Lesson Objectives€¦ · Web viewLesson Objectives. Students will be able to state general shape of a graph based on the exponent. Students will be able to determine how a positive

Lesson Objectives

Students will be able to state general shape of a graph based on the exponent Students will be able to determine how a positive or negative coefficient will affect the direction

of the graph Students will be able to determine why a higher exponent has more effect on a graph than a

smaller one Students will learn the skills of adding and subtracting polynomials. Students will gain ability to perform operations on polynomials becomes very important when

students get to calculus. Students will notice that there are different ways to write the equations to make them easier to

simplify. The students will review factoring integers, and connect the review to factoring

polynomials. The students will be able to factor quadratic equations. The students will be able factor third degree polynomials.

Students will apply the quadratic equation to solve polynomial functions. Students will define the steps of the quadratic formula. Students will describe the importance of the discriminant of the quadratic formula in finding the

solutions for x. Students will be able to find factors of special cubic polynomials Students will be able to find factors of cubic polynomials by grouping Students will be able to solve cubic equations by factoring

Students will identify the steps of the Rational Root Test. Students will define zero or root of a polynomial. Students will use the Rational Root Test to find the zeros of a polynomial. Students will identify other methods of finding the zeros of a polynomial such as by factoring or

graphing.Students will be able to identify how many possible roots a graph has

Students will be able to graph polynomial functions Students will be able to determine local minimums and maximums

Abissi, Pozolo, Schlinkert, Varney

Assessment Plan

Ideology

Our assessments will focus more on formative assessment than summative assessment With formative assessment, we can better help students as they progress through the unit

instead of just seeing how students scored as a whole. Summative assessment will be used so that the teacher can understand how well the

students learned the information, and make changes that are deemed necessary for the next time the unit is taught.

Types of Assessment

There will be a few different types of lesson assessments. We will assess the work in the exploration, and we will assess bookwork.

Through assessing exploration lessons, we can gain a better understanding of the students’ prior knowledge by the information they use as an aide for their explorations. We can also assess the main ideas presented in the lesson.

Bookwork will be designed to cover the main points of the lesson rather than covering given theorems and the students’ efficiency in using these theorems to solve large lists of problems.

Process of Assessment

Our unit will consist of daily homework, two quizzes, and one final unit test. Homework will be graded on accuracy to help understand how well the students

understood the information that was presented. If the majority of the students struggled in a specific area, we can represent that information or focus on that information in review sessions.

The quizzes will be no harder or easier than any of the information covered in the homework. The quizzes will cover the main ideas of the homework, and will give students a chance to see what homework lessons should be reviewed.

The final test will be a culmination of all the information that was presented throughout the unit. The test questions will not be much harder than quiz questions, but the test will cover a broader range of information. Students will be encouraged to use quizzes and homework as a study tool to understand what information will be covered.

Special Considerations

In order to promote fairness, every student will be given the same questions, but tests and quizzes can be adapted to suit the needs of special needs students. For example, a student that lacks fine motor skills may use a computer or any other approved device to give


answers. Another example would be a blind student may use brail, or have a para-pro help give questions, but para-pros are not allowed to help with answering questions.

Classmates can and will be encouraged to work together on homework problems, so that they may help each other for better understanding.

Students are encouraged to rework assignments that they did not do well on. If a student did poorly on a homework assignment, another assignment covering the same material can be given, and the teacher will give the student a grade averaging the two assignments if the second grade is better.

Quizzes and tests will be done individually, and any student caught cheating will be held accountable by the standards listed in the syllabus.

Quizzes and tests cannot be re-taken. An assignment, quiz, or test can be given to an individual student at a later date only if

that student’s absence is excused.


Michigan Merit Curriculum – Mathematics

High School Mathematics Content Expectations: Algebra II

Standard A1: Expressions, Equations, and Inequalities

A1.1 Construction, Interpretation, and Manipulation of Expressions

A1.1.1 Give a verbal description of an expression that is presented in symbolic form,

write an algebraic expression from a verbal description, and evaluate

expressions given values of the variables.

A1.1.4 Add, subtract, multiply, and simplify polynomials and rational expressions.

A1.1.5 Divide a polynomial by a monomial.

A1.2 Solutions of Equations and Inequalities

A1.2.2 Associate a given equation with a function whose zeros are the solutions of the

equation.

A1.2.9 Know common formulas and apply appropriately in contextual situations.

Standard A2: Functions

A2.1 Definitions, Representations, and Attributes of Functions

A2.1.1 Recognize whether a relationship (given in contextual, symbolic, tabular, or

graphical form) is a function, and identify its domain and range.

A2.1.2 Read, interpret, and use function notation, and evaluate a function at a value in

its domain.

A2.1.3 Represent functions in symbols, graphs, tables, diagrams, or words, and

translate among representations.


A2.1.6 Identify the zeros of a function, the intervals where the values of a function are

positive or negative, and describe the behavior of a function as x approaches

positive or negative infinity, given the symbolic and graphic representations.

A2.1.7 Identify and interpret the key features of a function from its graph or its

formula(s).

A2.2 Operations and Transformations with Functions

A2.2.1 Combine functions by addition, subtraction, multiplication, and division.

A2.2.2 Apply given transformations to parent functions, and represent symbolically.

A2.3 Representations of Functions

A2.3.1 Identify a function as a member of a family of functions based on its symbolic or graphical representation; recognize that different families of functions have different asymptotic behavior.

A2.3.3 Write the general symbolic forms that characterize each family of functions.

A2.4 Models for Real-World Situations Using Families of Functions

A2.4.1 Identify the family of functions best suited for modeling a given real-world

situation.

A2.4.2 Adapt the general symbolic form of a function to one that fits the specifications

of a given situation by using the information to replace arbitrary constants with

\ numbers.

A2.4.3 Using the adapted general symbolic form, draw reasonable conclusions about


the situation being modeled.

A3.6 Rational Functions

A3.6.1 Write the symbolic form and sketch the graph of simple rational functions.

A3.6.2 Analyze graphs of simple rational functions and understand the relationship

between the zeros of the numerator and denominator, and the function’s

intercepts, asymptotes, and domain.


Materials List

Technology:

Classroom computers equipped with Internet access

Ti-Nspire computer software

Ti-Nspire CAS handheld

Calculator Based Ranger (CBR) device

Overhead projector

Worksheets:

CBR “Walk the Graph” activity worksheet

Multiplying Polynomials worksheet

Synthetic Division worksheet

Steps to Creating an Excel Program for the Quadratic Equation worksheet

Factoring worksheet

Rational Zeros Test Exploration worksheet

Other:

Student White Boards

Meter stick

Masking tape

A Smile


Technology Plan

Our classroom will have the following technologies

Computers – We will have 15 computers around the perimeter of the classroom as well as one at the front of the class for the teacher’s instruction. Each computer will have TI- Nspire CAS software installed. We chose to use the Nspire CAS system over a different software because we are familiar with the syntax and more comfortable using it.

Calculators – Each student will have their own TI- Nspire CAS calculator, which they can take home with them at the end of the day. We chose to use calculators with a CAS system because of the dynamic nature of our lessons. We wanted students to be able to see how graphs change as they change the polynomial.

Projector/Visualizer – The classroom will have a projector which is connected to the computer at the front of the classroom. There will also be a visualizer to show paper documents and other items not on the computer. We chose to use this type of projector as opposed an overhead with transparencies for the convenience of not having to make transparencies for all of the overhead work.

Other – The classroom will contain a set of CBRs


Management Plan

Classroom Rules:

1. Be in your seat when the bell rings.

I want class to begin as soon as the bell rings. As soon as students come into the

classroom, they need to get their calculators. Other than that, students should be seated

and working on the warm-up when the bell rings. This rule also limits dead time created

by trying to make sure everyone is seated.

2. Bring all books and materials to class.

Students cannot learn if they do not have their materials and books. If a student comes to

class without a material, he/she wastes my time and his/her own time. Coming prepared

the first time allows students to begin their work as soon as possible and could limit the

number of interruptions during the class period.

3. Follow directions the first time they are given.

Students need to know how to follow directions. Following directions is a skill they will

be using on a daily basis in the real world. Students can easily follow directions, if they

are paying attention to me. Thus not only does this rule help reduce dead time because

everyone is doing what they are supposed to, but it also helps maintain the focus on the

task at hand.

Consequences:

When a student breaks a rule or does something else that might distract me or the other students, I

would first attempt to use the invisible discipline techniques. For example, if a student is not in his seat

when the bell rings, I might just give him “the look.” Another example is of a student who is popping her

gum in class. If gum is not allowed or the act is generally disruptive, I would say the student’s name and

make eye contact while continuing my lecture. Unfortunately, students do not always respond to the

invisible discipline techniques. Once I had exhausted these techniques, I would move on the

consequences of breaking the rules. I like the five step consequence plan.

1. First time → Write student’s name on the board. This lets the student know that a more severe

consequence is on the way if he does not follow the directions.

2. Second time → Place a check by the student’s name. The student must come in after school on

an assigned day for 15 minutes. During this time the student will be asked to fill out a form that


states what they have done wrong and how they plan to change their attitude in following class

periods. I will discuss the form with the student.

