Reflection Course Title: Pre-Calculus Grade level: 10-12 Quadratics in Factored Form
Focus: MA.912.A.2.6: The student will investigate the factors of a quadratic through its graph and through its equation. Vary the roots of the quadratic and examine how the graph and the equation change in response.
1. In the Gizmotm, use the slider to set a = 1. (To quickly set a slider to a specific number, type the number into the field to the right of the slider, and then press Enter.) Observe how the graph changes as you vary the values of r1 and r2. a. How does the graph change as the values of r1 and r2 are varied? b. What features of the graph do the values of r1 and r2 relate to? 2. Turn on Show probe. Find the y-value when x = r1 and x = r2 by dragging the probe to those x-values. a. What is the y-value at x = r1? At x = r2? b. Substitute x = r1 into the quadratic function y = a(x − r1)(x − r2) and then also substitute x = r2. What happens when the function is evaluated at its x-intercepts? 3. Find a quadratic function in factored form for each of the following pairs of x-intercepts. Check each of your answers by turning on Show x-intercepts and dragging the blue points to the given x-intercepts. a. (2, 0) and (−1, 0) b. (−3, 0) and (3, 0) c. (−2, 0) and (1, 0) 4. Set a = 1 and vary the values of r1 and r2 to find several parabolas that have only one x-intercept. a. What is the relationship between r1 and r2 when the graph only has one x-intercept? b. When there is only one x-intercept, what feature of the parabola does the x-intercept relate to? c. What is the factored form for a quadratic function with its vertex at the origin? Check your answer using the Gizmo. 5. Vary the value of a. a. What effect does the value of a have on the shape of the parabola? b. What does the parabola look like when a is positive? When a is negative? c. What effect does the value of a have on the x-intercepts of the parabola? d. What happens when a = 0? Explain why this occurs in terms of the equation y = a(x − r1)(x − r2). Factoring a Quadratic If you click on Show polynomial form, the function y = a(x − r1)(x − r2) is expanded and simplified so that you can see it written in standard form, y = ax2 + bx + c. Factoring a quadratic function algebraically entails performing the reverse operation: changing the quadratic from standard form to factored form. 1. With Show polynomial form turned on, set a = 1 and examine the standard form of the function as you vary the values of r1 and r2. a. How do the values of r1 and r2 relate to the value of c in the standard form y = ax2 + bx + c? b. How do the values of r1 and r2 relate to the value of b in the standard form of the function? c. On paper, expand (x − r1)(x − r2) algebraically and then simplify the expression. How does the result relate to the answers you just found? The lesson impacted student learning by showing improvement in the area of comprising the understanding of factoring quadratic functions. Gizmos simulations makes of factoring quadratic functions authentic, and meaningful. The student can observe, explore, recreate and receive immediate feedback about the process that would be otherwise complex and time consuming.
Danette Hernandez Math GIZMOS PD Follow-up Activity
GIZMOS Simulation Lesson: Slope – Activity A Sunshine State Standard: MA.912.A.3.10
REFLECTION: Describe how you used the GIZMO with your students: I used the GIZMO online simulation activity to review slopes with the students. The four sliders in the GIZMO simulation display allowed students to set the x- and y- coordinates of two different points. The graph at the right of the display showed the two points and the line that passes through them. The students adjusted the coordinates of two points in the plane and found the slope of the line. In addition, the students reviewed the concept of positive and negative slopes. Also, they were able to visualize and identify an undefined slope and a line with a slope of zero. The students were able to explore and visualize Rise and Run. They explained how the Rise is the vertical change between two points on a line, and the Run is the horizontal change between two points of a line. Furthermore, they were able to explain how the slope of a line is the ratio between Rise and Run.
Explain how well the lesson went from a teacher point of view: The lesson and simulation experience was stupendous. The students were able to grasp the concepts and master the skill.
Explain how the lesson impacted student learning: The lesson impacted student learning by increasing their understanding of slopes. The Gizmo simulations make it simple for students to explore and visualize the concepts. The students received immediate feedback throughout the learning process.
The Gizmo simulation lesson that I used was “Adding and Subtracting Integers.” The Next Generation Sunshine Standards addressed with this simulation lesson were: MA.912.D.7.1, MA.912.D.7.2, and MA.912.A.3.1.
