Pondering Parabolas
Dustin Boburka teaches Enriched Algebra at Orange Lutheran. The course provides a critical foundation for Honors Geometry and Algebra 2 and helps students master key concepts and important reasoning skills essential for mathematics. The enriched curriculum also challenges students to develop problem solving skills for life and encourages students to explore the role of mathematics in the world around them.
Parabolas and Problem Solving
One example of these important problem solving skills is distinguishing and effectively using standard and vertex form for quadratic equations. Quadratic equations refer to equations with at least one squared variable. The graph of a quadratic equation always gives you a parabola.
The most standard form is ax² + bx + c = 0. The letter x represents an unknown, a b and c are the coefficients representing known numbers, and the letter a is not equal to zero.
The vertex form is represented by the equation: f(x) = a(x - h)2 + k, where (h, k) is the vertex of the parabola.
Both forms produce a graph with a parabola, but the starting point for solving the problems are different. Standard form provides the axis of symmetry but requires mathematical steps to find the vertex, whereas vertex form provides the vertex from the outset (h,k values). The following graphs from Desmos.com (which Mr. Boburka used with his students) provide a visual representation of how these equations correspond to graphs on a coordinate plane.
Standard Form: Example Graph
Vertex Form: Example Graph
While teaching these lesson on quadratic forms, Mr. Boburka specifically wants to help students address the following questions:
How are quadratic equations used in everyday life to calculate change and variation of quantities? What are some examples?
How are quadratic equations different from the linear equations you learned previously (e.g., the rate of change)?
How does the form of the equation (standard or vertex) influence your starting point for solving the problem?
For each type of quadratic form, what are the characteristics of the graph (parabola), and what steps are required to figure out the characteristics?
How do I plot those characteristics on the coordinate plane?
After teaching this course for twelve years, Mr. Boburka continually finds these concepts and skills challenging to teach and difficult for students to master.
Part I: What are parabolas and what is standard form?
Recently Mr. Boburka modified his approach for addressing these important topics by constructing a two-part lesson on standard form and vertex form. We collaborated with Mr. Boburka as he developed and implemented this research lesson.
He designed the first day as an opportunity to familiarize students with quadratic equations, how they are different from linear equations, and the kinds of graphs and parabolas that quadratic equations produce.
Mr. Boburka felt it was important to help students recognize that quadratic equations are not just formulas we memorize to solve mathematical puzzles. Instead he wants student to experience them as methods for solving problems used in everyday life, such as calculating areas, determining a product's financial profit, or finding the speed of an object. To that end, he started the lesson by sharing examples of life situations from Sciencing.com where change and variation of quantities are important. Examples included finding the area of a room, calculating a profit, throwing or hitting objects in the air for athletics, or estimating the speed of a kayak.
For the next several exercises in this first lesson, Mr. Boburka facilitated a class discussion by asking students to analyze example graphs and equations using the web-based graphing application Desmos. Since students can easily manipulate the graphs and equations in Desmos, they were able to better visualize and ponder the characteristics of various graphs and better understand the functions each graph represents.
He started by building on students prior knowledge and showing them example graphs of linear equations students had previously studied. He then continued with a guided-exploration of additional graphs and parabolas generated from quadratic equations in standard form. He specifically focused on helping students discover the axis of symmetry and teaching this as the pivotal step for equations in standard form. He asked the class to describe what they notice about the left and right side of the graph. One student commented, “They are the same.” Mr. Boburka continued eliciting comments by asking, “What does he mean by that--they are both the same.” Another student compared the left and right side to the identical wings on both sides of a butterfly.
Click on the link below to view a short sequence of clips from these opening segments and the guided-exploration of graphs and parabolas.
From the axis of symmetry, the class learned how to find the vertex and how to use the vertex to identify the y-intercept. They also learned how to interpret the ‘a’ value of the formula to determine which direction a parabola will open (upward or downard).
Mr. Boburka wrapped up this first lesson by giving students several more examples in standard form with Desmos, allowing them to first visualize each graph and then make explicit connections back to the equation.
Part II: What is vertex form? How is it different from standard form?
On day 2, Mr. Boburka began with a review of standard form and specifically asked students to recall the line of symmetry as the starting point in standard form. He contrasted this with vertex form which (consistent with its name) begins with the vertex of the parabola rather than the line symmetry.
He then facilitated several exercises in pairs, asking students to type various equations into Desmos and estimate what the vertex was for each equation.
He followed each set of practice equations with discussion where students identified characteristics of graphing in vertex form. Students discovered how different equations and different components (h value and k value) of the equations affect the placement of the parabolas on the coordinate plane. A change in the h value causes a horizontal shift and a change in the k value causes a vertical shift. The class used x^2 and (x-2)^2 and (x+3)^2 to visualize a horizontal shift. And they used x^2 and x^2-2 and x^2+3 visualize a vertical shift.
Click on the link below and drag the slider for the h and k values to see how Desmos helped students visualize these characteristics.
Desmos - Visualizing h and k values
Building on these visual insights and connections between the equations and graphs, Mr. Boburka now increased the rigor of the task by asking students to study three new equations. Specifically, he instructed students to analyze the h and k values and identify the vertex before graphing them in Desmos. This time they used Desmos to check and confirm their answers rather than using it to identify the answers.
The next step of scaffolding was asking students to explain how they ascertained the vertex in these equations without using Desmos. The key stumbling block here was helping students see that the negative next to the h value in the equation causes a horizontal shift in the opposite direction of our intuition, while the positive next to the k value causes a vertical shift that is more intuitive.
f(x) = a(x - h)2 + k
Click below to view a clip where a student articulates this key point.
