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Section 2.1 Sets and Whole Numbers Mathematics for Elementary School Teachers - 4th Edition O’DAFFER, CHARLES, COONEY, DOSSEY, SCHIELACK How do you think the idea of numbers developed? How could a child who doesn’t know how to count verify that 2 sets have the same number of objects? That one set has more than another set? Sets and Whole Numbers - Section 2.1 A set is a collection of objects or ideas that can be listed or described A set is usually listed with a capital letter A set can be represented using braces {} A = {a, e, i, o, u} C = {Blue, Red, Yellow} A set can also be represented using a circle A= a u e o i C= Blue Red Yellow Each object in the set is called an element of the set C= Blue Red Yellow Blue is an element of set C Blue C Orange is not an element of set C Orange C Definition of a One-to-One Correspondence Sets A and B have a one-to-one correspondence if and only if each element of A is paired with exactly one element of B and each element of B is paired with exactly one element of A. Set A Set B 1 a 2 b 3 c The order of the elements does not matter Definition of Equivalent Sets Sets A and B are equivalent sets if and only if there is a one-to-one correspondence between A and B Set A Set B one two three Dog Cat Frog A~B Finite Set A set with a limited number of elements Example: A = {Dog, Cat, Fish, Frog} Infinite Set A set with an unlimited number of elements Example: N = {1, 2, 3, 4, 5, . . . } Section 2.2 Addition and Subtraction of Whole Numbers Mathematics for Elementary School Teachers - 4th Edition O’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 Addition Number Line Model - joining two continuous lengths 5+4=9 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) Modeling Subtraction Taking away a subset of a set. Suppose that you have 12 Pokemon cards and give away 7. How many Pokemon cards will you have left? o 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? o o o Comparing two sets of discrete objects. Suppose that you have 12 candies and someone else has 7 candies. How many more candies 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 Which of the properties of addition hold for subtraction? 1. Closure 2. Identity 3. Commutative 4. Associative 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) Section 2.3 Multiplication and Division of Whole Numbers Mathematics for Elementary School Teachers - 4th Edition O’DAFFER, CHARLES, COONEY, DOSSEY, SCHIELACK How are addition, subtraction, multiplication, and division connected? • Subtraction is the inverse of addition. • Division is the inverse of multiplication. • Multiplication is repeated addition. • Division is repeated subtraction. • “Amanda Bean’s Amazing Dream” Using Models and Sets to Define Multiplication Multiplication - joining equivalent sets 3 sets with 2 objects in each set 3 x 2 = 6 or 2 + 2 + 2 = 6 Repeated Addition Multiplication using a rectangular array 3 rows 2 in each row 3x2=6 Using Models and Sets to Define Multiplication Multiplication using the Area of a Rectangle width length Area model of a polygon Can be a continuous region Definition of Cartesian Product The Cartesian product of two sets A and B, A X B (read “A cross B”) is the set of all ordered pairs (x, y) such that x is an element of A and y is an element of B. Example: A = { 1, 2, 3 } and B = { a, b }, A x B = { (1, a), (1, b), (2, a), (2, b), (3, a), (3, b) } Note that sets A and B can be equal Problem Solving: Color Combinations for Invitations Suppose that you are using construction paper to make invitations for a club function. The construction paper comes in blue, green, red, and yellow, and you have gold, silver, or black ink. How many different color combinations of paper and ink do you have to choose from? Use a tree diagram or an array of ordered pairs to match each color of paper with each color of ink. Blue Green Red Yellow Gold Silver Black (B, G) (B, S) (B, Bk) (GR, G) (GR, S) (GR, Bk) (R, G) (R, S) (R, Bk) (Y, G) (Y, S) (Y, Bk) 4 x 3 = 12 combinations Using Models and Sets to Define Multiplication Multiplication by joining segments of equal length on a number line Number of segments being joined 4 x 3 = 12 Length of one segment Properties of Multiplication of Whole Numbers Closure property For whole numbers a and b, a x b is a unique whole number Identity property There exists a unique whole number, 1, such that 1 x a = a x 1 = a for every whole number a. Thus 1 is the multiplicative identity element. Commutative property For whole numbers a and b, a x b = b x a Associative property For whole numbers a, b, and c, (a x b) x c = a x (b x c) Zero property For each whole number a, a x 0 = 0 x a = 0 Distributive property of multiplication over addition For whole numbers a, b, and c, a x (b + c) = (a x b) + (a x c) Suppose you do not know the fact 9 X 12. A. How can you use other known facts to figure out the answer? B. Find as many different ways as possible and explain why your way works. Models of Division • Think of a division problem you might give to a fourth grader. Modeling Division (continued) How many in each group (subset)? There is a total of 12 cookies. You want to give cookies to 3 people. How many cookies can each person get? This is the Sharing interpretation of division. Models of Division How many groups (subsets)? You have a total of 12 cookies, and want to put 3 cookies in each bag. How many bags can you fill? This is the Repeated Subtraction or Measurement interpretation of Division. Division as the Inverse of Multiplication Factor Factor Product 9 x 8 = 72 Product Factor Factor 72÷8 = 9 So the answer to the division equation, 9, is one of the factors in the related multiplication equation. This relationship suggest the following