First Week (Problem Solving)

You learned how to work on a problem for which both methods and precise goals were not given. You began to learn to discuss and write mathematics in a comprehensive manner. Specific elements included trial and error, generalization, formulation of clear statements, and justification.

The Poison worksheet was intended primarily to get you working in groups, and get you to figure out something entirely new without being shown how. The mathemtacial content includes the definition of a strategy, and ability to state facts about modular arithmetic (remainders) both verbally and mathematically.

Probably the trickiest element in the Hippos problem concerns the role of variable assignment. When you assign variables to the weights of the individual hippos, in what ways are different choices equivalent and in what ways to different choices imply different physical scenarios? Another part of the Hippos problem is to actually solve the equations. More important than the specific solution is the idea that a system of algebraic equations, the exact type of which you've never seen before, can quite possibly be solved by applying some basic manipulations (solving for one variable, substituting). Keeping in mind what the variables represent will also help you stay on track when you are solving them.

Second Week (More problem solving)

The Dots and Patterns worksheet was designed to give you a chance to practice, without help from me, the same skills of pattern observation, verbalization, testing, sharpening and generalizing.

Later in the course we will be taking a close look at the four basic operations, addition, subtraction, multiplication and division. The Banned Book Survey gave you a preview of what it means to take a closer look at addition and subtraction. The hard part was knowing exactly what the given data represented and what it meant when you added or subtracted some of these quantities. Only once these uncertainties were settled, could you then figure out which operations you needed to do to get the "answer".

The handshakes problem was designed with two goals in mind. The first was to reinforce your knowledge of the "triangular numbers" 1, 3, 6, 10, and so on. The second was to give you a problem that required a combination of trial and error with knowledge of an algebraic formula: the formula helped you figure out numbers of handshakes among a group of over 50 people (too large a group to count all the handshakes), but you couldn't simply use the formula and solve, since one of the constraints was that you had to have a whole number of people on the float. As a class you did very well in mustering up the courage to mess around even when you thought you were stuck, and find a way (several ways actually) to the answer.

Third Week (Algebra, counting, sets, logic)

This week capped off the first segment of the course: the language of mathematics. Algebra is the language of generalization; our algebra segment concentrated on writing verbal information in algebraic form. Set notation is the basic list-making notation of mathematics, and goes along with logic, the language of making compound mathematical statements. And counting is the basic operation underlying the operations of addition, subtraction and multiplication that we will soon be looking at in depth.

The Chickens, rabbits, etc. worksheet emphasizes the two aspects of algebra that are essential to understanding and solving story problems. One, seen already in Hippos, is the precise and unambiguous assigment of variables to physical quantities. The other, not so prominent in Hippos, is the encoding of given information into equations. In the barnyard problem, this second part is quite straightforward; the oven problem teaches care in writing down ratio information (is a degree Farenheit double a degree Celsius, or is the temperature measurement in Farenheit double the temperaturee measurement in Celsius?). The driving problem gives a lesson in hidden equations: in order to compare the two car trips you need not only two speed variables, but a distance variable which is the same for both trips.

The Set Theory readings are designed to give you all the definitions of set theoretic operations (union, intersection, complement), to tell you the meaning of a standard Venn diagram, and to give you practice in translating between these two representations of sets and verbal represenatations. Here are a few more reasons that sets are important: (1) the complete solution to a math problem is often a set of possible correct answers; you need to know how to talk about this. (2) In order to talk about when two properties are the same (like being a square and having an odd number of factors - these are the same!), you need to know how to describe the sets of things with these properties and show the two sets are the same. (3) Logic is, of course, the precise language of compound mathematical statements. Operations on sets mirror the language of logic in that the set of objects for which properties p and q (e.g., being odd and over 100) are true, is just the intersection of the set for which p is true (the set of odd numbers) and the set for which q is true (the set of numbers over 100). Similarly, union captures logical "or" and complement captures logical "not".

