3.+Lesson+Plans

= Parabola Lesson Plan =

- 2.9.11.G: Solve problems using analytical geometry. || - Students will be able to identify if a parabola opens up or down. - Students will be able to graph parabolas by find the vertex, line of symmetry, and y-intercept of a quadratic equation. - Students will be able to identify the y-intercept, line of symmetry and vertex given a graphed parabola. || - As the students walk into the classroom, they will be given a pre-test to graphing parabolas. I will be standing at the door, handing out a piece of paper. On that paper I will have the equation y = x^2 + x – 2 and a graph depicting the graphed parabola. I will ask them if they can identify the y-intercept, the vertex, and the line of symmetry on both the graph and how they could find it in the equation. I will give them a few minutes to complete this warm-up. I will copy the exact worksheet on a transparency and using a marker, I will ask students if they can identify each section and ask why they chose what they did. As a class, I want to have a discussion about how and where the students found these concepts. This will be a prelude into the parabola lesson plan. || - Calculator - Project with transparencies - Whiteboard/blackboard with markers to draw equations - Algebra textbook || - I will then put some simple equations up on the board to have the students practice finding the y-intercept and the vertex, such as y = x^2 + 14x + 48. This exercise will be done orally, making sure that each student understands the concept. - Once the students have mastered finding the vertex and the y-intercept, I will start on graphing parabolas. On the board, I will put the equation y = -16x + 500. Before we can even begin to place the vertex and y-intercept on the graph, the students must learn the direction of a parabola. “When a person does not get something that he wants, he thinks it is negative and therefore he is sad.” This means that if the variable “a” in the function is negative, the parabola will open downward, like a frowning face. “When a person gets something they want, he thinks it is positive and therefore he is happy.” This means that if the variable “a” in the function is positive, the parabola will open upward, like a smiley face. Using a phrase that will, hopefully, click into the students’ head, they will understand which direction a parabola opens. Now that they know this equation will open downward (a = -16), they can plot the y-intercept and the vertex. The line o symmetry will go straight through the vertex. The final step is drawing a curve from the vertex to the y-intercept and repeating that step on the other side of the line of symmetry. - Together the class will graph two equations. y = 5x^2–6x – 5 and y = -12x^2+3x-13 - The students should then be able to also do the reverse. Instead of graphing the parabola using the quadratic equation, the student should also be able to find the vertex, the line of symmetry, and the y-intercept using a graph and convert it into a quadratic function. || - Are there any sayings that you can create to help you remember how to find the direction of a parabola? || - I want the first part of the class (warm-up) to be more of a discussion where students are able to express how they believe to find the points I have asked. I want these learners to come up with different methods to solving the equation, knowing that, at the start, no one student really knows how to solve them. || - I will give the students the last few minutes of the class to complete this. Before they leave the class, they are to drop them in a box by the door. At the end of the day, I will read through them and see what they understood and what I need to work on. || - While we are working through the steps of the problems, I will also be walking around the room to check to see if students are taking notes and completing the work at hand. During the discussion portions, I will also be calling on random students to make sure they understand each step. ||
 * ** Target Grade Level ** ||
 * - 9th grade Algebra I ||
 * ** Pennsylvania Content Standards ** ||
 * - 2.8.11.E: Use equations to represent curves (e.g., lines, circles, ellipses, parabolas, hyperbolas ).
 * ** Instructional Objectives ** ||
 * - Students will be able to calculate the vertex, line of symmetry, and y-intercept of a parabola using a quadratic equation.
 * ** Description of Introductory Activity and Discussion ** ||
 * - This lesson will be conducted after learning about quadratic formulas. The students should already know how to solve quadratic formulas. I will now be taking this one step further by introducing how to graph them and the concept of a parabola.
 * ** Materials Needed ** ||
 * - Graphing paper
 * ** Description of Learning Activities ** ||
 * - After the warm-up is completed, I want to introduce new vocabulary to the class. We will be discussing the terms parabola, vertex and line of symmetry. Since we just finished a lesson on quadratic functions, I will but the equation f(x) = ax^2 + bx + c on the board, indicating how to find the y-intercept (0,c) and the vertex (-b/2a, f(-b/2a)).
 * ** Discussion Questions ** ||
 * - What geographic shape would a quadratic function make if it were graphed?
 * ** Lesson Modifications for diverse learners ** ||
 * - I will be having students who show a faster understanding of the material come up to the front of the room to explain their solutions to the problems I have given to the class. This will allow those students to talk their way through the problem and it may also help those students who are having more difficulty, hear a student’s perspective on how he solved the problem. Sometimes students can learn more about a topic from someone that they can relate to, someone their age or someone that they already know, then a teacher, especially a teacher that he or she just met a few days ago.
 * ** Lesson Closure ** ||
 * - As the lesson comes to a close, I will pass out exit logs. On this sheet, I want the students to write down what they thought about the lesson. Students are not to put their names on the sheet. I believe that students tend to be more honest about a lesson when they do not have their name attached to their opinion. Do they have any questions about the lesson that I need to go over in more detail?
 * ** Formative Assessment ** ||
 * - I will be using the exit logs as a way to see how many students understood the lesson. Although the sheets will be anonymous, it will allow me to get a better understanding of how many people understand the concept and those who do not.