3. Third time → Another check is placed by the student’s name. The student must come in after

school for 30 minutes on an assigned day. During this time the student will fill out a form stating

why they continued to do the troublesome activity and how they will fix the problem. I will

discuss the form with the student. If the student plays a sport or is involved in an extracurricular

activity, the student will be required to explain the problems he is having in class. If there is extra

time, I will have math worksheets with extra problems from the homework for the student to

attempt.

4. Fourth time → Another check in placed by the student’s name. The student must come in after

school for 45 minutes on an assigned day. Not only will the student fill out a form and discuss it

with me, but the student will also be required to call a parent or guardian and explain to him/her

what we just talked about. Extra homework will be available if there is any time left in the 45

minutes.

5. Fifth time → Student is sent right to the office. At this point the student clearly is not

responding and needs to explain to the principle or vice principle what the problem is.

Rewards:

I do not believe that students need to be bribed to follow very basic rules. Once they are in high

school, students know that in the real world they will not get a prize just for coming to work on time. I

understand that certain rewards can benefit a classroom setting at times. My rewards would be as

follows:

1. Praise (everyday) → Acknowledging students for being helpful to others or following the rules

on a daily basis is an important reward. Without much work a teacher can make a student’s day

better just by noticing the good he did.

2. Positive notes home (random) → A positive note home to a parent is a good way to encourage a

student to do well in school. A positive note can also help a teacher to develop strong

relationships with parents.

3. Free time (biweekly) → This would be an entire class reward as opposed to the previous rewards

which were individual. If all of the students in the class managed to go two weeks without

breaking a certain amount of rules, then the class would be able to have a 30 minute free time to

quietly work on other homework or some other quiet work.

4. The joy of learning → Following rules means that students are creating a positive work space for

themselves. Students will be more likely to enjoy the class if the class follows the rules because


the rules are designed to help them. If I do my job and keep things interesting, students will

enjoy learning in my class.


Description of Students

o Subject Area → 11th grade Algebra IIo Student Characteristics →

Academic: Our class would consist of mostly tenth and eleventh grade students. It would be a basic class. All of our students would already have passed algebra I or the equivalent and some geometry. The class is a basic level so I would have a large range of intelligences within my class. I assume most of my students do not have a strong interest in mathematics. By eleventh grade many students have lost interest in school, but there will be some students who are still eager to learn. It will be our jobs to bring the interest back into the students.

Personal: Our school is in a suburban neighborhood. Many of our students would be involved in after-school activities. The school would offer after-school tutoring, but students who are involved in after-school activities might not be able to use those resources. Hopefully most of my students will have stable homes to live in with supportive parents. This, of course, would not be the case for all of my students. Most of my students will be able to drive or will be close to obtaining their driver’s licenses.


Algebra 2

East Great Falls High School- 2010 School Year

Course Syllabus

Welcome to 11th grade mathematics! We are looking forward to an exciting and productive year with you.

Class Materials

1. Pencils2. Paper (This will be provided for the students)3. Notebook: All students will need a notebook. The notebook will be used for taking notes

in class, in class questions, and self assessments. Students are encouraged to take notes. The teacher does not have a copy of all notes provided.

4. 3 Ring Binder or Folder5. Calculators (TI N-Spire Plus recommended, but need TI 83 or better)

Grading Policy

Six Week Math Grade Will Consist of the following

Tests 38%

Quizzes 26%

Homework 26%

Participation 10%

Total 100%

Semester Grades will be based on the following percentages

A 93-100 C 70-74

A- 90-92 C- 67-69

B+ 87-89 D+ 64-66

B 82-86 D 58-63

B- 78-81 D- 53-57

C+ 75-77 E 52 or below

Tests 38%


There will be a test at the end of every unit covered. Each unit will be approximately 3 weeks. There will be a review the day before the test to help students study. This review may be in the form of a game. Students will also be given a review sheet a day before the test to study from.

Quizzes 26%

All quizzes will be announced. Quiz dates will be given at the beginning of each unit. Quizzes will be similar to homework problems. If a student misses a quiz due to absence, they will be expected to make it up on the day they return. Their lowest quiz grade will be dropped per marking period.

Homework 26%

Homework will be given at the end of most lessons. Homework must be completed before the class starts. Homework will be graded by completion and accuracy.

No late homework assignments will be accepted unless the student was absent.

Your two lowest homework grades will be dropped

Participation 10% (Tardy Policy)

Students will start with 20 participation points. After every 3 tardies the student will lose 5 participation points. After every unexcused absence, the student will lose 10 participation points (5% of the total grade).

Cheating

Cheating will not be tolerated. If a student is caught cheating on a test or quiz, that student will receive a zero on that test or quiz. In the event of plagiarism, the situation will be handled according to school policy.

Current Grades

A detailed grade analysis sheet will be given to the students periodically. The sheet will list every test, quiz, homework, and participation grade from the current marking period. It will be the responsibility of the student to show this to their parents or legal guardian.

Student grades will be available for parents on the East Great Falls web site. Parents must use their child’s student number to access their grade.

A conference with the teacher, parents, and student will be required if the student fails any marking period. Students must have 4 years of Math to graduate (State Law). Failing any of the semesters will put the student behind in graduation requirements.

Other Information

I will be available anytime before school. I usually arrive at East Great Falls High School an hour before class. If you need to talk to me after class, I will be available most days after school until 4:00 pm.


Parents are welcome to call me at school and leave a message at my voice mail 867-5309.

Thank you


Internet Resources

For our unit, we created a hotlist website. This would be a resource for both parents and students. One nice aspect about creating a hotlist is that it can be updated with new and different links whenever needed. The parent section contains links to helpful mathematics websites. The students section contains links to sites used in the classroom and other sites that may help them if they have questions. There is also a teacher link which includes some of the websites we used in creating our lesson. The link to our hotlist is: http://www.kn.att.com/wired/fil/pages/listpolynomiva.html

Students:

o http://www.classzone.com/cz/books/algebra_2_2007_na/book_home.htm?state=NY The site above is the textbook website. There are links to homework problems, games, vocabulary builders, and other resources. Choose the chapter you are interested in viewing material for and explore from there!

o http://www.shodor.org/interactivate/activities/AlgebraFour/ Shodor is a fantastic website for being actively involved with the mathematics materials. This particular link leads to a helpful game for reviewing many of the skills we have discussed in our polynomials unit.

o http://www.quia.com/ba/28820.html This is another game which could be used as a review of adding and subtracting polynomials. The template is much like the board game, Battleship. Students can play against the computer set at different skill levels.

o http://nlvm.usu.edu/en/nav/category_g_4_t_2.html The National Library of Virtual Manipulatives is a great website to get hands-on experience with important mathematics concepts. In our unit, Kenny used the algebra tiles link in his launch explore summarize lesson plan.

o http://courses.wccnet.edu/~rwhatcher/VAT/ Another link that Kenny used in his launch explore summarize lesson plan was this website. This is another algebra tiles website, but some of the skills can be more advanced. This is a great resource for extra practice or review!

Teachers:

o http://nctm.org/ The National Council of Teachers of Mathematics website contains links to teaching mathematics articles and journals and the national mathematics standards. For full benefit of the website, teachers must join! But there are some resources for non-members as well.

o http://illuminations.nctm.org/LessonDetail.aspx?id=L798 Illuminations is a great source for mathematics activities for a wide range of topics. This particular website is a polynomial activity that we did not use in our unit. The activity has to do with multiplying polynomials and some factoring.


http://illuminations.nctm.org/LessonDetail.aspx?id=L798

http://nctm.org/

http://courses.wccnet.edu/~rwhatcher/VAT/

http://nlvm.usu.edu/en/nav/category_g_4_t_2.html

http://www.quia.com/ba/28820.html

http://www.shodor.org/interactivate/activities/AlgebraFour/

http://www.classzone.com/cz/books/algebra_2_2007_na/book_home.htm?state=NY

http://www.kn.att.com/wired/fil/pages/listpolynomiva.html

o http://www.thefutureschannel.com/algebra/polynomial_roller_coasters.php Jordan used this website to help him create his rollercoaster launch explore summarize lesson plan. Students create rollercoasters by changing the coefficients of the different terms in a given polynomial.

o http://classroom.4teachers.org/ This website is a handy tool for teachers in creating a classroom layout. Andrew used this website to create the layout for our classroom.

o http://www.murray.k12.ga.us/teacher/kara%20leonard/mini%20t%27s/games/games.htm Another great teacher resource that we used in our unit plan was this website which teachers or students can use to create review games. There are a number of different game templates to choose from. In our unit, we did a Jeopardy theme.

Parents:

o http://www.math.com/ This is a good reference website if you or your student are unsure about a mathematical concept.

o http://www.purplemath.com/ Purplemath is specifically an algebra website. It contains lessons on many topics we will cover during the school year. It is another great resource if you want to learn more about a mathematics topic.


http://www.purplemath.com/

http://www.math.com/

http://www.murray.k12.ga.us/teacher/kara%20leonard/mini%20t's/games/games.htm

http://classroom.4teachers.org/

http://www.thefutureschannel.com/algebra/polynomial_roller_coasters.php

Instructor: Kenny PozoloTime: 50 minute class periodTopic: CBR / Walking Graphs / Exponent ReviewClass: Algebra II

For this lesson, the class will be introduced to polynomials using the CBR. I will divide

the class into small groups and issue each one a CBR. Then I will draw three or four different

polynomials on the board. One student from each group will have to copy down the polynomials

on a sheet of paper. Another person from the group will have the honor of trying to successfully

“walk” the path of each polynomial. The final group member will be in charge of the CBR,

holding it steady so the “walker” can mimic the graph. This will introduce the basic shapes of

polynomials to the students, as well as review concepts like y-intercept, x-intercept, and slope.

