Understanding Best-fit with a Visual Mode Measuring Error in a Linear Model
Use this e-example to minimize the error in a line of best fit by toggling between four different measuring methods. Options include the ability to change the location of five points and the ability to rotate and translate the line of best fit.
Use the button on the lower left-hand corner to switch between four different measuring methods:
You can change the linear equation (line of best-fit) by clicking a dragging the blue point on the graph.
You can also change the location of each data point by clicking and dragging on any black point. Once activated, this point will turn purple.
In each mode, the total error, as well as how it was calculated, will be shown. In the comparison mode, these three values will be shown in a color-coded manner.
For the given data set and for each of the measures of error, find a line (a linear model) for which the error is as small as possible. Try various slopes and various y-intercepts before you settle on your line of "best fit." For each method, record the equation of the line.
Students should have experience graphing data generated by linear situations and writing equations for the lines that pass through such data points. Finding equations is relatively straightforward when the data all lie on a line. When the data are only approximately linear, however, no line will fit the data exactly and students must decide from among many possible linear models. This situation often arises when data come from real contexts and a model is desired from which predictions can be made. Before students engage with these interactive examples, they should be given a set of data that is somewhat, but not exactly, linear and asked to plot a line that they think fits the data well. They should be asked to defend their choice of linear model. Some might argue that their line is a good fit because it "passes through" many of the points. Others might argue that fitting well means that it is "closest to the most points" or that it is "in the middle of the points." Students could be asked to define statements such as "closest to the most points" numerically and to quantify their reasoning in other ways so that the effectiveness of two proposed models can be compared.Given a set of bivariate data, graphing calculators or spreadsheets may be used to find the least-squares regression line for the data set. This investigation may be used to help students develop an understanding that there is more than one way to define the "line of best fit" and to help them develop meaning for the approach they are most likely to encounter: the method of "least squares." The least-squares regression line minimizes the sum of the squares of the residuals, a criterion not often suggested by students. The interactive figure above provides a visual model for the sum of the squares and prepares students to approximate a least-squares regression line in subsequent examples.