Lane Smith 091408 EDU 357

1
Lane Smith09/14/08 EDU 357

• Lesson
• Learning to How to Graph and How to Analyze the
  Graphs of Polynomial Equations (Grades 8 and 9)

2
Lesson Objectives

• By the end of the next several class periods,
  students will be able to analyze and graph
  polynomial equations (in the form of yax2 bx
  c, where a,b, and c are real numbers and a does
  not equal zero) on paper. However, they will also
  learn how to graph these equations via graphing
  calculators (TI-83s, TI-84s, etc) and
  mathematical websites. Also, they will learn to
  dissect features of the polynomial graphs and
  will learn to answer questions accordingly.

3
Introduce the Learning Activity

• To introduce the lesson, I would first briefly
  review the definition of a polynomial, the
  properties of a polynomial, and how to solve
  polynomial equations. Then, I would say, Based
  on the properties of a polynomial, we know there
  are infinitely many solutions (values) to this
  polynomial when we are given various inputs.
  Therefore, we will try to illustrate this concept
  by seeing what kind of picture (graph) emerges
  when we plot several points on paper, and
  eventually, on the graphing calculator/website.

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Introduce the Learning Activity (cont)

• Because the idea of graphing polynomials might
  seem intimidating at first, I would break
  students up into groups, giving each group their
  own polynomial equation. Without giving the
  students too much to think about at one time and
  without giving away the lesson plan in one simple
  moment, I would prompt the students to try
  picturing what the graph looks like by doing
  techniques we've covered in previous days. For
  example, I would remark, What about putting an
  input (x-value) into your equation to see what
  kind of output (y-value) you get? Now, how about
  doing several of these input/output trials? Do
  you see a pattern emerge from these numbers? Is
  there a sequence? Does this pattern reveal any
  type of image in your minds?

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Introduce the Learning Activity (cont)

• Because it would be unreasonable to have too
  lofty of expectations for the students concerning
  this task, I would consider giving a handful of
  bonus points to the group who comes closest to
  seeing a pattern and to translate that pattern
  into a communicable image.

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Introduce the Learning Activity (cont)

• I believe this introduction would be beneficial
  because students will be motivated to foster
  learning atmospheres in their groups. By working
  in a group setting, students have a unique
  ability to encourage one another to do his/her
  best, not only because the students want to see
  their peers learn more, but also because the
  students want their group to have no weak links
  in order to achieve a high degree of productivity.

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Introduce the Learning Activity (cont)

• Also, with the lingering prospect of bonus
  points, students will be increasingly motivated
  to do this activity because most of the time they
  enjoy taking advantage of extra credit,
  especially if it is offered only sparingly.

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Introduce the Learning Activity (cont)

• Keeping this motivation in mind, I believe
  students will be interested in the topic because
  of the group work. In addition, if they did
  indeed see a pattern with the inputs/outputs,
  they will interested to see how I explain the
  pattern and why the pattern flows the way it
  does. Even further, if students do not see a
  pattern and are completely lost, they will be
  interested to see that a pattern does, in fact,
  exist and that there are methods to translate
  these patterns into graphs.

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Provide Information

• Keeping the properties of the polynomial in
  mind, I would teach the students how to graph the
  equation step-by-step on a piece of lined or
  graph paper.

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Provide Information (cont)

• First, I would give them the formula for how to
  find the turning point of a polynomial equation.
  Then, I would explain how to determine whether
  turning point is a maximum or a minimum. Because
  reasoning and proof are critical in the teaching
  and learning of mathematics, I would then give
  them a brief proof of why this formula exists and
  how we determine its existence.

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Provide Information (cont)

• Although memorization can be a pitfall while
  learning mathematics, I would urge the students
  to memorize this formula because of how important
  it is in the lesson.

