Math for Elementary Teachers Chapter 2 Sets Whole Numbers, and Numeration

Sets as a Basis for Whole Numbers Set – a collection of objects A verbal description A listing of the members separated by commas or With braces {} Set-builder notation Elements(members) – objects in a set.

Sets Sets are denoted by capital letters – A,B,C . indicates that an object is an element of a set . indicates that an object is NOT an element of a set . empty set (or null set) a set without elements.

Set Examples Verbal – the set of states that border the Pacific Ocean Listing A:{Alaska, California, Hawaii, Oregon, Washington} .Oregon A .New York A .The Set of all States bordering Iraq .

More on Sets Two sets are equal ( A=B) if and only if they have precisely the same elements Two sets, A and B, are equial if every elements of A is in B, and vice versa If A does not equal B then A B

Rules regarding Sets The same element is not listed more than once within a set the order of the elements in a set is immaterial.

One-to-One Correspondence Definition: A 1-1 correspondence between two sets A and B is a pairing of the elements of A with the elements of B so that each element of A corresponds to exactly one element of B, and vice versa. If there is a 1-1 correspondence between sets A and B, we write A~B and say that A and B are equivalent or match.

One-to-One Correspondence Four possible 1-1 Equal sets are always equivalent BUT equivalent sets are not necessarily equal {1,2}~{a,b} BUT {1,2} {a,b}.

Subset of a Set: A B Definition: Set A is said to be a subset of B, written A B, if and only if every element of A is also an element of B.

Subset examples: Vermont is a subset of the set of all New England states .

Subset examples continued If and B has an element that is not in A, we write and say that A is a proper subset of B Thus , since and c is in the second set but not in the first.

Venn Diagrams U = universe Disjoint Sets – Sets A and B have no elements in common Sets {a,b,c} and {d,e,f} are disjoint Sets {x,y} and {y,z} have y in common and are not disjoint.

Union of Sets: Definition: The union of two sets A and B, written is the set that consists of all elements belonging either to a or to b (or to both).

Union of Sets: . The notion of set union is the basis for the addition of whole numbers, but only when disjoint sets are used 2+3=5 .

Intersection of Sets: Definition: The intersection of sets A and B, written is the set of all elements common to sets A and B.

Complement of a Set: Definition: The complement of a set A, Written ,is the set of all elements in the universe, U, that are not in A.

Difference of Sets: A-B Definition: The set difference (or relative complement) a set B from set A, written A-B, is the set of all elements in A that are not in B.

Whole numbers and numeration Section 2.2 Whole numbers and numeration

Numbers and Numerals The study of the set of whole numbers W={0,1,2,3,4…} is the foundation of elementary school mathematics A number is an idea, or an abstractions, that represents a quantity. The symbols that we see, srite or touch when representing numbers are called numerals.

Three uses of whole numbers Cardinal number – whole numbers used to describe how many elements are in a finite set Ordinal numbers - concerned with order e.g. your team is in fourth place Identification numbers – used to name things – credit card, telephone number, etc it’s a symbol for something.

The symbol n(A) is used to represent the number of elements in a finite set A. n({a,b,c})=3 n({a,b,c,…,z})=26.

Ordering Whole Numbers (1-1 correspondences) Definition: Ordering Whole Numbers: Let a=n(A) and b=n(B) then a<b (read a is less than b) or b>a (b is greater than a) if A is equivalent to a proper subset of B.

Problem: determine which is greater 3 or 8 in three different ways Counting chant – one, two, three, etc Set Method – a set with three elements can be matched with a proper subset of a set with eight elements 3<8 and 8>3.

Problem: determine which is greater 3 or 8 in three different ways (cont) Whole-Number Line – since 3 is to the left of 8 on the number line, 3 is less than 8 and 8 is greater than 3.

Numeration Systems Tally numeration system – single strokes, one for each object counted. Improved with grouping.

The Egyptian Numeration System developed around 3400 B.C invovles grouping by ten. =? 321.

The Roman Numeration System Developed between 500 B.C. and A.D. 100 The values are found by adding the values of the various basic numerals MCVIII is 1000+100+5+1+1+1=1108 New elements Subtractive principle Multiplicative principle.

Subtractive system Permits simplifications using combinations of basic numbers IV – take one from five instead of IIII The value of the pair is the value of the larger less the value of the smaller.

Multiplicative System Utilizes a horizontal bar above a numeral to represent 1000 times the number Then means 5 times 1000 or 5000 and is 1100 System still needs many more symbols than current system and is cumbersome for doing arithmetic.

The Babylonian Numeration System Evolved between 3000 and 2000 B.C. Used only two numerals, one and ten for numbers up to 59 system was simply additive Introduced the notion of place value – symbols have different values depending on the place they are written.

Sections 2.3 The Hindu-Arabic System Digits 0,1,2,3,4,5,6,7,8,9 – 10 digits can be used in combination to represent all possible numbers Grouping by tens (decimal system) known as the base of the system – Arabic is a base ten system Place value (positional) Each of the various places in the number has it’s own value.

Models for multi digit numbers Bundles of Sticks – each ten sticks bound together with a band Base ten pieces (Dienes blocks) individual cubes grouped in tens.

The Hindu-Arabic System Additive and multiplicative The value of a Hindu-Arabic numeral is found by multiplying each place value by its corresponding digit and then adding all of the resulting products. Place values: thousand hundred ten one Digits 6 5 2 3 Numeral value 6x1000 + 5x100 + 2x10 + 3x1 Numeral 6523.

Observations about the naming procedure The number 0,1,…12 all have unique names The numbers 13,14, …19 are the “teens” The numbers 20,…99 are combinations of earlier names but reversed from the teens in that the tens place is named first e.g. 57 is “fifty-seven The number 100, … 999 are combinations of hundreds and previous names e.g. 637 reads “six hundred thirty-seven” In numerals containing more than three digits, groups of three digits are usually set off by commas e.g. 123,456,789 .

Learning Three distinct ideas that children need to learn to understand the Hindu-Arabic numeration system .

Base 5 operations We can express numeration systems as base systems The number 18 in Hindu-Arabic can be stated as 18ten 18 base ten To study a system with only five digits (0,1,2,3,4) we would call that a base 5 system e.g. base five 37five .