Planning Guide
Creating Learner-Focused
Schools
* Madison-Oneida BOCES- This document may not be
reproduced in any form without the expressed written consent of the District
Superintendent or his designee.
| LU Title: Polar Coordinates
& Graphing
|
Author(s): Carl Wheat
|
| Grade Level: 12
|
School Address: Clinton Central
School
|
| Subject Area: Math 12
|
School Phone/Fax:
315-853-5574
|
CONTENT
KNOWLEDGE
| Declarative
|
Procedural
|
- Describe the Polar Coordinate System
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- Evaluate the function of an angle
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- Describe the Rectangular Coordinate System
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- Plot points on the polar coordinate system
|
- State the formulas used to convert from polar to rectangular form and vice
versa
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- Convert radian measure to degree measure
|
- State and describe the differences between the three subtypes of the
Limacon
|
- Convert coordinates from polar to rectangular and vice versa
|
- Describe the effect of changing the value of a parameter in the equation of
a Rose, Limacon, or Lemniscate
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- Graph a given polar curve
|
- State the differences between a Lemniscate, Limacon, and a Rose curve
|
- Change the value of a parameter in a graphing template
|
| |
|
ESSENTIAL
QUESTIONS
- How does the critical analysis of polar graphing help you to
describe the universe around you?
INITIATING ACTIVITY
Students will go to the computer lab and be introduced to the graphing
utility Studyworks. They will be shown how to use four templates each of which
is designed to graph a type of polar curve.
Students will then be given a set of 21 equations to be graphed using the
templates mentioned above. They will be expected to copy a sketch of each graph
onto polar graph paper. As they proceed through the exercise they will be
attempting to answer questions about each type of graph. The questions are
aimed at getting them to begin to understand how to analyze a curve using a
graphing utility.
*Studyworks-Product of Mathsoft Inc.-101 Main Street, Cambridge, Mass. 02142
Studyworks is a full screen graphing and calculating tool, a math word
processor, a computer algebra system, and a reservoir of facts and formulas.
The technology part of this unit could be done using graphing calculators,
or other graphing packages that allow polar graphing.
Connection to State
Learning Standards
Content Area: Math
Level: High School 12th Grade: The performance indicators listed below are
from the Math B draft document published by the state called Mathematics Core
Curriculum.
| 4A-Represent problem situations symbolically by using algebraic
expressions, sequences, tree diagrams, geometric figures, and
graphs.
|
|
4C-Choose appropriate representations to facilitate the solving
of a problem.
|
| 4K-Determine the effects of changing parameters of the graphs
of functions.
|
|
5C-Derive and apply formulas relating angle measure and arc
degree measure in a circle.
|
Unit Theme:
Polar Coordinate & Graphing
| 5E-Define trigonometric functions in terms of a unit
circle.
|
|
7D-Analyze the effects of parametric changes on the graphs of
functions.
|
Learning
Experiences
Declarative Knowledge
| What declarative knowledge should e in the
process of acquiring & integrating? As a result of the unit, the student
will know or understand…
|
What experiences or activities will be used
to help students acquire & integrate this knowledge?
|
What strategies will be used to help
students construct meaning, organize and/or store the knowledge?
|
Describe what will be done.
|
| How to locate a point on a
polar graphing coordinate system.
How to describe the differences between the three types of Limacon curves.
How to describe the differences between the Rose, the Limacon, and the
Lemniscate.
How to describe the effect of changing the value of a parameter in a the
rose, the Limacon, and the Lemniscate.
The formulas that are used to convert from polar to rectangular form and
vice versa.
|
Close your eyes and imagine you
are in a control room with radar screens. Describe what you see on those radar
screens? Use think, pair, and share.
Students will graph many curves in the computer lab.
Students will graph many curves in the computer lab.
Students will graph many curves in the computer lab.
Students will superimpose a polar graph system over a rectangular graph
system.
|
Students will be able to compare
their results with their neighbors.
A class discussion of the their results will be held and then their results
will be compared with a prepared set of differences and similarities.
Students will draw a picture. Students will be asked to make connections to
the Pythagorean theorem and also to trig relationships from Math III.
|
Students will
locate points on polar graphing paper and check their answers with their
neighbors (pairs check).
Students will be asked to answer questions designed to help them see
differences and similarities.
Students will be given a summary sheet to compare with their results and
then asked to apply this new knowledge to identify and describe another set of
polar curves.
Students will have the opportunity to test their own values for the
parameters in the templates of each type of curve.
Students will be asked to draw a rectangular grid and locate a point in the
first quadrant. Then they will be asked to find the hypotenuse (the distance
from the origin to the point) of the triangle formed when a perpendicular is
drawn from the point to the x-axis and the origin and the points are connected.
|
Learning
Experiences
Procedural Knowledge
| What procedural knowledge will students be
in the process of acquiring & integrating? As a result of this unit,
students will be able to:
|
What will be done to help students
construct models, shape & internalize the knowledge?
|
Describe what will be done.
|
| Convert coordinates from
polar to rectangular and vice versa.
Evaluate the function of a given angle.
Plot the coordinates of a point in polar form.
Identify and sketch the graph of a given equation in polar form.
Convert an equation in rectangular form to an equation in polar form and
vice versa.
|
Students will superimpose a polar
graph system over a rectangular graph system.
Pairs check
Pairs check
Pairs check, Generate graphs on a graphing utility
Think/pair/share
Homework problems
|
Students will be
asked to draw a rectangular grid and locate a point in the first quadrant. Then
they will be asked to find the hypotenuse (the distance from the origin to the
point) of the triangle formed when a perpendicular is drawn from the point to
the x-axis and the origin and the points are connected.
