Investigating Quadratics

Name: Kristin Tuma & Jen ThomasSubject: AlgebraGrade: 9 – 12 Date: 12/08/2010Length of Class: 50 minutes

Prior Knowledge: Students should be relatively familiar with graphing linear equations and understanding/analyzing relationships between two variables. Furthermore, students have been introduced to the general form of the quadratic equation f(x) =ax2+bx +c; However, they have never graphed quadratic equations. They has been small association between quadratic equations and the fact that their graphs are parabolic. Students are know what a parabola looks like/can identify one.

I. Goals:

a. To recognize parabolic shapes/objects.b. To develop an understanding of quadratic equations in the standard form.

II. Objectives:

a. Students will be able to create a line of best fit for real-life parabolic situations.b. Students will be able to explain how the parameters a, h, and k affect the graph of the quadratic

equation f(x) = a(x−h)2+k.c. Students will be able to apply general knowledge of parabolas to identify parabolic

shapes/objects in real-life settings.

III. Materials/Resources/Technology:

a. Warm-Up sheet for each studentb. In-Class Worksheet for each studentc. Digital camera for each group of studentsd. GeoGebra accessibility e. Projector

ActivityStart of Class:

Begin by having each student take a Warm-Up Worksheet as they walk into class. Instruct the students to work on their Warm-Up Worksheet immediately at their assigned seats.

Give the students about 3-5 minutes work on the worksheet. While they are working, walk around the classroom and observe student work. Once the majority of students have completed the Warm-Up Worksheet, come back

together as a class in order to discuss the Warm-Up Worksheet. Questions for Discussion: Example 1: Would you use a straight line or an exponential line to demonstrate the

relationship? Why? Example 2: Would you use a straight line or an exponential line to demonstrate the

relationship? Why? Example 3: Would you use a straight line or a parabola to demonstrate the relationship?

Why?Introduction of Lesson:

“In our Warm-Up we reviewed how to analyze relationships between two variables. Our last example demonstrated a possible relationship that could be depicted by a parabolic curve. We have discussed how graphs of quadratic equations are parabolic. Today, we will be examining the standard form of quadratic equations. You will also be identifying parabolic-like shapes in the real-life settings.

Lesson Procedures: Distribute an In-Class Worksheet to each student Play half of the video clip from Dan Myer’s Blog. Have students work on Questions 1 and 2. Come back together as a class and take a quick pole: Who thinks the ball will go in hoop?

Who thinks the ball will not go in the hoop? Why? Transition: “You have now predicted whether or not the ball will go into the hoop. Now

we will test your predictions. Use the tools available in GeoGebra to move around the sliders, a, h, and k (these are the parameters for the standard form of a quadratic equation). Try to move the sliders in order to create a parabola that best fits the picture (motion of the basketballs).”

Have students open up Geogebra Have students work on Questions, 3 – 7. When most students seem to be done, come back together as a class for discussion. Question: “How does the parameter a affect the graph of a quadratic equation? What

happens when a is positive? Negative?” Question: “How does the parameter h affect the graph of a quadratic equation? What

happens when h is positive? Negative?” Question: “How does the parameter k affect the graph of a quadratic equation? What

happens when k is positive? Negative?” Question: What values of a, h, and k did you use to create a graph that best fits the curve

of the projectile motion of the basketballs? Discuss differences between vertex form and general form of the quadratic equation. Play the full video clip from Dan Myer’s Blog. (the ball goes in!) Transition: “Alright, so we have viewed the projectile motion of a basketball, which

appears to be parabolic. We derived a quadratic equation in vertex from that best fit the curve. Now it is your turn! In your groups, go around the building and try to find one image that appears to be parabolic. For example, we found a fire hydrant that was covered in show and appears to be parabolic. [Show picture]. In your groups, try to find an image that seems to be parabolic. Take a picture of that image and come back to class! We will proceed from there!”

Have students upload their pictures using the USB cords. Follow Steps 1 – 15.

Closure: Come back together as a class and have students share their pictures. Compare different

pictures and ask students which picture they believe best represents a quadratic function. That is, what picture is most parabolic?

Assessments/Checks for Understanding: As students work on their Warm-Up Worksheet, walk around the room and make sure they

are on task. Provide assistance and guidance where needed.

Students need to complete the first GeoGebra activity of the projectile motion of the basketball.

Students are held accountable for finding an object/image in real-life settings that appears to be parabolic and taking a picture.

Students need to complete the second GeoGebra activity using their picture

Reflection

In our lesson we used GeoGebra to discover parabolas and quadratic functions. After our warm-up, we began our lesson with a video of a person shooting a basketball from Dan Meyer’s Blog. We stopped the video when the ball is halfway to the hoop; this way the students will be motivated to figure out if the ball will go through the hoop or not. After the students find a quadratic formula and a parabola that fits the basketball motion, we will show the class a video. By actually seeing if the basketball goes through the hoop or not is a good way of confirming if their prediction was correct. Being able to see it is much more convincing that just having the teachers tell students whether it goes through or not.

We also used GeoGebra to find a graph and equation that predicted if the basketball will go through the hoop and for their own parabolic photo. For the basketball example, we used an applet from Dan Meyer’s blog that already had sliders. It was a picture of the video we just showed, except the ball is only half way to the hoop. Through the sliders we let the students discover how the different variables of the vertex form change the parabolic graph. We wanted technology to help guide our students to discover this by asking on our worksheet to move the a, k, and h slider to realize that they make the curve change or have horizontal or vertical shifts. By using GeoGebra, the students could immediately see how the graph changed when they moved the sliders. This way it is much more efficient than having the students graph multiple quadratic equations to discover the same concepts. It also allows the students to actually discover how the variables affect the graph, instead of making assumptions or just believing the teacher’s word.

For when the students graphed a quadratic equation on their own photos, we wanted them to determine the quadratic function by trying to find the equation then plotting it instead of using the sliders. This way they can receive immediate feedback if their equation is close or not. This is also a good assessment for us to determine if our students understood the lesson by being able to create a quadratic function that closely describes the picture they took. We could also assess if they understand what parabolas look like from the pictures they took. This is not only an immediate assessment, but a way for students to apply a new concept to the real world. It is a fun way to try to find a quadratic equation.

By creating this lesson we have come to realize that it takes a lot of time. Not only do you have to plan out the lesson, but you have to practice the technology and make sure you know it very well. At the same time, if this lesson with technology is successful it will be easy and much less time consuming repeating it the next year. Even though it is a lot of work, we think it is worth incorporating because, if done properly, can get students to be more engaged in mathematics.

Warm-Up

Directions: Draw a graph that you believe best fits the relationship among the following variables:

Exploring Quadratics

Part I: Predicting the Projectile Motion of a Basketball

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1. Based upon the short video clip and its corresponding picture above, do you think ball will go in the hoop?

2. Draw a parabola on the graph provided that best fits the projectile motion of the basketball. Does your answer for Question 1 change or remain the same?

3. Go to ci436jenthomas.weebly.com and download this image by clicking on the “Basketball Image.” It will open in GeoGebra. Notice that there are sliders a, h, and k above the image.

a. Move around slider a. How does a affect the shape of the graph? Be sure to look at cases when a is negative and when a is positive.

b. Move around slider h. How does h affect the shape of the graph? Be sure to look at cases when h is negative and when h is positive.

c. Move around slider k. How does h affect the shape of the graph? Be sure to look at cases when k is negative and when k is positive.

4. Parameters a, h, and k are used in the vertex form for the equation of a quadratic function:

f(x) = a(x−h)2+k

5. Move around the sliders to create a graph of a quadratic that best fits the curve created by the projectile motion of the basketballs. Record your values here:

a_________________________

h_________________________

k_________________________

Formula for the quadratic in vertex form a(x−h)2+k:

__________________________________Formula that GeoGebra produced in general form (located in top right corner):

__________________________________

6. Based upon this graph, do you think the ball will go in the hoop? Has your answer from Question 1 changed or does it remain the same?

7. After watching the whole video, was your guess correct or incorrect?

Part II: Find your Own Parabolic Shapes!

You have just taken a parabolic shape and derived a quadratic equation that best fits that parabola. Now, it’s your turn to find an object/image that appears to be parabolic.

(Note: The graph a quadratic function is a parabola)

Part III: Uploading your Picture into GeoGebra

Step 1: Upload/import your picture on your camera (taken from Part II) onto your Desktop.

Step 2: Open up GeoGebra

Step 3: Click on the following key:

Step 4: Select “Insert Image”

Step 5: Click on any white space in between the X-axes and the Y-axes

Step 6: Open up your picture

Step 7: Your picture should appear in GeoGebra.

Step 8: Right click on your picture and Select “Object Properties”

Step 9: Select the “Position” Tab

Step 10: Change the coordinates of the “Corners” so that your picture is facing upright and the bottom left hand corner is located at the origin.

For example, the “Original Image” was how GeoGebra uploaded our picture, and “New Image” shows how we changed the corners to be the following to get Image 2:

Original Image New Image

Corner 1: (10, 0)Corner 4 : (0, 0)

Step 11: Right click on your picture and select “Object Properties”

Step 12: This time, select the “Basic” Tab

Step 13: Check the box that says “Fix Object”

Step 14: Using what you know about the parameters a, h, and k of the standard form of a quadratic equation [ a(x−h)2+k ], try to create an equation that that will produce a quadratic graph that best fits your parabolic picture. Your equation can be entered into GeoGebra at the bottom of the screen in the “Input” toolbar.

Step 15: Continue to manipulate a, h, and k until you are fully satisfied with the quadratic graph produced. Record your equation in standard form!

_________________________________________

Note: Even though you will enter your “Input” in the standard form, GeoGebra distributes/foils the equation and will display the general form (i.e. ax2 + bx + c). Therefore, it is important to record what values you used for a, h, and k.