Instructional Leadership Initiative: Supporting Standards-based Practice
Unit:
Intermediate Algebra: Conics
School:
Mar Vista High School
Strengths:
The philosophy of instruction presented at the beginning of the unit offers readers
an insight as to how scaffolding of learning is envisioned.
The annotation of anchor papers provides clear explanations for the rational behind
each rating. The analysis presented in each of these annotations represents more
than a cursory examination of the work. These analyses could also become
informative tools for reteaching and for revamping of the unit.
Concerns:
The standard requires much more from students than does this assessment. All of
the equations presented on the assessment are in general form, whereas the
standard calls for identification and graphing of conics given equations in various
forms.
Students are being held accountable for identifying the conics and specifying their
characteristics although there are no explicit instructions requiring them to do so. It
is difficult to understand how students could be penalized for not doing something
that has not been explicitly stated as a requirement.
All items on the assessment require the same type of approach and analysis. With a
limited format of tasks and only one item per conic type, it is not apparent that
results from this assessment would offer reliable data for judging whether students
meet the standard.
Mar Vista High School – Intermediate Algebra: Conics
Revised 10/14/02
Western Assessment Collaborative at
Page 1
Instructional Leadership Initiative: Supporting Standards-based Practice
I.
BACKGROUND
Unit Title:
Grade Level: 10th
Conics
Unit Designers:
Steven Case, Mark James, Sandra Rivera
Discipline/Course Title:
Timeframe:
Intermediate Algebra (2nd Semester)
10 Days
Teacher to Teacher Notes:
After completing this unit, students will be able to recognize, identify
and graph the four major conics (parabola, circle, ellipse and hyperbola).
Degenerate conics and eccentricity will not be covered in this unit.
Each student will be able to work with general and specific forms of the
conics. Various instructional methods will be utilized to provide access
to different learning modalities, including auditory, visual and
kinesthetic. This topic is part of California Intermediate Algebra
Standard 17.0 and Sweetwater Standards 4.1 through 4.4, which covers
general and standard forms of conics.
Printed Materials Needed:
Text (McDougal Littell Algebra 2)
Section 10.2 – 10.6 (pg. 595 – 631),
Chapter 10 Resource Book,
Practice Worksheets, 2 Unit
Quizzes, Unit Exam, Practice
Worksheets (Parabolas & Circles,
Ellipses & Hyperbolas). Conic 3x5
cards
Resources (non-print):
Internet Resources:
Mar Vista High School – Intermediate Algebra: Conics
Revised 10/14/02
Western Assessment Collaborative at
Instructional Leadership Initiative: Supporting Standards-based Practice
II. CONTENT STANDARDS ADDRESSED
The required content knowledge
California State Standard 17.0, Sweetwater Union
State/District: High School District Standard 4.1 – 4.4
Title: Conics
California State Standard
17.0 Given a quadratic equation of the form Ax2 + By2 + Cx + Dy
+ E = 0, students can use the method for completing the square to
put the equation into standard form and can recognize whether the
graph of the equation is a circle, ellipse, parabola, or hyperbola.
Students can then graph the equation.
District Standard 4.1 – 4.4
4.1 Students are able to identify and graph parabolas (with the
equation given in various forms) and determine the zeros, vertex,
directrix, and axis of symmetry.
4.2 Students are able to identify and graph circles (with the
equation given in various forms) and determine the radius.
4.3 Students are able to identify and graph ellipses (with the
equation given in various forms) and determine the foci, major and
minor axis, and vertices and co-vertices.
4.4 Students are able to identify and graph hyperbolas (with the
equation given in various forms), determining the foci, vertices,
and asymptotes.
Due to the complexity of the conic standards, students will be
introduced to the standard by starting with simpler topics and
progressing to more complex conics. To maintain simplicity
throughout the unit, each new conic was introduced centered on the
origin.
Parabolas
Given students have been exposed to parabolic functions in previous
courses, parabolas would be the most logical conic to start with. In
order to rewrite the equation in standard form, students shall complete
the square. Then, according to the standard form of the equation,
students will be able to identify the major characteristics of the
parabola (vertex, directrix, and axis of symmetry) in order to graph it
correctly.
Stating the philosophy for how
the standard is to be presented
is valuable information;
however, what is missing is an
unpacking of the standard
which would delineate all of
the vocabulary, skills,
formulas, and concepts
subsumed within the focus of
the standard(s). The given
information restates the
standard(s) but does not
clarify what meeting the
standard would really require.
