See How Graphs Work

To understand how graphs work, two graphs are better than one. Watch animations. Explore change in variables. Sketch two different graphs to show the same relationship between variables.

• Key Understandings

  ✅ Graphs represent relationships between variables. ✅ Different looking graphs can represent the same relationship between variables.

• How can two different looking graphs represent the same relationship between variables?

  Students have many opportunities to use different types of representations to show the same relationship between variables (e.g., graphs, tables, equations). In these activities, students use different forms of the same type of representation (two different looking graphs) to show the same relationship between variables. Here is a research report on why this move makes sense. Students explore how each variable changes, then they sketch different graphs to represent the same relationship between variables. Students might be surprised that the sliders stay on the vertical and horizontal axes. When only one variable is changing, a line is sufficient to represent that change. Graphs can be more meaningful for students when they understand what variables are measuring. To introduce the activities, ask students to show what each variable is measuring. For example, students might have trouble understanding that Cannon Man’s total distance traveled is greater than zero, because Cannon Man ends up in the same place he started. Students can draw diagrams or use manipulatives (such as string) to show how they could measure Cannon Man’s total distance traveled. Sometimes students only look at a graph’s appearance to determine if it represents a relationship that is also a function. The phrase “____ as a function of ____” helps to emphasize that functions specify a relationship between two different variables. Here is an article with student data to support the effectiveness of this approach.

• Cannon Man
• Toy Car

• How can a point on a graph represent a relationship between variables?

  Often students plot points first, then sketch a graph. In the Ferris Wheel activities, students sketch different graphs, then reflect on what a point on those graphs could represent. The Ferris Wheel activities could complement a unit on trigonometric functions. However, students do not need to be familiar with trigonometric functions to work with the Ferris Wheel activities. The Ferris Wheel activities can provide opportunities for students to use relationships between variables to make sense of graph curvature. Questions can help students to focus on relationships. For example, “How is the width changing when the green car is nearing the top of the Ferris wheel?” The Ferris Wheel Height v. Width graph likely is different from graphs that students have previously encountered. Students can use relationships between the height and width to make sense of the graph. Questions can help students to focus on relationships. For example, “What is happening to the width and the height when the green car is moving from the top of the Ferris wheel to the bottom?”

• Ferris Wheel (Distance v. Height)
• Ferris Wheel (Distance v. Width)
• Ferris Wheel (Height v. Width)

• Can a single point on a graph represent different moments in an animation?

  Students have many opportunities to use graphs to represent relationships shown in animations. In these activities, the animations “keep going,” while the graphs trace over themselves. The activities help students build on the key understanding that points represent relationships between variables. If two variables have the same relationship in different moments of an animation, then the same point on a graph can represent both moments. For example, in the Dynamic Tent, the height gets larger and then gets smaller. When the height is getting larger, the relationship between the height and the base is the same as when the height is getting smaller. Therefore, a single point is enough to show the relationship in both moments. Even though the animation keeps going, the graph does not have a new trace to show it. In the Dynamic Tent, the height is a function of the base, and the base is a function of the height. In the Dynamic Kite, the length is a function of the width, and the width is a function of the length. In these activities, students can reason about relationships between variables to investigate inverse and one-to-one functions. However, students do not need to be familiar with inverse or one-to-one functions to use these activities. Here is an article with student data to support the effectiveness of this approach.

• Dynamic Tent
• Changing Kite

• Why are some endpoints included on graphs, while others are not?

  Students may notice that some endpoints are included on a graph, while others are not included. In this activity, students examine an object printed by a 3-D printer, then explore a relationship between the total amount printed and the height of the object. This activity helps students build on the key understanding that graphs represent relationships between variables. In the 3-D printer, students can reason about relationships between variables to investigate piecewise functions. However, students do not need to be familiar with piecewise functions to use this activity.

• 3-D Printer