Once they have completed the CBR activity, we’ll spend the remaining time reviewing rules of

exponents.


Lesson Objectives

Students will be able to state general shape of a graph based on the exponent Students will be able to determine how a positive or negative coefficient will affect the direction

of the graph Students will be able to determine why a higher exponent has more effect on a graph than a

smaller one

Materials

Computers with TI-Nspire CAS Roller Coaster Worksheet Properties of polynomials sheet

Standards and Benchmarks

I.1 Patterns I.2 Variability and Change III.3 Description and Interpretation III.4 Inference and Prediction

Launch

Give students brief over view of what lesson is about:

The shape of a roller coaster can be represented by a polynomial, such as

ax6+bx5+cx4+dx3+ex2+fx+g.

Depending on the different coefficients of each term we can get different shaped roller coasters or polynomials.


Here are two different roller coasters and their respective polynomials

-.65x6-.5x5+10.97x4+4.52x3-38.71x2-12.9x+58.06 -.97x6-.5x5+13.55x4+4.52x3-46.77x2-35.48x+109.68

Your job is to find out what parts of the polynomial affect particular parts of the roller coaster and how they do so.

Explore

Students will move into groups of 2-3 to a computer. Each student needs to have an answer sheet with explanations of the following questions.

Walk around the class to answer any questions and to monitor progress.

1.) Open RollerCoaster.tns on the TI-Nspire. You should see a screen similar to the one at the right. This roller coaster represents the polynomial

-1.61x6-1.94x5+16.13x4+11.61x3-35.48x2-19.35x+51.61


The sliders at the top of the page are attached to the different coefficients to each term in the polynomial. By moving each of the sliders, we can get a “new” roller coaster.

Let’s first look at g. Play with the g slider and see what happens with the roller coaster. How does g affect the graph? Why? Move g back to 51.61 after you have made your conjecture

G moves the roller coaster up and down on the y- axis. The higher the absolute value of g, the further up or down the roller coaster will be moved. This is because g is simply the constant of the polynomial and represents a straight line.

2.) Now let’s look at the f slider. Change the slider from -12.9 to -54.84. What do you notice happened to the roller coaster? (You may need to change to scale of your graph to see the full effect of this movement. To do so, position your mouse on the y-axis, press the shift key and drag your mouse down until the units of the y- axis is 20.)

Continue to change f to a lower and lower negative number. What happens to roller coaster as you do this? What do you think would happen if you changed f to -200? Why? What happens if you change f to a positive value? Does the sign of f affect the general direction fo the graph? Move f back to -12.9 once you have made your conjectures.

The roller coaster had a steeper drop to the right when you decrease the value of f. This is because f is attached to the x term, which is linear. This causes a positive/negative slope on the graph depending on the value of f. The greater the absolute value of f, the greater the slope will be. If f is negative the general direction of the graph will be from upper left to lower right and lower left to upper right if positive.

3.) Now let’s look at e. Change the slider from -35.48 to -80.65. What happened to the shape of the roller coaster? Move e all the way to -100, what happened to the shape of the graph? What would happen in you changed e to -200? Why? Change e to 100. How does e affect the graph now? Is this similar to when e was negative? Why does e behave this way? Move e back to -35.48 when finished.

As you move e to a higher and higher absolute value the graph will begin to take a more parabolic shape. This is because e is the coefficient to the x2 term. The higher the value of the coefficient the skinnier the parabola will be. When e is positive the parabola will open towards the top of the graph whereas if it is negative the graph will open towards of the bottom of the graph.


4.) Change slider d from 11.61 to 30.65. What happened to the shape of the roller coaster? Slide d to a negative number. What happened to the shape of the roller coaster? Why do you think this is? Look at the terms containing d and f, is there anything similar about their exponents? Did they affect the roller coaster in similar ways? Why do you think this is? Move d back to 11.61 when finished making conjectures.

As you move d to a higher and higher absolute value, the graph will have a steeper drop to the left or right depending on whether the value is positive and negative. D is attached to the x3 term which is a cubic graph. Both d and f are attached to x terms with odd exponents. This will cause the graph to change direction and steepness of slope depending on whether or not the value of the coefficient is odd or even.

5.) Change slider c from 16.77 to 24.19. What do you notice about the shape of the roller coaster? Move c to a negative number. Does the roller coaster react in the way you thought it would? How is this similar to problem 3? Move c back to 16.77. Instead of sliding c type in 26.77 and notice how much the roller coaster changes. Now move c back to 16.77 and change e from -35.48 to -45.48. What do you notice about how much the roller coaster changes when changing c as compared to e? Why do you think this is?

As you move c to a higher and higher absolute value the graph will begin to take a more parabolic shape. This is because e is the coefficient to the x4 term. The higher the value of the coefficient the skinnier the parabola will be. This is similar to problem 3 because both exponents attached to c and e are even. The effect that c has on the graph is more drastic than the effect that e has because c is attached to the x4 term which causes the values of x to change at a higher rate than those of x2.

6.) Instead of changing sliders a and b, make a conjecture as to how they will affect the shape of the roller coaster based off what you have found with the other sliders and their exponents. How will they affect the general slope of the graph? Will they cause the graph to be more parabolic? Will they affect the graph more drastically than the other sliders? Why?

B is attached to the x5 term so it will affect the graph in a similar fashion to d and f. The higher the absolute value of b, the greater the slope will be. If b is negative the graph will go from UL to LR and LL to UR if positive.


A is attached to the x6 term so it will affect the graph in a similar fashion to c and e. The higher the absolute value of a the graph will take a more parabolic shape. The higher the value, the skinnier the parabola will be.

Both will cause more drastic changes on the graph than the other terms because they are of a higher degree which causes the values of x to change faster than the other terms.

7.) Go to Problem 2.1. You should see a screen like the one at the right.

The solid roller coaster is controlled by sliders. Your job is to use what you have learned from the previous problems and manipulate the sliders so that the solid roller coaster matches the dashed roller coaster

The equation to the dashed graph should be similar to

−0.97x6-1.29x5+9.68x4-6.45x3-9.68x2+61.29x+83.87

Summarize

Have students present their final roller coaster and see if anyone could line up the two graphs. For students that could not line up the graphs discuss why there was a difficulty.

Discuss how a term with an odd/even exponent affects the graph. Discuss why a term with a larger exponent will affect the graph in a more drastic way than a term

with a lower exponent Discuss general shape of linear, parabola, cubic, etc. Complete Properties of Polynomial graphs worksheet


Properties of Polynomial Graphs

Exponent Type of GraphGeneral shape of Graph

Positive Negative

1 linear LL UR UL LR

2 parabola UL UR LL LR

3 cubic LL UR UL LR

4 UL UR LL LR

odd - LL UR UL LR

even - UL UR LL LR

Draw in a rough sketch of each graph


y= 3x2 y= -2x5

y= -x4 y= 3x3


Subject Area: Algebra 2

Grade Level: 11th grade

Unit Title: Polynomials

Lesson Title: Adding and Subtracting Polynomials

Michigan Objectives:

1. (A1.1.1) Give a verbal description of an expression that is presented in symbolic form, write an algebraic expression from a verbal description, and evaluate expressions given values of the variables.

2. (A1.1.4) Add, subtract, and simplify polynomials and rational expressions.

Materials/Resources Needed:

Computer Lab (one computer per student) http://www.quia.com/ba/28820.html (Adding & Subtracting Polynomials Battleship)

Anticipatory Set:

Ask: Do you know how to add and subtract polynomials?

Ask: What does “combining like terms” mean?

Say: Although this should be a review for most of you, adding and subtracting polynomials is an important skill to develop. It can also be challenging if you do not recognize “like terms.”

Objectives for Students:

1. The main objective for this lesson is for students to learn the skills of adding and subtracting polynomials.

2. The ability to perform operations on polynomials becomes very important when students get to calculus.

3. Also it is important for students to notice that there are different ways to write the equations to make them easier to simplify.

Input/Modeling:

Let’s begin with an example: (7 x2+3 x−4 )+(3 x2−2 x+11 )=?i. Do you know how to simplify this equation?

ii. What is the first step?



If the students do not mention combining like terms, I will bring it up here.i. Do you know what combining like terms means?

ii. In this equation what are the like terms? Like terms are the monomials that have the same variables raised to the same powers.

i. In this equation, they are: 7 x2and 3 x2

3 x and −2 x -4 and 11

ii. Now what do we do with these like terms?

The correct answer to this example is: (7 x2+3 x−4 )+(3 x2−2 x+11)=10 x2+x+7 Another way to set this problem up is to write it like a long addition problem. This will help you

line up like terms. It is important to realize that there are more than one different ways to set problems up.

i.

