Math for Elementary Teachers

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

2. 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.

3. 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.

4. 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 .

5. 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

6. 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.

7. 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.

8. 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}.

9. 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.

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

11. 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.

12. 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.

13. 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).

14. 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 .

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

16. 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.

17. 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.

18. Section 2.2 Whole numbers and numeration

19. 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.

20. 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.

21. 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.

22. 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.

23. 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.

24. 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.

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

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

27. 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.

28. 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.

29. 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.

30. 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.

31. 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.

32. 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.

33. 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.

34. 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 .

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

36. 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 .