= Ellipse Lesson Plan =

- 2.9.11.G: Solve problems using analytical geometry. || - Students will be able to define the term ellipse. - Students will be able to create a graph of an ellipse given an equation. || - They will then do the same thing for the second piece of cardboard, but this time it will be landscape oriented. This will emphasize to the students the difference between the major and the minor axis depending on the orientation of the ellipse. The students will keep these diagrams to help them throughout the ellipse lesson, especially when graphing the equations to locate the foci, vertices and major/minor axis.
 * ** Target Grade Level ** ||
 * - 9th grade Algebra I ||
 * ** Pennsylvania Content Standards ** ||
 * - 2.8.11.E: Use equations to represent curves (e.g., lines, circles, ellipses, parabolas, hyperbolas ).
 * ** Instructional Objectives ** ||
 * - Students will be able to diagram the foci, vertices, major axis and minor axis given an ellipse.
 * ** Description of Introductory Activity and Discussion ** ||
 * - Students will be grouped into pairs of two and given two pieces of cardboard (8x10) a piece of string, and four thumbtacks. Using these materials, the students will construct their own ellipse. Holding the first piece of cardboard portrait, the students will place the thumbtacks semi-close together. They will then tie each end of the string to a thumbtack and catch the loop that it makes with either their pencil or pen. While keeping the string tight, they will draw an ellipse. They will draw a line down the center of the ellipse vertically and then horizontally. Since the page is portrait oriented, the line that is vertical will be labeled as the major axis (since it is longer) and the horizontal line is the minor axis. The two points where the major axis hit the ellipse is called the vertex. The final part of the ellipse the students will label is the foci, which are where they put the thumbtacks into the cardboard.

|| - Thumbtacks - Cardboard || - We will first start the lesson off by providing definitions of ellipse, foci, vertices, major axis, and minor axis. An ellipse is the set of all points in a plane whose sum of the distance between two fixed points (our thumbtacks for example) is constant. The two fixed points are called the foci and the line that goes through the foci and intersects with the ellipse at two points called the vertices. The major axis and the minor axis are perpendicular to each other and the major axis is the one whose endpoints connect the vertices. As the students can see from our diagrams, an ellipse can be elongated in any directions, but we will only be focusing on horizontally or vertically. - Once the students understand the definitions of the terms (hopefully using the diagram as a reference will help) we will move on to using the standard form of an equation of an ellipse. Lets first look at our ellipse that is elongated horizontally. We will be using the variables a, b, and c to help represent the distances for each term. Although the definition of the ellipse is given in terms of its foci, they are actually not a part of the graph. A complete graph of an ellipse can be obtained without graphing the foci at all. “a” represents the distance of a vertex from the origin, “b” represents the distance from the origin the minor axis intersects the ellipse, and “c” represents the distance a focus is from the origin. So looking at out diagram of the horizontal ellipse, the vertices would have a coordinate of (-a,0) and (a,0) the minor axis or hit of the ellipse would be (0,b) and (0, -b) and the foci would be (-c,0) and (c,0). We will do the same for the vertical ellipse as well. For both equations, b^2 = a^2 - c^2 so equivalently c^2 = a^2 - b^2. If the denominator the x^2 term is larger than the denominator of the y^2 term, the major axis is horizontal. So given the equation x^2/9 + y^2/4 = 1, the x^2 terms is 9, which is bigger than the y^2 terms, so this ellipse is horizontal. So, the standard form of the ellipse equation is x^2/a^2 + y^2 + b^2 =1. In this case, the x^2 term is 9 which is equation to a^2 so to get that alone, we need to take the square root of that number to just get a. a = ±3 so the vertex coordinates would be (-3,0) and (3,0). To find the height of the ellipse, we take the term associated with the y^2 term which is 4, so b^2 = 4 according to the standard equation. To get b alone we take the square root of 4 and we get ±2. The height of the ellipse is (0,2) and (0,-2). The final part that we need to find is the foci which is our c terms. In the previous section we learned that c^2 = a^2 - b^2 so we plug in the number and we get c^2 = 9 - 4. c^2 = 5 and to get c alone we take the square root of each side and end up with c = ±√5.
 * ** Materials Needed ** ||
 * - String
 * ** Description of Learning Activities ** ||
 * - Students will return to their individual seats to start on the actual content of the lesson. They should keep the diagrams of the ellipse out to use as a reference as they are taking notes.

We will do another example together on the board using the vertical ellipse and then I will give them their homework to start on if there is any time left in the period. || - How and where can an ellipse be found in every day life? - How can you differentiate between a horizontal ellipse and a vertical ellipse? || - Students with learning disabilities will also be given regular graph paper that they can construct their ellipses on rather than created one by hand because it is easier to find the exact lines when using graph paper and it will be beneficial to them. || - X^2/16 + Y^2/4 = 1 - X^2/49 + Y^2/81 = 1 - X^2/64 + Y^2/100 = 1 - X^2/25 + Y^2/64 = 1 - X^2/49 + Y^2/36 = 1 ||
 * ** Discussion Questions ** ||
 * - What is the purpose of the foci when they do not need to be graphed to create an ellipse?
 * ** Lesson Modifications for diverse learners ** ||
 * - Students will disabilities will be paired with another student that he or she works well with and will help them complete the diagram of the ellipse. By using a hands on experience, I am hoping to have not only learning support students but all students in general get a better understanding of the lesson.
 * ** Lesson Closure ** ||
 * - Once again I will be using exit slips to make sure the students understand the concept of find the foci given an equation. I will ask them to describe on a piece of paper how to locate the foci of an ellipse given the equation X^2/25 + Y^2/16 = 1 ||
 * ** Formative Assessment ** ||
 * - I will be using their diagrams as a form of assessment for this lesson. If I can see that the students are having difficulty identifying the points on the diagram, I will spend more time going over their definitions and locating them on a graph. ||
 * ** Homework ** ||
 * - Students will graph the ellipse and locate the foci on the following problems: