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Section 2.1Sets and Whole NumbersMathematics for Elementary School Teachers - 4th EditionODAFFER, CHARLES, COONEY, DOSSEY, SCHIELACK

How do you think the idea of numbers developed?How could a child who doesnt 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 2.1A set is a collection of objectsor ideas that can be listed or describedA = {a, e, i, o, u}C = {Blue, Red, Yellow}A set is usually listed with a capital letterA set can be represented using braces { }A set can also be represented using a circle

Each object in the set is called an element of the setBlue is an element of set COrange is not an element of set C

Definition of a One-to-One CorrespondenceSets 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.The order of the elements does not matter

Definition of Equivalent SetsSets A and B are equivalent sets if and only if there is a one-to-one correspondence between A and B

Finite SetA set with a limited number of elementsExample: A = {Dog, Cat, Fish, Frog}

Section 2.2Addition and Subtraction of Whole NumbersMathematics for Elementary School Teachers - 4th EditionODAFFER, CHARLES, COONEY, DOSSEY, SCHIELACK

Using Models to Provide an Intuitive Understanding of AdditionJoining two groups of discrete objects3 books + 4 books = 7 books

Using Models to Provide an Intuitive Understanding of AdditionNumber Line Model - joining two continuous lengths5 + 4 = 9

Properties of Addition of Whole NumbersClosure PropertyFor whole numbers a and b, a + b is a unique whole numberIdentity PropertyThere 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 PropertyFor whole numbers a and b, a + b = b + aAssociative PropertyFor whole numbers a, b, and c, (a + b) + c = a + (b + c)

Modeling SubtractionTaking 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?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 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 NumbersIn 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.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.Restating the definition substituting whole numbers:

Comparing Addition and Subtraction Properties of Whole Numbers

Which of the properties of addition hold for subtraction?ClosureIdentityCommutativeAssociative

Properties of Addition of Whole NumbersClosure PropertyFor whole numbers a and b, a + b is a unique whole numberIdentity PropertyThere 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 PropertyFor whole numbers a and b, a + b = b + aAssociative PropertyFor whole numbers a, b, and c, (a + b) + c = a + (b + c)

Section 2.3Multiplication and Division of Whole NumbersMathematics for Elementary School Teachers - 4th EditionODAFFER, 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 Beans Amazing Dream

Using Models and Sets to Define Multiplication

Multiplication using the Area of a RectangleUsing Models and Sets to Define Multiplication

Definition of Cartesian ProductThe 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

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.Problem Solving: Color Combinations for Invitations4 x 3 = 12 combinations

GoldSilverBlackBlue(B, G)(B, S)(B, Bk)Green(GR, G)(GR, S)(GR, Bk)Red(R, G)(R, S)(R, Bk)Yellow(Y, G)(Y, S)(Y, Bk)

Multiplication by joining segments of equal length on a number line4 x 3 = 12Using Models and Sets to Define Multiplication

Properties of Multiplication of Whole NumbersClosure propertyFor whole numbers a and b, a x b is a unique whole number

Suppose you do not know the fact 9 X 12.

How can you use other known facts to figure out the answer?

Find as many different ways as possible and explain why your way works.

Models of DivisionThink of a division problem you might give to a fourth grader.

Modeling Division (continued)This is the Sharing interpretation of division.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?

Models of DivisionThis is the Repeated Subtraction or Measurement interpretation of Division. You have a total of 12 cookies, and want to put 3 cookies in each bag. How many bags can you fill?How many groups (subsets)?

Division as the Inverse of MultiplicationThis relationship suggest the following definition:So the answer to the division equation, 9, is one of the factors in the related multiplication equation.

Definition of DivisionIn the division of whole numbers a and b (b0): 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

Think of 36 as the product and 3 as one of the factors

What factor multiplied by 3 gives the product 36When asked to find the quotient 36 3 =You can turn it into a multiplication problem:

x 3 = 36

Then ask,

Division does not have the same properties as multiplicationDoes the Closure, Identity, Commutative, Associative, Zero, and Distributive Properties hold for Division as they do for Multiplication?

Division by 0a. 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.

When you look at division as finding the missing factor it helps to give understanding why zero cannot be used as a divisor.There is no solution!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 = 0Why Division by Zero is Undefined

Section 2.4NumerationMathematics for Elementary School Teachers - 4th EditionODAFFER, CHARLES, COONEY, DOSSEY, SCHIELACK

The word symbol for cat is different than the actual catA symbol is different from what it represents

Here is another familiar numeral (or name) for the number twoNumeration SystemsJust as the written symbol 2 is not itself a number.The written symbol, 2, that represents a number is called a numeral.

Definition of Numeration SystemAn accepted collection of properties and symbols that enables people to systematically write numerals to represent numbers. (p. 106, text)

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 system10 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 numeralUses a zero in its numeration system

Definition of Place ValueIn a numeration system with place value, the position of a symbol in a numeral determines that symbols 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 ValueBase-Ten Blocks - proportional model for place valueThousands cube, Hundreds square, Tens stick, Ones cubeorblock, flat, long, unittext, p. 1102,345

Expanded Notation:1324 = (11000) + (3100) + (210) + (41)1324 = (1103) + (3102) + (2101) + (4100)Example (using base 10):orThis is a way of writing numbers to show place value, by multiplying each digit in the numeral by its matching place value.

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(a) form groups of 10we can group these tiles into two groups of ten with 7 tiles left over(b) form groups of 5we 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.27No group of 5 is left over, so we need to use a 0 in that position in the numeral: 102five102five

Find the base-ten representation for 1324fiveFind the base-ten representation for 344sixFind the base-ten representation for 110011twoExpressing Numerals with Different Bases:

Find the representation of the number 256 in base six1(216) + 1(36) + 0(6) + 4(1) = 1104six1(63) + 1(62) + 0(61) + 4(60)Expressing Numerals with Different Bases:

Roman Numeration SystemDeveloped between 500 B.C.E and 100 C.E.Group partially by fivesWould need to add new symbolsPosition indicates when to add or subtractNo use of zero900 + 90 + 9 = 999Write the Hindu-Arabic numerals for the numbers represented by the Roman Numerals:

Egyptian Numeration SystemDeveloped: 3400 B.C.EGroup by tensNew symbols would be needed as system growsNo place valueNo use of zero

Babylonian Numeration SystemDeveloped between 3000 and 2000 B.C.EThere are two symbols in the Babylonian Numeration SystemBase 60Place value42(601) + 34(600) = 2520 + 34 = 2,554Zero came laterWrite the Hindu-Arabic numerals for the numbers represented by the following numerals from the Babylonian system:

Mayan Numeration SystemDeveloped between 300 C.E and 900 C.EBase - mostly by 20Number of symbols: 3Place value - verticalUse of ZeroWrite the Hindu-Arabic numerals for the numbers represented by the following numerals from the Mayan system:0(200) = 06(201) = 1208(20 18) = 28802880 + 120 + 0 = 3000

Summary of Numeration System Characteristics

SystemGroupingSymbolsPlace ValueUse of ZeroEgyptianBy tensInfinitely many possibly neededNoNoBabylonianBy sixtiesTwoYesNot at firstRomanPartiallyby fivesInfinitely many possibly neededPosition indicates when to add or subtractNoMayanMostlyby twentiesThreeYes, VerticallyYesHindu-ArabicBy tensTenYesYes

The EndChapter 2