Pondering Parabolas

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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.butterfly

Day 1 Video Segments

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.

Day 2 Opening Video Segment

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.worksheet

Day 2 – Student Insight

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.

Day 2 Closing Segments

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.

Reflectionsrunner

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.

 

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You’re free! What’s your next move?

A HyFlex journey through the Civil War Reconstruction

 By OLu research partner, Brad Ermeling

Mr. Peter Lark has taught US history for 19 years, long enough to witness changes and additions to the latest chapter of our nation’s story. Throughout his career, Mr. Lark has consistently grappled with the challenge of covering this full breadth of expanding content while also constructing rich and memorable learning opportunities where students can discover, apply and transfer ideas about history to their everyday lives.

Mr. Lark views US History as a critical course for learning not only the storyline and content of our nation’s journey but also for helping students understand how actions and events of the past influence beliefs and issues we face today. It also helps them learn that people will hold diverse opinions about life and politics. These opinions, while different from their own, may be supported by equally good reason and rationale.

This year at Orange Lutheran, Mr. Lark is teaching his first HyFlex US history class. A HyFlex course blends face-to-face (f2f) learning with a flexible learning session (FLS) that is technology enhanced and primarily self-directed. It allows students to set the pace for a portion of their learning while still providing opportunities for face-to-face collaboration and guaranteeing access to individual or small group assistance from their teacher within the school day. Teachers can also use flex sessions to reduce class size and optimize facilitation of group work by bringing in one group or a smaller set of groups. Mr. Lark has been working to leverage the unique design of HyFlex to foster what he describes as a “US History Lab.” One insightful example is a recent lesson he planned on the Civil War Reconstruction.

More than a Series of Battles

When students first think of the Civil War they often think of generals and soldiers, guns and military strategy. They think of images so often captured in popular films or TV series about the North and South with men dressed in blue and grey uniforms, officers on horses, cannons and bayonets. Mr. Lark’s goal in teaching the Civil War, and particularly the reconstruction period following the war is to help students look beyond the military battles and victories and grapple with the deeper issues framing the period. Key questions Mr. Lark wants student to ponder include

  • What caused this war between North and South?
  • What was resolved and accomplished at the war’s conclusion?
  • What complexities did the nation face in bringing the South back into the Union?
  • What does this teach us about our government, society, and culture?
  • How do the outcomes of this war still affect our contemporary lives and government?
  • How do the failures of this reconstruction period linger on in the lives of African Americans today?

Part I: The US History Lab

To aid comprehension of these profound historical themes, Mr. Lark constructed a two-part lesson that guided students through the emotional journey and complex challenges of this critical period. Using the unique structure of his HyFlex class, he designed the first day as a US History Lab where students worked in teams of three to four and traveled through a series of stations focused on the reconstruction era. He divided the flex session into half (approximately 35 minutes each) and assigned a few teams to each half of the period. Click here to view a one minute introduction to the flex session and the station exercise Mr. Lark designed.

Students then spent seven to eight minutes in each of the stations described below.

Station 1: Setting the Stage: Who was Roger Taney? (7-8 minutes)

Station format: Students review a news clipping from the recent statue removal of Roger Taney and provide a written description of Taney’s decision about Dred Scott.

Reflection Questions: Who was Roger Taney? What did he decide in the famous Dred Scott Decision (1858)? When did the Civil War begin?

Station 2: The Civil War: What was won? (7-8 min)

Station format: Students watch a short documentary video describing the historical significance of the Emancipation Proclamation.

Reflection Questions: What was won as an outcome of the Civil War? Watch the clip, review the images and think as a group. With your group, think about and rank the most important outcomes of the war? Why did you rank each this way?

Station 3: What did war leave behind? (7-8 min)

Station format: Students review assorted images: Newspaper headlines on Lincoln’s death, pictures of people with amputated limbs, burned path through Georgia, and bent rails on train tracks.

Reflection Questions: What was left behind by the war? Use the images to aid your thinking about what the war left behind. What was left behind in the North? What was left behind in the South?

Station 4: You’re free! What’s your next move? (7-8 min)

Station format: Students study contrasting images of a freed slave with a jubilant expression juxtaposed to the image of freed slave with a despondent expression.