7 x2+3 x−43 x2−2 x+11

Now let’s look at an example of subtraction: (10 x2−3 x+12)−(2 x2+5 x−2)=?i. How is this problem similar to the one we just completed?

ii. What are the like terms in this equation?

The like terms in this equation are: 10 x2and 2 x2

−3 x and 5 x12 and -2

i. How is this problem different than the other example we looked at?ii. What must we do with the like terms in this example?

In this example we must subtract the like terms instead of adding them together. So the correct

answer to this example is: (10 x2−3 x+12)−(2 x2+5 x−2)=8 x2−8 x+14 It is also to note that we can write this example just like the addition example:

i.

10 x2−3 x+122 x2+5 x−2

Check for Understanding:

I will write 5 examples on the board. Here are five examples of polynomial addition and subtraction. Please work on these while I walk

around to see how you are doing. If you have any questions, raise your hand, and I will assist.

i. (2 x3+5)+( x2−6 x+12)=

ii. (5 x2−12 x+4 )−(11 x4−3 x3+15)=

iii. (8 x4−3 x3+2 x−1)+(5 x2+5 x3+7 x )=

iv. ( x5−x 4−x3)−(3 x5+6 x3−x2)=

v. ( x4−5 x3− x+10 )−(4 x3+3 x2+10 x−6 )= The answers are:

i. (2 x3+5)+( x2−6 x+12)=2 x3+x2−6 x+17


ii. (5 x2−12 x+4 )−(11 x4−3 x3+15)=−11 x4+3 x3+5 x2−12 x−11

iii. (8 x4−3 x3+2 x−1)+(5 x2+5 x3+7 x )=8 x4+2 x3+5 x2+9 x−1

iv. ( x5−x 4−x3)−(3 x5+6 x3−x2)=−2x5−x4−7 x3+x2

v. ( x4−5 x3− x+10 )−(4 x3+3 x2+10 x−6 )=x 4−9 x3−3 x2−11 x+16 I will address any mistakes individually unless many students are having the same problem.

Although I do not expect this, I will be prepared to discuss like terms more in depth or explain to students that they might need to rearrange the equations in order to match the like terms together. The other problem students may face is forgetting to distribute the negative 1 throughout the polynomial.

Guided Practice:

Students will all sign on to a computer. They will be instructed to go to this website, http://www.quia.com/ba/28820.html. This is a review game for adding and subtracting polynomials.

The rules are very simple. They are based on the game of battleship. Each time you strike one of the computer’s battleships you must solve a polynomial addition or subtraction problem. The problems are multiple choice.

Once again I will walk around to make sure the students are not having difficulties. I will answer any individual questions about how to play the game or about the actual addition or subtraction.

Independent Practice:

Section 5.3 (eWorkbook) http://www.classzone.com/cz/books/algebra_2_2007_na/resources/htmls/ml_hsm_alg2_eWorkbook/index.html (Only addition and subtraction problems)


http://www.classzone.com/cz/books/algebra_2_2007_na/resources/htmls/ml_hsm_alg2_eWorkbook/index.html



Instructor: Kenny PozoloTime: 50 minute class periodTopic: Multiplying PolynomialsClass: Algebra II

Instructional Objectives:

1. The students will multiply polynomials, using a number of different methods, including algebra tiles, the distributive property, FOIL, and multiplying vertically.

2. The students will use polynomial multiplication in real-life situations, such as determining area.

Warm-Up: [5 minutes]

1. Write down the distributive property. a(b + c) = ab + ac

Launch: [5 minutes]1. Begin area of rectangle review.

Give some examples of rectangles with side lengths of positive whole integers. The students should breeze through those examples. Next, offer up side lengths of x and y. Still, the students should have no trouble.

2. Replace the side lengths with binomials: (x + 1) and (x + 2). Ask the students to speculate what the area would be. Discuss some possible solutions.

Explore: [35 minutes] Part 1 – Algebra Tiles: [10 minutes]

1. Have the students take up stations at the classroom computers.2. Direct them to the following website:

http://nlvm.usu.edu/en/nav/frames_asid_189_g_4_t_2.html?open=teacher&hidepanel=true&from=category_g_4_t_2.html

3. Go over the basic directions of the applet. Familiarize them with the applet. Go over which button does what. Show them how to create the binomials using algebra tiles.

4. Work through the Launch problem: (x + 1)(x + 2).

Part 2 – Distributive Property: [10 minutes]

1. Go back to the Warm-Up.2. Write (a + b)(c + d) on the board and ask the students for suggestions on how to solve.3. Cover up the (a + b) portion with a piece of paper. Tell the class to think of the covered

piece as a variable, and to apply the distributive property. (paper*c + paper*d)


4. Give them an example to work through. (x + 2)(x + 3) x(x + 3) + 2(x + 3) Use distributive property. (x2 + 3x) + (2x + 6) Combine like terms. x2 + 5x + 6

Part 3 – FOIL: [10 minutes]

1. Note-taking: FOIL is a mnemonic device: First – Outer – Inner - Last Great to use when multiplying binomials. (i + j)(m + n)

o First: multiply the first number of each binomial – imo Outer: multiply the outside numbers – ino Inner: multiply the inner number – jmo Last: multiply the last numbers of each binomial – jn

2. Give an example to work through. (3x – 4)(2x + 1) Multiply the First numbers: 3x*2x = 6x2

Multiply the Outer numbers: 3x*1 = 3x Multiply the Inner numbers: -4*2x = -8x Multiply the Last numbers: -4*1 = -4 Combine like terms to get 6x2 – 5x – 4

Part 4 – Multiplying Vertically: [5 minutes]1. Note-taking

Easiest to do when one of the polynomials is long. Just like traditional integer multiplication. Put longer polynomial on top. Example:

o (x – 2)(-x2 + 3x + 5)o –x2 + 3x + 5

x – 2

2x2 – 6x – 10-x3 + 3x2 + 5x-x3 + 5x2 – x – 10

Summarize: [5 minutes]1. Distribute Multiplying Polynomials worksheet.

Multiplying Polynomials Worksheet:


Directions: Use the given method to find the product (2x +3)(4x +1).

1. Use an area model (algebra tiles).

2. Use the distributive property.

3. Use the FOIL method.

Directions: Use multiplying vertically to find the product.

4. (2x2 + 4x – 8)(x +1) ` 5. (x3 - x2 + 5x + 3)(2x +1)

Directions: Find the area of the football field.



(3x + 10) ft

(x – 6) ft

Instructor: Kenny PozoloTime: 50 minute class periodTopic: Synthetic DivisionClass: Algebra II

Instructional Objectives:

1. The students will divide a higher degree polynomial by a binomial through synthetic division. 2. The students will relate synthetic division to long division to observe similarities and differences

between the two methods.3. The students will explain how synthetic division works.

Warm-Up: [10 minutes]

1. Put the polynomials in standard form. List the coefficients of each term. 5x – 2x2 + 9 + 4x3

o 4x3 – 2x2 + 5x +9o 4, -2, 5, 9

-4x4 – 7x2 + x3 + x4

o -3x4 + x3 – 7x2

o -3, 1, -7, 0, 0

Input / Modeling: [30 minutes]

1. Have the class recall the process of long division.2. Bring up synthetic division. Define it. Point out the similarities between the two.3. Point out that the polynomial must be in standard form.4. Go through the process with the students:

Omit all variables and exponents and focus solely on the coefficients. Reverse the divisor, which allows us to add instead of subtract. List the coefficients. Add the first column together. Multiply the sum by the divisor. Place the product under the top number of the next column. Repeat steps 4 – 6 until you run out of columns. The final sum is the remainder. Fill in the missing variables and exponents.


5. Go through an example, step by step, with the students: 3x3 – 6x + 2 divided by (x – 2) The numbers representing the divisor and the dividend are placed into a division-like

configuration:

The first number in the dividend (3) is put into the first position of the result area (below the horizontal line). This number is the coefficient of the x3 term in the original polynomial:

Then this first entry in the result (3) is multiplied by the divisor (2) and the product is placed under the next term in the dividend (0):

Next the number from the dividend and the result of the multiplication are added together and the sum is put in the next position on the result line:

This process is continued for the remainder of the numbers in the dividend:


The result line reads “3, 6, 6, 14.” The first three numbers are the coefficients for our quotient and the last number is the remainder.

The final answer: 3x2 + 6x + 6 + 14/(x – 2)

6. Give an example for them to work through on their own and check for understanding.7. Ask a few follow-up questions:

By switching the divisor, what can we do instead of subtracting? If the remainder is zero, what that that mean? If we use long division, will we get the same answer?

Closure: [10 minutes]

1. Have the students respond to the following question: How does long division work?2. Have the students also write one thing they learned and one thing they would like to know more

about regarding synthetic division.


Andrew Abissi

MTH 462

4/9/10

LES on Factoring Polynomials

Standards and Benchmarks

EXPECTATION L2.1.2 Calculate fluently with numerical expressions involving exponents; use the rules of exponents; evaluate numerical expressions involving rational and negative exponents; transition easily between roots and exponents.

EXPECTATION A1.1.3 Factor algebraic expressions using, for example, greatest common factor, grouping, and the special product identities

EXPECTATION A1.1.4 Add, subtract, multiply, and simplify polynomials and rational expressions. EXECTATION A1.1.5 Divide a polynomial by a monomial

Objectives

1. The students will review factoring integers, and connect the review to factoring polynomials.2. The students will be able to factor quadratic equations.3. The students will be able factor third degree polynomials.