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Provide Information (cont)

• Then, I would provide the students with a fairly
  simple polynomial equation (for example, y x2
  2x 1). Next, I would ask the students to
  identify the a, b, and c components of this
  specific equation. They should identify that a
  1, b 2, and c 1. Next, I would explain that
  the turning point of this equation is a minimum
  because a gt 0 (they will have learned this fact
  during the reasoning and proof stage of the
  lesson). Then, I would explain the turning point
  occurs at x ( 2 / (2 multiplied by 1)), which
  equals x -2/2, which equals x -1. Then, I
  would point out that we have only found out the
  x-value where our minimum occurs.

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Provide Information (cont)

• I would then instruct them to find the y-value
  where the minimum occurs. To find this, I would
  say, We merely substitute in our x-value where
  the minimum occurs into our equation. The
  resulting answer is the y-value where the minimum
  occurs. Therefore, in this example, 0 would be
  the y-value where the minimum occurs (0)2
  (2) multiplied by (-1) 1) 0 1 1 0 .
  Thus, in parenthetical form, our minimum value
  occurs at (-1, 0).

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Provide Information (cont)

• After we have found the turning point of this
  equation, I would plot this point on a Cartesian
  coordinate plane. Afterward, I would say, Let's
  pick three x-values to the left of the turning
  point (examples x - 4, x -3, and x -2) and
  three x-values to the right of the turning
  point (examples x 0, x 1, and x 2) and
  substitute them into our equation. This way,
  we'll have several points on our graph and we
  will be able to see how graph behaves.

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Provide Information (cont)

• After we have substituted the x -4, x -3,
  and x -2 into our equation, we should see the
  following points exist to the left of our turning
  point (-4,9), (-3,4), and (-2,1). Similarly,
  after we have substituted x 0, x 1, and x 2
  into our equation, we should see the following
  points exist to the right of our turning point
  (0,1), (1,4), and (2,9).

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Provide Information (cont)

• Because we have seven points on our graph, I'd
  remark, We should have a good idea of what our
  graph looks like. Then, I'd tell them that we
  connect the dots in a logical order, much in the
  same way as when we graph linear equations (a
  lesson that would be taught much earlier than
  this one). At this point, students should see the
  points get higher and higher to the left of our
  minimum. They should note a similar pattern to
  the right of our minimum. After this, I would
  explain that because we've already established
  that polynomial factors have infinitely many
  solutions, our seven points are not enough to
  draw a definitive graph of this equation.

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Provide Information (cont)

• To make up for this problem, I would explain to
  them that we must extrapolate the graph by
  extending arrows from our outer-edged points
  (-4,9) and (2,9) to indicate the graph goes on
  forever, in other words, we must show there
  exists more points on the graph than the seven
  we've plotted. Students should already have an
  understanding of this concept, as an analogous
  concept would be used in graphing a linear
  equation. Because linear equations have
  infinitely many solutions, we must draw arrows on
  our outer-edged points to indicate the infinite
  nature of the linear equation. The idea is no
  different for polynomials.

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Provide Information (cont)

• After the students have drawn the graphs on
  paper, I'd illustrate how to graph the equations
  on the graphing calculator. First, I would tell
  the students to turn their calculators on. Then,
  I'd tell them to press y , which is a key
  that provides a forum to type in an equation.
  Then I'd tell them to press the 'X' key, followed
  by the 'Squared' key. Next, I'd tell them to
  press the addition key, then I'd tell them to
  press the '2' key followed by the 'X' key.
  Finally, I'd tell them to press the addition key
  followed by '1'. Then, I'd tell them to press
  graph. In this example, the students should see
  the following

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Provide Information (cont)
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Provide Information (cont)

• Just like what they should have seen on their
  papers, we have a clear minimum value at (-1,0)
  and the points from both the left and right of
  this point go higher and higher. Just like when
  we graphed the equation on the board/paper, the
  students should not be surprised that no points
  exist in quadrants three or four on the graph
  after all, our minimum occurs at (-1,0). If there
  is a minimum at y 0, then obviously there
  should be no other points below y 0. That is
  why, of course, we see no values at y -1, y -
  2, y - 3, and so forth.

