each group a different change. During summary
discussion make sure students compare nets and demonstrate the changes in surface area
on their nets.
November 23, 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 2. Quadratics and Solutions;
Topic Notes
Task 3.2.2. Solutions Chart—Quadratic Solutions During the summary discussion of
this task, make sure participants not only complete the chart, but make connections
among the different representations. Questions to include in the discussion:
1.
2.
3.
4.
5.
6.
What do you notice about the x-intercepts, the zeros, and the factors?
What do you notice about the x-intercepts and the values on the table?
How does the value in the table for x is 0 relate to the graph?
What happens if 0 is substituted into the polynomial form?
How do the answers to the quadratic formula relate to the factors?
How do the factors relate to the algebra tile representation of the
polynomial form?
Task 3.2.3. Parameter Changes from Tables—Look at the Moves
Part 1 Changing C
• Provide every participant with a copy of “Look at the Moves”. Working
in pairs, allow them 3 or 4 minutes to finish question #1, then convene for
a whole group discussion. The “big idea” is to relate the shift of a line to
the increasing values of y.
• Allow participants 5 to 10 minutes to work in pairs on problems 2 through
5. Reconvene for a whole group discussion. Make sure all participants
understand and can demonstrate how the shift in the graph is related to
each change in values of y.
• Participants work problems 6 and 7 with their partners. Return to the
whole group for discussion.
• Participants work in pairs to complete questions 8, 9, and 10. They will
share their solutions, strategies, and paragraph explanations.
Part 2 Changing A
• This section can be used in a way similar to Part 1. The “big idea” here is
that the coefficient of the argument not only makes the graph steeper or
less steep but also increases the y-values by a constant factor.
Task 3.2.4. Patterns of Projectiles—What Goes Up Divide participants into groups of
3 or 4; each group needs to complete the full task and be prepared to discuss with the
group as a whole.
Task 3.2.5. Parameter Changes from Graphs—The Autobiography of a Cat The
presenter will read the Cat story using the graphing calculator program as an illustration.
Participants then will create their own stories and illustrate their stories using parameter
changes with quadratic functions by modifying the graphing calculator program.
Summary
The topic was constructed to strengthen the relationships among the different
representations of quadratics, solutions for quadratics, and parameter changes.
November 23, 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 2. Quadratics and Solutions;
Topic Notes
Extensions
#1 Changing Height – Assign each group of participants a different solid (square
pyramid, rectangular pyramid, triangular pyramid, cylinder, etc.). Follow the
same procedure used in Changing Height to create models of surface area
changing as height changes.
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.
Algebra I Assessments (Dana Center, 2001) includes several tasks that can be
used as extensions or open-ended assessments for the quadratics strand.
Fireworks Celebration, Golf, and How Much Paint are examples of problem
solving situations that provide excellent assessments for this topic.
November 23, 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.