Proportional Reasoning - Expanding Functional Reasoning

Expanding Functional Reasoning

The four functional relations noted above, constant sum, constant difference, constant product, and constant ratio, are based on the four arithmetic operations students are most familiar with, namely, addition, subtraction, multiplication and division. Most relations in the real world do not fall into one of these categories. However, if students learn simple techniques such as thought experiments and plotting graphs, they will be able to apply these techniques to more complex situations.

Again, consider Newton’s equation for the force of gravity:

If a student understands the functional relation between the variables, then he/she should be able to answer the following thought experiments.

What would happen to the force of gravitational attraction if:

• one of the masses is doubled?
• one mass doubled and the other mass halved?
• both masses doubled?
• both masses halved?
• the distance between the masses doubled?
• the distance between the masses halved?

Generally, thought experiments must be confirmed by experimental results. Many children and adults when asked to perform a thought experiment on the mass of an object and the velocity with which it falls to the earth might say that when the mass is doubled then the object will fall twice as fast. However, experimental results do not back up this “logical” thought experiment so it is always essential that theoretical results agree with experimental data.