3 Chapter Numeration Systems and Whole Number Operations Copyright © 2016, 2013, and 2010, Pearson Education, Inc.

3-2 Addition and Subtraction of Whole Numbers Number relationships including comparing and ordering. The meaning of addition and subtraction by studying various models and in turn learn addition and subtraction facts. Properties of addition and subtraction and how to use them to develop computational strategies. The inverse relationship between addition and subtraction.

Addition of Whole Numbers Counting On Counting on is an addition strategy where addition is performed by counting on from one of the numbers, for example, 5 + 3 can be computed by starting at 5 and counting 6, 7, 8.

Addition of Whole Numbers Set Model Suppose Jane has 4 blocks in one pile and 3 in another. If she combines the two groups, how many objects are there in the combined group? Note that the sets must be disjoint (have no elements in common) or an incorrect conclusion can be drawn.

Definition Addition of Whole Numbers Let A and B be two disjoint finite sets. If n(A) = a and n(B) = b, then a + b = n(A U B).

Number-Line Model Josh has 4 feet of red ribbon and 3 feet of white ribbon. How many feet of ribbon does he have altogether? One day, Gail drank 4 ounces of orange juice in the morning and 3 ounces at lunchtime. If she drank no other orange juice that day, how many ounces of orange juice did she drink for the entire day?

Number-Line Model Students need to understand that the sum represented by any two directed arrows can be found by placing the endpoint of the first directed arrow at 0 and then joining to it the directed arrow for the second number with no gaps or overlaps. The sum of the numbers can then be read.

Definition Less Than: For any whole numbers a and b, a is less than b, written a < b, if, and only if, there exists a natural number k such that a + k = b. a ≤ b means a < b or a = b. a > b is the same as b < a.

Whole Number Addition Properties Closure Property of Addition of Whole Numbers If a and b are whole numbers, then a + b is a whole number. The closure property implies that the sum of two whole numbers exists and that the sum is a unique whole number. For example, 5 + 2 is a unique whole number, 7.

Whole Number Addition Properties Commutative Property of Addition of Whole Numbers If a and b are any whole numbers, then a + b = b + a.

Whole Number Addition Properties Associative Property of Addition of Whole Numbers If a, b, and c are any whole numbers, then (a + b) + c = a + (b + c).

Whole Number Addition Properties Identity Property of Addition of Whole Numbers There is a unique whole number, 0, the additive identity, such that for any whole number a, a + 0 = a = 0 + a.

Example Which properties are illustrated in each of the following? b. 1001 + 733 is a unique whole number. Commutative property of addition Closure property of addition

Example (cont) Which properties are illustrated in each of the following? c. (3 + 5) + 7 = (5 + 3) + 7 d. (8 + 5) + 2 = 2 + (8 + 5) = (2 + 8) + 5 Commutative property of addition Commutative and associative properties of addition

Mastering Basic Addition Facts Counting on: Start with the greater addend then count on the smaller addend. For example: 4 + 2, start with 4, then count on another two, 5, 6. Doubles: After students master doubles (such as 3 + 3), doubles + 1 and doubles plus 2 can be learned easily. For example, if a student knows 6 + 6 = 12, then 6 + 7 is (6 + 6) + 1 = 12 + 1 = 13.

Mastering Basic Addition Facts Making 10: Regroup to form a group of 10 and a leftover. For example: 8 + 5 can be added as follows:

Mastering Basic Addition Facts Counting back: Usually used when one number is 1 or 2 less than 10. For example, because 9 is 1 less than 10, then 9 + 7 is 1 less than 10 + 7 or 16.

Subtraction of Whole Numbers Subtraction of whole numbers can be modeled in several different ways: Take-Away Model – views subtraction as a second set of objects being taken away from the original set Missing Addend Model – an algebraic-type of reasoning is used where students compute a difference by determining the value of an “unknown” addend.

Subtraction of Whole Numbers Comparison Model – students determine “how many more” of one quantity exists than another. Number-Line Model – subtraction is represented by moving left on the number line a given number of units.

Subtraction of Whole Numbers Take-Away Model

Subtraction of Whole Numbers Missing-Addend Model 8 − 3 = This can be thought of as the number of blocks that must be added to 3 in order to get 8. The number 8 – 3 is the missing addend in the equation 3 + = 8 5

Subtraction of Whole Numbers Missing-Addend Model

Definition Subtraction of Whole Numbers: For any whole numbers a and b, such that a ≥ b, a − b is the unique whole number c such that b + c = a.

Subtraction of Whole Numbers Comparison Model Juan has 8 blocks and Susan has 3 blocks. How many more blocks does Juan have than Susan? 8 − 3 = 5

Subtraction of Whole Numbers Number-Line (Measurement) Model 5 − 3 = 2

Properties of Subtraction It can be shown that if a < b, then a − b is not meaningful in the set of whole numbers. Therefore, subtraction is not closed on the set of whole numbers.

Introductory Algebra Using Whole-Number Addition and Subtraction Sentences such as 9 + 5 = ☼ and 12 − ◊ = 4 can be true or false depending on the values of ☼ and ◊. For example, if ☼ = 10, then 9 + 5 = ☼ is false. If ◊ = 8, then 12 − ◊ = 4 is true. If the value that is used makes the equation true, it is a solution to the equation.