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

CHAPTER OVERVIEW 1
CHAPTER 5 Understanding Integer Operations and Properties
Major Objectives of the Chapter
Chapter Overview
Preservice elementary teachers will be able to:
use real world situations to illustrate/model the meaning of an integer;
use equations to demonstrate the mathematical need for integers;
use models and mathematical reasoning to explain integer operations;
identify the properties of integer addition and multiplication;
compare two integers and order a set of integers; and
choose an addition or multiplication equation and verify it using the basic properties.
Meeting Some Challenges
One challenge associated with the teaching of Chapter 5 is that students often feel that they already know
how to add, subtract, multiply, divide, and order integers, so why study it again. To meet this challenge,
you need to help students see that the purpose of relooking at integers is to understand why the skills they
have learned are correct. When students are asked, for example, to explain why a negative integer times a
negative integer is a positive integer, they are often at a loss to give any plausible reasons. This can
motivate a discussion of the role of models, patterns, and properties in verifying that the operations behave
as we have been taught.
A second challenge is to help students see the important role that the basic properties play in verifying
integer operations. While it is important not to overdo “proof,” it is equally important not to ignore it.
Staying with a proof such as the one on p. 240, p.363, or even p. 264 long enough to enable students to
understand what is going on can give them an insight into mathematics that they cannot get otherwise. It
is helpful for them to understand that while models can help us understand and feel comfortable with the
operations, they do not constitute a proof that the operations are correctly performed. Only the basic
properties, coupled with logical reasoning, can do that.
Another challenge involves connecting the integers with other number systems. One way to make this
connection is to continually compare and contrast the integer operations and properties with those of the
whole numbers. Also, be sure to point out that the inability to find an integer solution to an equation such
as -3x = 8 suggests the need for another type of number.
Teaching Section 5.1 Integers: Addition, Subtraction, and Order Properties
The emphasis in this section is on understanding what integers are and how they are used, using models to
understand addition and subtraction, using properties and reasoning to verify that the operations have been
performed correctly, and developing an understanding of comparing and ordering integers.
First, focus on the idea that there are both real-world and mathematical reasons for needing integers.
Encourage students to look for real-world situations that model integers. As you look at the operations of
addition and subtraction, emphasize the idea that there are several vehicles for giving meaning to these
operations. These vehicles include models (counters, number line, and so forth), patterns derived from

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