Since many of my students had never used Gizmo before, I wanted to teach them how the Gizmo simulations work so that in the future they will be able to do a lesson on their own or with a partner. I used this Gizmo with my student as a whole class instruction and as an introduction to the concept of adding & subtracting integers and rational numbers. This also served as a review of how to graph rational numbers on a number line. I first told the students what all the buttons were and I modeled how to move things around using the activpen from the Promethean board. I called on different students to go up to the Promethean board, they would then follow the instructions from the guide as I read it to them. Then I asked questions to the student manipulating the Gizmo, to the entire class and/or to individual students. At the same time the students in their desks were taking notes. They copied several of the things that were shown on the Gizmo, took notes on the concepts that they were learning and solved sample problems. Along with the sample problems I called on different students to share their responses to the problems and check it using the Gizmo value sliders and number line. I believe that my lesson turned out to be very good and I will definitely continue to do more lessons like this one. This was an excellent way to have the students take notes without complaining, get them to participate, and do something hands-on which gave them the opportunity to understand the concepts covered in a different way. The students understood the concepts and how the Gizmo is used. They even told me that they would feel comfortable doing this on their own and that they really enjoyed doing this as a class because it was more hands-on than the other things they did in their previous math classes. I felt a tremendous amount of joy to see that they were using higher order thinking skills. The discussions that the class held were good and I really felt that I acted as a facilitator, not as the person telling them that they were right or wrong. They came up with great questions for me and other students. I also came up with great questions as I went along when I noticed that some students were having trouble understanding what I felt that they knew but still needed to expand their knowledge. The entire class was alert and it gave most of the students in the class a chance to participate by going up to the board and clicking on something, reading and answering questions. The students were also able to practice how to combine their prior knowledge with the new material that they were seeing in order to answer some of the questions that they encountered. This lesson gave me an opportunity to address the different learning styles, such as visual, auditory and kinesthetic. Since this lesson addressed those learning styles, I think that in turn, it was more valuable and/or meaningful to them. I am hopeful that they will be able to remember the lesson objectives.
GIZMOS: Systems of Linear Equations Sunshine State Standards: MA.912.A.3.13, MA.912.A.3.14
REFLECTION: The students used the GIZMO Activity to solve systems of equations by graphing. The tabs at the top of the activity allowed students to see their graphs as lines on the coordinate plane, as a matrix and as a table of values simultaneously. The sliders in the GIZMO simulation display allowed students to change the y-intercepts and the slopes of their lines and observe the changes. One of the changes they were able to understand better is the concept of positive, negative, zero and undefined slopes. They were able to see the types of lines that produce these slopes and how those differences affected the graphed lines. They were able to differentiate visually between a line with an undefined slope and a line with a slope of zero. Also, they now better understand when a systems of equations has one solution, infinitely many solutions, or no solution at all.
Explain how well the lesson went from a teacher point of view: The GIZMOS added to student understanding of graphs and slopes by producing changes right in front of their eyes. This is something that cannot be achieved just by doing classwork or homework from a textbook. The GIZMOS add the benefit of visual impact.
Explain how the lesson impacted student learning: The lesson impacted student learning by increasing their understanding of the solution of a system of graphed lines and of slopes. Students are able not only to observe changes immediately, but also to create their own changes and see how those changes affect the graph.
Adam Nehme 10/14/2010
ReplyDeleteMath Gizmos
Reflection
Course Title: Pre-Calculus
Grade level: 10-12
Quadratics in Factored Form
Focus: MA.912.A.2.6: The student will investigate the factors of a quadratic through its graph and through its equation. Vary the roots of the quadratic and examine how the graph and the equation change in response.
1. In the Gizmotm, use the slider to set a = 1. (To quickly set a slider to a specific number, type the number into the field to the right of the slider, and then press Enter.) Observe how the graph changes as you vary the values of r1 and r2.
a. How does the graph change as the values of r1 and r2 are varied?
b. What features of the graph do the values of r1 and r2 relate to?
2. Turn on Show probe. Find the y-value when x = r1 and x = r2 by dragging the probe to those x-values.
a. What is the y-value at x = r1? At x = r2?
b. Substitute x = r1 into the quadratic function y = a(x − r1)(x − r2) and then also substitute x = r2. What happens when the function is evaluated at its x-intercepts?
3. Find a quadratic function in factored form for each of the following pairs of x-intercepts. Check each of your answers by turning on Show x-intercepts and dragging the blue points to the given x-intercepts.
a. (2, 0) and (−1, 0)
b. (−3, 0) and (3, 0)
c. (−2, 0) and (1, 0)
4. Set a = 1 and vary the values of r1 and r2 to find several parabolas that have only one x-intercept.
a. What is the relationship between r1 and r2 when the graph only has one x-intercept?
b. When there is only one x-intercept, what feature of the parabola does the x-intercept relate to?
c. What is the factored form for a quadratic function with its vertex at the origin? Check your answer using the Gizmo.
5. Vary the value of a.
a. What effect does the value of a have on the shape of the parabola?
b. What does the parabola look like when a is positive? When a is negative?
c. What effect does the value of a have on the x-intercepts of the parabola?
d. What happens when a = 0? Explain why this occurs in terms of the equation y = a(x − r1)(x − r2).
Factoring a Quadratic
If you click on Show polynomial form, the function y = a(x − r1)(x − r2) is expanded and simplified so that you can see it written in standard form, y = ax2 + bx + c. Factoring a quadratic function algebraically entails performing the reverse operation: changing the quadratic from standard form to factored form.
1. With Show polynomial form turned on, set a = 1 and examine the standard form of the function as you vary the values of r1 and r2.
a. How do the values of r1 and r2 relate to the value of c in the standard form y = ax2 + bx + c?
b. How do the values of r1 and r2 relate to the value of b in the standard form of the function?
c. On paper, expand (x − r1)(x − r2) algebraically and then simplify the expression. How does the result relate to the answers you just found?
