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 Z_n (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) a^m+n = a^m aⁿ.

d) (a^m)ⁿ = a^mn = (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	b	c	d
a
b
c
d

q Prove Theorems 7,9 and 10

· Subgroups

· Theorems

11. If H is any subgroup of a group G, then the identity element of G is in H and is the identity element of H.

12. A subset H of a Group G is a subgroup if and only if:

a) H is not the empty set.

b) For every pair of elements a and b in H, the product ab ^–1 is an element of H.

13. If a is an element of a group G, the set H = {a^k| k is any integer} is a subgroup of G.

14. If a cyclic group G with generator a has order n, then aⁿ = e and the distinct elements of G are the elements {a, a²,a³, …, aⁿ = e}.

HW:

q Prove that the set of even integers is a subgroup of the additive group of the integers.

q Find all subgroups of the symmetric group S₃.

q Find the order of each element in each group S₃ and Z₇ under addition.

q If H and K are subgroups of G, prove that HÇK is also a subgroup of G.

· Definitions: set multiplication, one to one correspondence, partition

· cosets

· Theorems

15. If H is a subgroup of the group G and if a and b are elements of G, the following conditions are equivalent.

a) aÎ Hb b) Ha = Hb. c) ab ^-1ÎH

16. If H is a subgroup of G, the right cosets of H in G form a partition of G.

17. (LaGrange) If H is a subgroup of a finite group G, then the order of H is a divisor of the order of G.

HW:

q Prove theorems 15 and 16 for left cosets.

q Let G be the additive group Z₂₄. Let H be the subgroup H = { 0,6,12,18}. Determine whether each of the following statements id true or false.

a) 7º11 mod H b) 10º 5 mod H c) 13º19 mod H d) 3º15 mod H

q Let G and H be as above. For each of the following congruences, find three distinct values of x for which the congruence is true.

a) 7ºx mod H b) xº17 mod H c) xº12 mod H d) 10ºx mod H

q Consider the following information concerning groups W, X, Y, and Z.

Group W contains 22 elements.

Group X contains 23 elements.

Group Y contains 24 elemetns.

Group Z contains 25 elements.

Use LaGrange’s theorem to answer the following questions.

a) Which group has the most different sizes of possible subgroups, and which has the fewest?

b) What sizes of subgroups are possible in group W ? in Group Z ?

q Suppose G is a 50 element group.

a) What are the possible sizes for subgroups of G?

b) For each possible size, assume there is a subgroup of that size, and specify how many distinct cosets that subgroup has.

c) Answer a) and b) if the group has 55 elements.

Connection to lessons in ALGEBRA 1 (Glencoe): 3-1,3-5,5-6

Laboratory 2:

Ø Hands on Equations (single operation).

Ø Cups and counters

Laboratory 3:

Ø Modular Arithmetic.

III Rings (4 weeks)

· Definitions and examples including Z₆, Z₇

· Theorems

1. For any a,x,y in a ring R, if a+x = a+ y then x = y and if x+a = y+a then x = y.

2. There is only one additive identity in any ring. Each element of a ring has a unique additive inverse.

3. For any a, b in a ring R, the equation a + x = b has one and only one solution in R.

4. In any ring, ^- (x+y) = ^- x + ^- y.

HW:

q Find all solutions to the following equations in Z₇. (a) x – 3 = 6, (b) 5x = 4, (c) x² + 2x + 6 = 0, (d) x³ = 6, (e) 3x + 2 = 6.

q Solve the following equations in the ring of 2x2 matrices over the set of integers and check your solution.

q b)

q

q Explain why theorems 1-3 hold in the context of what you know about groups and the definitions of a ring.

· integral domains

· fields

· Theorems

5. There is only one multiplicative identity in any field. Each non-zero element of a field has a unique multiplicative inverse.

6. For any nonzero a in a field F and any elements x and y in F, if ax = ay then x = y and if xa = ya then y = y.

7. For any nonzero a in a field F and any element b in F, the equation ax = b has one and only one solution in F.

8. In any field, if x and y are non-zero then (xy) ^–1 = x ^–1 y ^–1.

9. For any element a in a ring R, a0=0.

10. For any element in a ring R, 0a = 0.

11. In any field F, 0 ¹ 1.

12. In any field F, 0 has no multiplicative inverse.

13. For any elements a,b in a ring R, ^- ab = ^- (ab).

14. For any elements a,b in a ring R, ^- a ^– b = ab.

15. For any nonzero a in a field F and any elements b and c in F, the equation ax + b = c has one an only one solution in F.