Mar Vista High School – Intermediate Algebra: Conics
Revised 10/14/02
Western Assessment Collaborative at
Instructional Leadership Initiative: Supporting Standards-based Practice
Circle
Due to the familiarity of its shape, the circle is the next logical conic to
study. After correctly completing the square to write the equation in
standard form, students shall be able to identify center and radius in
order to graph it correctly.
Ellipse
The next logical conic to cover is the ellipse given its similarity to the
circle. After correctly completing the square to write the equation in
standard form, students shall be able to identify the center, foci,
vertices, and major and minor axes in order to graph it correctly.
Hyperbola
Hyperbolas are covered last due to their complexity. After correctly
completing the square to write the equation in standard form, students
shall be able to identify the center, foci, vertices, transverse axis, and
the equations of the asymptotes in order to graph it correctly.
Enabling Prerequisite Skills:
•
•
•
•
•
Understand the equations and graphs of parabolas and lines.
Understand and use the Pythagorean theorem.
Capable of completing the square and factoring perfect square
trinomials.
Simplify square roots.
Able to graph conic sections centered on the origin given its
standard form.
The last prerequisite skill listed
seems misplaced since it is
clearly part of the identified
standard.
Teacher to Teacher Notes:
Throughout this conic unit, the specific standard of completing the
square and graphing is embedded in the instruction of the prerequisite
skills needed for student success. Due to the nature of the instruction
of this unit, the 10 days stated to cover this standard are not
consecutive. Days required to develop prerequisite skills are
interspersed throughout the unit. As the standard addresses
completion of the square to graph each conic, each shape will need to
be presented first centered at the origin in standard form.
Various forms of instructional strategies need to be used in order to
accommodate students’ different learning styles. This can include the
following:
• Lecture and board presentations. (visual and auditory)
Mar Vista High School – Intermediate Algebra: Conics
Revised 10/14/02
Western Assessment Collaborative at
Instructional Leadership Initiative: Supporting Standards-based Practice
•
•
“Living Graph” – using actual students to represent major
characteristics like the vertices or foci on a coordinate plane on a
set of axes. Students can visualize concepts of the symmetry of
each conic. (kinesthetic and tactile)
Visual demonstrations of certain conic sections:
Ellipse demonstration by posting two tacks on a board and
encircling them with a loose string.
Flashlight(s) demonstration for all four conics
Standard mathematical three-dimensional models of the conic
sections.
3x5 card activity for kinesthetic learners (see teacher-to-teacher
notes core lesson 9).
Mar Vista High School – Intermediate Algebra: Conics
Revised 10/14/02
Western Assessment Collaborative at
Instructional Leadership Initiative: Supporting Standards-based Practice
III. THE ASSESSMENT
What students will need to do to provide evidence that
they have met the standard.
Type(s) of Evidence Required to Assess the
Standard(s):
Students should demonstrate the ability to:
•
•
•
Identify specific conic families given a mixture of all kinds.
Change from the general form of a conic to specific standard form
for all families of conics.
Identify the standard form of each conic family.
Assessment Method(s):
Both the California and district
standards require graphing.
This skill seems absent in what
students must do to
demonstrate evidence of
meeting the standard.
1 Unit Exam
Teacher to Teacher Notes:
For Student-Ready Assessment, see attached (Standard 4 Exam 1
Form A & B). The assessment prompts were kept minimal for the
exam. There are no directions to list the major characteristics of each
graph, or even to identify each graph. In order to master the standard,
a student must identify and list information pertinent to each graph.
Prompts will have already been given to the students in previous
opportunities to succeed in this standard. For each Core Lesson, a quiz
is included (Standard 4 Quiz 1 Form A & B and Quiz 2 Form A & B)
to measure how well the students understand the difference between
each conic.
Holding students accountable
for unspecified expectations
hardly seems justified. Explicit
instructions on each
assessment are usually
assumed to be standard
elements
Assessment Prompt(s):
See attached
Mar Vista High School – Intermediate Algebra: Conics
Revised 10/14/02
Western Assessment Collaborative at
Instructional Leadership Initiative: Supporting Standards-based Practice
Standard 4 Exam 3 Form A
Period ___ Name_____________________________
Row ____
Intermediate Algebra Teacher ____________________ Date ______
SUHSD Objectives:
4.1 Students are able to identify and graph parabolas (with the
equation given in various form) and determine the zeros,
vertex, directrix, and axis of symmetry.
4.2 Students are able to identify and graph circles (with the
equation given in various forms) and determine the radius.
