ENRICHMENT
FOR SECONDARY MATH
This part of the site gives access to constructivist learning resources involving enrichment ideas for secondary level mathematics. Constructivist Learning Resources describes what is meant by concept construction and discusses various types of constructivist learning resources.
The initial focus of most secondary school mathematics courses is on ordinary algebra. Ordinary algebra focuses on the study of laws for various numerical systems and the application of these laws in solving equations and in transforming algebraic expression. More advanced secondary courses use ordinary algebra with functions whose domain is the field of real numbers. The emphasis in all of these courses tends to be more on techniques than on concepts and conceptual relationships. This gives a very limited appreciation of contemporary mathematics.
These secondary enrichment units are designed to provide a broader perspective on mathematics than that which is usually encountered in secondary school mathematics courses. Some of the ideas may be subtle, but the mathematical background needed to understanding it involves nothing more than a good understand of ordinary algebra. What is most needed is an understanding of the use of variables and various function concepts.
Intro to Cyclic Rings The initial focus of most secondary school mathematics courses is on ordinary algebra. Ordinary algebra focuses on the study of laws for various numerical systems and the application of these laws in solving equations and in transforming algebraic expressions. More advanced secondary courses use ordinary algebra with functions whose domain is the field of real numbers. The emphasis in all of these courses tends to be more on techniques than on concepts and conceptual relationships. This gives a very limited appreciation of contemporary mathematics. This Type 4 resource briefly indicates a slightly broader perspective by introducing some structures whose algebra is similar in some ways but different in others from those usually studied in secondary mathematics.
Diophantine Roots of Separable Factor Equations This is a Type 3 enrichment resource for ordinary algebra. It’s main purpose is to provide an application of ordinary algebra which most people would not usually encounter in their formal education. The intended audience is secondary teachers of mathematics, home schooling parents, as well as anyone else who thinks algebra is fun. So the presentation is often more abstract than the presentation found in most books on ordinary algebra.
Polynomial Functions Suppose we start a sequence of number with 1, then add 3, then add 5, and continue adding successive odd numbers. We would obtain the numbers 1, 4, 9, 16, etc. It is easy to see that the function f defined by f(x) = x2 gives a formula for element number x. Why does this happen? How can this example be generalized? Exploring such questions is the topic of this unit.
Combination Functions If a team has 5 outfielders {A, B, C, D, E} then outfield there 10 combinations available for the outfield. One way to see this is to list them.
{A,B,C},{A,B,D},{A,B,E},{A,C,D},{A,C,E},{A,D,E},{B,C,D},{A,C,E},{B,D,E},{C,D,E}
The purpose of this unit is to examine a number of situations involving combinations and to abstract from them both a recursive and an explicit descriptions for to functions related to combinations.
Exponential Functions Rabbits multiply. This is not a comment on their ability to do arithmetic, but an observation that rabbits produce more rabbits. Without external checks on their increase, the number of rabbits at time x+1 is some multiple of the number of rabbits at time x. Starting with a population of 18 rabbits, suppose they increase by a factor of 10 each year. In 1 year there will be 180, in 2 years 1800, in 3 years 18000, etc. Letting f(x) represent the population expected from uninhibited growth in x years, the conditions giving rise to this function can be described recursively. Using the recursive description, we can easily calculate the values above, and since we are repeatedly multiplying by 10 we see that each answer is 18 times a power of 10.
Recursive Description: f(0) = 18, f(x+ 1) = 10f(x) Explicit
Description: f(x) = 18(10)x
In general, biological growth tends to have an exponential component because the population at the end of a time period tends to be some multiple of the population at the beginning of the period. Certain other types of growth tend to be exponential for similar reasons. This unit explores exponential growth situations.
Permutation Puzzles This unit is a simplified version of the first part of the unit Permutation Groups. It provides simple examples and applications of an algebraic structure which we classify as a non-commutative group. Specifically it introduces the concept of a permutation and of a group of permutations. It then applies these concepts to the solution of various puzzles.
Something Out of Nothing or Modeling with Sets.htm: This paper is a simplified and somewhat pictorial version of some ideas created by mathematical logicians as part of a program to show how all of mathematics can be modeled in set theory. The reasons for doing this are rooted in a desire for a unified conceptual net. Such reasons are much more esthetic and hedonic than prudential, and while most mathematicians appreciate the fact that mathematics has a multitude of scientific and ordinary applications, our reasons for the way we structure mathematics are largely conceptual. The last chapter of my book on entitled A Personal Net for Understanding Mathematics contains a discussion of these reasons along with a discussion of the role of set theory in contemporary mathematics. This paper only suggests a small part of this role, focusing primarily on how arithmetic can be modeled in set theory. If this material seems extremely artificial, that is because it is. In fact, its artificial nature is what recommended it, since the primary purpose here is to illustrate in an extreme manner the extent to which mathematics has become almost pure conceptual study. It begins by indicting a conceptualization of set theory in a way that may seem strange to most people, namely that the only objects that can be elements of sets are other sets. Moreover the main set used to build other sets from is the empty set.
Supernatural Binary Frog.htm This is a brief Type 4 resource. It gives an imaginative way to think about infinite bit string names for the real numbers between 0 and 1. The only mathematical background needed is an understanding of fractions. Altho an understanding of the sum of a geometric progression will allow you to verify certain claims that I only classify as plausible.
Thinkadot Algebra The purpose of this resource is to provide a broader perspective on algebraic equations and operations than the one usually encountered in ordinary algebra. Specifically it will involve activities that show how an understanding of either function algebra or the algebra of the ring Z8 can be used to analyze questions related to Thinkadot puzzles. You should try some of these puzzles before using this resource.
Using Remainder Concepts The initial focus of most secondary school mathematics courses is on ordinary algebra. Ordinary algebra focuses on the study of laws for various numerical systems and the application of these laws in solving equations and in transforming algebraic expressions. More advanced secondary courses use ordinary algebra with functions whose domain is the field of real numbers. The emphasis in all of these courses tends to be more on techniques than on concepts and conceptual relationships. This gives a very limited appreciation of contemporary mathematics. This present resource is an enrichment topic for ordinary algebra, designed to provide a broader perspective on mathematics than that which is usually encountered in secondary school mathematics courses. The intended audience is secondary teachers of mathematics, home schooling parents, as well as anyone else who thinks algebra is fun. So the presentation is often more abstract than the presentation found in most books on ordinary algebra. Some of the presentation may seem aimed at a reader with less sophistication than indicated in the prerequisites. This is because some readers may want to adapt some of these ideas for use with students.