Glicks and Glucks was a straightforward reinforcement of your reading on propositional logic. The notations of propositional logic (the symbols $\wedge, \vee, \neg$ and $\rightarrow$) are relatively unimportant, but the idea of making a compound statement by connecting two statements with an "and" or an "or" is central to mathematical communication. It is vital to know what the precise mathematical meaning of a compound statement is, and how to write a precise mathematical statement that reflects the content of a given English sentence.

Last week you learned to count two-element subsets of a set. We extended this to counting all subsets of a set ("Comparing without counting" problem 2) and clubs and squads of size k chosen from a set of size n ("Ballots, squads and Monsieur Pascal, to be completed later). These counting operations, along with the idea of one-to-one correspondence ("Comparing without counting problem 1) are related to the arithmetical operations of multiplication and exponentiation.

Fourth Week (Factors)

The fundamental theorem of arithmetic, which you are responsible for knowing, says that prime numbers behave like chemical elements (at least as far as multiplication is concerned: you can "take apart" any number into a product of primes, and when you are done, no matter how you did it, you always get the same number of each prime factor.

All in the Timing is designed to lead you to the fundamental theorem and to show you what you can learn from the factorization of a number. The last two problems on this worksheet deal with applications of factoring, to which we will return later.

The locker problem is supposed to lead you to see how factors come up in an unexpected context, and how to reason out which numbers have an odd number of factors. Both this worksheet and "All in the Timing" will be reinforced next week by an additional worksheet "Counting Factors".

Fifth Week (Operations, more factors)

The mathematical operations familiar to school children (addition subtraction, multiplication and division) have properties, some of which are intuituively known and understood by children and others of which need explicitly to be taught. In this part of the course, we discuss them in the general context of binary operations on sets, and relate abstract properties, such as commutativity and associativity, to facts about numbers, computations and solutions of problems. The final goal is to be able to relate these properties to algorithms you will teach that accomplish these operations.

The readings on operations and their properties are designed to give you the language to discuss binary operations on sets. It is tailored particularly to arithmetical binary operations and their properties. You are supposed to understand what an operation is, and what an abstract property, such as commutativity, is. Of particular interest later will be the notion of an inverse, which underlies the definitions of negative numbers and of fractions.

Tarzan was designed to make connections between properties you already intuitively understand and their formal definitions.

Counting Factors was a worksheet to reinforce what you learned in "All in the Timing". The sequence of problems culminates in a rule to determine how many factors a number has, given its prime factorization. The rule, along with a mental picture of why, should become second nature to you.

Sixth Week (Negative numbers, bases, base 10 numeration)

Students learn about negative numbers mainly by working with them (procedural knowledge). Nevertheless, teachers need to know their definitions and properties (declarative knowledge). The material on negative numbers begins with the physical basis for negative numbers, continues with statements of rules for using negative numbers arising from their physical properties, and ends with formal justifications for these manipulations (we didn't do this last part). The other major topic we started this week was our base 10 numeration system. To test whether you understand facts about base 10 numeration, you will be asked to understand and explain things in other bases. Working in other bases is also a way to defamiliarize yourself with base 10, and better grasp what it is like to learn numeration for the first time.

The first worksheet on negative numbers was really an interactive reading. The main object was to see what it means when negative numbers are used to represent physical quantities, and to see how laws of manipulation follow from their uses.

The worksheet "Why, part II" was designed to test and improve your understanding of modeling with negative numbers, specifically of modeling the product operation, and to give you a stock answer to a question you will surely face about multiplication of negative numbers. Equally important to understanding what constitutes a good model for explaining a concept is understanding what story problems are NOT models for a concept, despite their having some superficial resemblance to the concept.

Stupid number tricks draws a connection between base 10 representations of numbers and an algebraic representations. The important thing here is to use algebra to analyze how the a base 10 representation will change as you perform various operations.

Throwing Yourself Off Base was the first in a short series of worksheets in other bases. It is designed to give you a better understanding of base 10 numeration by showing you how an unfamiliar base works. In particular, it helps you come to see that the familiar 1's, 10's, 100's places and so on are really special cases of a generalization, namely the 1, X, X^2, ... places. Analysis of operations in base 10 often requires this point of view (in particular, understanding of divisibility tests).