Reflection Questions: The war is over and you are now a freed-slave. What would you choose to do next? How would you go about doing that?

Click below to watch two example student groups contemplating the images and reflection questions at Station 4 (You’re free! What’s your next move?”).

Group #1: “I think it would be scary.”

Group #2: “Slavery under a new name.”

Unlike his traditional class, the flex session enabled Mr. Lark to focus his attention on just three groups of students at a time (half of the class) and doubled the amount of energy and attention he could invest with each group as they progressed through the stations. In the clip from Group #1, for example, Mr. Lark was able to monitor the distribution of talk among team members and strategically draw out important insights from one girl in the group who had been quietly listening but not contributing much to the discussion. Similarly, in the clip from Group #2, he was able to listen-in during a pivotal moment, reinforce the group’s emerging insights about the South “reinventing slavery,” and build some anticipation for key terms and ideas they would discuss in the subsequent class period.

One of the goals of small group work is to help “make students’ thinking visible” so the teacher can better understand, probe, and nudge forward student thinking as they wrestle aloud with important questions. Mr. Lark believes these types of exchanges are critical opportunities for helping students make memorable connections with the content. Combining the HyFlex format with the station design increased the probability of those exchanges and removed the pressure Mr. Lark normally faces in managing the entire class while also circulating to facilitate deeper thinking. The result was deeper reflective discussions among groups at each station which prepared students for deeper analysis and study in the subsequent class period.

Click here to watch Mr. Lark wrap up the station work and build a bridge of anticipation for the upcoming f2f lesson.

Part II: Bringing it All Together

During the next regular class period following this flex lesson, Mr. Lark organized the classroom into groups based on notes they recorded in their station work. He strategically distributed students with classmates that were on different teams during the previous lesson so they might gain new insights and perspectives. He reframed and reintroduced the key questions from each station and facilitated a whole group discussion interspersed with opportunities for small group sharing and exchanges. Before and after reintroducing each question he elaborated on key events and themes from the period to deepen their understanding and insight and to aid students in making connections to the present day.

Click here to watch a video clip from Day 2 where Mr. Lark guides the class through a deeper analysis of the questions from Station 1: Roger Taney and the Dred Scott decision.

Using this same approach, Mr. Lark continued working through the key themes and ideas for each of the four stations. He wrapped up the exercise by asking students to ponder one additional question. “If you were an African American living at this time, what would you hope for?” Students talked about equality, mobility to get out of the South, more diversity within communities, and more help from the government. Mr. Lark pointed out that the government, up until this point, had applied a very strict interpretation of the constitution and had not played an active role as a change agent in society. The Civil War changed that, he explained, resulting in new funding and initiatives such as the transcontinental railroad, the homestead act which opened up the Western Territory for settlement, and the establishment of many state universities.

Finally, Mr. Lark transitioned from this discussion to a more detailed explanation of the key terms and historical milestones of the era, elaborating on the political, social, and economic factors Americans faced as they struggled to reunify the nation.

Student Interviews

Observations of students during station work and throughout the f2f lesson on Day 2 revealed a significant level of reflection, introspection, and empathy for the challenges Americans, and specifically African Americans, faced during this time period. The excerpts from two students interviewed below provide additional evidence of that introspection and insight. Both students (Avery Seagren and Kyle Hill) were members of Group 2 featured in the previous video clip.

I: As you traveled through the stations during that first lesson, what were some of your thoughts, feelings, and reactions?

A: I thought it was interesting the way we went about it. I like that we got to hear other people’s opinions…I thought it was nice to have the small groups…just to condense it…you feel like there is more discussion going on between you and the teacher.

K: I found the Roger Taney article very interesting…I saw how a guy was just following the trends of that time and technically making the right decisions became a villain for that one thing he did. He was not at the forefront of racism…but because of that one trial, he became this figure that people had painted to be a huge racist.

I: As you worked through the station about “What was won?” what were some of your impressions?

A: I think there were split opinions…back then…I think it was fantastic…especially the 13th Amendment. It was interesting to think about the Emancipation Proclamation. I didn’t know it was only the rebelling states’ slaves that were freed…

K: Out of all the war and violence what was born was the hope that all types of people might be treated equally without the fear of being targeted by bigotry and racism…the end of the Civil War is a start of a new beginning…what was won was hope that everyone might dwell together in unity.