Materials List

1. Whiteboard2. Pencil3. Paper4. Calculator (not needed, but could help for arithmetic)5. HW handout

Launch.

1. The students will need to have had a lesson in expanding factored polynomials for this lesson The students will also need to know how to find factors of integers. In order to connect with prior knowledge, ask the following questions as a warm up exercise. These questions can be posted on the board, and role can be taken while the students answer.

Questions: (5 min)


Expand the following problems.

(x+4)*(x+5) x2+9 x+20

(x+6)*(x+7) x2+13 x+42

(x+21)*(x+12) x2+33 x+252

2. After the students expand the previous expressions, the teacher will give these questions about finding factors.

Questions: (10 min)

What are the factors of 12? + - 1,2,3,4,6,12

Which 2 factors sum to 7 with a product of 12?



What are the factors of -30? + - 1,2,3,5,6,10,15,30

Which 2 factors sum to 29 with a product of -30?

Which 2 factors sum to -13 with a product of -30?



What are the factors of 105? + - 1,3,5,7,15,21,35,105


Explore (20 min)

The teacher should have the students work in small groups (2 or 3) and have them factor the following expressions.

1. Consider the factors of the constant of the following equations. Given the following format, fill in the blanks of the equation. If time permits, you can create more equations.

x2+6 x+9 = (x+_)*(x+_) (x+3)*(x+3)

x2+7x+6 = (x+_)*(x+_) (x+6)*(x+1)

x2+4 x−12 = (x+_)*(x-_) (x+6)*(x-2)

x2+ x−90 = (x+_)*(x-_) (x+10)*(x-9)


x2−11 x+24 =(x-_)*(x-_) (x-8)*(x-3)

x2−9 x+20 = (x-_)*(x-_) (x-5)*(x-4)

2. Factor the following quadratic expressions

x2+6 x+5=¿ (x+5)*(x+1)

x2+4 x−5=¿ (x-1)*(x+5)

x2−2 x−35=¿(x-7)*(x+5)

x2−9 x+14=¿ (x-7)*(x-2)

x2+24 x+144=¿ ( x+12 )2

3. Consider the factors of the constant of the following expressions and the constant multiplied by x2. Given the following format, fill in the blanks of the equation.

x3+15 x2+66 x+105 = (x+_)*(x+_)*(x+_) (x+5)*(x+3)*(x+7)

4. Factor the following expression

x3+10 x2+31 x+30 (x+5)*(x+2)*(x+3)

Summarize (9 min)

1. When factoring quadratic polynomials, the roots of the polynomial always divide the constant. To find out which divisors of the constant are roots of the polynomial, make sure the roots sum to the constant for the x term.

EX: x2+7x+6 = (x+6)*(x+1) because both 6 and one divide 6, and the sum of 6+1=7.

2. When factoring third degree polynomials, the roots of the polynomial also divide the constant. To find out which divisors of the constant are roots of the polynomial, make sure all three roots sum to the constant multiplied by x2.

EX: x3+15 x2+66 x+105=(x+3)∗(x+5)∗(x+7) and 3, 5, and 7 divide 105, and 3+5+7=15

3. This method is can also be used to factor fourth or any degree polynomial. The roots will divide the constant, and the roots will sum up to the constant multiplied x^k, where k is the degree of the polynomial minus one.

Ex: ( x+3 )∗(x+2 )∗( x+5 )∗( x+1 )∗( x+4 )=x5+15 x4+something+120 and 3,2,5,1, and 4 divide 120, and 1+2+3+4+5 = 15

4. Finally, considering the polynomial a x2+bx+c, the factors of c only sum to equal b if a=1.


Example: 2 x2+17 x+21 = (2x+3)(x+7) it’s still true that 7 and 3 are factors of 21, but 7 and 3 do not sum to 17. However, if you factor out the 2 in the expanded expression, the sum of the factors of the constant will equal the linear term.

2 x2+17 x+21=2(x2+8.5 x+10.5)= 2((x+1.5)*(x+7))

Estimated time (49 min)


Homework

Factor the following quadratic expressions.

1. x2+6 x+52. x2+7x+123. x2−10 x+254. x2−x−305. x2−1446. x2+4 x−217. x2−15 x+50

Factor the following trinomial expressions.

8. x3+15 x2+66 x+1059. x3+10 x2+31 x+3010. x3−x2−17 x−15

Factor the fourth and fifth degree polynomial

11. x4+4 x3+6 x2+4 x+112. x5+6 x4+14 x3+16x2+9 x+2


Subject Area: Algebra II

Grade Level: 11th


Lesson Title: The Quadratic Equation

Standards:

A1.2.9: Know common formulas (such as the quadratic equation) and apply them in contextual situations.

A1.2.5: Solve polynomial equations and justify the steps in the solution.

Objectives:

Students will apply the quadratic equation to solve polynomial functions. Students will define the steps of the quadratic formula. Students will describe the importance of the discriminant of the quadratic formula in finding the

solutions for x.

Materials Needed:

Class set of computers with Excel Steps to Creating an Excel Program for the Quadratic Formula Source: http://twt.borderlink.org/30667/5066/5066.html

Input/Modeling:

Today we will be learning about the quadratic formula. There are many ways to solve quadratic equations. One of these is the quadratic formula.

o What are some ways you have already learned to solve quadratic equations? Graphing and factoring

The quadratic equation is: x=−b±√b2−4 ac

2 a whenax 2+bx+c .o I will write this on the board as I say it.o I want students to notice the different components of the quadratic formula.

I will give a brief lecture about the discriminant of the equation and its importance.o You use the discriminant to determine how the number and type of answers to the

quadratic equation.o If the discriminant is:

0 There is one root, and it is real. Positive There are two roots, and they are real. Negative There are two roots, and they are complex.


http://twt.borderlink.org/30667/5066/5066.html

Although it is important to know the quadratic formula, it is also important to learn how to use technology to solve problems. Today we will go to the computer lab to learn how Excel can help us solve for x.

o In the computer lab, I will pass out the program steps worksheet.o I will demonstrate the first few steps on the worksheet.o If the students have trouble after this, I will be walking around to assist them.

Guided Practice:

Students will work to create an Excel program for solving a quadratic equation using the quadratic formula.

Then students will be asked to complete a set of problems using their program. As the students work, I will walk around to make sure they are staying on task and are not having

trouble.

Check for Understanding:

Ask students the following questions:o What are the different ways we have discussed for solving quadratic equations?o What is the importance of the discriminant of the quadratic equation?o When is the quadratic formula appropriate to use as opposed to other methods?

Closure:

Students will have an opportunity to work with the quadratic equation again in the next lesson. Students will be asked to begin to consider the most appropriate ways of solving quadratic

equations.


Standards and Objectives

I.1 Patterns IV.1 Concepts and Properties of Numbers V.1 Operations and their properties V.2 Algebraic and Analytic Thinking

Introduction

Lesson Objectives

Students will be able to find factors of special cubic polynomials Students will be able to find factors of cubic polynomials by grouping Students will be able to solve cubic equations by factoring

Anticipatory Set

Have student’s factor polynomials using quadratic formula

a.) x2 + 6x – 8 b.) x2 – 15x – 17

( x – 4) (x – 2)

Materials

Computers Student White Boards Factoring worksheet

Development


( x− √157 +152

) ( x − 15 − √1572

)

Input: Have students give responses to their anticipatory set. Clear up any confusion about finding the factors of equations by using the quadratic formula

Sample student responses

Model: Explain to students that when we deal with cubic polynomials there are several ways to find their factors and that there are clues as to which method to use.

The first method is identifying the polynomial as a sum of two cubes or a difference of two cubes.

A sum of two cubes takes the form of a3 + b3 and can be broken down into

(a + b) (a2 – ab + b2)

Make sure students understand that a and b can be variables or constants or a combination of both

Ex. x3 + 27 = (x)3 + (3)3 = (x +3) (x2 – 3x + 9)

Ex. x3 + 64 = (x)3 + (4)3 = (x + 4) (x2 – 4x + 16)

Ex. 8x3 + 1 = (2x)3 + (1)3 = (2x + 1) (2x2 – 2x + 1)

A difference of two cubes takes the form of a3 - b3 and can be broken down into

(a - b) (a2 + ab + b2)

Make sure students understand that a and b can be variables or constants or a combination of both


Ex. x3 - 125 = (x)3 - (5)3 = (x - 5) (x2 + 5x + 25)

Ex. x3 - 8 = (x)3 - (2)3 = (x - 2) (x2 + 2x + 4)

Ex. 216x3 - 125 = (6x)3 - (5)3 = (6x - 5) (36x2 + 30x + 25)

Check for understanding:

Place several problems on the visualizer have students come to the front and solve a problem. The rest of the class will “correct” the problem by using a white board which they all have. Students will draw a check mark if the problem is solved correctly or a x if the problem is solved incorrectly. If a larger number of students are incorrect, make sure to correct any misunderstandings.

x3 + 125 = (x + 5) (x2 – 5x + 25)

343x3 – 216 = (7x – 6) (49x2 + 42x +36)

Model: Explain to students that if the polynomial doesn’t form sum or difference of two cubes then we can also look to group the terms of the polynomials so that we can pull out a common item.

