Notes:

The Proportional Reasoning Concept Builder is an adjustable-size file that displays nicely on smart phones, on tablets such as the iPad, on Chromebooks, and on laptops and desktops. The size of the Concept Builder can be scaled to fit the device that it is displayed on. The compatibility with smart phones, iPads, other tablets, and Chromebooks make it a perfect tool for use in a 1:1 classroom.

Teaching Ideas and Suggestions:

Most of us who teach Physics will at some point emphasize the skill of analyzing a data set in order to determine a relationship or a relationship type. Being able to inspect data for two variables and determine whether the variables exhibit a direct, an inverse, or a quadratic relationship is mission critical for most Physics classes. And for many Physics classes, once such a relationship is determined, being able to use it to predict the value of the dependent variable for any given change in the independent variable is equally important. This Concept Builder addresses these skills. We refer to such skills as proportional reasoning.

Each of the three activities in this Concept Builder have a slightly different flavor. The goal of the collection of three activities is for students to gain skill in recognizing relationships from tabular data and to use the relationship and proportional reasoning to predict the value of the dependent variable if the independent variable is doubled, tripled, or quadrupled. The Concept Builder was intended as an in-class activity. Teachers are encouraged to preview the activity (or view the Questions in the separate file) in order to judge which activities would be most appropriate for their students. Our summary of the three activities is as follows:

• Which Two Rows: Question Groups 1-4 ... Given a Data Table representing generic variables X and Y, students must identify two rows of the table that demonstrated a stated pattern - such as doubling the value of X will double the value of Y
• Predict the Value: Question Groups 5-8 ... Given information about the relationship between generic variables X and Y and given the values of X and Y, students must predict the value of Y if the value of X is doubled, tripled, or quadrupled.
• Extrapolate: Question Groups 9-14 ... Given a Data Table, students must inspect the values and determine the relationship type (linear, inverse, or quadratic), identify the manner in which Y varies with a given change in X, and extrapolate beyond the domain of the data set to predict the value of for a stated value of X.

While we recommend all three activities in the order provided, there is no reason why one activity needs to be done as a pre-cursor to another activity. The questions are shown below. Teachers are encouraged to view the questions in order to judge the appropriateness of each activity for their courses.

In order to complete an activity, a student must correctly analyze each question in that activity. If a student's analysis is incorrect, then the student will have to correctly analyze the same or very similar question twice in order to successfully complete the activity. This approach provides the student extra practice on questions for which they exhibited difficulty. As a student progresses through an activity, a system of stars and other indicators are used to indicate progress on the activity. A star is an indicator of correctly analyzing the question. Once a star is earned, that question is removed from the que of questions to be analyzed. Each situation is color-coded with either a yellow or a red box. A red box indicates that the student has incorrectly analyzed the question and will have to correctly analyze it twice before earning a star. A yellow box is an indicator that the question must be correctly analyzed one time in order to earn a star. Once every question at a level has been analyzed, the student earns a Trophy which is displayed on the Main Menu. This system of stars and trophies allows a teacher to easily check-off student progress or offer credit for completing assigned activities.

The most valuable (and most overlooked) aspect of this Concept Builder is the Help Me! feature. Each question group is accompanied by a Help page that discusses the specifics of the question. This Help feature transforms the activity from a question-answering activity into a concept-building activity. The student who takes the time to use the Help pages can be transformed from a guesser to a learner and from an unsure student to a confident student. The "meat and potatoes" of the Help pages are in the sections titled "How to Think About This Situation:" Students need to be encouraged by teachers to use the Help Me! button and to read this section of the page. A student that takes time to reflect upon how they are answering the question and how an expert would think about the situation can transform their naivete into expertise.

Related Resources

There are very few resources at The Physics Classroom website that pertain solely to the topic of proportional reasoning. Most of the emphasis on proportional reasoning is spread across all the topics. Proportional reasoning becomes more of a tool that is used than a topic that is expressed. As such, we are not currenlty able to recommend one specific resource on the site to prepare students for this Concept Builder. Sorry.