Sets and Whole Numbers 2.1 Sets and Operations on Sets 2.2 Sets, Counting, and the Whole Numbers 2.3 Addition and Subtraction of Whole Numbers 2.4 Multiplication and Division of Whole Numbers

2.1 Sets and Operations on Sets Slide 2-3

THREE WAYS TO DEFINE A SET 1. Word Description: The set of even numbers between 2 and 10 inclusive. 2. Listing in Braces: {2, 4, 6, 8, 10} 3. Set Builder Notation: { x | x = 2n, 1 ≤ n ≤ 5, n N} Slide 2-5

Example 2.1 Describing Sets Each set that follows is taken from the universe N of the natural numbers and is described either in words, by listing the set in braces, or with set-builder notation. Provide the two remaining types of description for each set. a. The set of natural numbers greater than 12 and less than 17. {13, 14, 15, 16}; listing Slide 2-4

Example 2.1 continued b. {3, 6, 9, 12,…} The set of all natural numbers that are multiples of 3; word description. c. The set of the first 10 odd natural numbers. {1, 3, 5, 7, 9, 11, 13, 15, 17, 19} Slide 2-5

VENN DIAGRAMS Sets can be represented pictorially by Venn diagrams, named for the English logician John Venn. The universal set U is represented by a rectangle. Any set A within the universe is represented by a closed loop lying within the rectangle. Slide 2-6 A U

DEFINITION: INTERSECTION OF SETS The intersection of two sets A and B, written, is the set of elements common to both A and B. That is, Slide 2-7

DEFINITION: DISJOINT SETS Two sets C and D are disjoint if C and D have no elements in common. That is, C and D are disjoint means that Slide 2-8

DEFINITION: UNION OF SETS The union of sets A and B, written, is the set of all elements that are in A or B. That is, Slide 2-9

2.2 Sets, Counting, and Whole Numbers Slide 2-10

TYPES OF NUMBERS Nominal numbers A number can be an identification, or nominal number, such as a ticket number. Ordinal numbers Ordinal numbers communicate location in an ordered sequence, such as first, second, third. Cardinal numbers A cardinal number communicates the basic notion of “how many,” such as four may tell us how many tickets we have for an event. Slide 2-11

Example 2.8 Showing the Order of Whole Numbers Use (a) sets, (b) tiles, (c) rods, and (d) the number line to show that 4 < 7. a. Slide 2-12

Example 2.8 continued Use (a) sets, (b) tiles, (c) rods, and (d) the number line to show that 4 < 7. b. c. With rods, the order of whole numbers is interpreted by comparing the lengths of the rods. Slide 2-13

Example 2.8 continued Use (a) sets, (b) tiles, (c) rods, and (d) the number line to show that 4 < 7. d. On the number line, 4 < 7 because 4 is to the left of 7. Slide 2-14

2.3 Addition and Subtraction of Whole Numbers Slide 2-15

TWO CONCEPTUAL MODELS FOR ADDITION OF WHOLE NUMBERS Number-Line (Measurement) Model Slide 2-16

THE MEASUREMENT MODEL OF WHOLE NUMBER ADDITION On the number line, whole numbers are geometrically interpreted as distances. Slide 2-17

PROPERTIES OF WHOLE NUMBER ADDITION CLOSURE PROPERTY If a and b are any two whole numbers, then a + b is a unique whole number. COMMUTATIVE PROPERTY If a and b are any two whole numbers, then a + b = b + a. Slide 2-18

PROPERTIES OF WHOLE NUMBER ADDITION ASSOCIATIVE PROPERTY If a, b and c are any three whole numbers, then ADDITIVE-IDENTITY PROPERTY OF ZERO If a is any whole number, then a + 0 = 0 + a = a. Slide 2-19

Example 2.11 Using Properties of Whole-Number Addition Which property justifies each of the following statements? (i) = (ii)(7 + 5) + 8 = 7 + (5 + 8) (iii)A million plus a quintillion is not infinite. (i) Commutative Property (ii) Associative Property (iii) The sum is a whole number by the closure property and is therefore a finite value. Slide 2-20

Example 2.12 Illustrating Properties What property of whole-number addition are shown on the following number lines? a. The commutative property: = Slide 2-21

Example 2.12 continued What property of whole-number addition are shown on the following number lines? b. The associative property: (3 + 2) + 6 = 3 + (2 + 6). Slide 2-22

Example 2.12 continued What property of whole-number addition are shown on the following number lines? c. The additive-identity property: = 5. Slide 2-23