4.3 Students are able to identify and graph ellipses (with the
equation given in various forms) and determine the foci,
major and minor axis, and vertices and co-vertices.
4.4 Students are able to identify and graph hyperbolas (with
the equation given in various forms), determining the foci,
vertices, and asymptotes.
Text Sections: 10.2 through 10.6
SUHSD
Standard
4.
14.
4
1.
Change each equation from general form to standard
form and graph in the coordinate plane provided.
9x2 + 25y2 – 54x – 50y – 119 = 0
Text
Sec.
10.6
25 pts
The standard requires much
more from students than does
this assessment. All of the
equations are presented here
in general form, whereas the
standard calls for identification
and graphing of conics given
equations in various forms.
The directions do not require
students to identify the conic
nor specify characteristics of
the conic. It is difficult to
understand how students
could be penalized for not
doing something that has not
been explicitly stated as a
requirement. All items require
the same type of approach and
analysis. With a limited format
of tasks and one item per
conic type, it is not apparent
that results from this
assessment would offer
reliable data for judging
whether students meet the
standard.
Mar Vista High School – Intermediate Algebra: Conics
Revised 10/14/02
Western Assessment Collaborative at
Instructional Leadership Initiative: Supporting Standards-based Practice
SUHSD
Standard
4.
14.
4
2.
4.
14.
4
3.
4.
14.
4
4.
Change each equation from general form to standard
form and graph in the coordinate plane provided.
x2 + y2 – 8x + 6y + 24 = 0
Text
Sec.
10.6
25 pts
x2 – 4y2 + 10x – 16y + 5 = 0
10.6
25 pts
x2 – 6x + 4y + 5 = 0
10.6
25 pts
Mar Vista High School – Intermediate Algebra: Conics
Revised 10/14/02
Western Assessment Collaborative at
Instructional Leadership Initiative: Supporting Standards-based Practice
IV. CRITERIA FOR SUCCESS
What will be expected of the students on the
assessment?
Characteristics of a High Quality Response to the
Assessment:
•
•
•
•
Change the equation from the general form to the standard form.
Identify all four conics correctly.
Graph the standard form of the equation correctly.
Graph the major characteristics of the conics correctly.
There is no stated requirement
on the assessment that
students identify the conics
correctly. Without that explicit
requirement it is difficult to
justify that a high quality
response would include this
characteristic.
Mar Vista High School – Intermediate Algebra: Conics
Revised 10/14/02
Western Assessment Collaborative at
Instructional Leadership Initiative: Supporting Standards-based Practice
V.
OPPORTUNITIES TO LEARN AND PERFORM
Instructional plan to assure that every student has
adequate opportunities to learn and practice what is
expected.
Opportunities to Learn:
•
•
•
•
•
Students will graph conic sections (parabola, circle, ellipse,
hyperbola) centered at the origin after Lecture and board
presentations. (visual and auditory).
Students will identify characteristics of each conic section given
the standard form or the graph through activities such as the
“Living Graph” and visual representations of each conic.
Students will graph conic sections that have been translated off the
origin.
Students will complete the square to change from a conic’s general
form to its standard form.
Students will recognize and identify the difference between each of
the four conic sections given its graph or equation (general or
standard) after 3x5 card activity for kinesthetic learners (see
teacher-to-teacher notes core lesson 9).
Students will graph conic
sections, but based upon what
given type of information?
Opportunities to Perform:
•
•
•
Students will practice graphing, writing and identifying conic
equations in class and for homework through activities in the text
and worksheets attached.
Students will complete a series of 2 Quizzes (see attached).
Students will complete 1 Unit Exam
Mar Vista High School – Intermediate Algebra: Conics
Revised 10/14/02
Western Assessment Collaborative at
Instructional Leadership Initiative: Supporting Standards-based Practice
VI. THE PERFORMANCE STANDARD
Rubric or other form of scoring guide
Scoring Guide – Conics
Mar Vista High School (Intermediate Algebra)
Meets the Standard
Level 4
• Identify all four conics correctly.
• Completes the square correctly with no conceptual
errors and no arithmetic errors.
• Graphs all major characteristics of all four conics
correctly.
Level 3
• Identify all four conics correctly.
• Completes the square correctly with no conceptual
errors with standard arithmetic errors.
• Graphs the conics according to their standard
equation (graphs correlate with the incorrect
completion of square).
• Graphs may have some graphical errors or
misunderstandings.
There is no explicit direction
for students to identify the
conic in each item.
What is a “conceptual error” as
described here? Does this
mean using an incorrect
algorithm?