The lesson impacted student learning by showing improvement in the area of comprising the understanding of factoring quadratic functions. Gizmos simulations makes of factoring quadratic functions authentic, and meaningful. The student can observe, explore, recreate and receive immediate feedback about the process that would be otherwise complex and time consuming.
Danette Hernandez
ReplyDeleteMath GIZMOS
PD Follow-up Activity
GIZMOS Simulation Lesson: Slope – Activity A
Sunshine State Standard: MA.912.A.3.10
REFLECTION:
Describe how you used the GIZMO with your students:
I used the GIZMO online simulation activity to review slopes with the students. The four sliders in the GIZMO simulation display allowed students to set the x- and y- coordinates of two different points. The graph at the right of the display showed the two points and the line that passes through them. The students adjusted the coordinates of two points in the plane and found the slope of the line. In addition, the students reviewed the concept of positive and negative slopes. Also, they were able to visualize and identify an undefined slope and a line with a slope of zero. The students were able to explore and visualize Rise and Run. They explained how the Rise is the vertical change between two points on a line, and the Run is the horizontal change between two points of a line. Furthermore, they were able to explain how the slope of a line is the ratio between Rise and Run.
Explain how well the lesson went from a teacher point of view:
The lesson and simulation experience was stupendous. The students were able to grasp the concepts and master the skill.
Explain how the lesson impacted student learning:
The lesson impacted student learning by increasing their understanding of slopes. The Gizmo simulations make it simple for students to explore and visualize the concepts. The students received immediate feedback throughout the learning process.
The Gizmo simulation lesson that I used was “Adding and Subtracting Integers.” The Next Generation Sunshine Standards addressed with this simulation lesson were: MA.912.D.7.1, MA.912.D.7.2, and MA.912.A.3.1.
ReplyDeleteSince many of my students had never used Gizmo before, I wanted to teach them how the Gizmo simulations work so that in the future they will be able to do a lesson on their own or with a partner. I used this Gizmo with my student as a whole class instruction and as an introduction to the concept of adding & subtracting integers and rational numbers. This also served as a review of how to graph rational numbers on a number line. I first told the students what all the buttons were and I modeled how to move things around using the activpen from the Promethean board. I called on different students to go up to the Promethean board, they would then follow the instructions from the guide as I read it to them. Then I asked questions to the student manipulating the Gizmo, to the entire class and/or to individual students. At the same time the students in their desks were taking notes. They copied several of the things that were shown on the Gizmo, took notes on the concepts that they were learning and solved sample problems. Along with the sample problems I called on different students to share their responses to the problems and check it using the Gizmo value sliders and number line.
I believe that my lesson turned out to be very good and I will definitely continue to do more lessons like this one. This was an excellent way to have the students take notes without complaining, get them to participate, and do something hands-on which gave them the opportunity to understand the concepts covered in a different way. The students understood the concepts and how the Gizmo is used. They even told me that they would feel comfortable doing this on their own and that they really enjoyed doing this as a class because it was more hands-on than the other things they did in their previous math classes. I felt a tremendous amount of joy to see that they were using higher order thinking skills. The discussions that the class held were good and I really felt that I acted as a facilitator, not as the person telling them that they were right or wrong. They came up with great questions for me and other students. I also came up with great questions as I went along when I noticed that some students were having trouble understanding what I felt that they knew but still needed to expand their knowledge. The entire class was alert and it gave most of the students in the class a chance to participate by going up to the board and clicking on something, reading and answering questions.
The students were also able to practice how to combine their prior knowledge with the new material that they were seeing in order to answer some of the questions that they encountered. This lesson gave me an opportunity to address the different learning styles, such as visual, auditory and kinesthetic. Since this lesson addressed those learning styles, I think that in turn, it was more valuable and/or meaningful to them. I am hopeful that they will be able to remember the lesson objectives.
Y. Valdes
ReplyDeleteMath GIZMOS
PD Follow-up Activity
GIZMOS: Systems of Linear Equations
Sunshine State Standards: MA.912.A.3.13, MA.912.A.3.14
REFLECTION:
The students used the GIZMO Activity to solve systems of equations by graphing. The tabs at the top of the activity allowed students to see their graphs as lines on the coordinate plane, as a matrix and as a table of values simultaneously. The sliders in the GIZMO simulation display allowed students to change the y-intercepts and the slopes of their lines and observe the changes. One of the changes they were able to understand better is the concept of positive, negative, zero and undefined slopes. They were able to see the types of lines that produce these slopes and how those differences affected the graphed lines. They were able to differentiate visually between a line with an undefined slope and a line with a slope of zero. Also, they now better understand when a systems of equations has one solution, infinitely many solutions, or no solution at all.
Explain how well the lesson went from a teacher point of view:
The GIZMOS added to student understanding of graphs and slopes by producing changes right in front of their eyes. This is something that cannot be achieved just by doing classwork or homework from a textbook. The GIZMOS add the benefit of visual impact.
Explain how the lesson impacted student learning:
The lesson impacted student learning by increasing their understanding of the solution of a system of graphed lines and of slopes. Students are able not only to observe changes immediately, but also to create their own changes and see how those changes affect the graph.