Ex. x3 - 3x2 - 36x + 108 = (x3 - 3x2) + (- 36x + 108)

From here we can pull out a common term from each group. The first group has an x2 in common and the second group has a -36.


This might be difficult to see that the 108 has a negative factor. Show students that since the sign in the first grouping is negative, we most likely want the sign in the second grouping to be negative as well.

x2 (x – 3) – 36 (x – 3)

From here we can pull (x – 3) out of each term. This leaves us with

(x – 3) (x2 – 36)

Ask students if they think the problem is completely finished. If they say yes ask them what the factors of the original polynomial are. If they say no, ask why the problem is not finished. Students should say that x2 – 36 is not factored completely as it is a difference of two squares.

(x – 3) (x – 6) (x + 6)

Ex. x3 - 10x2 - 49x + 490 = (x3 - 10x2) + (-49x + 490)

x2(x – 10) – 49(x – 10)

(x2 – 49) (x – 10)

(x – 7) (x + 7) (x – 10)

Ex. x3 - 3x2 - 81x + 243 = (x3 - 3x2) + (-81x + 243)

x2(x – 3) – 81(x – 3)

(x2 – 81) (x – 3)

(x – 9) (x + 9) (x – 3)


Check for understanding:

Place several problems on the visualizer and have different students than before come to the front and solve the problems. Students use their whiteboards in the same fashion as they did earlier to check the students work. If a larger number of students are incorrect, make sure to correct any misunderstandings.

4x3 – 36x2 – 80x + 320 = (4x3 – 36x2) +(–80x + 720)

4x2(x – 9) – 80(x – 9)

(4x2 – 80) (x – 9)

4(x2 – 20) (x – 9)

Guided Practice: Pass out Factoring Worksheet and have students complete individually. Walk around class to provide support where needed, allow students to bounce ideas off one another and help each other, but each student is responsible for his or her own worksheet.

Independent Practice: Section 5.2 (eWorkbook)


Closure: 10 minutes prior to the end of class ask class if there is anything unclear about they have learned today and if I need to clarify anything. Tell students what their independent practice is and remind them about the quiz tomorrow and that they will have time to review prior to the quiz. Have students hand in the Factoring worksheet whether they finished it or not.

Assessment: Look over the students factoring worksheet to see if they have an understanding on the concepts. Correct what they have done. Pass them back tomorrow as part of the review for the quiz.




Name

Factoring Worksheet

1. x3 – 9x2 + 8x 2. x3 – 5x2 + 5x + 3

3. x3 +3x2 + 3x - 7 4. x3 – 3x2 – 5x +4


5. x3 – 9x2 + 8x 6. x3 + 3x2 – 8x – 10

7. x3 - 125 8. x3 + 8x2 – 7x – 2


9. x3 + 2x2 + x 10. 512x3 + 216

11. x3 + 4x2 +4x +1 12. x3 + 6x2 +7x



Lesson Title: The Rational Root Test

Michigan Standards:

o A1.2.2 Students will associate a given equation with a function whose zeros are the solutions of the equation.

o A1.2.5 Students will solve polynomial equations and equations involving rational expressions and justify steps in the solution.

o A2.1.6 Students will identify zeros of a function.

Objectives:

o Students will identify the steps of the Rational Root Test.o Students will define zero or root of a polynomial.o Students will use the Rational Root Test to find the zeros of a polynomial.o Students will identify other methods of finding the zeros of a polynomial such as by factoring or

graphing.

Materials Needed:

o CAS Nspireo Rational Zero Test Exploration Worksheet

Launch:

o Say: Today we will be exploring an aspect of polynomials that will both teach you something new and allow you to use your prior knowledge.Ask: What are some things you can tell me about polynomials?

o Say: There are a few vocabulary words I would like to briefly discuss before we begin the exploration today. Ask: Can someone tell me what a constant is?Ask: How about a leading coefficient?Ask/ Write: Given a polynomial. Could you label the constant and the leading coefficient?

o Say: There also is a new vocabulary word that you will need to know today. The zero or root of a

polynomial is the solution of the polynomial, P( x )=0 . It is the value of x that make the polynomial equal to zero.

o Say: The following worksheet will allow you to explore the zeros of polynomials and how they relate to what you have previously about polynomials.

Explore:

o These are the questions on the exploration worksheet. Given a list of 10 polynomials. Find the factors of the leading coefficient for each of the polynomials. Find the factors of the constant for each of the polynomials.


Create as many fractions as possible using the leading coefficient factors as the denominators and the constant factors as the numerators for each polynomial.

Plug the fractions back into the polynomial. Graph the polynomial. Where does the graph of the polynomial intersect the x-axis? What connections can you make between the graph and the other numbers you have

found?o The goals of this exploration are for:

Students to discover the Rational Root Test. Students to discover the connection between where the graph of a polynomial intersects

the x-axis and the zeros of that polynomial. Students to discover that some of the numbers they find in the Rational Root Test do not

work as the zeros of the polynomial.

Summarize:

o Although the Rational Root Test is an aspect of finding the zeros of a polynomial, graphing and factoring are other important aspects of understanding polynomials. Both of these techniques can be used to find the zeros of a polynomial. These are techniques that students also have already learned. I plan to cover these ways of checking for correctness in the summary.

o I will give concrete demonstrations for the students involving factoring and graphing and how they relate to the zeros of a polynomial. I will have a calculator visualizer for the graphing aspect of finding the zeros. I will also factor one of the examples from the worksheet on the whiteboard.

o I will expect students to take notes on this portion of the discussion. I will write important points (mine or the students) on the whiteboard to keep everyone’s thoughts organized. The following bullets include the way I would try to guide the class into summarizing what they discovered and what I wanted them to get out of the lesson:o Ask: Are there any questions?

Feedback: I will walk around the room, but if any students still have questions, I will address them first.

o Ask: What connections did you find in your observations?Feedback: I want the students to realize that the fractions that create an answer of zero are the same as the numbers in the coordinates of the intersection points in the graphs of the polynomials.

o Tell: The numbers you found that made the equation equal are actually the zeros of a function. In the first part of the activity, you performed what is called the Rational Zero Test. In many polynomials, the zeros are not nice numbers like the ones you found. This is one way to determine whether a polynomial had rational roots.

o Ask: Can you think of another way we might be able to find the zeros of a polynomial?Feedback: I would like students to discuss the prior knowledge they have gained about factoring. Hopefully by using the graphing the students will see the connection. If they do not, I would take one of the polynomials from the assignment and factor it. Then I would have the students graph it.

Independent Practice:


o Section 5.6 (eWorkbook) http://www.classzone.com/cz/books/algebra_2_2007_na/resources/htmls/ml_hsm_alg2_eWorkbook/index.html


Exploration: Zeros of Polynomials

1. Here is a list of 10 polynomials:

a) x2+9 x+20

b) x2−11x+24

c) x2−4 x−12

d) 2 x2+11 x+12

e) 3 x2+25 x−18

f) 4 x2−29 x+7

g) 12 x2−11 x−5

h) 10 x2+54 x+56

i) 42 x2−33 x+6

j) 21 x2−37 x−28

2. Find the factors of the leading coefficient for each of the polynomials.

Polynomial Leading Coefficient Factors of the Leading Coefficient

x2+9 x+20

x2−11x+24

x2−4 x−12

2 x2+11 x+12

3 x2+25 x−18

4 x2−29 x+7

12 x2−11 x−5

10 x2+54 x+56

42 x2−33 x+6

21 x2−37 x−28


3. Find the factors of the constant for each of the polynomials.

Polynomial Constant Factors of the Constant

x2+9 x+20

x2−11x+24

x2−4 x−12

2 x2+11 x+12

3 x2+25 x−18

4 x2−29 x+7

12 x2−11 x−5

10 x2+54 x+56

42 x2−33 x+6

21 x2−37 x−28

4. Create as many fractions as possible using the leading coefficient factors as the denominators and the constant factors as the numerators for each polynomial. (All of these fractions can be positive or negative.)

Polynomial Fractions

x2+9 x+20

x2−11x+24

x2−4 x−12

2 x2+11 x+12

3 x2+25 x−18

4 x2−29 x+7

12 x2−11 x−5


10 x2+54 x+56

42 x2−33 x+6

21 x2−37 x−28

5. Plug the fractions into the corresponding polynomials.

a) x2+9 x+20

b) x2−11x+24

c) x2−4 x−12

d) 2 x2+11 x+12

e) 3 x2+25 x−18

f) 4 x2−29 x+7


g) 12 x2−11 x−5

h) 10 x2+54 x+56

i) 42 x2−33 x+6

j) 21 x2−37 x−28

6. Using your graphing calculator, graph each of the polynomials.

7. Where do the graphs of the polynomials intersect the x-axis? What are the coordinates for each polynomial where it intersects the x-axis?

What connections can you make between the graphs of the polynomials and the answers you got when you substitute the fractions into the polynomials?

