Chapter 3 Whole Numbers and Their Operations Copyright

3 -1 Addition and Subtraction of Whole Numbers § Addition of Whole Numbers § Whole-Number Addition Properties § Mastering Basic Addition Facts § Subtraction of Whole Numbers § Properties of Subtraction § Introductory Algebra Using Whole-Number Addition and Subtraction Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Definition Addition of Whole Numbers Let A and B be two disjoint finite sets.

Number-Line Model 1. Josh has 4 feet of red ribbon and 3 feet of white ribbon. How many feet of ribbon does he have altogether? 2. 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? Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Ordering Whole Numbers Children compare and order whole numbers (at least to 100) to develop an understanding of and solve problems involving the relative sizes of these numbers. They think of whole numbers between 10 and 100 in terms of groups of tens and ones (especially recognizing the numbers 11 to 19 as 1 group of ten and particular numbers of ones). They understand the sequential order of the counting numbers and their relative magnitudes and represent numbers on a number line. NCTM grade 1 Curriculum Focal Points, p. 13 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Whole Number Addition Properties Commutative Property of Addition of Whole Numbers If a and

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). Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 3 -1 Which properties are illustrated in each of the following? a. 5 + 7 = 7 + 5 Commutative property of addition b. 1001 + 733 is a unique whole number. Closure property of addition Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 3 -1(cont) Which properties are illustrated in each of the following? c. (3 + 5) + 7 = (5 + 3) + 7 Commutative property of addition d. (8 + 5) + 2 = 2 + (8 + 5) = (2 + 8) + 5 Commutative and associative properties of addition Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Mastering Basic Addition Facts Making 10: Regroup to form a group of 10 and

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Subtraction of Whole Numbers Take-Away Model Copyright © 2013, 2010, and 2007, Pearson Education,

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 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Subtraction of Whole Numbers Missing-Addend Model Copyright © 2013, 2010, and 2007, Pearson Education,

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Subtraction of Whole Numbers Comparison Model Juan has 8 blocks and Susan has 3

Subtraction of Whole Numbers Number-Line (Measurement) Model 5− 3=2 Copyright © 2013, 2010, and

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Introductory Algebra Using Whole-Number Addition and Subtraction Sentences such as 9 + 5 = x and 12 − y = 4 can be true or false depending on the values of x and y. For example, if x = 10, then 9 + 5 = x is false. If y = 8, then 12 − y = 4 is true. If the value that is used makes the equation true, it is a solution to the equation. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.