I: What else did you realize about the time period and circumstances as you also reflected on “What was lost?” And “You’re free. What’s your next move?”

A: Definitely there was a lot more lost in the South than the North. It was interesting to learn that one of the Northern tactics was burning everything in their wake when they were sweeping through the South…it was sad to see all the destruction, because that was their economy.

K: I’m sure most of the slaves were excited to be free, but I’m sure many were also feeling pessimistic about leaving the plantation where they had shelter, food, and clothing. Now they are thrown into a world that doesn’t see them on the level of the rest of society. So what are they going to do? How are they going to survive? I’m sure some of them still felt enslaved…I think fear was a huge thing.

I: How does this lesson about the Civil War Reconstruction relate to the way you think about life in America today?

A: I think all of the violence over racism today…I feel like we should have learned from history. I definitely reflects the same pattern.

K: During the [second] class period, Mr. Lark talked about how the churches were segregated. I think it’s interesting how that has translated over time…kept that same thing with one race being predominant in a church. I think we have that time period to blame. There is still a separation of races.

I: One of the things you learned in these lessons was that the Civil War Reconstruction Era was a time period that was focused on change. What can we learn from this time period about how change happens in society?

A: Mr. Lark had said that change sometimes takes people dying, a generation that has a certain opinion to go away…I definitely agree with that. People just are so firm in their belief that they aren’t open to another opinion…those people are the ones blocking progress.

K: Well…it takes forever. I think that’s something Mr. Lark was also making clear…that change doesn’t happen in a few years. It takes…decades.

Reflections

As Mr. Lark reflected on the lesson, he shared his own observations of students’ journey through the lesson content. He was gratified to see how students connected with the emotions of the period, how they stepped out of their own world view to consider the complexities from multiple perspectives. He was excited to see their level of reflection on the questions he posed as they made connections between the challenges of the Reconstruction era and the pressing issues in society today. A week later, after looking at their essay responses on a Civil War test, he was thrilled to see the highest class average he has ever experienced in this course.

Mr. Lark also reflected on his new “Lab” experiment with the HyFlex model and the opportunity this afforded for more focused interaction with a smaller number of students. He is looking forward to adapting that design for future units throughout the year as he continues to balance presentation of key facts and content with rich and memorable learning opportunities for discovery and application.

Carpe Diem: Seize the Games

Latin might technically be a “dead language,” but the mood of Miss Sampson’s Latin classes is one of boisterous enthusiasm. This high energy teacher engages her students from bell to bell by challenging them through fast paced activities with ample support. We all know that when someone is engaged in the process of learning, they are more likely to take ownership over their own education. This is so apparent in the Latin 1 classroom. Here, Miss Sampson peppers her direct instruction with exciting iPad “Space Races” where students work in teams to demonstrate mastery over the latest segment of instruction. Freshmen love the competition and comradery, and Miss Sampson loves the immediate assessment information: at a glance, she can see which kids “got” the grammar concept, and which ones need some additional help. Click here to read more about Miss Sampson’s unique and engaging way of giving direct instruction.

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Let’s Have S’more Chemistry: Marshmallows, Chocolate, Grams, and Moles

By OLu research partner, Brad Ermeling

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Tanya Grasz teaches Honors Chemistry at Orange Lutheran. Honors Chem serves as a foundation for and a gateway to other advanced coursework at OLu. Students who enroll for this class have demonstrated solid mastery of biology and algebra as well as a capacity for problem solving and critical thinking. The subject matter is demanding. Even well-prepared students often struggle with the advanced content as concepts build and converge over time with increasingly complex applications.

Tanya approaches this challenge with the same spirit of inquiry she expects from her students as they engage in ongoing investigation of scientific phenomenon. She is energized when students grasp an important lesson or idea but also perplexed when they struggle to master key concepts after careful teaching and assistance. One example is her ongoing efforts to help students use mole ratios when solving stoichiometry problems.

