Verifying Results from CAS - More About

The following example shows how use of mode can impact on the types of answers that students obtain. The example requires students to find the area of the given triangle. First they use Pythagoras’ Theorem to find the length of the unknown side and then the area can be determined.

The first solution has a CAS set in exact mode and the values for the unknown side are given as surds. The positive x-value is then used in the area formula to obtain an exact value in surd form.

The second solution uses the same method as above, but the CAS mode is set to provide approximate answers, rather than exact. In this case the x value is 5.65685 and the area is 19.799.

Some students might then think that they will set their mode to exact to provide the “exact answer”. In the following CAS display the mode has been reset to exact at this point. The solution uses the number displayed in the previous line of calculation, which is the x-value displayed to six decimal places, rather than the x-value stored within the CAS.  In this case the CAS has changed the 5.65685 into a fraction and then simplified the result.

Students should understand why this is not an exact answer to the problem even though the display is a fraction and it seems very precise. This can be shown by subtracting this approximation for the area from the exact value determined initially. It is important for students to understand when an answer is exact or an approximation and mode is important here.