The goal of this lesson is to help students use curve fitting to determine mathematical expressions for the relationships between physical quantities based on the shape of a graph. This lesson addresses the HSA-CED.A.2 standard because it asks students to create mathematical expressions using data that connects two physical quantities. It aligns with the NGSS Practices of Using Mathematical Reasoning (SP5) and Developing and Using Models (SP2) for Science because students will use mathematical logic to create visuals that demonstrate their understanding of the mathematical and graphical models for the relationship between two physical quantities.
Within this lesson, students will obtain information for a non-fictional text on graphical data analysis using a school-wide annotation strategy that identifies important information, models, and puzzling information within a text. Students use this text to help identify relationships between two physical quantities based on the shape of the two-dimensional graph of the dependence of a dependent physical quantity on an independent physical quantity. Students will then use their understanding of mathematical models to complete a graphing practice worksheet. Finally, students work in teams of 2-4 to create visuals that illustrate ways to use a spreadsheet in order to fit curves to mathematical models. I assess student understanding throughout the lesson using informal check-ins, and will assess each student's work at the end of the school day.
Many students bemoan the idea that physics is another mathematics class. I attempt to negate these frustrations by asking students to treat mathematics as a language that helps us model our interactions with the world around us. In this lesson, I talk about the idea that math models are awesome because of how effective and reproducible they are; math models are ideal for students who are focused on precision and accuracy.
At the beginning of each lesson, I have a quick Bell-ringer to get students focused on the tasks for today's lesson. There is a slide with the date, the objective and an additional prompt projected on the interactive whiteboard with a red label that says "COPY THIS" in the top left-hand corner. Sometimes the additional prompt is a BIG IDEA for the lesson, or the Quote of the Day or a Quick Fact from current events that is related to the lesson. The red label helps my students easily interact with the information as soon as they enter the room and avoids losing transition time as students enter the classroom.
Today's lesson has the big idea about the characteristics of math models that I want students to remember and readily be able to explain as we progress through the course. I choose this big idea because I want students to learn the value in making connections across subject areas as they learn more complex subjects.
I am a member of the American Modeling Teachers Association and believe that authentic learning is driven by connecting hands-on activities with robust content. One of the first assignments in this units is to annotate the first reading called reading from the modeling physics curriculum on common graph templates. I ask students to annotate by adding a * to important ideas, underlining key facts, boxing equations and adding question marks to concepts that students still find puzzling. At each lab station, a team of four students split the reading into chunks, annotate and share their annotations with their lab mates.
I use this annotation method because it is a schoolwide practice across disciplines and grade levels that has been proven to raise the level of understanding of complex content for both English Language Learners and students academically in the bottom third of each class. The reading-writing team at my school suggest that chunking and annotating complex content into smaller sections helps struggling students process this material in a way that make sense to them and helps speed up the processing time these students take to analyze this type of information. Later on in this lesson students take the information from this reading to create a short 1-2 page summary for 9th-grade physical science students to use during a similar unit in their science class.
In this section, I ask students to find commonalities between the graph types on the summary page found on page 15 of the reading from the previous section. We tease out the ideas that graphs require an x-axis label that typically corresponds to an independent variable and a y-axis label that typically corresponds to a dependent variable. Then we work on creating appropriate titles which can either be of the "tweet-able" variety or in the general form of "The Dependence of a dependent variable on an independent variable". I use this general form of title creation because it helps students make connections to skills they acquired in physical science which is a part of the spiraling science curriculum. Students then spend 15 minutes completing a Graphing Practice activity where they create graphs of two physical quantities, identify the independent and dependent variables and determine a mathematical expression for the data.
After our discussion, I ask students to create a summary in their notebooks that would be easy for a 9th-grade student to understand. I then circulate and ask students questions to push their understanding, like "Why do we care about the naming convention of graphs?" and "Why do you say that?". Some student responses include, "The way we name graphs help us make sense of information" and "Good names helps others understand how to read the information." I ask these types of questions because I want students to focus on why they think what they do and to evaluate whether their written words communicate these ideas to a specific audience.
After students have completed their summaries, I use my interactive whiteboard to project a short video:
The video shows students how to create graphs in Excel and how to fit add a linear function to those graphs. I choose an instructional video because I want students to see each of the steps in real time by modeling the process for them. I then ask the entire class if they have any questions, which I answer to give clarification to the task. After I have answered student puzzles, I ask students to check their school email or the class website for the video so that can follow along with and then implement in Microsoft Excel.
After students use Microsoft Excel to create graphs and created a linear fit to the curve, I ask them to create visuals about the process. I ask students to use a visual of their choice, the popular choices are Prezi, Popplet, Coggle and Mini Posters. I use visuals as a strategy because I have several students whose learning styles are aligned with visual intelligence.
Students create a visual that illustrates "How to use Excel to Linearize a Graph".
The visuals must include:
Students work in teams of 2-4 to create this visual. Each team member writes a percentage estimate of how much of they each contribute to the final work product. Students assess and edit their work using a rubric while trying to maximize their scores.
I grade the visuals on a five-point scale from not yet to highly proficient. I want students to consider multiple viewpoints on topics within physics and to use the information they gather from their peers to broaden their understanding of a topic. This rubric is one that my students are familiar with because it is similar to the 9th-grade physical science rubric. One recurring point of focus for my curriculum is an emphasis on positive collaboration. In this part of the lesson, students discuss their presentations with their station partners. Click here to see an example of student work. Students edit their peer's work using a rubric.
I provide students with an Exit Slip with a set of writing prompts for a routine called compass points. Students spend about ten minutes writing responses on a slip of paper that demonstrate the current level of understanding of topics from today's lesson. Some student responses include:"Knowing common graph types is important", "Learning how to copy functions in Excel is important". While other student responses include suggestions of "Watch the video and follow along with a Chrome book or you'll be lost", "Write your summary in short sentences so that it's easy for a 9th grader to follow along with."
After ten minutes elapse, I collect the slips to grade them. I use this type of closure activity because it encourages students to consider multiple viewpoints on a scientific topic. To wrap up the lesson, I remind students that I will grade and return the exit slips at the beginning of the next lesson.