| LU Title: Graphing Quadratics | Author(s): Michael Loveric |
| Grade Level: 10 | School : Whitesboro High School |
| Topic/Subject Area: Course II - Mathematics | Address: 6000 Rte
291 Marcy, NY 13403 |
| Email: mlover@whitesboro-high.moric.org | Phone/Fax: (315)768-9800 |
Overview
Graphing quadratics in the cartesian plane will be limited to parabolas of the form y = ax2 + bx + c where a, b, and c Î Â . Also, the students will solve systems of equations algebraically and graphically that involve lines and parabolas. The use of the web site www.webmath.com will be integrated to check solutions of quadratic equations for points of intersection on the x-axis, if they exist, and to check graphing techniques. In addition, the TI-81 (graphing calculator) will be used to confirm the results found algebraically and graphically by hand. This unit will take approximately six days.
Content Knowledge
Declarative Knowledge
Procedural Knowledge
Essential Questions
How many solutions can a system of equations have? Show your knowledge using diagrams of the three types of systems we have studied.
What are the differences between solving a quadratic equation algebraically and graphically?
Connection to NYS Learning Standards
MST 1: Key Idea 3: Students will use critcal thinking skills in the solution of
mathematical problems.
Students will apply algebraic and geometric concepts and skills in the
to the solution of problems.
MST 3: Key Idea 3: Students will use mathematical operations and relationships among them to
understand mathematics.
Students will addition, subtraction, multiplication, division, and
exponentiation with real numbers and algebraic expressions.
MST 3: Key Idea 4: Students will use mathematical modeling/multiple representation to provide
a means of presenting, interpreting, communicating, and connecting mathematical information and relationships.
Students will use graphing utilities to create and explore geometric and
algebraic models. (TI-81, www.webmath.com)
Initiating Activity
The students will use a TI-81 to complete a review worksheet of how to graph linear equations. Then, they will graph a system of equations and find the roots algebraically and graphically.
Using the TI-81 graphing calculator, graph the following lines and sketch your graphs in the boxes below. Label the axes, origin and y-intercepts for each graph in the form y = mx + b.
1. y = x + 2 2. y = x + 4 3. y = x + 6
What changes occurred on the graph when the value of b increased? _____________________________
4. y = 2x + 2 5. y = 3x + 4 6. y = -x + 6
7. Looking at the graphs of 1 and 4, what changed when the coefficient of x went from 1 to 2?
8. Looking at the graphs of 2 and 5, what changed as the coefficient of x increased?
9. Looking at the graphs of 3 and 6, what are the results in terms of slope when there is a -1 in front of the x?
10. How can you graph a line like 3y = 6 - 9x ? (Notice when you input values on the calculator what is displayed on the left side of the equal sign?)
Now, let's graph something a little different. Sketch your graph in the box provided, pay attention this time to the x-intercepts and the vertex (where the graph turns).
11. y = x2 + 6x + 8 12. y = x2 - 4
note: the vertex is where the graph turns up or down
The graphs of 11 and 12 have a pattern. What is similar about these two graphs?
Now we should graph the following using a table, and the graph paper below.
13. y = 2x - 4 -3 £ x £ 3 14. y = x2 - 4 -3 £ x £ 3
Learning Experiences
Day 1: Students will complete the initiating activity with a partner to review the techniques of graphing linear equations of the form y = mx + b and the effects of changing the slope and y-intercept. The use of the TI -81 graphing calculator will help the students graph the lines quickly so time can be spent on the interpretive questions. These questions will help construct meaning from prior knowledge and link it to new information. The three-minute pause will be used after a series of graphs have been sketched from the graphing calculator to answer questions. Also, the three-minute pause will be used at the end of class to summarize what was learned. Assessment will be done by observation of the student groups.
Day 2: Students will graph parabolas of the form y = ax2 + bx + c using the graphing calculator
and by completing a table. Particular attention will be focused on parabolas where the value of a is negative. With just a couple of graphs, the students will discover whether a parabola opens up or down. The five phase inquiry process will be established using examples and non-examples of parabolas that open up and down. Assessment will be accomplished by observation and by collecting homework.
Day 3: Students will be asked a series of questions to stimulate prior knowledge on solving quadratics (i.e. how many solutions are there, how did we solve them previously?) Then, the reciprocal teaching technique will be used to help students remember this prior knowledge. After a three-minute pause for student groups to write a list of important ideas associated with solving quadratics, a student leader will be chosen for each group to discuss the topics that his or her group remembered. After summarizing, questioning, and clarifying, the student leaders will ask the class for predictions about how this prior knowledge can be reinforced with our graphical methods of parabolas. From this point, parabolas will be graphed that either intersect the x-axis twice, once, or not at all; this experience will help the students connect the algebraic and graphic nature of parabolas. Assessment will be done with a quiz and the class will use the rest of the time practicing solving quadratics by both methods.
