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Solving Polynomial Equations
Objectives:
§
The students will be able to factor a second degree polynomial
(quadratic) equation with a leading coefficient of 1.
§
The students will be able to solve a factored second
degree polynomial equation using the Zero-Product Property.
§
The students will be able to graph on a graphing
calculator a quadratic equation and find the x-intercepts.
§
The students will recognize that the x-intercepts of the
graph of a quadratic equation are the solutions.
Download
as Microsoft Word Documents or Inspiration File:
Instructional Plan Template
Mathematics Governor’s Institute 2005
Names of
group members:
John Bradica,
Topic/Theme:
Solving a Polynomial Equation by Factoring
Level:
Algebra 1
Time
Element:
2 Periods or 1 Block
NCTM
Standards Addressed:
- Understanding Patterns, Relations
and Functions
- Represent and Analyze Mathematic
Situations and Structures Using Algebraic Symbols
PA Math
Standards Addressed:
- 2.8.11.E – Use Equations to
Represent Curves
- 2.8.11.N – Solve Quadratic
Equations Both Symbolic and Graphic
Math
Assessment Anchors Addressed:
- M.11.A.1.2.1 – Factoring Algebraic Expressions
- M.11.D.2.1.5 – Solve Quadratic Equations Using Factoring
- M.11.D.2.2.1 – Add, Subtract and Multiply Polynomial Expressions
- R.11.A.2 – Demonstrate The Ability
to Understand and Interpret Nonfiction Text Including Textbooks
Objectives:
1. The students will be able to factor a
second degree polynomial (quadratic) equation with a leading coefficient of 1.
2. The students will be able to solve a
factored second degree polynomial equation using the Zero-Product Property.
3. The students will be able to graph on a
graphing calculator a quadratic equation and find the x-intercepts.
4. The students will recognize that the
x-intercepts of the graph of a quadratic equation are the solutions.
Instructional
Strategies and Plan (include strategies used to help different types of
learners, i.e. auditory, visual, etc):
1. Model the procedure for the sling-shot
experiment.
2. Complete the guided worksheet for the
sling-shot data.
3. Class discussion or analysis of data
generated.
4. Frayer Model for new vocabulary review.
5.
Free
response question for summative evaluation
Materials/Resources:
- Graphing Calculators
- Guided worksheet and answer key
- Large rubber bands
- Ping-pong balls
Interdisciplinary
Connections:
·
Students will read the story included in the mathematical model
worksheet.
·
Technology
The students will use the graphing calculators to graph the quadratic
functions.
·
Other
Vocabulary – The students will complete a
vocabulary assignment using a word bank and a Frayer
Model vocabulary review.
Writing - The students will complete a
reflective writing.
Assessment
Strategies:
·
Formative
Evaluation (checking student understanding during the lesson):
Student worksheet (see Instructional Plan above)
·
Summative
Evaluation (How will it be determined that the objectives were achieved?):
Free Response Question see file Quadratic Free Response.doc
Correctives/Remediation:
Extensions/Enrichment:
See the Extra Credit question in the file Quadratic Free Response.doc
Special
Accommodations (special needs students)
·
Description
of the Special Needs student selected:
·
Accommodations
to use with this student:
|
Definition:
|
Facts/Characteristics: |
|
Definition:
|
Facts/Characteristics: |
|
Examples: |
Non-Examples: |
|
Examples: |
Non-Examples: |
|
Definition:
|
Facts/Characteristics: Has x˛ x˛ is
highest term Must have = One variable (x) Can have x term Can have constant |
|
Definition:
|
Facts/Characteristics: |
|
Examples:
|
Non-Examples:
Cubed 2 variables no x˛ |
|
Examples: |
Non-Examples: |
John Bradica,
Solving Polynomial
Equations by Factoring
What is the relationship between polynomial factors, the solutions to polynomial equations and their graphs?
- What is the difference between a polynomial expression and polynomial equation?
- What is the zero product property and how and when to apply it?
- How do you find the solutions to factored polynomial equations?
- What is standard form for a polynomial?
- How do you find the solutions to polynomials in standard form by factoring?
- How do you find the solutions to polynomial equations in non-standard form?
- How do you graph a simple quadratic function?
Vocab
Equation
Expression
Standard Form
Zero-Product property
Solution
Factor
Zero of a polynomial
Polynomial
Monomial
Binomial
Trinomial
Culminating activity
To be developed
Anchors
M11.A.1.2.1 – Factoring Algebraic Expressions
M11.D.2.1.5 – Solve Quadratic Equations using factoring
M11.D.2.2.1 – Add, subtract and multiply polynomial expressions
Once upon
a time in Mathemagic land, there was a
knight, Sir Doofus, who gave togas to members of the
followers of Pythagoras once a year at the height of the full moon closest to
the summer solstice. Due to budget cuts
beyond his control, this year he is giving out T-shirts by using a sling-shot.
In order to receive
a T-shirt, it must land directly at your feet, and remember that the sling-shot
always shoots to the right, never to the left.
Since you really
want one of the coveted Pythagoras Rocks! T-shirts, you need to figure out
where they will land. Fortunately, we
have inside information on the possible flight paths of these T’s. One of these is
. Using your
mathematical minion, the graphing calculator, graph this equation to find the
point(s) where the T-shirt will land.