Must students also graph the
complete conic to attain a
Level 4?
What is a “standard arithmetic
error?”
Does Not Meet The Standard
Level 2
• Identifies all four conics correctly.
• Completes the square with conceptual errors.
• Missing or attempted graphs (parts of curves but
not the curves themselves).
• Graphs have graphical errors or
misunderstandings.
Level 1
• Incorrectly identifying any of the four conics.
• Incomplete attempt to complete the square.
• Major conceptual and arithmetic errors, if any
attempt at all.
• Severely illogically defined graphs, if any at all.
Mar Vista High School – Intermediate Algebra: Conics
Revised 10/14/02
Western Assessment Collaborative at
Instructional Leadership Initiative: Supporting Standards-based Practice
VII. SAMPLES OF STUDENT WORK WITH
COMMENTARY
Commentary – Overview:
Paper B: Score 4
Meets the Standard
As you can see, this paper can be used as the key as there are
absolutely no conceptual errors in completing the square or arithmetic
errors throughout the process of transforming from general to standard
form in any of the conic equations. To obtain a level 4 score a student
is not allowed any errors, conceptual or arithmetic. Obviously, all the
conics were identified correctly, which is also necessary throughout a
level 3 or even level 2 score. Each of the graphs is also essentially
correct. All the conics are centered correctly. The ellipse and
hyperbola are both horizontal with correct vertices, co-vertices, and
asymptotes. The circle has the exact radius. Finally, the parabola is
facing down and of correct width.
Mar Vista High School – Intermediate Algebra: Conics
Revised 10/14/02
Western Assessment Collaborative at
Instructional Leadership Initiative: Supporting Standards-based Practice
Mar Vista High School – Intermediate Algebra: Conics
Revised 10/14/02
Western Assessment Collaborative at
Instructional Leadership Initiative: Supporting Standards-based Practice
Paper D: Score 3
Meets the Standard
This paper might seem deserving of a level 4 score if you merely
consider the Algebra. There are no conceptual or arithmetic
misunderstandings concerning the algebra, just the same as Paper B.
However, if you consider the graphs, there are obvious graphical
misunderstandings. All of the vertices, co-vertices, asymptotes,
centers and major characteristics of each graph are correct in
accordance with the algebraic standard form. This student has an
apparent misconception about the shape of conics. The ellipse comes
together with little curve and too ‘pointy’ at the vertices. The
hyperbola curves away from the asymptotes, the very opposite
meaning of an asymptote. Finally, the parabola, although with correct
width, curves so sharply after the two points not on the axis of
symmetry, it makes two vertical lines. Although these graphical
misunderstandings might seem picky, remember that standard
arithmetic errors, for example, adding or subtracting the constant
incorrectly or forgetting a sign while factoring, are allowed through
the completion of the square in a level 3 score. To further draw the
distinction between a level 3 and level 4 score we must look past the
algebraic process and also consider the graphical process. Obviously,
graphical errors would follow any faulty completion of square in
obtaining the standard form. If a student graphs a conic according to
their erroneous standard form, that does not prove any graphical
misunderstanding, it just reinforces their algebraic mistakes.
However, the standard explicitly states that a student not only has to
complete the square and distinguish between each conic, but also must
graph each of them. Other graphical misunderstandings include
misconceptions of the verticalness or horizontalness of each conic or
errors concerning the dimensions of each conic. It is the graphical
misinterpretation that keeps this paper at a level 3 score.
Mar Vista High School – Intermediate Algebra: Conics
Revised 10/14/02
Western Assessment Collaborative at
Instructional Leadership Initiative: Supporting Standards-based Practice
Mar Vista High School – Intermediate Algebra: Conics
Revised 10/14/02
Western Assessment Collaborative at
Instructional Leadership Initiative: Supporting Standards-based Practice
Mar Vista High School – Intermediate Algebra: Conics
Revised 10/14/02
Western Assessment Collaborative at
Instructional Leadership Initiative: Supporting Standards-based Practice
Paper L: Score 2
Does Not Meet The Standard
This paper will show how the lack of conceptual knowledge in either
completing the square or factoring can keep a student from meeting
the standard. The student identifies all four conics correctly, which is
required for a level 2 score. However, the student shows no evidence
that what is added to one side must also be added to the other while
completing the square – a major conceptual misunderstanding. The
standard forms for each equation seem ‘forced’, although semi-correct
(aside from a few sign errors) for the circle and ellipse with no
identification of balancing either equation. The hyperbola proves this
fact without a doubt as two non-square coefficients lie underneath each
variable. There are obvious graphical misconceptions as well with the
hyperbola evidenced by the nonsymmetrical dimensions. Without
mastery of the algebraic or graphical concepts this can earn no higher
than a level 2 score.
