Recently, our Art students had a wonderful opportunity to come to an after school workshop with a live model to practice their figure drawing. Mrs. Doss and Mrs. Mooney brought in 2 different models during 2 different after school sessions. Students from a variety of art classes could choose to attend these sessions to practice their figure drawing skills. Figure drawing is a tricky skill to develop to get the proportions correct (among other things) on a human body. As Mrs. Hamby, art department chair, says “imagine 6 more semesters of students having these workshops and how strong their figure drawing will be!” Below is some pictures of students hard at work during this workshop.
In English 1, Mrs. Bowers tried something new with her freshmen students, having a Socratic Seminar. This process is not new to Orange Lutheran teachers as many use it in upper level English and Theology courses. But Mrs. Bowers thought she would give it a try in her English 1 course. She writes:
Normally in choir, the students start the day with warm-ups, which include breathing exercises & the students standing in a specific order. However, for this exercise Mrs. Carvale asked the students to disperse to different parts of the room where they could sit away from their peers. She turned the lights off and walked them through a meditation exercise that prompted the students to close their eyes and focus on their breathing. After a few minutes of focused breathing, the students kept our eyes closed and listened to one of the songs that they are currently working on.
Prior to this day, Mrs. Carvale met with our yoga teacher, Mrs. Lark, so she could work on a mediation and breathing exercises. Her goal was also to work on posture and emotion of the song, which are also both found in yoga and mediation practices. When asked about her experience Mrs. Carvale stated, “It was a great change to take a moment to LISTEN, rather than just always singing our songs. This exercise was great for the choir to hear & experience the impact that they can have on their audiences.”
After the students completed this, they reflected in their journals. Below are a few quotes about their experiences:
“Yoga today was balancing. I enjoyed sitting at school and focusing on something other than my routine external pressures. Putting pressure aside and putting school in perspective helped me balance my emotions for a moment.” Susanna Bolin, 10th grade Honors Choir student
“As a singer, my body is my instrument. In yoga, breath is important when moving and is the foundation of all practices. Similarly breath is the foundation of singing. Meditation has helped me so much. I am glad that we got to spend this time in quiet, just being.” Dani Buschini, 10th grade Honors Choir student
“Taking a few minutes to sit, relax and focus on breathing was a refreshing addition to my day. It allowed me to pause, be mindful and forget about my daily causes of stress. Yoga correlates to singing as both heavily rely on breathing and relaxing your body. Caring for your body and allowing for moments of reflection and revitalization can give your body a nice break and make a great vessel for using your instrument to sing.” Danny Smith, 10th grade Honors Choir student
Recently in Señora Holness’ Spanish 4 class, students were assigned a Spanish speaking country to study and learn about different working practices. The students had to research topics such as the length of a working day, how many work hours per week, benefits, vacation days per year, and length of maternity leave. Throughout this experience, students studied the working culture of their various countries while practicing the vocabulary of working practices. Once they compiled the information, they had to give a short presentation about their country to the rest of the class speaking entirely in Spanish. This allowed the chance for students to practice both their speaking skills and their listening skills. They also completed a graphic organizer about the different countries as they were listening. This allowed the students to compare the different countries working practices in one location. Students really enjoyed the idea of a siesta in the afternoon and were fascinated by the length of maternity leave for new mothers.
Also, as a break throughout the presentations, they had to work with a partner create a skit of different working practices and also practice the vocabulary that they are learning. One partnership did an interview, while another modeled a good versus a bad employee. All skits were a fun and engaging break from the presentations and also was another opportunity to practice vocabulary, speaking, and listening skills.
Ava Dunwoody, senior, said “This project allowed me to utilize the skills I have been learning in the classroom in a practical and real-life example. Not only did it help me more readily understand the vocabulary of this unit, but it also helped me articulate my speaking skills in a way that will prepare me to speak Spanish more naturally. It was also a lot of fun.”
In our last blog post of the school year, we are spotlighting our Honors Engineering end of the year project! The students were tasked with building a boat that can hold 1 or 2 members of their team and move across the pool 25 yards. The only building materials that they were given was glue, paint, small amount of plywood and paper. The students were tasked with keeping a journal of their building experience. The students met early this past Saturday to attempt their trek across the pool. For some it was a dry success, others a wet failure but either way they had fun, and learned engineering skills!
Check out some pictures of the event, including a Little Lancer learning about the boat.
By OLu research partner, Brad Ermeling
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.
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 downward).
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.
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.
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.
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.
Recently in Mr. David Dekker’s Physics class, he had his students build a Wave Machine to better visualize a wave. Read what Mr. Dekker said about his Wave Machine below:
The purpose of the wave machine is to help students visualize and understand Simple Harmonic Motion. SHM includes pendulums, mass/spring systems and waves! The wave machine is always a big hit in class because the students are able to experience first-hand the concepts that are being taught in class. We are able to use this wave machine to demonstrate a transverse wave, a wave pulse vs. a continuous wave, wave interference, wave reflection and much more. At the end of my planned demo I allow the students to ask questions of their own and test it on the wave machine. Overall it is just a simple, fun way to experience waves in the classroom. The wave machine is made out of kabob sticks with jelly candies at the end to provide weights. It took about 10 or 15 minutes to build with the help of my class and can easily be rolled up and stored for use in future years.