16. For all a,b,c,d,e,f in a field F, if ae + ^- (bd) ¹ 0 then the system of equations given by ax + by = c AND dx + ey = f has one and only one solution .

17. For any a, c in a field F and for any nonzero b,d in F, (ab ^–1)(cd ^–1) = (ac)(bd)^–1.

18. Let a and b be elements of an arbitrary ring R, and let m be a natural number. Then, for any natural number n,

a) n0 = 0 (0ÎR is the zero of R) and 1ⁿ = 1 (the latter assuming R has a unity 1).

b) For any a and b in R, n(a + b) = na + nb and (ab)ⁿ = aⁿbⁿ (the latter if an only if ab=ba).

c) (m + n)a = ma + na and a^m+n = a^maⁿ.

d) nm(a) = n(ma) = m(na) and (a^m)ⁿ = a^mn = (aⁿ)^m.

19. Let F be any field, then F is an integral domain.

HW:

q Use arithmetic mod 6 to show that theorem 7 DOES NOT HOLD in every ring.

q Show that 3 and 4 have no multiplicative inverses in the set Z₆.

q Explain why theorems 5-7 hold in the context of what you know about groups and the definitions of a ring.

q Prove Theorems 10, 15, 16 and 17.

q From arithmetic mod 7, find three examples to illustrate each of theorems 13, 14 and 17.

q Use arithmetic mod 6 to show that theorem 15 DOES NOT HOLD in every ring.

q Assume that all of the coefficients in the following are from Z₇ and all operations are modulo 7. Perform the indicated operations and rewrite your answers leaving no minus signs in them.

q (2x – 3)²

q (x – 2)(x + 5)

q Assume that R is the ring Z₆ with operations modulo 6. Compute each of the following for the element a = 3ÎR.

q 3a b) 10a c) a³ d) a⁷ e) (-5)a f) (-10)a

q Assume that R is the ring Z₆ with operations modulo 6. Compute each of the following for the element a = 3ÎR.

q 3a b) 10a c) a³ d) a⁷ e) (-5)a f) (-10)a g) a^-2 h) a^-5

q Assume that R is the ring of 2x2 matrices over Q with the usual matrix operations. Compute each of the following for the element .

q 3a b) 10a c) a³ d) a⁴ e) (-3)a f) (-10)a g) a^-2 h) a^-3

q

· polynomial rings

· Theorems

20. Let R be any ring, then R[x] is a ring.

21. Let R be any ring with nonzero zero divisors, then R[x] also has nonzero zero divisors.

22. If R is an integral domain then R[x] is an integral domain.

23. Let R be an integral domain. If p(x) and q(x) are any polynomials in R[x] then

a) Either p(x) + q(x) = 0 or Deg( p(x)+q(x)) £ Max( Deg(p(x),Deg(q(x))

b) Either p(x)q(x) = 0 or Deg(p(x)q(x)) = Deg(p(x)) + Deg(q(x))

24. Let R be an integral domain. If p(x) and q(x) are any polynomials in R[x] with degree greater than or equal to one then Deg(p(x)) < Deg(p(x)q(x)) and Deg(q(x) < Deg(p(x)q(x)).

HW:

q Prove that the polynomials over each of the rings, the integers, the rational numbers and the real numbers form an integral domain.

q Find the sum and product of each of the polynomials in the given polynomial ring.

q f(x) = 2x² + 3x + 4, g(x) = 3x² + 2x + 3 in Z₆[x].

q f(x) = 2x³+ 4x² + 3x + 2, g(x) = 3x⁴ + 2x + 4 in Z₇[x].

· The division algorithm

· Theorems

25. The Division Algorithm. Let F be a field. For any f(x) in F[x] and for any nonzero g(x) in F[x], there exist elements q(x) and r(x) in F[x] such that f(x) = q(x)g(x) + r(x) where either r(x) is identically zero or else 0£Deg(r(x))<Deg(g(x))

HW:

q Find q(x) and r(x) as described in the division algorithm so that f(x) = q(x) g(x) + r(x) with r(x) either equal to zero or else Deg(r(x)) < Deg(g(x)) in the given polynomial ring.

a) f(x) = x⁶ + 3x⁵ + 4x² – 3x + 2 and g(x) = x² + 2x – 3 in Z₇[x].

b) f(x) = x⁶ + 3x⁵ + 4x² – 3x + 2 and g(x) = 3x² + 2x – 3 in Z₇[x].