This argument supports the
notion that students who have
not mastered prerequisite
algebraic skills may learn new
concepts but be unable to
demonstrate understanding.
Mar Vista High School – Intermediate Algebra: Conics
Revised 10/14/02
Western Assessment Collaborative at
Instructional Leadership Initiative: Supporting Standards-based Practice
Mar Vista High School – Intermediate Algebra: Conics
Revised 10/14/02
Western Assessment Collaborative at
Instructional Leadership Initiative: Supporting Standards-based Practice
Paper I: Score 1
Paper I: Score 1
Does Not Meet The Standard
This student starts by identifying all four conics correctly, although the false
identification of any one of the conics would earn a level 1 score. The attempt to
complete the square is seen in the ellipse, circle and hyperbola, however major
misunderstandings are present. In the ellipse the student starts to group, factor and
complete the square, even remembering to add the correct amount on the opposite
side for balancing. However, the constant on the right side remains, as the
coefficients on the left side that were factored to begin with mysteriously disappear
and nothing is divided to finish the standard form. The completion of square for the
hyperbola is incomplete evidenced by the lack of balance in the equation of the
second step. The parabola was not even attempted. The standard form for the circle
is achieved, however it is a combination of algebraic and complete graphical
misconception that leaves this paper as a good example of a level 1 score.
Mar Vista High School – Intermediate Algebra: Conics
Revised 10/14/02
Western Assessment Collaborative at
Instructional Leadership Initiative: Supporting Standards-based Practice
Mar Vista High School – Intermediate Algebra: Conics
Revised 10/14/02
Western Assessment Collaborative at
Instructional Leadership Initiative: Supporting Standards-based Practice
Teacher to Teacher Notes:
We realize that our final assessment needs minor adjustments. As the
standard explicitly states, students must first identify each conic. This
instruction prompt was missing from the assessment. As the assessment
stands now, it is implied but not stated that to graph each of the four conics
the students must first identify each of the conics. In addition, the assessment
needs to specify to list all the major characteristics for each conic. Although
the state standard does not include writing equations (just graphing them), to
increase the rigor, a question about writing the equation either in standard or
general form given its graph or pertinent information can also be included in
the assessment.
The inclusion of a more
rigorous type of item would
help to discern whether
students had real
understanding of the concepts
and could apply them.
Mar Vista High School – Intermediate Algebra: Conics
Revised 10/14/02
Western Assessment Collaborative at
Instructional Leadership Initiative: Supporting Standards-based Practice
Student Forms
(Assessment)
Mar Vista High School – Intermediate Algebra: Conics
Revised 10/14/02
Western Assessment Collaborative at
Standard 4 Exam 3 Form A
Period ___ Name___________________
Row ____
Intermediate Algebra Teacher ____________________ Date ______
SUHSD Objectives:
5.1 Students are able to identify and graph parabolas (with the equation given in
various form) and determine the zeros, vertex, directrix, and axis of symmetry.
5.2 Students are able to identify and graph circles (with the equation given in various
forms) and determine the radius.
5.3 Students are able to identify and graph ellipses (with the equation given in various
forms) and determine the foci, major and minor axis, and vertices and co-vertices.
5.4 Students are able to identify and graph hyperbolas (with the equation given in
various forms), determining the foci, vertices, and asymptotes.
Text Sections: 10.2 through 10.6
Change each equation from general form to standard form and graph in the
coordinate plane provided.
SUHSD
Standard
4.14.4
1.
9x2 + 25y2 – 54x – 50y – 119 = 0
Text
Sec.
10.6
25 pts
4.1
4.4
2.
x2 + y2 – 8x + 6y + 24 = 0
10.6
25
pts
Standard 4 Exam 3 Form A
Change each equation from general form to standard form and graph in the
coordinate plane provided.
SUHSD
Standard
4.1
4.4
3.
x2 – 4y2 + 10x – 16y + 5 = 0
Text
Sec.
10.6
25
pts
4.1
4.4
4.
x2 – 6x + 4y + 5 = 0
10.6
25
pts
Instructional Leadership Initiative: Supporting Standards-based Practice
Teacher Forms
(Answer Key)
Mar Vista High School – Intermediate Algebra: Conics
Revised 10/14/02
Western Assessment Collaborative at