Following these introductory exercises for vertex form, the class spent the rest of the period working to identify other characteristics of the graphs including the y-intercept, maximum, minimum, range, domain, width, and mirror points.
Mr. Boburka concluded class by emphasizing the value of Desmos as a tool for confirming answers and obtaining feedback during individual homework and practice.
Click on the link below to view a few clips from these last segments of Day 2.
Student Interviews
The excerpts from two students interviewed below provide helpful evidence of the student introspection and insight fostered by these learning opportunities. Both students, Logan Mills and Christa Barksdale were also featured in the video clips. Logan shared the analogy of the butterfly for the axis of symmetry and Christa is the student next to him in the video clips who answered a question about the vertex in one of the sample problems.
I: Describe some of your thoughts at the beginning of this lesson? What was your initial impression of quadratic equations?
C: At first I was kind of confused. I was thinking about how they incorporate into our everyday lives and I didn’t understand how all of that worked. But when we used the app Desmos that helped show me. Seeing it on the screen and putting in the equation and calculating it and seeing how it worked helped me understand a little bit better. The golf ball made sense to me...that put a picture in my head...there is one [specific] point where it starts to come down.
L: I was a little bit overwhelmed...that we have to learn a bunch of new formulas...how to solve for the parabola and how to find the vertex...I remember he said they are used for business like charting profit...I like how he was able to relate it to real world jobs.
I: How does the form of the equation (standard or vertex) influence your starting point for solving the problem?
C: For standard form, you look to find the a, b, & c and you have to figure out the vertex. For vertex form, you just find the vertex in the parenthesis with the x. The vertex form seems much easier to me.
L: You build the graph a different way. One thing that really helped me tell them apart was the vertex form had the parenthesis...You go straight to find the coordinates, and then find the vertex.
I: Tell me your impression of the Desmos app. What did you find helpful about that?
C: I found it really helpful to put in the exact equation that was given to you and it shows you what it looks like...I could pinpoint where the line intersects with each point. I think it’s a good app to check...it gives you confidence if you did get the right answer.
L: I love Desmos because when you are done graphing you can put the original problem into Desmos and see what the perfect graph should look like...If you got it wrong you can go back and fix it. When I look at Desmos I can see where the line [of symmetry] is and I can make sure I find the exact coordinates...I also see where the min and max point are...and if it’s in the wrong spot I know I have a problem.
I: Do you recall any particular moments in the lesson (either Day 1 or Day 2) where you felt like you reached a new level of insight or understanding about these concepts?
C: I remember when learning the vertex form, how the h is always opposite and the c, k, you just put them down. I also remember one day I was working with Logan. We were working with standard form. We were trying to find the y-intercept and we didn’t square it. I had to sit back and look at it before I could understand.
L: The butterfly...it just popped into my head...because each wing is identical if you split it down the middle. What I had to do was fine the [line of symmetry] and the min and max point on the parabola had to line up. It wouldn’t have the butterfly effect...if it wasn’t on the [line].
I: Take a moment to read these examples again from the beginning of the lesson. What insights do you have now about how quadratic equations are used in everyday life?
C: Reading about the sports again...like throwing the ball to your friend, how far away they are and how high you throw it so it comes down...you have to know the height so it will come down like a parabola or curve. And also like building...for an area of the room. You have to know the length, width, height and how big it is. I always thought that was just geometry...that isn’t algebra...but it talked about figuring out how big the wood is to make sure it will fit. If you only have four square feet of wood, you have to do 2x squared and if it’s less than or equal to 4 you could use it, but if was greater, then it wouldn’t fit.
L: The first picture has the throwing of the javelin and I think that’s an even better analogy because a javelin is like a long stick that you have to throw up...and you have to calculate the angle...If you throw it straight up, it will come right back down. Knowing that--it does help, because I’m a triple jumper...and so I have to take off at three different jumps and end up in the pit. The first jump I take has to be a little bit more vertical than the second and I have to take off at about a 45 degree angle...the second one I go directly forward so I get more momentum and the third one is like a very big parabola...I have to go really high to get a better a distance.
Reflections
As Mr. Boburka reflected on these lessons and the student results, he shared his own observations and insights as well as plans for future instruction. During pair work and class discussions he was pleased to find evidence of students correctly using terminology to describe graphing characteristics and solutions. Understanding and utilizing key mathematics vocabulary is an important first step in grasping concepts of graphs and parabolas.
During both lessons (standard form and vertex form), he also observed how the large number of characteristics associated with each quadratic form presented a significant cognitive load for students. While identifying and describing each characteristic is important, he felt it would be beneficial, especially at the Algebra II level, to feature more prominently the vertex and axis of symmetry as key components for analysis regardless of which form students are using.
Another insight he identified related to ways he might be more intentional about engaging students with central ideas in mathematics. Going forward, Mr. Boburka hopes to find key points in each chapter where he might intentionally facilitate opportunities for students to explore, think, and contribute to classroom discussion about the use of math concepts in everyday life. As opposed to just sharing examples, he wants to help students generate and analyze their own prior knowledge, observations, and ideas.
Finally, Mr. Boburka reflected on the pivotal moment in this lesson when Logan brought up the analogy of the butterfly for the axis of symmetry. He expressed the importance of being patient and providing time for students to contribute, struggle, and share their thinking. Too often we rush rush to cover content while students rush to solve problems. Recognizing how significant this was for both Logan and the class, Mr. Boburka plans to be more intentional about creating similar opportunities for open-ended dialogue and discussion.