Connection to lessons in ALGEBRA 1 (Glencoe): 3-5,3-69—1,9-2,9-5,9-6,9-7,10-1.

Laboratory 4:

Ø Hands on Equations (two operations).

Laboratory 5:

Ø Polynomial models

Ø Algeblocks

IV Factors and solutions to equations (2 weeks)

· the evaluation mapping

· Theorems

1. A polynomial equation f(x) = 0 with f(x) in F[x] has a solution x =a in F if an only if (x-a) is a factor of f(x).

HW:

q Compute the values in Z₇ for each of the following evaluation mappings.

q f₂(x²+ 3) b) f₃((x⁴ + 2x)(x³ – 3x² + 3)) c) f₃(x⁴ + 2x) f₃(x³ – 3x² + 3)

q For each of the following equations in Z_7, either solve the equation or show the equation has no solution in Z₇ using arithmetic modulo 7.

q 2 – x = 5 b) 3x + 2 = 5 c) x² = 1 d) 3x² = 6 e) x³ = 6

q f) x² + 2x + 2 = 0 g) (x – 4)² = 4

q For each of the following equations in Z₅, either solve the equation or show the equation has no solution in Z₅ using arithmetic modulo 5.

q x – 3 = 2 b) 2 – x = 4 c) 3x = 4 d) 2x = 3 e) 4x = 2

q f) x² = 4 g) x² = 3 h) x² + 4x + 2 = 0.

q Find all zeros in the indicated field of the given polynomial with coefficients in that field.

q x² + 1 in Z₂ b) x³ + 2x + 2 in Z₇

q The polynomial x⁴ + 4 can be factored into a product of linear factors in Z₅[x]. Find this factorization.

q The polynomial x³ + 2x² + 2x + 1 can be factored into a product of linear factors in Z₇[x]. Find this factorization.

q Find a polynomial f(x) in Z₇ which has no linear factors. What does this say about the solvability of the equation f(x) = 0?

q Show that x² – 2 = 0 has no solutions in Q.

· LCM and GCD

· Euclid’s Algorithm

· Theorems

2. Let F be a field. The Euclidean Algorithm holds in F[x].

3. Let F be a field. The integral domain F[x] can be extended to a field by constructing, for each nonzero f(x), a multiplicative inverse denoted by and extending the operations of addition and multiplication to these new elements so that the properties of a field will hold. This is called the field of quotients for F[x].

HW:

q Find the g.c.d. of each of the following pairs of polynomials over the field Q of rational numbers.

(a) 2x³ – 4x² + x – 2 and x³ – x² – x – 2

(b) x⁴ + x³ + x² + x + 1 and x³ – 1

(c) x⁵ + x⁴ + 2x³ – x² – x - 2 and x⁴ + 2x³ + 5x² + 4x + 4

(d) x³ – 2x² + x + 4 and x² + x + 1

q Let f(x) = x² – x - 2 and g(x) = 1/(3x² – 9) and h(x) = 3x³ –6x² +3x + 9 be in the field of quotients for Q[x]. Calculate each of the following.

(a) f(x) + g(x)

(b) f(x)g(x)

(c) f(x) + g(x) + f(x)g(x) – h(x)

Connection to lessons in ALGEBRA 1 (Glencoe): 10-4.

Laboratory 6:

Ø Algeblocks and factoring

Ø Developing the sum and product method for factoring a general trinomial

V Graph theory (3 weeks)

· Cayley graphs

· connected graphs

· edge paths

· vertex degree

· terminal vertices and terminal edges

· vertex paths

· polygonal graphs

· crossing curves

· dual graphs

· Hamiltonian graph

· critical vertex

· H-edge

· simple graph

· nth order H-edges

· regular solid

· Theorems

1. If a graph has more than two odd vertices, then it has no edge path.

2. A graph that has an edge path is nearly connected.

3. If a graph is nearly connected and has at most two odd vertices then it has an edge path.

4. If a graph has a vertex path then it is connected.

5. If a graph has a vertex path then it has at most two terminal vertices.

6. If a graph has a closed vertex path then is has no terminal vertices.

7. A polygonal graph has a crossing curve if and only if its dual graph has an edge path.

8. If a polygonal graph has a crossing curve, the either all faces have even order, or there are exactly two faces of odd order and the curve begins in one odd face and ends in the other.

