Addition and Subtraction of Whole Numbers Section 2.2 Addition and Subtraction of Whole Numbers Mathematics for Elementary School Teachers - 4th Edition O’DAFFER, CHARLES, COONEY, DOSSEY, SCHIELACK

Provide an Intuitive Understanding of Addition Using Models to Provide an Intuitive Understanding of Addition Joining two groups of discrete objects 3 books + 4 books = 7 books

Provide an Intuitive Understanding of Addition Using Models to Provide an Intuitive Understanding of Addition Number Line Model - joining two continuous lengths 5 + 4 = 9

Sets are used to provide a proper definition of Addition Definition of Union of Two sets The 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 B A B A=B AUB AUB AUB A and B have no elements in common A and B have some elements in common A and B are the same set

Definition of Intersection of Two Sets The 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: Example 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 B A B A=B 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

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 Numbers In 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 sum Example: 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 Property For whole numbers a and b, a + b is a unique whole number Identity Property There 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 Property For whole numbers a and b, a + b = b + a Associative Property For 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. 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. A student had 12 letters. 7 of them had stamps. How many letters did not have stamps? Comparing two sets of discrete objects. 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) 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 Numbers In 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. Restating the definition substituting whole numbers: 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. 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 End Section 2.2 Linda Roper