CHAPTER 2. Understanding Whole-Number Operations and ...

2 CHAPTER 5 UNDERSTANDING INTEGER OPERATIONS AND PROPERTIES
using the calculator, and basic mathematical properties and relationships. These vehicles can be
individually selected to meet the needs of persons with different types of learning styles.
As the basic properties are introduced, focus on their value in mathematically verifying that the procedures
we derived from models for operating on integers are correct. This provides a good opportunity to discuss
the tandem use of inductive and deductive reasoning in mathematics--that is, how we discover a possibly
true relationship using examples and inductive reasoning, and then use deductive reasoning and basic
assumptions or properties to actually prove that the discovered relationship is true.
Some key questions to stimulate discussion are:
How would you respond to someone who claimed that integers are not very important?
How are the operations addition and subtraction related?
What are the roles of counter and number line models in understanding integers, integer addition
and subtraction, and integer ordering?
In addition to the Reinforcing Concepts and Skills exercises in section A, you may wish to give some extra
emphasis to Exercises 36, 44, 52, and 67, which are designed to extend the student’s understanding of
some important integer properties and relationships.
Teaching Section 5.2 Integer Multiplication and Division and Other Properties
In this section, the goal is to use models to give understanding to integer multiplication, develop integer
division as it relates to multiplication, and present some additional properties of integer opposites and
integer multiplication and division. Since the idea that “a negative times a negative is a positive” often
seems to be just a memorized response for many students, it is useful to take plenty of time to give
meaning to this using (a) a counters model, (b) a charged field model, (c) a number line model, (d)
patterns and a calculator, and (e) basic properties of integer multiplication. Because of the complexity of
using models to make integer division plausible, the modeling of division is often more complicated than
it is worth. For this reason, we focus primarily on the inverse relationship between division and
multiplication and develop division using this relationship. Be sure to take enough time to discuss the use
of deductive reasoning and the basic properties to verify simple statements about integer multiplication
and other generalizations about integers. Students should learn to view division such as 36 ÷ -4 as asking
the question, “What factor multiplied by
-4 gives the product 36?” This type of thinking can also help students understand why we don’t divide by
zero. Finally, discuss with students the value of opposites and distributive-type properties of integers in
helping them read and understand mathematical development. Some key questions to stimulate discussion
are:
What method do you feel would be most effective in helping someone understand how to find
the product of two negative integers?
How are the four integer operations related?
How are the following alike? How are they different?
a. the operations addition and multiplication
b. the relationship between addition and subtraction and the relationship between multiplication
and division