Sketch the graph for this equation below and label the landing point(s).
Complete
the following table:
|
EQUATION |
POINT WHERE
T-SHIRT TOUCHES THE GROUND |
FACTOR THE
EQUATION |
|
|
|
|
These are the additional possible T-shirt flight
paths. Please find the landing points
for these as well:
|
EQUATION |
POINT WHERE
T-SHIRT TOUCHES THE GROUND |
FACTOR THE
EQUATION |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Fill-in
the blanks using the words provided below.
|
Curve |
Solutions |
X-Intercept |
|
Factors |
Expressions |
Quadratic |
|
Linear |
Zero-Product Property |
Y-Intercept |
1.) The
path of each t-shirt is described as a ____________________.
2.) The
curve is the graph of a ____________________ equation.
3.) (x-2)
and (x+3) are ____________________ of 
.
4.) The
____________________ tells us that x – 2 = 0 or x + 3 = 0 for the
equation .
5.) The
values 2 and –3 are the ____________________ of the equation 
.
6.) The
point where the t-shirt lands is called the ____________________ of the graph
of each quadratic equation.
7.) The point on the graph where the sling-shot releases
the T-shirt is the ____________________.
Reflection
What
relationships can you find within the table above?
Once upon
a time in Mathemagic land, there was a
knight, Sir Doofus, who gave togas to members of the
followers of Pythagoras once a year at the height of the full moon closest to
the summer solstice. Due to budget cuts
beyond his control, this year he is giving out T-shirts by using a sling-shot.
In order to receive
a T-shirt, it must land directly at your feet, and remember that the sling-shot
always shoots to the right, never to the left.
Since you really
want one of the coveted Pythagoras Rocks! T-shirts, you need to figure out
where they will land. Fortunately, we
have inside information on the possible flight paths of these T’s. One of these is. Using your
mathematical minion, the graphing calculator, graph this equation to find the
point(s) where the T-shirt will land.
Sketch the graph for this equation below and label the landing point(s).
NOTE: Depending on
students familiarity with the graphing calculator, you may have to coach them
on entering the first equation on the “y = “ screen
and using the “Zoom Standard” option to get the best picture.
Complete
the following table:
|
EQUATION |
POINT WHERE
T-SHIRT TOUCHES THE GROUND |
FACTOR THE
EQUATION |
|
|
(2,0) |
-1(x – 2)(x + 3) |
These are the additional possible T-shirt flight
paths. Please find the landing points
for these as well:
|
EQUATION |
POINT WHERE
T-SHIRT TOUCHES THE GROUND |
FACTOR THE
EQUATION |
|
|
(2,0) |
-1(x – 2)(x + 3) |
|
|
(1,0) |
-1(x – 1)(x + 4) |
|
|
(5,0) |
-1(x – 5)(x + 1) |
|
|
(7,0) |
-1(x – 7)(x + 4) |
Fill-in
the blanks using the words provided below.
|
Curve |
Solutions |
X-Intercept |
|
Factors |
Expressions |
Quadratic |
|
Linear |
Zero-Product Property |
Y-Intercept |
7.) The
path of each t-shirt is described as a _______ Curve____________.
8.) The
curve is the graph of a __ Quadratic
________________ equation.
9.) (x-2)
and (x+3) are ______ Factors ___________
of .
10.)
The Zero-Product Property tells
us that x – 2 = 0 or x + 3 = 0 for the equation .
11.)
The values 2 and –3 are the __ Solutions_________ of
the equation.
12.)
The point where the t-shirt lands is called the __X-Intercept__________
of the graph of each quadratic equation.
7.) The point on the graph where the sling-shot
releases the T-shirt is the
____ Y-Intercept ______.
Reflection
What
relationships can you find within the table above?
Students should conclude that the
solutions of a quadratic equation are the X-intercepts.
Water Balloon Target Practice
Laura, Kim, Bob and John are on different floors in the same building. They are each armed with one water balloon. They all want to hit Bernie who is 12 feet in front of the building. They each throw their balloons with a different arc given below:
Bob throws
in an arc with the equation 
.
John throws
in an arc with the equation 
.
Kim throws in an arc with the equation 
.
Laura
throws in an arc with the equation 
.
(a)Which one of them will hit Bernie??? (b) Explain how you arrived at this answer.
(Bernie)
Extra Credit: (a) Of
the four friends, which is at the highest window of the building? (b)Explain how you arrived at this answer
also.
Water Balloon Free Response Rubric
(a) 2
pts –answer is correct
1 pt – attempted to answer, but made a computational error
(b) 3 pts
– method correctly, completely and clearly explained
2 pts – method is correct, but is either incomplete or unclear
1 pts - method is mostly correct,
complete and clearly explained
0 pt - no explanation
Extra Credit
(a) 2
pts –answer is correct
1 pt – attempted to answer, but made a computational error
(b) 3 pts
– method correctly, completely and clearly explained
2 pts – method is correct, but is either incomplete or unclear
1 pts - method is mostly correct,
complete and clearly explained
0 pt - no explanation
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