A number of problems will be done in class that require students to evaluate
the function of a given angle.
Students will be asked to plot points by first locating the terminal ray and
then counting the distance from the pole.
Students will be asked to graph a number of curves in polar form with and
without their notes. Homework will be assigned.
Students will be asked to use the conversion formulas to convert and
equation from one system to the other.
|
Learning
Experiences
Extending and Refining
| What knowledge will students be extending
and refining? Specifically, they will be extending and refining their
understanding of…
|
What reasoning process will they be
using?
|
Describe what will be done.
|
| 1) Graphing using the polar
coordinate system by making connections between it and the rectangular
coordinate system.
2) How to determine the effect of replacing parametric values in equations
with constants.
|
 Classifying
- Inductive Reasoning
- Deductive Reasoning
- Error Analysis
- Analyzing Perspectives
- Constructing Support
Abstracting
|
1) Students will use their prior
knowledge of trigonometry and the Pythagorean theorem to develop a connection
between polar coordinates and rectangular coordinates.
2) Students will use computer software and prepared templates to generate
graphs of many curves. They will then be asked to analyze and classify the
results of their findings.
|
Learning
Experiences
Meaningful Use Tasks
| What knowledge will students be using
meaningfully? Specifically, they will be demonstrating their understanding of
and ability to…
|
What reasoning process will they be
using?
|
Describe what will be done.
|
| Determine the effect of changing
the values of a with respect to b and of changing c in the polar equation
r=_____c_____
a + b sin 0Ú
|
Experimental Inquiry (generating
an explanation for a phenomenon and testing the explanation)
|
Students will be given a template
in Studyworks set up with the given equation.
Students will be given a problem set that asks them to analyze the effects
of making c negative or positive, to analyze the effect of changing the values
of a and b. In particular they will be asked to predict what will happen when
a<b, a>b, and a=b.
They will be asked to make connections if possible with prior curves they
have studied.
As an extension they will be asked to show how to get from the rectangular
form of the equation to the polar form.
|
| Step 1
|
Step 2
|
Step 3
|
| What knowledge will students be
using meaningfully? Specifically, they will be demonstrating their
understanding of and ability to...........
|
What reasoning process will they be
using?
|
Describe student's products and
performances and the criteria for evaluation.
|
| |
[ ] Decision Making
(selecting from seemingly equal alternatives or examining the decisions of
others)
[ ] Problem Solving
(seeking to achieve a goal by overcoming constraints or limiting conditions)
[ ] Invention
(creating something to meet a need or improve on a situation)
[ ] Experimental Inquiry
(generating an explanation for a phenomenon and testing the explanation)
[ ] Investigation
(resolving confusions or contradictions related to a historical event, a
hypothetical past or future event, or to the defining characteristics of
something)
[ ] Systems Analysis
(analyzing the parts of a system and how they interact)
[ ] Other:
|
Products/Performances
|
| Criteria for evaluation
|
Rubric:
Key Questions:
What are the key elements, traits, or dimensions that will be evaluated?
Are the identified elements of equal importance or will they be weighed
differently?
|
|
Element #1
|
Element #2
|
Element #3
|
Element #4
|
| Elements
Scale
|
Makes
Prediction
|
Tests Prediction
|
Data Collection
|
Learning Outcome
|
| Weights
|
1
|
1
|
2
|
2
|
| 4
|
The
student makes a prediction regarding the relationship of a to b for each case,
and a prediction about the effect of c.
|
The
students sets up 4 examples for each case and then tests each case. Negative
values for b are included.
|
The
student writes down the data for each case tested in an organized method and
makes a sketch or drawing.
|
The student makes
a conclusion from the data about each case. The conclusion is complete and is
stated in good form. The conclusion includes some generalization.
|
|
3
|
The
student makes a prediction but does not relate all cases of a and b, or does
not make a prediction about c.
|
The
student sets up 2 examples for each case and then tests each case. Negative
values for b are not included.
|
The
student does not collect the data from all the cases that were prepared or the
sketches do not agree with the examples.
|
The student
provides a conclusion with no important errors. Algebraic results are stated
with no comment as to connection with tests or prediction.
|
|
2
|
The
student makes a prediction about the parameters that shows a misunderstanding
of the problem
|
The
student sets up 1 example for each case and then tests each case.
|
The
student produces errors in collecting the data.
|
The student
provides an incomplete explanation.
|
|
1
|
The
student makes a prediction that is not verifiable by changing the given
parameters.
|
There
is some evidence of preplanning.
|
The
inaccurate or incomplete results are due to omissions in the process.
|
The student
provides an incorrect interpretation of the relationship between a and b or the
value of c.
|
|
0
|
No
prediction is made.
|
No evidence of
test cases or preplanning.
|
No
evidence of tests being made.
|
No
conclusion or interpretation is stated.
|
NOTE: Rubric or other
performance assessment instruments may be used.
Constructing a Holistic
Scoring Tool
(Rubric or Activity Specific Key)
Key Questions:
* How many score points are needed to discriminate among the full range of
different degrees of understanding, proficiency, or quality?
This response, product, or performance provides evidence of
understanding of
concept/principle/generalization or
proficiency in
skill/process/strategy.
| Score Point
4
|
Score Point
3
|
| [ ]
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[ ]
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| [ ]
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[ ]
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| [ ]
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[ ]
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| [ ]
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[ ]
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| Score Point
2
|
Score Point
1
|
| [ ]
|
[ ]
|
| [ ]
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[ ]
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| [ ]
|
[ ]
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| [ ]
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[ ]
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Have You Considered These Yet?
Learn to Learn Skills:
Assessment Modifications:
Unit Schedule/Time Plan:
Written Overview: