Teacher Name: Douglas Harrington

Class and Grade Taught: Algebra II Extended – 11th grade

Lesson Date: 14 October 2011

Lesson Logistics and Setting:

Unit Topic: Working with Quadratic Functions

- Previous Lesson Topic: Modeling Data with Quadratic Functions

- Current Lesson Topic: Properties of Parabolas

- Next Lesson Topic: Translating Parabolas

Lesson Objectives:

1. The learner will develop the vertex form of the equation for a parabola through observation.

2. The learner will determine how a, b, and c affect the shape and location of parabolas in the form .

3. Students will identify quadratics (specifically parabolas) as the family of functions best suited for modeling a thrown object.

4. Students will sketch the graph of a quadratic function given appropriate information.

Standards Addressed:

- A3.3.2: Identify the elements of a parabola (vertex, axis of symmetry, direction of opening) given its symbolic form or its graph,

and relate these elements to the coefficient(s) of the symbolic form of the function.

- A3.3.5: Express quadratic functions in vertex form to identify their maxima or minima and in factored form to identify their zeros.

- A2.1.7: Identify and interpret the key features of a function from its graph or its formula(e).

- A2.3.1: Identify a function as a member of a family of functions based on its symbolic or graphical representation; recognize that

different families of functions have different asymptotic behavior.

Proposed Evidence of Successful Completion of Objectives:

1. In order to generalize a formula from the task, students must combine observations into a single form. Furthermore, students

must provide constants to represent the values which affect quadratic functions in vertex form. Evidence of such completion

may be evidenced in the students, organic production of the generalized vertex form of a quadratic function in answering

Question 5. Furthermore, student-led discussion may highlight the combination of factors influencing the behavior of the

quadratic function to which the teacher may introduce the vertex-form equation. By examining an arbitrary quadratic function

written in vertex form, students can determine the graphs behavior and identify the elements of a parabola when asked.

2. In order to successfully determine how a, b, and c affect the shape and location of the parabola, students must produce

generalizations (or, more accurately, deductions) from the four functions examined:

Students who connect the maximum height of each bird to the c value for each parabola have successfully met the objective

(Question 2). In order to push those students further, the teacher should ask the students where the minimum value is on the

equation and allow students to make conjectures about maximum and minimum values of parabolas open up. If such

student-student talk occurs, the students clearly demonstrate some knowledge about c as the maximum or minimum value

depending on the direction of opening. Successful completion of the task is evident in talk which explores why the maximum

value is c. This entails students discussing that the value being squared must be zero as any other value subtracts from c

(students can rearrange the equation and highlight the output is the squared amount subtracted from the maximum height). In

symbolic form, this is a connection to the maximum value occurring at (x, c) where x – b = 0.

Student talk or work that occurs above also completes some successful exploration of the effect of b on the location of the

parabola. For full success, students must identify the vertex position depends on the value of b. Furthermore, before x

reaches b, the value of the quadratic is increasing and when b is greater than x, the value is decreasing. In order to answer

Question 3 fully, students must identify that the vertex occurs halfway between the zeroes of the function. Furthermore, the

parabola is symmetric around the line x = b.

Effectively answering Question 4 requires students to connect the parabola opening down with the value of a. At this point,

students may or may not notice that the value of a changes the rate of the parabola’s increasing (resulting in a “squeezed” or

“stretched” look), but this is not a focus of the task. Instead, student-student and teacher-student discourse should highlight

whether a is positive or negative determines if a(x-b)^2 should be added to or subtracted from the value c. This results in a

parabola opening up or down.

3. In the warm-up students must identify which graph best models a field-goal kick. Students must rely on observed

circumstances to identify the right trajectory. Students will then be pushed to pick the graph most like a street racer’s

acceleration – this time students cannot rely on a physical path to choose the correct graph. Success hinges on the students’

ability to highlight non-linear change as the reason for the shape of the graph.

4. In order to reach any of the lesson’s objectives, students must be able to accurately produce graphs – a skill sharpened

during the students’ introduction to functions. Although this is less an objective than pre-requisite skill, the task allows

students opportunity to refine graphing skills based on provided function. Evidence of successful completion of this objective

is simply accurately sketched graphs for Question 1.

Materials Needed: Interactive Projector, Angry Birds Game, Colored Pencils, Handout

Introductory Routines:

- As students enter the class, the teacher walks around the room and engaging in conversation.

- Once the bell rings, the teacher passes out the warm-up activity and students are to be working silently and alone for the

duration of 5 minutes.

- As students complete the warm-up activity, the teacher finalizes the attendance and corrects any outstanding issues.

- After 5 minutes is passed, the teacher and students discuss the ideas of the warm-up activities. Upon completion, students pass

in the warm-up activity and last night’s homework – help is offered to students who seek it.

- The launch is ready to begin.

Lesson Activities:

Name: _____________________________________________________________ Hour: _______

Angry Birds: Exploring Parabolas

John has just started the new set of levels for the game Angry Birds. Drawing on his Algebra II knowledge, he correctly identified the path of the birds as

a parabola. In order to maximize his score by not missing a shot, John has produced four possible trajectories for the launched bird. The four pathways are

represented by the following equations:

1. Using the above information, decide which equation will result in a hit. Use any method you like, but be sure to explain why you know or believe your

chosen equation results in a hit:

Besides finding the most effective way to knock the pig out of the sky, graphing each equation can help us generalize about the effects changing

variables has on the shape of parabolas. Fill in the following questions based on your graph, table, or other representations:

2. Record the point at which the bird reaches its maximum height from the ground. What connections do you notice between the maximum height for

each shot and its equation? What do you notice about the point?

3. Record the distance each bird travelled horizontally. What connections do you notice between this distance and the equation? What connections do you

notice between the length travelled horizontally and the point of maximum height?

4. What natural force causes the bird’s ascension to slow and then ultimately results in the bird’s fall? What mathematically causes the values to rise and

then fall? Why does the parabola open down?

5. Create a generalized equation for a parabola using constants a, b, and c to represent the factors we just observed:

Forward Thinking: Find the value for the question mark so that each statement is true.

1. hits the pig at (3, 15).

2. hits the pig at (5, ¾).

Lesson Students are working… Anticipated Student

Thinking/Questions

Teacher Moves

Launch Road Map: (written on board)

1. Warm-up/Homework (7-10 minutes)

2. On Your Own Work (7 minutes)

3. Group Discovery (10-15 minutes)

4. Group Discussion (10-15 minutes)

5. Summary/HW (5 minutes)

“In this unit, we will be turning our focus toward quadratic functions. Today, we are going to make some observations between the equation for a parabola and its graph. How many of you have played the popular game Angry Birds? Do you know what mathematical idea/shape the launched birds follow through the air? A parabola – which is the shape of a graphed quadratic function. As you can see on our task, our job today is to help John decide which equation to use in order to hit the pig. In doing so, we will be making observations about the ways in which the equations

determine the shape of the graph. For example, in the linear case, we took a look at why y=mx+b made sense mathematically. We are going to try to develop a similar form for quadratics. For the first 7 minutes, I want you to work alone and silently. Graph each equation and write down as many observations as you can. There are colored pencils at the front of the room. Then for the next 15 minutes, you will work in groups discussing your observations, making new ones, and generating a form for any quadratic function.”

Explore Students will be interacting with the handout for most of the time. It provides students with a framework to make deductions from their observations of four different equations in vertex form. Although all students will create a graph, making sense of each equation before graphing is on the individual student. There is a classroom set of graphing

Students will remain arranged in their rows; however, students will be given group work time (not a classroom norm given disruptive behavior patterns). No more than four per group and only those students sitting on either side or in front and behind. For the first 7 minutes, students should be working silently alone. The goal is

Question 1: For the student who finds the vertex and plots 4 points – 2 on either side: Where the inside is zero, the output is the highest. Well, one equation is y=-2(x-3)^2+18. So when x = 3, you don’t subtract anything from 18 so that’s the highest

Question 1: How did you know where the point you chose would result in the highest point? What do you mean by inside?

calculators available. Students can create a table of different values. Students can even use note that each equation is in vertex form and use that information throughout the remainder of the assignment (most students have seen vertex form, but only briefly in Algebra I). There are no true manipulatives, but students can use contextual, graphical, and algebraic contexts to bridge understanding.

that every student should have all or part of the graphs drawn on the handout and attempted observations. For 3 minutes, students will generate a quick list of observations and questions. The students will then work in groups for 15 minutes. Students should be synthesizing their observations into distinct points and begin pushing towards a generalization. All work must be recorded on the handout individually. Different equations should be drawn in different color.

point. Well every point around that has a mirror on the other side. New Line of Questioning: I wanted to see how much it curved so I made sure I had enough points to guess. Pretty confident. I’m a pretty good graph maker. I’m not sure? I can graph it more carefully. It’s at about (18, 3)?

What else is significant about that point – let’s call it the vertex? [Introduce vocabulary] Why does that mirroring happen? Does it occur around the point or about a line? [Push for axis of symmetry. Possible moment to introduce terminology] Why did you choose 5 points to sketch your graph? I see you have the first equation landing a hit on the pig. How confident are you that your choice is accurate? Is there another way to justify whether your choice really lands a hit and is not a near miss? No, I was thinking a whole different representation. Where is the pig located? Is there a way we could

We can plug in x = 18 to the equation and see if we get 3! For the student struggling to make a graph: Mr. H, I’m having a hard time graphing. The points are not lining up in a straight line so I keep moving up one at a time. Is the graph just a curve that stops growing somewhere? [Misconception: Students who have found a few points but are all on the wrong side of the vertex may create a curve somewhat like a radical function] The y-values go up as x-values increase and then after a certain point they start decreasing. The graph looks like a frown. The graph looks like an upside down “U”

check to see if that point lies on our line or is at least close to it since the pig is much bigger than a point? What about graphing do you seem to be having a hard time with? Sometimes graphing these by hand can be difficult. Why don’t we break out a graphing calculator? What do you notice is happening? I want you to think about

For the student who chose the fourth equation because it has an 18 at the end and the pig has an x-value of approx. 18:

why that is happening. Questions 2 through 5 ask you more about this. * Use a line of questioning similar to the student who was confident that the pig was on the first equation simply by graphing. More though, challenge this student to back his reasoning up with a visual/graphical representation. Good Questions to Ask: Does it make sense to extend the parabola below the x-axis? What about to the left of the y-axis? Why would we not include those values? Do any of the functions have a minimum value? Why do you think that? What about in the context of our problem? Why does each function have a maximum value? Do you notice your height in

Question 2: Where do I begin measuring along the x-axis because when I use my calculator it looks like the x-values just keep going and going? Because lower and lower x-values create even lower y-values? We are standing on the ground. So we are measuring the distance from the ground up. From the point the bird was shot to the point it hit the ground.

the equation? Do you notice your highest point in the equation? Why do you think that is? Is there a lowest point for each equation? What about if we removed the context – Is there a minimum value? * Requires much more unpacking to get at the concept of axis of symmetry. Why do you think that is? That’s very good and that will help us in later sections. What about our context suggests that maybe we should limit our domain? That’s good. So where should we measure from to find the length across the x-axis?

The distance to the x-coordinate of the vertex is half the total distance. When you take the square root of a number, there are always 2 solutions. So it’s like the same thing but backwards – 2 numbers squared can equal the same number!

Good, so let’s measure that. What do you notice about the distance that was travelled horizontally and the distance from the origin to the x-value of your vertex? Based on your graph, the point bird was launched from and the point the bird stopped have the same y-coordinate. Are those the only pair of x-coordinates with the same y-coordinate? Why do you think that is? Good. What point is always in the middle of the pairs? How can we show that with an equation? Good Questions to Ask: Where should we start our measurements based on the context of the problem? Without the context, would these parabolas carry on forever? How and where?

Question 4: Well the bird went up and came back down because that’s where we were aiming to hit the bird. Because gravity will pull it down. The kicker hits it strongly so the ball is moving fast, but as the ball continues it begins to slow – this is gravity bringing it back down. I’m not sure I’m following you Mr. H.

I see pairs of x-coordinates with the same y-coordinate. How far are they from the x-coordinate of the vertex? Is the parabola symmetric? How so? Why did we need to aim higher than the bird in order to hit it? Good connection. So if gravity is working against it the whole time why does the graph flatten and curve at the top instead of change direction immediately. Think of a kicked football – why does it follow a curved path? So what mathematically is acting like gravity in terms of our equations? What causes our parabola to begin going up and come

Hmm… They all have a negative coefficient! Well yeah, there is a limit as to how high something can go given its speed. Oh! The negative coefficient causes us to take away from the maximum height. So the vertex is the highest point because we aren’t taking anything away! Question 5: * My students have struggled to develop equations/functions from appropriate information during the linearity unit so writing a generalized form may be beyond they’re

back down? Each equation shares at least one thing in common – they go up and come back down. What does each algebraically have in common? Do you think that has anything to do with why there is a maximum value? Let’s think mathematically though. You identified a highest point. Why is that the highest point instead of another? How does the negative coefficient play into that?

abilities. However, I want to challenge them in ways they are not use to – especially in reasoning and deduction skills. Therefore, I may not expend too much energy exploring student solutions in this format. I may simply take over as teacher and help synthesize their observations. Mr. H what are you talking about using a, b, and c to describe the way each affects the parabola? m was the slope and b represented the y-intercept.

Remember slope-intercept form: y = mx+b? What did m represent and tell us? What about b? See the form unlocked some meaning for us. I want you to use the observations you made in questions 1 through 4 to generalize what factors affected the parabola’s shape. Then, I want you to try and create the formula. Look at one equation and compare your observations to that equation to find out which values did what.

Summarize/Share and Discuss The share and discuss portion will focus on drawing

I want to touch on topics in this order:

out how each number affects the shape of the parabola. All four equations will be written on the white board and their graphs will be on the interactive whiteboard.

The value at the end in each equation was the highest value. The distance the bird travelled was x the distance to the vertex along the x-axis. I think the same?

1. Maximum value and a connection to minimum value of an equation.

2. Vertex location. 3. Axis of symmetry and

connection to vertex. 4. Direction of opening.

Maximum value: What did you notice about the maximum height’s relationship to the equation? Teacher circles the value and writes, “determines the maximum value.” What about the minimum value? I also want you to think what will determine a minimum or maximum value. Vertex location/Axis of symmetry: What connections did you notice in Question 3? If we picked a value two units from the right of the vertex and another two units from the left, what would the outputs look like?

Right in the center. Well, there’s a negative sign in each equation. Well, with a negative, we are always subtracting.

Could we reflect one side onto the other? Where would we put that mirror? So which number is a connection to that? Direction of opening: Why do you think the parabola opens down? Yeah, that’s good. Do you think that the negative is at all linked to the reason there is a maximum value? Why? Good, when we square a number the resulting number is always… positive, right? So we are always subtracting from the initial value. As we approach the x coordinate the difference decreases so we subtract a smaller number. After we pass it the number grows again. What if we didn’t have a negative number what would be the maximum value? “So, let’s synthesize this data. What we actually discussed is the vertex form equation for a quadratic.”

Teacher writes the form and discusses a, b, and c in terms of the observations discussed.

Summary Statement: Today, we unknowingly unpacked the equation, y = a (x – b)^2 + c. The coefficient a determines the direction of opening, the squeeze or stretch of the parabola, and whether there is a maximum or minimum value. The coefficient b determines the x coordinate of the vertex as well as the equation for the axis of symmetry, x = b. The coefficient c determines the y coordinate of the vertex as well as the maximum/minimum value depending on the sign of a. Any questions before the homework?

Homework: Forward Thinking questions and CME Algebra II Page 800.