Algebra for Middle School Teachers

Homework exercises have been liberally chosen from the following texts.:

 

Principles of Modern Algebra by J. Eldon Whitesitt      

A  Modern Introduction to Basic Mathematics by Mervin L. Keedy     

A First Course in Abstract Algebra by John B. Fraleigh

Our Current Math 103 Textbook

 

The topics selected for coverage are based on the curriculum in ALGEBRA I, published by GLENCOE and are specifically related to the SOL’s for the State of Virginia.

 

There are six laboratories scheduled during the course.  Each laboratory will involve hands on tools and demonstration of how the topics of the course arise in the middle school curriculum and how they are taught.  A seventh laboratory may be scheduled if time permits.

 

I           Algebraic Structures (2 weeks)

·        Definitions:  Binary Operation, Associative property, commutative property, identity property, inverse property, distributive property

·        Counting Numbers

·        Whole Numbers

·        Integers

HW:

q       Is the operation subtraction closed when considered as an operation

a)      on the set of all integers?

b)      on the set of positive integers?

c)      on the set of all integers which are divisible by three?    

d)      In each case explain your answer.

q       Is the operation of subtraction on the set of all integers (a) commutative? (b) associative? In each case explain your answer.

q       Show that division by non-zero elements is right distributive, but not left distributive over addition in the set of rational numbers.

q       For the set of integers, define the operation * as follows:  a*b = a + b – ab where and b are any integers and + and -  are the usual addition and subtraction and ab is the usual product of integers.  Find the identity element in the set relative to this operation.  Let n be an arbitrary integer and find the inverse of n relative to the operation *.

 

·        Rational Numbers

HW:

q       For the set of rational numbers, we define two binary operations $ and # (in terms of ordinary arithmetic operations) as follows: a $ b = 2ab and a#b = a + 2b for any two rational numbers a and b. 

a)   Is $ commutative?  If so, explain your answer and if not, give a counter example. 

b)   Is # commutative?  If so, explain your answer and if not, give a counter example.

c)   Prove or disprove that $ is associative.

d)   Prove or disprove that # is associative.

e)   Prove or disprove that $ is left distributive over #.

f)     Prove or disprove that # is left distributive over $.

 

·        Matrices

 

HW:

q       Given the matrices  find (a) A+B, (b) AB, (c) BA, (d) 5A, (e) A-B, (f) B-A.

q       Given the matrices  find (a) X(YZ), (b) (XY)Z, (c) X(Y+Z), (d) XY + XZ, (e) (X+Y)Z, (f) XZ + YZ

q       Find a nonzero divisor of zero in M the set of all 2x2 matrices with integer entries.

q       Let E be the set of all English words and let S be the set of all “letter strings” [finite lists of letters, possibly repeated] of our alphabet.  Notice that E is a subset of S.  In each instance below a process *  that makes sense on  each of these two sets is described.  [The Greek letters a and b represent arbitrary elements of these sets.]  Answer each of the following questions for each exercise:

1.   Give two more examples to show how * works.

2.   Is * a binary operation on S? on E?  Justify each answer with a reason or counterexample.

3.   If * is a binary operation on the set, say whether * is associative and/or commutative, supporting your answers with reasons or counterexamples.

a)      a*b is formed by putting a and b next to each other, forming a single string.  For example, moon*glow = moonglow.

[This is like the password process used by CompuServe and other electronic services.]

b)      a*b is the word or string that in alphabetical order comes first.  For example, trip*trap = trap.

      [This is a basic part of any word-sorting algorithm.]

c)      a*b is the string of letters, including repeated letters, that are common to a and b, arranged in alphabetical order.  For example, beaker*knee = eek.

d)      a*b is the number of letters in the longer word or string, or if they have the same number of letters, it’s that number.  For example, movie*theater = 7.

 

 

Connection to lessons in ALGEBRA 1 (Glencoe): 1-6, 1-8, 5-1.  Here will be an opportunity to identify the whole numbers, integers, rational numbers and matrices as distinct algebraic structures by virtue of the properties of the binary operations in each structure.

 

Laboratory 1:

Ø      Venn Diagrams

Ø      Integer card game

Ø      Use of counters to show +,- and ´ for the integers.