9. If a polygonal graph has more than two odd faces, then it has no crossing curve.

10. A Hamilton graph contains no critical vertices.

11. In a Hamilton graph no vertex is incident with more than 2 H-edges.

12. If the H-edges of a graph G include a closed path among some, but not all, vertices then G is not a Hamilton graph.

13. An edge incident to a vertex of degree two is an H-edge.

14. Starting with a single vertex and then expanding by successively performing one of two types of construction one may construct any polygonal graph. Type 1. Add a new vertex and join it by an edge to an existing vertex. Type 2. Add a new edge [possibly a loop] by joining two [not necessarily distinct] vertices already present.

15. Euler’s Formula. Let G = (V,E) be a polygonal graph and let n = #(V), e= #(E) and j be the number of faces of G. n + j - e = 2.

16. In any regular solid, S, where D is the degree of each vertex, R is the order of each face and e is the number of edges in the flat graph that represents S.

HW:

· Consider the following drawings.

		D
						I
										J
								(iii)
	(i)
					(ii)

						(v)
			(iv)					(vi)
				Z
			(vi)
	q Which of the drawings as labeled above represent graphs? q Which of the graphs above are connected? q Which of the graphs above are flat? q Which of the graphs above contain a loop?

q Referring to figure (vii) above:

a) Identify a path from X to X that uses no edge more than once.

b) Identify a path from W to X that uses every edge at least once.

c) Identify a path from W to Z that is an edge path.

d) How many different edge paths are there from W to X?

e) How many different edge paths are there from X to Y?

f) How many different paths are there from X to Y?

q If we define the distance between two vertices on a graph as the number of edges along a path connecting the two vertices which has the smallest number of edges:

a) In figure (vi), what is the distance from P to T?

b) In figure (vi), what is the distance from P to S?

c) In figure (vii), What is the distance from W to Y?

d) In figure (vii), what is the distance from W to Z?

q For each of the following sets of conditions, give an example of a graph with four vertices that satisfies all of the given conditions. If this is not possible, explain why not.

a) Disconnected and nonflat.

b) Disconnected, with an edge path.

c) Disconnected, with a vertex path.

d) Connected, with no vertex path.

e) With and edge path, but no vertex path.

f) With a vertex path, but no edge path.

g) With an edge path and a vertex path.

q Consider the figures below:

1. Find the degree of each vertex in figure (c).

2. Which of the six graphs have no edge path?

3. Which have an open edge path?

4. Which have a closed edge path?

5. Which are Euler graphs?

6. Identify the faces and the orders of the faces of graphs (a), (b), (e) and (f).

7. Draw the duals of graphs (a), (b), (e) and (f).

8. Which of the graphs (a), (b), (e) and (f) have crossing curves?

q Consider the figures below:

1. Which graphs have terminal vertices?
2. Which graphs have critical vertices?
3. Which graphs are not simple?
4. Identify the H-edges of graphs (c), (d), (e) and (f).
5. Identify the secondary H-edges of graphs (c), (d), (e) and (f).
6. Which graphs are not Hamilton Graphs?
7. Which graphs are Hamilton graphs?
8. Identify the number of vertices, the number of edges and the number of faces for graphs (c), (e) and (f).

q Which of the graphs of the tetrahedron, hexahedron, octahedron and icosahedron are Euler graphs? Which are Hamilton graphs?

q A certain construction job has several phases to it and each phase requires a certain amount of time to complete. The table below summarizes the time required for each phase of the job.

Task	Days
A. Frame the walls	3
B. Panel walls	4
C. Hang acoustic ceiling	3
D. Install wiring	2
E. Install electrical fixtures	1
F. Do Plumbing	3
G. Install wet bar	2
H. Lay carpet	2
Total	20

There are always at least two workers assigned to each job and so it will take 10 days to complete the job with a two-person crew. A further examination of the project suggests that certain phases must be completed before other phases begin. The table below summarizes the precedence requirements of the job.

Before Starting Task	We Must Complete Task
B	A,D,F
D	A
E	A,B,C,D
G	F
H	A,B,C,D,E,F,G

Use this information to construct a directed graph to summarize the information and then study that digraph to determine if a two-person crew can complete the job in less than 10 days. What about a three-person crew or two two-person crews? Assume that a crew will have to be paid for a whole day whether it works the whole day or not. What option would be cheapest? For example, if a three-person crew can do the job in 5 days that is only 15 person days of labor compared to 20 person days of labor if a two-person crew could do it in 10 days.