definition: Definition of Division • In the division of whole numbers a and b (b≠0): a ÷ b = c if and only if c is a unique whole number such that c x b = a. In the equation, a ÷ b = c, a is the dividend, b is the divisor, and c is the quotient. Division as Finding the Missing Factor When asked to find the quotient 36 ÷ 3 =? You can turn it into a multiplication problem: ?x 3 = 36 Think of 36 as the product and 3 as one of the factors Then ask, What factor multiplied by 3 gives the product 36 ? Does the Closure, Identity, Commutative, Associative, Zero, and Distributive Properties hold for Division as they do for Multiplication? Division does not have the same properties as multiplication Division by 0 a. Is 0 divided by a number defined? (i.e. 0/4) b. Is a number divided by 0 defined? (i.e. 5/0) Explain your reasoning. Why Division by Zero is Undefined When you look at division as finding the missing factor it helps to give understanding why zero cannot be used as a divisor. 3 ÷ 0 = ?No number multiplied by 0 gives 3. There is no solution! 0 ÷ 0 = ?Any number multiplied by 0 gives 0. There are infinite solutions! Thus, in both cases 0 cannot be used as a divisor. However, 0 ÷ 3 = ? has the answer 0. 3 x 0 = 0 Section 2.4 Numeration Mathematics for Elementary School Teachers - 4th Edition O’DAFFER, CHARLES, COONEY, DOSSEY, SCHIELACK A symbol is different from what it represents The word symbol for cat is different than the actual cat Numeration Systems Just as the written symbol 2 is not itself a number. The written symbol, 2, that represents a number is called a numeral. Here is another familiar numeral (or name) for the number two Definition of Numeration System An accepted collection of properties and symbols that enables people to systematically write numerals to represent numbers. (p. 106, text) Egyptian Numeration System Babylonian Numeration System Roman Numeration System Mayan Numeration System Hindu-Arabic Numeration System Hindu-Arabic Numeration System • Developed by Indian and Arabic cultures • It is our most familiar example of a numeration system • Group by tens: base ten system •10 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 • Place value - Yes! The value of the digit is determined by its position in a numeral •Uses a zero in its numeration system Definition of Place Value In a numeration system with place value, the position of a symbol in a numeral determines that symbol’s value in that particular numeral. For example, in the Hindu-Arabic numeral 220, the first 2 represents two hundred and the second 2 represents twenty. Models of Base-Ten Place Value Base-Ten Blocks - proportional model for place value Thousands cube, Hundreds square, Tens stick, Ones cube or block, flat, long, unit text, p. 110 2,345 Expanded Notation: This is a way of writing numbers to show place value, by multiplying each digit in the numeral by its matching place value. Example (using base 10): 1324 = (1×103) + (3×102) + (2×101) + (4×100) or 1324 = (1×1000) + (3×100) + (2×10) + (4×1) Expressing Numerals with Different Bases: Show why the quantity of tiles shown can be expressed as (a) 27 in base ten and (b)102 in base five, written 102five we can group (a) form groups of 10 these tiles into two 27 groups of ten with 7 tiles left over (b) form groups of 5 102five we can group these tiles into groups of 5 and have enough of these groups of 5 to make one larger group of 5 fives, with 2 tiles left over. No group of 5 is left over, so we need to use a 0 in that position in the numeral: Expressing Numerals with Different Bases: Find the base-ten representation for 1324five 1324five = (1×53) + (3×52) + (2×51) + (4×50) = 1(125) + 3(25) + 2(5) + 4(1) = 125 + 75 + 10 + 4 = 214ten Find the base-ten representation for 344six Find the base-ten representation for 110011two Expressing Numerals with Different Bases: Find the representation of the number 256 in base six 256 - 216 40 -36 4 1(63) + 1(62) + 0(61) + 4(60) 1(216) + 1(36) + 0(6) + 4(1) = 1104six 0 6 =1 1 6 =6 62 = 36 3 6 = 216 64 = 1296 Roman Numeration System Developed between 500 B.C.E and 100 C.E. •Group partially by fives •Would need to add new symbols •Position indicates when to add or subtract •No use of zero Write the Hindu-Arabic numerals for the numbers represented by the Roman Numerals: ⅭⅯⅩⅭⅼⅩ 900 + 90 + 9 = 999 ⅼ (one) Ⅴ (five) Ⅹ (ten) Ⅼ (fifty) Ⅽ (one hundred) Ⅾ (five hundred) Ⅿ(one thousand) Egyptian Numeration System Developed: 3400 B.C.E reed Group by tens New symbols would be needed as system grows No place value heel bone One Ten coiled rope One Hundred lotus flower One Thousand bent finger Ten Thousand burbot fish One Hundred Thousand No use of zero kneeling figure or astonished man One Million Babylonian Numeration System Developed between 3000 and 2000 B.C.E There are two symbols in the Babylonian Numeration System Base 60 Place value one ten Zero came later Write the Hindu-Arabic numerals for the numbers represented by the following numerals from the Babylonian system: 1 42(60 ) + 0 34(60 ) = 2520 + 34 = 2,554 Mayan Numeration System Developed between 300 C.E and 900 C.E •Base - mostly by 20 •Number of symbols: 3 •Place value - vertical •Use of Zero Symbols =0 =1 =5 Write the Hindu-Arabic numerals for the numbers represented by the following numerals from the Mayan system: 8(20 ×18) = 2880 6(201) = 120 0(200) = 0 2880 + 120 + 0 = 3000 Summary of Numeration System Characteristics System Grouping Symbols Place Value Use of Zero No No Egyptian By tens Infinitely many possibly needed Babylonia n By sixties Two Yes Not at first Roman Partially by fives Infinitely many possibly needed Position indicates when to add or subtract No Mayan Mostly by twenties Three Yes, Vertically Yes By tens Ten Yes Yes Hindu- The End Chapter 2