Section 2.1 Sets and Whole Numbers Mathematics for Elementary School Teachers - 4th Edition...

Section 2.1

Sets and Whole Numbers

Mathematics for Elementary School Teachers - 4th EditionO’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 2.1A set is a collection of objectsor ideas that can be listed or

described

A = {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

A = oi

eua C =

BlueRed

Yellow

Each object in the set is called an element of the set

C = BlueRed

YellowBlue is an element of set C

Blue C

Orange is not an element of set C

Orange 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.

Set A

1

2

3

Set B

c

b

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

Set A

onetwo

three

Set B

FrogCat

Dog

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 EditionO’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 AdditionNumber Line Model - joining two continuous

lengths

5 + 4 = 9

Properties of Addition of Whole Numbers

Closure PropertyFor whole numbers a and b, a + b is a unique

whole number

Identity 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 + a

Associative PropertyFor whole numbers a, b, and c, (a + b) + c =

a + (b + c)

Modeling Subtraction Taking away a subset of a set.o 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.o A student had 12 letters. 7 of them had stamps. How many letters did not have stamps?

Comparing two sets of discrete objects.o 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)o 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?

1.Closure

2.Identity

3.Commutative

4.Associative

Properties of Addition of Whole Numbers

Closure PropertyFor whole numbers a and b, a + b is a unique

whole number

Identity 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 + a

Associative PropertyFor 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 EditionO’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”

Multiplication - joining equivalent sets

3 sets with 2 objects in each set3 x 2 = 6 or 2 + 2 + 2 = 6

Repeated Addition

Multiplication using a rectangular array

3 rows2 in each row

3 x 2 = 6

Using Models and Sets to Define Multiplication

Multiplication using the Area of a Rectangle

width

lengthArea model of a polygon

Can be a continuous region

Using Models and Sets to Define Multiplication

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

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 Invitations

GoldGold SilverSilver BlackBlackBlueBlue (B, G) (B, S) (B, Bk)

GreenGreen (GR, G) (GR, S) (GR, Bk)

RedRed (R, G) (R, S) (R, Bk)

YellowYellow (Y, G) (Y, S) (Y, Bk)

4 x 3 = 12 combinations

Multiplication by joining segments of equal length on

a number line

4 x 3 = 12

Length of one

segment

Number of segments

being joined

Using Models and Sets to Define Multiplication

Properties of Multiplication of Whole Numbers

Closure propertyFor whole numbers a and b, a x b is a unique whole number

Identity propertyThere 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 propertyFor whole numbers a and b, a x b = b x a

Associative propertyFor whole numbers a, b, and c, (a x b) x c = a x (b x c)

Zero propertyFor each whole number a, a x 0 = 0 x a = 0

Distributive property of multiplication over additionFor 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)

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 Division

This 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 Multiplication

Factor Factor Product

9 x 8 = 72

÷72 8 = 9

Product Factor Factor

This 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 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

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

What factor multiplied by 3 gives the product 36 ？

When 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 multiplication

Does the Closure, Identity, Commutative, Associative, Zero, and Distributive

Properties hold for Division as they do for 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.

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

Why Division by Zero is Undefined

Section 2.4

Numeration

Mathematics for Elementary School Teachers - 4th EditionO’DAFFER, CHARLES, COONEY, DOSSEY, SCHIELACK

The word symbol for cat is different than the actual cat

A symbol is different from what it represents

Here is another familiar numeral (or

name) for the number two

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.

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

Egyptian Numeration System

Babylonian Numeration System

Roman Numeration System

Mayan 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 ValueIn 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

orblock, flat, long, unit

text, p. 110

2,345

Expanded Notation:

1324 = (1×1000) + (3×100) + (2×10) + (4×1)

1324 = (1×103) + (3×102) + (2×101) + (4×100)

Example (using base 10):

or

This 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 5 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.

27

No group of 5 is left over, so we need to use a 0 in that position in

the numeral: 102five

102five

Find the base-ten representation for 1324five

Find the base-ten representation for 344six

Find the base-ten representation for 110011two

= 1(125) + 3(25) + 2(5) + 4(1)

1324five = (1×53) + (3×52) + (2×51) + (4×50)

= 125 + 75 + 10 + 4= 214ten

Expressing Numerals with Different Bases:

Find the representation of the number 256 in base six

64 = 129663 = 21662 = 36

60 = 161 = 6

256- 216

40-36

4

1(216) + 1(36) + 0(6) + 4(1) = 1104six

1(63) + 1(62) + 0(61) + 4(60)

Expressing Numerals with Different Bases:

Roman Numeration SystemDeveloped between 500 B.C.E and 100 C.E.

ⅬⅭⅮⅯ

Ⅹ

ⅼⅤ

(one)

(five)

(ten)

(fifty)

(one hundred)

(five hundred)(one thousand)

•Group partially by fives•Would need to add new symbols

•Position indicates when to add or

subtract•No use of zero

Ⅽ Ⅿ Ⅹ Ⅽ ⅼ Ⅹ

900 + 90 + 9 = 999

Write the Hindu-Arabic numerals for the numbers represented by the Roman

Numerals:

Egyptian Numeration SystemDeveloped: 3400 B.C.E

One

Ten

One Hundred

One Thousand

Ten Thousand

One Hundred Thousand

One Million

reed

heel bone

coiled rope

lotus flower

bent finger

burbot fish

kneeling figureor

astonished man

Group by tens

New symbols would be needed as system grows

No place value

No use of zero

Babylonian Numeration SystemDeveloped between 3000 and 2000 B.C.E

There are two symbols in the Babylonian Numeration System

Base 60Place value one ten

42(601) + 34(600) = 2520 + 34 = 2,554

Zero came later

Write 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.E

•Base - mostly by 20•Number of symbols: 3•Place value - vertical•Use of Zero

Symbols

= 1= 5

= 0

Write the Hindu-Arabic numerals for the numbers represented by the following numerals from the

Mayan system:

0(200) = 0

6(201) = 120

8(20 ×18) = 2880

2880 + 120 + 0 = 3000

Summary of Numeration System Characteristics

SystemSystem GroupinGroupingg

SymbolsSymbols Place Place ValueValue

Use of Use of ZeroZero

EgyptiaEgyptiann

By tens

Infinitely many

possibly needed

No No

BabyloniBabylonianan

By sixties Two Yes Not at first

RomanRoman Partiallyby fives

Infinitely many

possibly needed

Position indicates

when to add or subtract

No

MayanMayan Mostlyby twenties

ThreeYes,

VerticallyYes

Hindu-Hindu-ArabicArabic

By tens Ten Yes Yes

The EndChapter 2