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Provide Information (cont)

• After this, I would briefly show the students
  how to manipulate the graph on their calculators.
  For example, I would show them how to find the
  value of the function given a complex input such
  as x 3.764674. This type of illustration would
  be somewhat trivial because students have the
  ability to compute this on paper, however, I
  believe it's important to demonstrate this
  because it familiarizes the students with their
  technology.

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Provide Information (cont)

• Not only will this method ensure a more accurate
  answer (because it would be easier to approximate
  this answer with technology than by hand), but it
  will also show the students they can save time
  while solving a problem. This, of course, is
  useful because it gives students more time to
  practice more problems.

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Provide Information (cont)

• Following this demonstration, I would show them
  useful websites where they can graph equations at
  home, in the event they've misplaced their
  calculators or they have run out of batteries.
  For example, I would show them this website
• http//www.coolmath.com/graphit/index.html.
• Luckily, I will not have to spend much class
  time on how to use this particular website as
  instructions are provided. Regardless, I would
  spend time graphing our y x2 2x 1 function
  on this website, simply to familiarize them with
  it. The students show witness the following

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Provide Information (cont)
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Provide Information (cont)

• Again, the students will see a minimum occurs at
  (-1, 0) and from both the left and the right, the
  graph increases and goes on forever.
• Because of limited class time, I would wait
  until a later time to show them other useful
  websites (so as not to overwhelm them) and/or I
  would hand out a bibliography of other useful
  websites to use.

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Provide Information (cont)

• After going through more examples (including
  more advanced problems), I would show the
  students the following website as a nice and fun
  way to wrap up the lesson
• http//www.teachertube.com/view_video.php?viewkey
  bfca2da721ca61a6e956.
• This website, I would say, Shows us how to
  graph polynomial equations. Because some of the
  mathematical language used in the piece might
  confuse students, I would either explain the
  meanings to them, I would urge them only to look
  at the graphing part, or both.

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Provide Information (cont)

• Finally, I would stress that the students should
  contact me through e-mail or the phone if they
  need clarification on a certain idea or ideas.

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Provide Practice

• Because there is quite a bit to this lesson,
  clearly it will take more than one day to fit all
  of this in. To give them practice, I would first
  split the students back up into the groups that
  they were in during the introduction to the
  learning activity. Subsequently, I would give
  each group the equation I assigned to them and
  tell them first to graph their problems on paper.
  Then, I would have them graph their equations on
  their calculators to verify they've graphed their
  problems correctly.

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Provide Practice (cont)

• Finally, if time permits, I would have each
  group go to the board and present their
  problems that is, I would have them graph them
  on the board. This way, the students will have
  further practice with seeing a variety of
  problems.

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Provide Practice (cont)

• At the end of the lesson, I would give the
  students several problems to do for homework.
  This way, the students will earn a great deal of
  practice. Also, they will have their
  calculators/websites at their disposals to verify
  their solutions.

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Provide Knowledge of Results

• First, the above exercise of the groups of
  students presenting their problems on the board
  will show me whether or not the students know
  what they're doing. If they get something wrong,
  I would take the time to explain what is
  incorrect, not only for the particular group's
  benefit, but for the class as a whole.

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Provide Knowledge of Results (cont)

• If the students did the problem wrong, I would
  be mindful not to embarrass them instead, I
  would say, Group A did a good job of trying to
  solve the problem, but there are a couple of
  mistakes present... Subsequently, like I've
  stated already, I would illustrate where they
  went wrong. I'd make sure, however, to praise
  them for their efforts.

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Provide Knowledge of Results (cont)

• Also, I would collect the students' homework
  assignment to see whether or not they did the
  problems correctly. If a student did the majority
  of the problems correctly, I would provide
  written feedback at the bottom of their homework,
  giving them positive feedback. If the student
  made any mistakes or did a problem incorrectly,
  I'd provide feedback in the sense of showing them
  the correct method somewhere in the margins of
  the paper.