Exploration: Zeros of Polynomials

Answer Key


8. Here is a list of 10 polynomials:

a) x2+9 x+20

b) x2−11x+24

c) x2−4 x−12

d) 2 x2+11 x+12

e) 3 x2+25 x−18

f) 4 x2−29 x+7

g) 12 x2−11 x−5

h) 10 x2+54 x+56

i) 42 x2−33 x+6

j) 21 x2−37 x−28

9. Find the factors of the leading coefficient for each of the polynomials.

Polynomial Leading Coefficient Factors of the Leading Coefficient

x2+9 x+20 1 1

x2−11x+24 1 1

x2−4 x−12 1 1

2 x2+11 x+12 2 1, 2

3 x2+25 x−18 3 1, 3

4 x2−29 x+7 4 1, 2, 4

12 x2−11 x−5 12 1, 2, 3, 4, 6, 12

10 x2+54 x+56 10 1, 2, 5, 10

42 x2−33 x+6 42 1, 2, 3, 6, 7, 14, 21, 42

21 x2−37 x−28 21 1, 3, 7, 21


10. Find the factors of the constant for each of the polynomials.

Polynomial Constant Factors of the Constant

x2+9 x+20 20 1, 2, 4, 5, 10, 20

x2−11x+24 24 1, 2, 3, 4, 6, 8, 12, 24

x2−4 x−12 12 1, 2, 3, 4, 6, 12

2 x2+11 x+12 12 1, 2, 3, 4, 6, 12

3 x2+25 x−18 18 1, 2, 3, 6, 8, 9, 18

4 x2−29 x+7 7 1, 7

12 x2−11 x−5 5 1, 5

10 x2+54 x+56 56 1, 2, 4, 7, 8, 14, 28, 56

42 x2−33 x+6 6 1, 2, 3, 6

21 x2−37 x−28 28 1, 2, 4, 7, 14, 28

11. Create as many fractions as possible using the leading coefficient factors as the denominators and the constant factors as the numerators for each polynomial. (All of these fractions can be positive or negative.)

Polynomial Fractions

x2+9 x+20 ± 1, 2, 4, 5, 10, 20

x2−11x+24 ±1, 2, 3, 4, 6, 8, 12, 24

x2−4 x−12 ± 1, 2, 3, 4, 6, 12

2 x2+11 x+12±1, 2, 3, 4, 6, 12,

12 ,

32

3 x2+25 x−18±1, 2, 3, 6, 8, 9, 18,

13 ,

23 ,

83

4 x2−29 x+7±1, 7,

12 ,

72 ,

14 ,

74


12 x2−11 x−5±1, 5,

12 ,

52 ,

13 ,

53 ,

14 ,

54 ,

16

,

56 ,

112 ,

512

10 x2+54 x+56±1, 2, 4, 7, 8, 14, 28, 56,

12 ,

72 ,

15 ,

25 ,

45 ,

75 ,

85 ,

145 ,

285 ,

565

,

110 ,

710

42 x2−33 x+6±1, 2, 3, 6,

12 ,

32 ,

13 ,

23 ,

16 ,

17

,

27 ,

37 ,

67 ,

114 ,

314 ,

121 ,

221 ,

142

21 x2−37 x−28±1, 2, 4, 7, 14, 28,

13 ,

23 ,

73 ,

143 ,

283 ,

17 ,

27 ,

47 ,

121 ,

221

12. Plug the fractions into the corresponding polynomials.Although I expect the students to actually do the work for all of the fractions, the ones that are equal to zero are the most important. I will specifically look for those.

a) P( x )=x2+9 x+20P(−4 )=(−4 )2+9(−4 )+20P(−4 )=16−36+20=0

P(−5)=(−5)2+9(−5 )+20P(−5)=25−45+20=0

b) P( x )=x2−11 x+24P(3 )=32−11(3 )+24P(3 )=9−33+24=0

P(8 )=82−11 (8)+24P(8 )=64−88+24=0


c) P( x )=x2−4 x−12P(−2)=(−2)2−4 (−2)−12P(−2)=4+8−12=0

P(6 )=(6)2−4(6 )−12P(6 )=36−24−12=0

d) P( x )=2 x2+11 x+12

P(−4 )=2(−4 )2+11(−4 )+12P(−4 )=32−44+12=0

P(−32

)=2(−32

)2+11(−32

)+12

P(−32

)=92

−332

+12=0

e) P( x )=3 x2+25 x−18

P(−9 )=3(−9 )2+25(−9 )−18P(−9 )=243−225−18=0

P(23

)=3(23

)2+25 (23

)−18

P(23

)=43

+503

−18=0

f) P( x )=4 x2−29 x+7

P(7 )=4(7 )2−29(7)+7P(7 )=196−203+7=0

P(14

)=4 (14

)2−29 (14

)+7

P(14

)=14

−294

+7=0

g) P( x )=12 x2−11x−5

P(−13

)=12(−13

)2−11(−13

)−5

P(−13

)=43

+113

−5=0

P(54

)=12(54

)2−11 (54

)−5

P(54

)=754

−554

−5=0


h) P( x )=10 x2+54 x+56

P(−4 )=10(−4 )2+54 (−4 )+56P(−4 )=160−216+56=0

P(−75

)=10(−75

)2+54 (−75

)+56

P(−75

)=985

−3785

+56=0

i) P( x )=42 x2−33 x+6

P(27

)=42(27

)2−33 (27

)+6

P(27

)=247

−667

+6=0

P(12

)=42(12

)2−33(12

)+6

P(12

)=212

−332

+6=0

j) P( x )=21 x2−37 x−28

P(73

)=21(73

)2−37(73

)−28

P(73

)=3433

−2593

−28=0

P(−47

)=21(−47

)2−37(−47

)−28

P(−47

)=487

+1487

−28=0

13. Using your graphing calculator, graph each of the polynomials.

14. Where do the graphs of the polynomials intersect the x-axis? What are the coordinates for each polynomial where it intersects the x-axis?

In this question, I want them to just list the coordinates of the polynomial where it meets the x-axis. The x-coordinates will be the same as the fractions that make the polynomials zero.

15. What connections can you make between the graphs of the polynomials and the answers you got when you substitute the fractions into the polynomials?

Answers may vary. The main point I want the students to discover is the Rational Zero Test. I want them to realize that they can go through the process to find numbers which can potentially be the zeros of a polynomial. Once they have those numbers, they can plug them into the polynomial to find if they actually are zeros or not. This also shows that not all numbers from the test are the zeros of the polynomial.

I also would like students to come to the realization that they can find the zeros using graphing as well. Hopefully they will pick up that the numbers that make the polynomials zero are also the numbers in the coordinates of where the polynomial graph intersects the x-axis.

As an extension, students might also see that they could just factor the polynomial to find the zeros as well.


Andrew Abissi

MTH 462

4/21/10

Complex roots and the fundamental theorem of algebra

I. Objectives1. The students will learn about the number i, and about the real and imaginary parts

of complex numbers.2. The students will learn that polynomials can have real roots, and/or non-real roots.3. The students will learn the fundamental theorem of algebra.

II. Benchmarks1. L2.1 Calculation Using Real and Complex Numbers2. L2.1.2 Calculate fluently with numerical expressions involving exponents; use the

rules of exponents;evaluate numerical expressions involving rational and negative exponents; transition easily between roots and exponents

3. L2.1.4 Know that the complex number i is one of two solutions to x2=−1.4. A1.1.3 Factor algebraic expressions using, for example, greatest common factor,

grouping, and the special product identities.III. Materials List

1. Pencil2. Paper3. Whiteboard4. Calculator (TI N-Spire CAS)

IV. Anticipatory Set1. The students will need a good understanding of factoring for this lesson. The

teacher can take role while having the students factor the following expressions and give the zeros of the expressions.

i. x2+6 x+5 ( x+1 ) ( x+5 )ii. x2−15 x+50 ( x−10 )(x−5)

iii. x2−10 x+25 ( x−5 )(x−5)iv. x2−144 ( x+12 )(x−12)v. x3+15 x2+66 x+105 ( x+7 ) ( x+3 )(x+5)

2. Have the students try to factor and find the zeros of the following expressions and ask if it can be done. Students are not expected to know about complex numbers, so they should say that none of these expressions are solvable.

i. x2+1 Roots: √−1 ,−√−1ii. x2−6 x+10 Roots: 3−√−1,3+√−1

iii. x2−8 x+20 Roots: 4+√−2 , 4−√−2


V. InputA. Relaying Information

i. Teacher will say “Each of the previous expressions can be factored, and they all have roots. The roots just happen to not be real numbers. They fall in a domain known as imaginary numbers.”

ii. The square root of a negative number is known as an imaginary number: √−6 ,√−13 ,etc. The most well known imaginary number is √−1 denoted as i. All imaginary numbers are commonly stated as a product of i. For example, √−7=7 i ,∧√−87=87i .

iii. A complex number is a sum of a real and imaginary number. For example, 4+6 i is a complex number. 4 is the real component. 6i the imaginary component.

iv. Have the students solve the following equations to get used to imaginary and complex numbers.