Brief Intro to Stoichiometry

Equations are a chemist’s recipe. Equations tell chemists what amounts of reactants to mix and what amounts of products to expect. When you know the quantity of one substance in a reaction, you can calculate the quantity of any other substance consumed or created in the reaction. For example, if we have the reaction N2 + 3H2 → 2NH3, we know that nitrogen and hydrogen molecules will react in a 1:3 proportion. The calculation of quantities in chemical reactions is called stoichiometry and mole ratios are the bridge for converting between various units of chemical quantities.

Teaching and Learning Challenge

Tanya found that students typically struggle with several key aspects of solving stoichiometry problems as well as general principles of dimensional analysis:

  • Interpreting and setting up problems
  • Understanding where the problem resides on the “mole map” (realizing what they are given and understanding where they need to end up)
  • Understanding and using mole ratios as the critical bridge for converting units of one substance to units of another.
  • And, most importantly…moving from a formulaic pattern (calculations of units) to a conceptual understanding of mole ratios.

Bridging the Gap

Tanya teamed up with another science colleague, Jill Ronstadt, to investigate instructional solutions. They planned and implemented a research lesson with the following components.

First, they used a cooking analogy to help students grasp the proportional conversion process with something familiar and accessible from everyday life. “A recipe is like a balanced chemical equation,” Tanya explained. “Ingredients are the reactants. Cookies or cake or burritos are the products.”

She then asked students to think through an example recipe for S’mores. The goal was for the students to understand that there is no way to know how many S’mores could be made given a certain number of ingredients if there is no recipe to follow or refer to.

Video Clip #1 – Chemistry is Like Cooking

Moving from food to chemistry, Tanya helped students review what the coefficients in the chemical equations represent using a relatively simple equation for ammonia (NH3) production. They discussed why the coefficients do not represent grams, but instead represent moles and molecules or sometimes liters (for problems using gases at standard temperature and pressure). This is why using mole ratios are so important–they bridge the gap between various units in the conversion process.

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Next, Tanya asked students to work on a stoichiometry problem. The problem asked students to move from grams to moles. “How many grams of ammonia are produced from 0.60 moles N2? “ She gave students time to work individually. While they were working, she took inventory of students that solved the problem successfully and those that made a wrong turn somewhere in the process. She found that 8 of 27 students had solved the problem correctly. She then divided the class into eight groups and asked students who were successful to teach and discuss with others how they solved the problem. Tanya wrapped up the lesson by reminding students that the equation is their “recipe” and the numbers for the mole ratio are coming from “the coefficients of a balanced chemical equation.” She then assigned additional problems for homework, including several questions about a recipe for chocolate chip cookies, and two problems involving chemical equations.

Video Clip #2 – Practice with Stoichiometry

Day 2 Results

The results from homework showed that nearly all students were able to solve problems about chocolate chip cookies, but most still had questions about problems using chemical equations. “They knew the mantra about using their equation to find the mole ratios,” Tanya explained, “but had not connected all the dots yet.”

Tanya spent the next class period persisting to help all students “connect the dots” and learn to use the balanced chemical equation as their “recipe” for solving stoichiometry problems. She guided the class through the homework examples. She paused at each phase of the problems to elicit student thinking and asked them to direct the solution process. She then gave students a new problem to solve independently and circulated around the room to document student progress. While circulating, she strategically placed a dot on individual papers to keep track of where students were struggling or asking questions. She then collected the papers to better diagnose the primary areas of difficulty. During this second round of work, Tanya’s records showed that of 27 students in the class, all but 8 solved the problem successfully and 5 of these 8 failed to correctly use the mole ratio. This was a substantial improvement from Day 1 where only eight students solved the Ammonia problem correctly. Tanya continued with several more problems at the end of the period until the entire class could use the “recipes” effectively.

Assigned Problems # of Students with Correct Answers
Day 1 Practice Problem 8/27 students
Day 2 Practice Problem 19/27 students (5 struggled with mole ratio)
Day 2 Additional Problems 27/27 (all students were successful)

The excerpts below from student interviews describe this journey of discovery and deeper understanding from two students’ point of view. Both students were members of the same small group featured in the second video clip above. In the video, Cove was the “teacher” for the group. Brandon was the student on the far right, opposite side of the table.