SUBTRACTION OF WHOLE NUMBERS Let a and b be whole numbers. The difference of a and b, written a – b, is the unique whole number c such that a = b + c. That is, a – b = c if, there is a whole number c, such that a = b + c. Slide 2-24

TERMINOLOGY: SUBTRACTION OF WHOLE NUMBERS Slide 2-25

FOUR CONCEPTUAL MODELS FOR SUBTRACTION OF WHOLE NUMBERS Take-Away Model Missing Addend Model Comparison Model Number-Line (Measurement) Model Slide 2-26

TAKE-AWAY MODEL Joel has 10 cookies. He gives 3 of the cookies to his little sister. How many cookies does Joel have left? To answer, the student uses subtraction. 10 – 3 Slide 2-27

COMPARISON MODEL Roger has read 9 books this week. His friend Jake has read 6 books this week. How many more books has Roger read than Jake? To answer, the student uses subtraction. 9 – 6 Slide 2-28

MISSING-ADDEND MODEL Sarah has saved $15. She needs $22 to buy a blouse she noticed last week. How much more money does she need to buy the blouse? To answer, the student uses subtraction. $22 – $15 (Note: $15 + ? = $22.) Slide 2-29

NUMBER-LINE MODEL Illustrate 8 – 5 on the number line. Slide 2-30

2.4 Multiplication and Division of Whole Numbers Slide 2-31

FOUR CONCEPTUAL MODELS FOR MULTIPLICATION OF WHOLE NUMBERS Array Model Rectangular Area Model Skip Count Model Multiplication Tree Model Slide 2-32

ARRAY MODEL Leah planted 3 rows of tomato plants with 4 plants in each row. How many tomato plants does she have planted? TTTT Slide ROWS 4 PLANTS PER ROW 3 x 4 array

RECTANGULAR AREA MODEL Mr. Hu bought an 8 ft by 6 ft rug for his house. How many square feet are covered by this rug? Slide ft x 6 ft = 48 sq ft

SKIP COUNT MODEL “Skip” by the number 2 six times: 2, 4, 6, 8, 10, 12 “Skip” by the number 3 five times: 3, 6, 9, 12, 15 Slide 2-35

MULTIPLICATION TREE MODEL Timmy has 3 tops (blue, red and green) and 2 shorts (jean and khaki) that can be mixed and matched. How many outfits does he have? Slide 2-36 CHOOSE TOP CHOOSE SHORT SIX OUTFITS

THREE CONCEPTUAL MODELS FOR DIVISION OF WHOLE NUMBERS Repeated-Subtraction Model Partition Model Missing-Factor Model Slide 2-37

REPEATED-SUBTRACTION MODEL Ms. Rislov has 28 students in her class whom she wishes to divide into cooperative learning groups of 4 students per group. Slide 2-38

PARTITION MODEL Slide 2-39

Example 2.16 Computing Quotients with Manipulatives Suppose you have 78 number tiles. Describe how to illustrate 78 ÷13 with the tiles, using each of the three basic conceptual models for division. a. Repeated subtraction. Remove groups of 13 tiles each. Since 6 groups are formed 78 ÷ 13 = 6. b. Partition. Partition the tiles into 13 equal-sized parts. Since each part contains exactly 6 tiles, 78 ÷13 = 6. c. Missing factor. Use the 78 tiles to form a rectangle with 13 rows. Since it turns out there are 6 columns n the rectangle, 78 ÷13 = 6. Slide 2-40

DEFINITION: THE POWER OPERATION Let a and m be whole numbers where m ≠ 0. Then a to the m th power, written a m, is defined by and Slide 2-41

Example 2.18 Working with Exponents Compute expressing your answers in the form of a single exponential expression a m. a = ( ) (7 7) = = 7 6 b = (3 3) (5 5) (4 4) = (3 5 4) 2 = 60 2 c. (3 2 ) 5 = (3) 2 (3) 2 (3) 2 (3) 2 (3) 2 = (3 3) (3 3) (3 3) (3 3) (3 3) = = 3 10 Slide 2-42

MULTIPLICATION RULES OF EXPONENTIALS Let a, b, m, and n be whole numbers where m ≠ 0 and n ≠ 0. Slide 2-43

ZERO AS AN EXPONENT Let a be any whole number, a ≠ 0. Then a 0 = 1. Slide 2-44

DIVISION RULES OF EXPONENTIALS Let a, b, m, and n be whole numbers where Slide 2-45

Example 2.19 Working with Exponents Rewrite these expression in exponential form a m. a = = 5 20 b = = 3 15 c. (3 5 ) 2 /3 4 = = 3 6 Slide 2-46