Section 2.2

Addition and Subtraction of Whole Numbers

Mathematics for Elementary School Teachers - 4th EditionO’DAFFER, CHARLES, COONEY, DOSSEY, SCHIELACK

Using Models to Provide an Intuitive Understanding of Addition

Joining two groups of discrete objects

3 books + 4 books = 7 books

Using Models to Provide an Intuitive Understanding of AdditionNumber Line Model - joining two continuous

lengths

5 + 4 = 9

Sets are used to provide a proper definition of Addition

Definition of Union of Two setsThe union of two sets A and B is the set containing every element belonging to Set A as well as every element belonging to Set B

Symbol: AUB (The union of Set A and Set B)

Example of the union of two sets:If you have collected a set of three books in art, A =

{a, b, c} and a set of four books in biology, B = {d, e, f, g}, the total set of these books that you have collected, {a, b, c, d, e, f, g} is called the union of sets A and B

Venn diagrams illustrating the union of two sets

A and B have no elements in

common

A and B have some elements in

common

A and B are the same

set

AUBAUB AUB

A B A B A=B

Definition of Intersection of Two SetsThe intersection of two sets A and B is the set of all elements common to both set A and set B

Symbol for the intersection of Set A and Set B:

ABExample of the intersection of

two sets:If you have collected a set of three books in education, A = {a, b, c} and a friend has collected a set of five books in education B = {a, b, c, d, e}, the set of books that you have both collected, {a, b, c}, is called the intersection of sets A and B

Venn diagrams illustrating the intersection of two sets

A and B have no elements in

common

A and B might have some elements in

common

A and B have

exactly the same

elements

A B A B A=B

AB

AB

AB

Definition of Disjoint Sets:Two sets are said to be disjoint if and only if their intersection is the empty set; that is, the sets have no elements in common.

Definition of Addition of Whole NumbersIn the addition of whole numbers, if A and B are two disjoint sets, and n(A) = a and n(B) = b, then a + b = n(A∪B). In the equation a + b = c, a and b are addends, and c is the sumExample: If you have collected a set of four books in math, a = 4, and you have collected a set of five books in education, b = 5, then there are 4 elements in set A and there are 5 elements in set B. Thus 4 + 5 = 9 is the number of elements in the union of set A and set B.

Properties of Addition of Whole Numbers

Closure PropertyFor whole numbers a and b, a + b is a unique

whole number

Identity PropertyThere exist a unique whole number, 0, such that 0 + a = a + 0 = a for every whole number a. Zero is the additive identity element.

Commutative PropertyFor whole numbers a and b, a + b = b + a

Associative PropertyFor whole numbers a, b, and c, (a + b) + c =

a + (b + c)

Definition of Less Than ( < ) and Greater Than ( > ) for Whole Numbers

Given whole numbers a and b, a is less than b, symbolized as a < b, if and only if there is a whole number k > 0 such that a + k = b. Also, b is greater than a, ( b > a ), whenever a < b.

Example: Let a = 7, b = 10, and k = 3. Let’s read the definition again using some numbers. Given whole numbers 7 and 10, 7 is less than 10, symbolized as 7 < 10, if and only if there is a whole number 3 > 0 such that 7 + 3 = 10. Also, 10 is greater than 7 ( 10 > 7 ), whenever 7 < 10.

Modeling Subtraction Taking away a subset of a set.o Suppose that you have 12 stamps and give away 7. How many stamps will you have left?

Separating a set of discrete objects into two disjoint sets.o A student had 12 letters. 7 of them had stamps. How many letters did not have stamps?

Comparing two sets of discrete objects.o Suppose that you have 12 stamps and someone else has 7 stamps. How many more stamps do you have than the other person.

Missing Addend (inverse of addition)o Suppose that you have 7 stamps and you need to mail 12 letters. How many more stamps are needed?

Geometrically by using two rays on the number line

Definition of Subtraction of Whole NumbersIn the subtraction of the whole numbers a and b, a – b = c if and only if c is a unique whole number such that c + b = a. In the equation, a – b = c, a is the minuend, b is the subtrahend, and c is the difference.

In the subtraction of the whole numbers 10 and 7, 10 – 7 = 3 if and only if 3 is a unique whole number such that 3 + 7 = 10. In the equation, 10 – 7 = 3, 10 is the minuend, 7 is the subtrahend, and 3 is the difference.

Restating the definition substituting whole numbers:

Comparing Addition and Subtraction Properties of Whole Numbers

Even though subtraction of whole numbers is closely related to addition of whole numbers, the properties of addition do not hold for subtraction.

The EndSection 2.2

Linda Roper