Thinking with Mathematical Models unit goals:
- Develop skill in collecting data from experiments and systematically recording that data in tables
- Construct coordinate graphs to represent data
- Make predictions from data tables or graph models
- Use patterns in data to find equations that model relationships between variables
- Use tables, graphs, and equations to model linear and nonlinear relationships between variables
- Distinguish between linear and nonlinear relationships
- Identify inverse relationships and describe their characteristics
- Use a graphing calculator to find and study graph models and equation models of relationships between variables
- Use intuitive ideas about rates of change to sketch graphs for, and to match graphs to, given situations
- Use intuitive ideas about rates of change to create stories that fit given graphs
The Mathematics in Thinking with Mathematical Models
In this unit, students explore the advantages of using algebraic models, in the form of graphs and equations, to describe situations. A table of data is often a good starting point for deciding what type of relationship is suggested by the data.
For example, table 1 below shows values for the linear relationship represented by the equation y = 3x + 5. As x increases by increments of 1, y increases by increments of 3.
Table 2 shows values for the linear relationship represented by the equation y = 20 — 2x (or y = —2x + 20). As x increases by increments of 1, y decreases by increments of 2.
Table 3 shows values for the nonlinear relationship represented by the equation y = 40/x (or xy=40). As x increases by increments of 1, y decreases but not at a constant rate.
Table 4 shows values for the nonlinear relationship represented by the equation y = (l.O6)x, which gives the amount y in a bank account after x years at an interest rate of 6% compounded annually. From the table, we can see that the doubling time for the investment is 13 years: it takes about 13 years for $1 to increase to $2 at a 6% interest rate compounded annually
Fitting a Curve to Given Data
Real situations usually generate “messy” data, and we cannot expect a line or a curve to fit such data exactly What we do look for is a graph model that fits the data pattern well enough to be useful as a predictive tool. The process of curve fitting is technically complex, but students can informally understand the goals of the process by drawing a simple curve to fit plotted points.
At the simplest level, for plotted data that suggest linear relationships, students can eyeball the data and, using a straightedge as a guide, try several modeling lines until they find one that seems to be a good fit. In this process, students often try to hit as many points as possible.
However, this strategy sometimes results in lines that do not give very good predictions. The best rule of thumb is to look for a line that seems to catch the overall trend in the data. Then, students can write an equation for the line.