This lesson is optional. It offers opportunities to look at multiple representations (equations, graphs, and tables) for some different contexts.
This final lesson on relationships between two quantities examines situations of constant area, constant volume, and a doubling relationship. Students have an opportunity to engage in MP7 as they notice the similar structures of the situations in the Making a Banner and Cereal Boxes activities, as well as connecting the Multiplying Mosquitoes activity to prior work with exponents and the Genie's coins situation from earlier in the unit. They may use those observations and knowledge to more easily solve the problems in the activities.
Consider offering students a choice about which one they work on. Then, in the lesson synthesis, invite students to share their work with the class and compare and contrast the representations of the different contexts.
- Coordinate (orally and in writing) graphs, tables, and equations that represent the same relationship.
- Create an equation and a graph to represent the relationship between two variables that are inversely proportional.
- Describe and interpret (orally and in writing) a graph that represents a nonlinear relationship between independent and dependent variables.
Let’s use graphs and equations to show relationships involving area, volume, and exponents.
- I can create tables and graphs that show different kinds of relationships between amounts.
- I can write equations that describe relationships with area and volume.
The coordinate plane is a system for telling where points are. For example. point \(R\) is located at \((3, 2)\) on the coordinate plane, because it is three units to the right and two units up.
The dependent variable is the result of a calculation.
For example, a boat travels at a constant speed of 25 miles per hour. The equation \(d=25t\) describes the relationship between the boat's distance and time. The dependent variable is the distance traveled, because \(d\) is the result of multiplying 25 by \(t\).
The independent variable is used to calculate the value of another variable.
For example, a boat travels at a constant speed of 25 miles per hour. The equation \(d=25t\) describes the relationship between the boat's distance and time. The independent variable is time, because \(t\) is multiplied by 25 to get \(d\).