Turning Routine Topics Into Mathematics Enrichment For The Gifted
The following are algebra/pre-algebra topics that I adapt for my gifted seventh grade students:
For gifted students I define infinite set in terms of one-to-one correspondence, i.e. an infinite set is one that can be put into one-to-one correspondence with a proper subset of itself. I introduce the notion of power set and from there we extend the notion of infinity to include different levels. We then prove that the set of natural numbers and the set of rationals have the same cardinality. We then have a discussion of real numbers including transcendentals and actually prove, using Cantors diagonal method that reals cannot be put into one-to-one correspondence with the set of natural numbers and hence the cardinality of natural numbers is less than that of reals. We of course discuss the definition of countable sets. By the end of the unit, the students have an understanding of the mathematical notion of infinity even though there are still lively arguments relating to its overall notion. They realize that mathematically there are an infinite number of higher levels of infinities which is quite fascinating for a seventh grader, even one who is capable of understanding the abstract concepts involved. I also include this problem in the unit on sets:
Prove that given a decade of two digit numbers; at least two subsets will have the same sum. I don’t know who I should credit this problem to but I believe it was on a past elite high school math contest, perhaps a USAMO problem.
For seventh grade gifted students, whether taking algebra or pre-algebra, we extend the unit by studying infinite geometric sequences and series. If students are able to solve multi-step equations, we study both arithmetic and geometric sequences and series. The most fascinating exercise in this extension is intuitively deriving the formula for finding the sum of an infinite geometric series i.e. S=a/(1-r). The students are also amazed when we convert a repeating decimal into a common fraction via the formula S= a/(1-r) and this strengthens their belief that the formula produces the correct mathematical result. Even though sequences and series is technically a pre-calculus topic, gifted seventh graders seem to understand the topic quite well when it is introduced as part of or as an extension of a standard rational numbers unit. I would like to include a problem that I assign each year from The Art of Problem Solving by Sandor Lehoczky and Richard Rusczyk. Richard actually spoke at our Stanford meeting about his concern for mathematically gifted students. The problem is as follows and involves an infinite geometric series as well as probability:
Ashly, Bob, Carol and Doug (not our Doug) are rescued from a desert island by a pirate who forces them to play a game. Each of the four, in alphabetical order by first names, is forced to roll dice. If the total on the two dice is either 8 or 9, the person rolling the dice is forced to walk the plank. The players go in order until one player loses: A, B, C, D, A, B, C, ……. What is the probability that Doug survives?
Richard gives credit for that problem to Mu Alpha Theta from their 1990 math contest. Gifted seventh graders find this problem very challenging but doable. This also brings up another point. When we (usually one or more students) present solutions to this problem the day after it is assigned for homework, almost every student understands the process and explanations. I would venture to say that average seventh graders would have a difficult if not an impassable road to understanding this problem, not to mention the solution. This is an excellent example of a problem that can be given and discussed with gifted math students but not regular students even though the roots of its solution can be traced back to the topic of rational numbers.
PROPERTIES OF REALS
For seventh grade gifted students we approach this topic via mathematical systems and groups. We develop the properties and introduce many of the definitions by studying operation tables. We eventually define a “group” and do basic proofs using group language. For instance we prove that the inverse of the inverse of A is equal to A. This is actually a difficult proof for even gifted seventh graders who have never been exposed to formal proofs. We translate between group, multiplicative, and additive languages and we extend our knowledge of group to define “field” and “ordered field”. We are then able to understand and appreciate the importance of the “group” concept by discovering groups in different branches of mathematics, i.e. the rational number field, modular arithmetic groups and fields, and permutation groups. We also have an excellent discussion of “the problem of zero”. Students walk away with the understanding that many mathematical structures are not commutative but many useful ones are for instance Abelian groups.
So we extend our unit on properties of reals and solving equations to include an important topic from number theory, the study of groups albeit at an elementary level. I now include a problem that I assign each year that can be made into a much simpler problem by using modular arithmetic.
Given two sequences:
Xo =1 Yo =1
X1 = 1 Y1 = 7
XN+1 = Xn + 2XN-1 YN+1 = 2YN + 3YN-1
The first few terms are:
XN = 1,1,3,5,11,21,43, …
YN = 1,7,17,55,161,487,…
Show that except for 1, the two sequences have nothing in common.
MIDDLE SCHOOL GEOMETRY
For gifted math students I extend and enrich the geometry unit by presenting the non-Eclidian geometries of Lobachevski and Riemann in relation to Euclid’s Parallel Postulate. We also cover special triangles, i.e. 30-60-90, 45-45-90, and derive the relationship of the lengths of their sides to their corresponding angles. We derive the formula for the area of an equilateral triangle and we have a special after school derivation of Heron’s formula for finding the area of an oblique triangle given its three sides. This also leads to a discussion to the triangle inequality theorem. We also cover some trigonometry including the law of cosines. We extend the concept of angle to include radian measure and their representation on the Cartesian coordinate plane.
The following are examples taken from The Art of Problem Solving of problems that I would typically assign during this unit:
1. Find the ratio of the area of a circle inscribed in a regular hexagon to the area of the circle circumscribed about the same hexagon.
2. If the sum of all the angles except one of a convex polygon is 2190, then how many sides does the polygon have?
3. What is the volume of a regular octahedron whose vertices are the centers of the faces of a cube whose edge has length 6? (This is actually a Mathcounts problem from 1985)
There are a few more but they require diagrams.