Day 4: Students will find the axis of symmetry x = -b/2a for a parabola in the form y = ax2 + bx
+ c by taking notes of examples. A written set of steps will be presented to help students find an interval for the x values if they are not given one. The students will then practice finding the axis of symmetry, an appropriate interval and finally graphing some parabolas where the equation is given but the interval is not. Students will be taken to the computer lab to use www.webmath.com to check the x-intercepts of their parabolas, if they exist. Assessment will be by observation of the class-work.
Extending and Refining Experience
Days 5 - 6: Students will work in cooperative groups to solve systems of equations involving
lines and parabolas. The modeling process will be used to help solve the system by both methods, algebraically and graphically. In pairs, one student will complete the task algebraically and the other will complete it graphically. They will share their answers to see if both are correct and compare pitfalls of both methods during a three-minute pause. Then, they will solve another system of equations by switching rolls. The person who solved the first system algebraically will use the graphical method and vice versa for the other member of the group. At the end of solving the second system, the group of two will compare answers and start to analyze any errors, and construct support for their own answer.
Culminating Performance
Each group of three students will produce a PowerPoint presentation of a system of equations that they invent and explain how or why they invented that system. The system of equations must contain a line and a parabola. The presentation will include a title, equations and graph of a line and parabola, explanation of the related vocabulary, explanation of the solution of the system, as well as, explanation of how and why they invented that system of equations within two to four minutes.
Rubric
Group Members: _____________________________________________________
Score: ____
|
Presentation
|
Time Limit and Presentation |
Vocabulary |
Appropriate System |
Graphical Solution |
|
3 The background and foreground color, text and overall appearance is readable and understandable.
|
3 Time to view the presentation is between 2 and 4 minutes. Ability to explain presentation is apparent. |
3 Effective use of vocabulary on graphs and correctly labeled. |
3 Quadratic system is appropriate for a course II student. Able to explain how or why this system was invented. |
3 Graphs are accurate and labels are correct. |
|
2 The background and foreground color, text and overall appearance is hard to read and difficult to understand.
|
2 Time to view the presentation is less than 2 or more than 4 minutes. Ability to explain presentation has errors but does not interfere with communication of concepts. |
2 Vocabulary used on graphs but not in all the correct places or missing key terms. |
2 Quadratic system lacks mental challenge. Able to explain how or why system was invented with a little difficulty. |
2 Graphs and labels are present with minor errors. |
|
1 The background and foreground color, text and overall appearance is not readable and barely comprehendible.
|
1 Time to view presentation is less than 1 minute or more than 5 minutes. Ability to explain the presentation exhibits lack of preparation. |
1 Little vocabulary used and incorrectly labeled. |
1 Quadratic system is missing key components. Not able to explain how or why the system was invented. |
1 Graphs and labels are present but have major errors. |
A score of zero will be assigned for any part of this project not completed.
Pre-requisite Skills
The students should have successfully completed solving quadratics by factoring and use of the quadratic formula. Also, the students should be familiar with the Internet site www.webmath.com, as well as, the computer software Powerpoint.
Modifications/Adaptions
Modifications will be made according to I.E.P.'s currently in place and any reasonable request for students of different learning styles. The student groups will be assigned by the teacher in order to foster productivity and to meet the changing needs of students in the classroom and computer lab.
Unit Schedule/Time Plan
The unit should take six class periods and one additional class period for presentation of the culminating performance task. The students will be expected to work during study halls or free time to complete the performance task with their group members.
Technology Integration
The students will use the TI -81 graphing calculators in the classroom. Also, they will have access to a computer lab with Internet connections to use www.webmath.com. In addition, the students will use PowerPoint to create a presentation for class.
Reflections
The students seemed to like using a calculator to graph linear equations as a review of course I material. They quickly remembered the affects of changing slopes and y - intercepts. The new information gained with technology took some of the negative energy that some students had about graphing and turned it into a positive experience for all. The time spent on the calculators gave students an insight as to what the graphs should look like from certain characteristics, which they remembered when graphing by hand. As the topics were covered, the students always had a clear picture of what the final graphs would look like. During the refining experience, students were amazed that they ended up with the same answers even though two different methods were used. It was an interesting experience with the students; I think it would be nice to add graphing circles to this unit also.
References
Rising, Gerald, et al. Unified Mathematics Book 2. Boston: MA:
Houghton Mifflin Company, 1985.
World Wide Web
"Webmath." 1997 - 2000.
http://www.webmath.com (13 march 2000)