Chapter 2Notes for InstructorsContentThe second chapter of Long and DeTemple is geared toward defining the whole numbersand the arithmetic operations on the whole numbers. The first section provides an intro-duction to set theory. The second section focuses on equivalence, cardinality, and orderingthe whole numbers. The third and fourth sections define whole number addition, subtrac-tion, multiplication, and division. These sections also introduce several models for theseoperations.Notes and SuggestionsNotes on Section 2.1: Sets and Operations on Sets• Because the definition for addition is based on the union of disjoint sets, it is importantfor students to have a basic understanding of some set theory. In particular, studentswill need to understand the following ideas:– Union of sets– Intersection of sets– Disjoint sets– Transitivity of inclusion– Commutativity of union– Associativity of union– Properties of the empty set– The inclusion-exclusion principle– Subset– Proper subsetYou can do a lot of other things in this section, but these are the essentials. Studentsshould be able to understand Venn diagrams and perhaps draw a few on their own, butI do not know that this skill is necessary to understand the key ideas in the remainderof the chapter.• You will notice that I have placed a special emphasis on the union of sets. I have donethis because addition on the whole numbers is defined using the union of disjoint sets.Since the main objective of this chapter is to define arithmetic operations on the wholenumbers, the union of sets seems to be one of the most important operation on sets.For example, the commutative property of addition follows from the commutativity ofunion.1• You will need to define subset and proper subset so that you can order the wholenumbers. These ideas can be slightly confusing for students, but I think they arenecessary for students to fully understand the relations ≤, ≥, <, and >. If A ⊂ B,students should be able to clearly explain why this is true. That is, they should arguethat each element of A is also an element of B and that there is an element of B thatis not an element of A. Moreover, they should understand that A ⊆ B and B ⊆ Aimplies that A = B.• If time permits, you should also define the complement of a set and the Cartesianproduct. The latter is used as a model for multiplication (and, in fact, can be usedto provide an alternate definition for multiplication). The former could be used toformally define subtraction by the take-away model. Moreover, students will need tobe familiar with the complement of a set when they study the chapter on probability.• You can make a lot of good True/False questions from the material in section 2.1.These questions can teach students to read definitions carefully. You can also teachstudents how to construct good counterexamples for those statements which are false.Moreover, you will need to stress that you can show that a statement is false byproviding a counterexample, but you (usually) cannot show that a statement is trueby providing an example.• An Activity Note: When introducing the operations on sets, I found that it washelpful to involve the students. Specifically, I told the students with brown hair tostand on the right side of the room. Then I told the students with brown eyes tostand on the left side of the room. At this point they should see the need to have anintersection. You can also talk about complements at this point because some studentsshould not be in either set. Moreover, you can address the inclusion exclusion principle,since it is likely that there will be students in the intersection of the sets. You cancertainly make this activity more elaborate if you like. I do believe that the activitywas, for my class, more interesting and effective than the traditional lecture approachfor teaching set theory.Notes on Section 2.2: Sets, Counting and the Whole Numbers• In this section, Long and DeTemple define one-to-one correspondence, equivalent sets,the whole numbers and the ordering of the whole numbers.• Manipulatives: On pages 87–89, Long and DeTemple discuss manipulatives that canbe used to represent the whole numbers. We do have the cubes discussed at the top ofpage 88 and Cuisenaire rods which are similar to the number strips shown on page 88.• Pacing: In retrospect, I do not think it was necessary to spend a whole lot of timeon section 2. I also think you should skip the Hamming codes unless you really have alot of extra time. You will probably need quite a bit of time on sections 3 and 4. TheHamming code discussion at the end of section 2 appears to be an attempt to show2the student that there are applications for set theory. I do believe that applicationsare important, but it is difficult to include them because of the pace of this course.I found that it was more important to devote extra time to the central ideas of thecourse which students will actually be teaching themselves.Notes on Sections 2.3 and 2.4: Addition and Subtraction of Whole Numbers and Multi-plication and Division of Whole Numbers• In sections 3 and 4 Long and Temple define addition, subtraction, multiplication anddivision of whole numbers. It is important that students understand the differentmodels of arithmetic discussed in these sections. Moreover, it is important that youdefine the arithmetic operations as they have done in the book because these definitionsgeneralize easily.• Addition:– I like to discuss the importance of the word “disjoint” in the definition for addition.This can be done by providing an example.An Activity Note: Sometimes it is useful to involve the class. Determine howmany students have brown hair and how many students have green eyes. Askthem to determine how many students have brown hair or green eyes. If you arelucky, your sets will not be disjoint. Ask them how the problem would be differentif we wanted to determine the number of students who have green eyes or blueeyes.– Students should use the set model when combining two groups and the numberline model when looking at distances.– The properties on page 101 follow (easily) from the definition of whole numberaddition and the properties of sets given on page 78.• Subtraction:– Students may think it is strange to define whole number subtraction by the missingaddend model. I find that it is useful to remind them that they first learneddivision by the missing factor model.–