Ø      Modeling the distributive property

Ø      Fold over proofs

Ø      Matching game

Ø      SOL questions

 

II          Groups (4 weeks)

·        definitions and examples including Zn (specifically Z6, Z7) and symmetric groups

·        Theorems

1.      For any elements a, x and y in a group G, if a¨x = a¨y then x = y.

2.      For any elements a, x and y in a group G, if x¨a = y¨a then x = y.

3.      For any elements a and b in a group G, the equation a¨x = b has one and only one solution (for x) in G.

4.      For any elements a and b in a group G, the equation x¨a = b has one and only one solution (for x) in G.

5.      In any group G, there is only one identity.

6.      In any group G, for each element x in G, there is only one inverse element

HW:

q       Given the permutations  find (a) ab, (b) ba, (c) a -1, (d) (ab)g, (e) a(bg), (f) the solution to ax = b, (g) the solution to bx = g.

q       Write out each element of S3.  Label the identity as r1 and each of the other elements as r2, r2, r3, r4, r5, r6.  Fill out the complete “Multiplication Table” for S3 using your elements as labeled in the previous problem

q       Consider the Subset A3 of S3 as follows: .  Show that A3 is closed for the operation of multiplication and that each element in A3 has a multiplicative inverse in A3.

q       Prove Theorems 2,4 and 6

q       The converse of theorem 1 is: For any elements a,x and y in a group G, if x = y then a¨x = a¨y.

a)      Does this require proof as a result in group theory? Why or why not? (Hint:  consider the definition of a  binary operation and use this in your discussion.)

b)      Euclid stated this same idea by saying “If equals are multiplied by equals, the results are equal.”  Discuss the appropriateness of saying “multiply both sides of an equation by the same thing” or “multiply one equation by another” in the context of Group theory.

q       Discuss what each of theorems 1-4 says about elements of (a) the integers and (b) the rational numbers in the context of the kinds of equations with (a) integer and (b) rational number coefficients that are guaranteed to have unique solutions for x.

 

·        Theorems

7.      In any group G, the identity is its own inverse.

8.      For any elements x and y in a group G, (x¨y) -1 = y -1 ¨a –1.

9.      For any element x in a group G, (x –1) –1 = x.

10.  Let a and b be elements of an arbitrary group G, and let m be a natural number.  Then, for any natural number n,

a)      en = e where e is the identity of G.

b)      For any a and b in G, (ab)n = anbn if an only if ab=ba.

c)      am+n = am an.

d)       (am)n = amn = (an)m.

·        Definitions: order of a group, order of an element

·        Cayley tables

HW:

q       Factor 3 in two different ways in Z6.  Can 5 be factored into factors in Z6 other than by using 1 as a factor?

q       Find two equations of the form ax = b, with a and b in Z6 and with b ¹ 1, which cannot be solved in Z6.

q       Let the following table define a binary operation on the set {2,4,6,8}

*

2

4

6

8

2

4

8

2

6

4

8

6

4

2

6

2

4

6

8

8

6

2

8

4

Answer each exercise below in reference to  *

a)      Determine 4*8

b)      Which element, if any, is the identity?

c)      Which element, if any, is the inverse of 8?

q       Each table below defines a binary operation on the indicated set..

ª

p

q

r

s

t

 

à

0

1

2

3

4

p

s

r

t

p

q

 

0

0

1

2

3

4

q

t

s

p

q

r

 

1

1

2

3

4

0

r

q

t

s

r

p

 

2

2

3

4

0

1

s

p

q

r

s

t

 

3

3

4

0

1

2

t

r

p

q

t

s

 

4

4

0

1

2

3

 

      Answer each exercise below with reference to the appropriate table.

a)      Calculate (qªr)ªp and 1à(3à2).

b)      Is there an identity for à?  If so what is it?

c)      Does q have an inverse?  If so, what is t?  If not, why not?

d)      Does 1 have an inverse?  If so, what is t?  If not, why not?

e)      Solve for x: 2àx = 3.

q       Fill in the table below so that you create a binary operation on the set {a,b,c,d} with the following three properties:

                                                   i.            c is the identity element,

                                                 ii.            b is the inverse of d, and

                                                iii.            the operation is commutative.

*

a