1) (6+2i)+(4+3i)= 10+5i2) i2=¿ -13) (3 i )2=¿ -94) (2+3 i )∗(3+4 i )=¿ 6+17i-12

B. Task Analysisi. With the use of complex numbers, we can easily factor and find roots for

polynomials that seem to have no roots.ii. Have the students find the roots of the previous expressions using

imaginary and complex numbers.1) x2+1 Roots: i ,−i2) x2−6 x+10 Roots: 3−i , 3+i3) x2−8 x+20 Roots: 4+2 i , 4−2 i

iii. Finally, use these roots to give the factorization.1) x2+1 ( x+i )(x−i)2) x2−6 x+10 ( x−3−i )(x−3+i )3) x2−8 x+20 ( x−4+2 i )(x−4−2 i)

C. Modelingi. Go over the following questions with the students,

1) Write a second degree polynomial with real coefficients and the given roots2i, -2i x2+4=0

2) Write a third degree polynomial with real coefficients and the given roots3, -i, i x3=3 x2+x−3=0

3) Write a fourth degree polynomial with real coefficients and the given roots


3, 2, -i, i x4−5x3+7 x2−5 x+6=0ii. Note that repeated roots are considered multiple roots. For example

1) x2+2x+1=0 Factored: (x+1)(x+1) Roots: -1 and -12) x4−4 x3+4 x2−4 x+1 Factored: (x-1)(x-1)(x-1)(x-1) Roots:

1, 1, 1, and 1iii. How many roots do each of the following equations have. How many

possible real roots, and how many possible non-real roots.1) x2+4 2, 2 non-real2) x10−1=0 10, 10 real3) x10+x8+x6+x4+x2+1=0 10, 10 non-real4) x4−x2=0 4, 4 real

VI. Checking for Understandingi. Teacher will ask “How many roots does an nth degree polynomial have” n

rootsii. The Fundamental Theorem of Algebra states, If P(x) is a polynomial with

a degree n>1, n is an integer, then P(x) has exactly n roots. The theorem also states that n has an even number of complex zeros.

VII. Guided Practicei. Find the complete solution set of the following equations

1) x3+3x2−13 x−15=0 {-1, -5, 3}2) x4−5x3+7 x2−5 x+6=0 {2, 3, i, -i}

3) x5+4 x4+4 x3−x2−4 x−4=0 {-2, -2, 1, 12

± √32

i}

VIII. Closurei. Today, we learned about imaginary and complex number.

i=√−1 ,i2=−1, i3=−√−1 , i4=1 , i5=i=√−1ii. a+bi is a complex number if a , b do not equal 0. If a=0 and b does not equal 0, then

the number is imaginary. If b=0 and a does not equal 0, then the number is real.iii. If a+bi is a root of an expression, then a-bi is a root of the same expression.iv. The Fundamental Theorem of Algebra states, If P(x) is a polynomial with a

degree n>1, n is an integer, then P(x) has exactly n roots.IX. Assessment/Reflection

i. The teacher should ask the students if they have any questions. The number i is new information, so it could take longer for students to grasp.

ii. The teacher should try re-assessing the lesson and making sure that the transition to more complex questions is sufficient.

X. HomeworkGive the factorization of the following expressions, give the number of roots, and list all roots


1) x2+6 x+5 2) 3 x3+30 x2+88 x+903) x2+164) x3+3x2−13 x−155) x4−5 x3+7 x2−5 x+66) x5+4 x4+4 x3−x2−4 x−47) x10−1


Standards

I.1 Patterns IV.1 Concepts and Properties of Numbers V.1 Operations and their properties V.2 Algebraic and Analytic Thinking

Lesson objectives

Students will be able to identify how many possible roots a graph has Students will be able to graph polynomial functions Students will be able to determine local minimums and maximums

Materials

TI – Nspire Calculator

Development

To develop student skills in graphing, give students several polynomial equations. Give them a table of x values and have them find the corresponding y values. From here have them graph the points and draw the corresponding line or curve.

After students have established confidence and skill in graphing the polynomials, quickly review what a root is from previous lesson. Ask students how they would identify a root on the graph. Make sure to ask students why some of the roots might be visible. What is a double root? What is a triple root? How do these come about?

Inform students of the idea of a local minimum and maximum. Give the students examples of each and have them identify these on graphs that they have drawn.

Guided Practice

Have students work in groups on problems from section 5.8 (eWorkbook). What they don’t finish will need to be completed for independent practice. Walk around room to monitor progress and provide help where necessary.

Individual Practice

Finish all of section 5.8


Quiz Review:

Add or subtract the following polynomials

a.) (x + 5) + (x2 – x - 3) b.) (2x3 + 13) – (3x4 - 11x – 3)

c.) (x3 + 2x2 – 11) – (x3 + 5x2 - 9) d.) (4x5 – 2x4 + 3) + (67x3 + 4x2 – 12x)

Multiply the following polynomials

a.) (x + 3) (2x – 7) b.) (x2 – 4x + 8) (6x3 - 4)


Quiz 1

Identify which equation is represented by the graph.

1.

a.) –x5 + 9x2 - 2x + 4b.) 5x4+8x3- x2 - xc.) 7x4 – 10x2 +.5x + 13d.) x2- x - 15

2.

a.) −0.5x3 - 3x2 – x + 6

b.) x5 + 3x2 +10

c.) 2x4 – 5x3 – 7x

d.) 2x3 + 15x2 + .5x - 10


3. Add or subtract the following polynomials

a.) (3x + 5) + (x2 – 6x +15) b.) (x3 – 5x2 + 13) – (x4 + 6x2 - 11x – 3)


c.) (4x3 + 5x2 – 7) – (4x3 + 5x2 - 7) d.) (x5 – 4x4 + 31) + (6x3 + 4x2 – 8x)

4. Multiply the following polynomials

a.) (3x + 4) (4x – 7) b.) (2x2 – 4x + 5) (x3- 5)


Quiz 1

Identify which equation is represented by the graph.

1.

e.) –x5 + 9x2 - 2x + 4f.) 5x4+8x3- x2 - xg.) 7x4 – 10x2 +.5x + 13h.) x2- x - 15

2.

a.) −0.5x3 - 3x2 – x + 6

b.) x5 + 3x2 +10

c.) 2x4 – 5x3 – 7x

d.) 2x3 + 15x2 + .5x - 10



a.) (3x + 5) + (x2 – 6x +15) b.) (x3 – 5x2 + 13) – (x4 + 6x2 - 11x – 3)

(x2 – 6x + 20) (-x4 + x3 – 11x2 – 11x +16)


c.) (4x3 + 5x2 – 7) – (4x3 + 5x2 - 7) d.) (x5 – 4x4 + 31) + (6x3 + 4x2 – 8x)

0 x5 – 4x4 + 6x3 + 4x2 – 8x + 31

4. Multiply the following polynomials

a.) (3x + 4) (4x – 7) b.) (2x2 – 4x + 5) (x3- 5)

12x2 – 5x – 28 2x5 – 4x4 +5x3 – 10x2 +20x +25


Week 2 Quiz

Show all of your work!

1. Factor the following polynomials. Use two different methods for each.

a. x2−7x−30=0

b. x2−16 x+48=0

c. x2+8 x+12=0

d. 4 x2−17 x−15=0

2. Use synthetic division to divide the following polynomials:

a. (16 x3−12 x2+48x )÷( 4 x )

b. (35 x2+61 x+24 )÷(5 x+3 )


c. (10 x2+67 x−21)÷( x+7 )

3. Factor the following cubic polynomials:

a. x3−4 x2−45 x=0

b. x3−64=0

c. x3+512=0


Week 2 Quiz

Show all of your work!

4. Factor the following polynomials. Use two different methods for each.

a. x2−7x−30=0 (x-10)(x+3)

b. x2−16 x+48=0 (x-12)(x-4)

c. x2+8 x+12=0 (x+2)(x+6)

d. 4 x2−17 x−15=0 (4x+3)(x-5)

5. Use synthetic division to divide the following polynomials:

a. (16 x3−12 x2+48x )÷( 4 x ) 4x2-3x+12

b. (35 x2+61 x+24 )÷(5 x+3 ) 7x+8

c. (10 x2+67 x−21)÷( x+7 ) 10x-3

6. Factor the following cubic polynomials:

a. x3−4 x2−45 x=0 x(x-9)(x+5)

b. x3−64=0 (x-4)(x2+4x+16)

c. x3+512=0 (x+8)(x2-8x+64)


Unit Test

Please show all work!

Directions: Use the given method to find the product (5x - 2)(-x + 1).

5. Use an area model (algebra tiles). (Hint: draw a picture)

6. Use the distributive property.

7. Use the FOIL method.


4. Identify which equation is represented by the graph

i.) –x5 + 9x2 - 2x + 4j.) 3x3 + 6x2 – 10x – 3 k.) 7x4 – 10x2 +.5x + 13l.) x2- x - 15


a.) (3x + 5) + (x2 – 6x +15) b.) (x3 – 5x2 + 13) – (x4 + 6x2 - 11x – 3)

6. Factor each polynomial


x3 – 125 x3 - 3x2 - 81x + 243

x2+1 x2−6 x+10

x3+3x2−13 x−15=0 x4−5 x3+7 x2−5 x+6

Directions: Use synthetic division to find a solution.

7. 2x3 + 3x -11 divided by (x – 3)


8. 8x4 – 2x2 + 1 divided by (x +7)


Documents

Lesson Objectives€¦ · Web viewLesson Objectives. Students will be able to state general shape of a graph based on the exponent. Students will be able to determine how a positive