The first four questions of this homework set are process oriented. In class we had just finished the derivation of the solution to the general cubic equation. We cover the general cubic equation in this class for several reasons. Perhaps most importantly, solving equations is an important historical topic in mathematics. During the Renaissance, Italian mathematics began to blossom as they attacked methods for finding roots of cubic and quartic equations. Moreover, their solutions to these equations show how numbers were thought of quite differently by mathematicians at that time. In particular, Cardano in his Ars Magna solves a quartic equation earlier than solving a linear equation with a negative solution. One reads from this that Cardano considered finding the general (positive real) solution to a quartic equation to be easier than finding a negative solution of a linear equation. This allows us to enter into a discussion of different understandings of numbers, and how numbers which today seem “easy” to deal with, might not have been so in the past. Mathematically, solving the cubic also allows us a natural entry into the complex numbers. In particular, a common fallacy of high school texts is that the complex numbers were developed to solve quadratic equations that didn't have real roots. In reality, however, the complex numbers were first used to find real solutions to cubic equations. Moreover, even to this day, finding these real solutions as sums of radicals appears to require using complex numbers. Thus solving the cubic allows us to enter into a historical discussion of the complex numbers.
In terms of bringing higher mathematics to the high school curriculum, the cubic also allows us to make several profitable investigations. In deriving the solution to the cubic, we are called upon to use several different mathematical moves. Our solution, which mimics one in Cardano's book, begins by trying to associate a cubic of the form x^3 + px = q (p,q positive) to a cube. In starting this process, we can address a similar question involving a square and discover a geometric derivation of the quadratic formula, allowing students to encounter a geometric understanding of the term “completing the square.” When we move to cubics involving a quadratic (x^2) term, we then address another method of solving a quadratic equation, namely by translating the parabola so that its vertex is on the y-axis. This gives the students a third derivation of the quadratic formula, and links together graphs and the algebraic solutions of equations. Having done all of this allows us to reflect upon a great deal of mathematics in a natural way. Moreover, students appear to appreciate these linkages in the class.
The questions from this set, are to help the students internalize how we solved the cubic, and to ask them to go through the derivation of the general solution on their own with actual numbers, as opposed to doing it in the abstract as we do in class. Questions 3 and 4, in fact are set up to have the students work through solving equations in the way that historically the types of equations were solved. Questions 1 and 2, on the other hand, were designed to get the students practice with the ideas discussed in trying to solve the equations along the way. Finally, question 5 is aimed at getting the students to start on the next topic, the understanding of algebraic numbers and how they differ from transcendental numbers. This problem is also aimed at getting the students to write a more complicated proof. Finally, problem 6 is where the students are supposed to reflect on how this higher level mathematics can inform high school teaching.
The second set of problems on numbers, were included so that we could assess whether the students understand how the various classifications of numbers fit together. The last question on the NCTM standards is then a valuable exercise for these students, as it forces them to read the standards with an eye for higher level mathematics. As is seen in some student responses show on this, when students read the standards, I have found that they do not always pay attention to the higher level mathematics that they suggest should be studied. Thus, in terms of long term work, this problem is meant to encourage students to read documents of this sort with an eye for higher mathematics.