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Provide Knowledge of Results (cont)

• For the students who did the majority of the
  homework incorrectly but clearly gave it an
  effort, I would praise them for giving an effort,
  but I'd also giving them feedback in the sense of
  showing them proper methods for the ones they did
  wrong (or at least some of the ones they did
  wrong). Even though the students gave it their
  best efforts, I would have them meet with me
  after school and go over the concepts again,
  giving them another opportunity to learn the
  ideas and practice them in my presence. For the
  students who gave a poor effort and/or didn't do
  the assignment at all, I would have them meet
  with me after school as well.

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Review the Activity

• After however many class periods it takes us to
  finish the activities (most likely no more than
  two full class periods), I would spend an entire
  class period reviewing what we've learned. That
  is, I would provide a couple of example problems
  and I would have the students graph them on
  paper. Then, we graph them on our calculators and
  we would then go on our website(s) and graph them
  on those for further verification.

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Review the Activity (cont)

• Then, I would give the students a review
  homework assignment which would cover problems
  analogous to those that will appear in the
  assessments. Unlike the unit homework
  assignments, though, I would give students answer
  sheets for the review assignment because we will
  have very little time to go over it the next
  class (the next class will be the beginning of
  assessment). All things considered, I would
  manipulate the unit so this review day falls on a
  Friday, leaving some days next week for
  assessment.

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Methods of Assessment

• In order to assess whether or not students have
  mastered the concept of graphing polynomial
  equations, I would do several things, mainly
  because how important of a concept this is in
  mathematics. First, say the following Monday, I
  would bring my students down to a computer lab
  (given one is free) so they could all sit at
  separate computers. Then, I would assign
  different polynomial equations to each student.
  Next, I would have each student access the
  graphing software and/or website I've requested
  them to use.

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Methods of Assessment (cont)

• Then, I would have each student graph his/her
  equation using the provided software or website
  in doing so, he/she should also be able quickly
  to answer a couple of general questions I have
  asked all of them (without using paper, etc),
  such as, What is the maximum/minimum point on
  your graphed equation? With my answer sheet, I
  would then scour the room, making sure the
  students have graphed their equations correctly,
  have answered their questions correctly, and
  would grade them according to how well they've
  followed instructions.

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Methods of Assessment (cont)

• The same day, I would give them a take-home quiz
  to assess even further whether or not they've
  mastered the technique of graphing polynomials
  (during this quiz, of course, students would be
  allowed to use their graphing calculators and/or
  websites). As an example problem, I would give
  them a two-part question. In the first part, I
  would ask them to graph a given equation on a
  piece of graph paper using the techniques we've
  learned in class so far.

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Methods of Assessment (cont)

• Then, on the second part, I would ask them to
  approximate the value of the function at a given
  input (For example, say I give them the equation
  y 2x2 3x 5. I would ask them to give me
  the function's value at perhaps x 2.456). The
  second part of the question might seem out of
  place, however, it will show me whether or not
  the student knows how to use his or her
  calculator and/or the website.

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Methods of Assessment (cont)

• Finally, I would give the students an exam at
  the end of the week. By the time of the exam
  date, the students should be able to master how
  to graph polynomials on paper, how to graph them
  via calculators and other technological devices
  (such as software/websites), and how to
  manipulate and understand what polynomial
  equations truly reveal to us (Example What does
  the maximum/minimum point of a polynomial
  equation actually mean? How do the notions of
  maximum/minimum points apply to real life? How
  does one find a maximum/minimum point?). Students
  should be able to answer questions like these
  using methods they've learned in and out of class.

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Methods of Assessment (cont)

• While two combined days of assessment might seem
  rather much (one day of an in-class quiz and an
  out-of-class quiz and one day of an exam), it
  will show me whether or not the students truly
  understand what they're doing. Also, because the
  assessments will be fairly close in a span of
  time, the information will remain fresh in their
  minds, thus allowing the students a greater
  chance for success.