Interview with honors chemistry student, Brandon Washiashi

I: When you were first asked to work on that ammonia problem, what process did you use to solve the problem. Where did you start?

B: We were coming off the previous test which covered moles and if I remember correctly, I think I tried taking ammonia and turning into moles…

I: How did that work for you?

B: Well, I got the wrong answer.

I: What were you were struggling with?

B: The numbers…I didn’t know the ratio and it’s kind of hard to calculate when you don’t know how much to calculate.

I: What did you understand about the s’mores analogy at that point?

B: I didn’t think to look there…It wasn’t on my radar until after the revelation of the ammonia problem.

I: What did you learn from the conversation with Cove and the group work?

B: That I need the correct amount of coefficients to make the correct number of ammonia.

I: After that, you had some homework problems. How were those?

B: The first one I was still getting into…I still had a couple of questions…after that I’ve been fine.

I: Was there anything that Cove said, that Mrs. Grasz said…where did it start to “click” for you?

B: I used the map…that was a big help…making sure it’s a balanced equation, of course, plug it in, and follow the bridge. For me, it was pretty straight-forward after that.

I: Where do you think your biggest gap was prior to that?

B: All goes back to the ratio…I can can’t cross unless I know the ratio.

I: How is this class affecting the way you think about science?

B: Well, I’m definitely thinking about it more mathematically now…Knowing all of the previous stuff is helping, because if I didn’t know how to look at a periodic table, I wouldn’t be able to find the atomic mass of Oxygen. If I didn’t know Oxygen was a diatomic molecule I would just look at it as…one atom instead of two. If I didn’t know about moles, I couldn’t convert into moles and cross the bridge and figure out the problem.

 

Interview with honors chemistry student, Cove Carlson

I:  What were you thinking as you listened to Mrs. Grasz explain the s’mores analogy?

C: We just finished a unit on conversion factors…I was comparing it to [moles]…I was piecing together as well, the balanced equation–the coefficients played a part…It was easy to picture the ratio. I make s’mores…so I can picture how each one corresponds to the other.

I: When you were first asked to work on that ammonia problem, tell me about your thinking.

C: I’m not a very good math person, so I know I was a little nervous. I wasn’t expecting to get it right, but I did have a sensation that “this makes sense”….I was looking at what I was given and thinking back to what she taught us with s’mores…There’s 0.5 chocolate bars to two graham crackers…there’s two ammonia to whatever it was in the equation…

I: Was there anything in particular that Mrs. Grasz said that helped you?

C: It really stuck with me that the coefficients are an important part of the balanced chemical equation.

I: During the group work segment, how did you feel about teaching the group?

C: …I felt a little proud…I got this. It makes sense to me.

I: Can you remember more about what was making sense to you?

C: …Before it had always been one mole. This equals one mole; one mole equals this. The biggest thing I understood was that now it wasn’t one mole. It was two, three, four, however many–the coefficients of the chemical equation.

I: How is this class affecting the way you think about science?

C: Before going into this class, I was thinking, “I’m not a science person…” After this, I was grounded. This is a solid base for me. I know how to do this. I can keep building off this…You got to be persistent…You might spend three or four hours on an assignment, but if that helps you get it, everything else becomes so easy, because it all builds off each other.

Reflections

As illustrated in the student comments above, successful classroom teaching is rarely a silver bullet or single magical strategy. More often, it’s the hard work of carefully stringing together a sequence of subtle, incremental innovations that help students advance toward deeper understanding. Tanya’s stoichiometry work is a great example. She helped students access prior knowledge and make connections to the work done in previous chapters. She developed a compelling analogy with the s’mores recipe. She strategically grouped students so they could learn from each other’s problem-solving efforts. She gave them multiple opportunities for practice and feedback. She diagnosed where they were struggling and scaffolded their learning through guided questions and targeted assistance. She helped students remember to start with their recipe–the balanced chemical equation–and she helped them learn to use the “mole ratio” as the bridge between reactants and products.  

Tanya’s persistence with teaching fueled a parallel persistence with learning, enabling students to grapple with difficult problems and master core concepts of stoichiometry. Tanya and Jill are now mapping out their next project to study and improve teaching and learning. Stay tuned for s’more examples and findings from teaching and learning at OLu.