Weebly for Education
Graduate Project
6/16/2015
UTB Mathematics
Elvira Buenavista-Vera
Weebly for Education 2015
Weebly for Education
Weebly for Education is a free website and blog hosting service for teachers
and their students. The website I have created can be found at the following
address: mrsebvera.weebly.com
Weebly for Education provides a graphical user interface that allows a
teacher to quickly and easily create a website and/or blog by dragging and
dropping website components. Component options include the ability to
post course notes and assignments, share a calendar, assign and accept
homework, communicate with students and parents.
There are three tabs on the website: Home, Syllabus, and Algebra I.
I.
The Home tab has a brief introduction of the teacher. The Home
tab also has a drop down menu with Contact Me and Rivera Early
College High School. The Contact Me tab can be utilized by both
students and parents to communicate with me. The Rivera Early
College High School tab takes the visitor to the campus website.
II.
The Syllabus tab allows the visitor to access the syllabus for the
Algebra I course. The syllabus includes information regarding the
course outline, important testing dates, the grading procedure, etc.
III.
The Algebra I tab allows the visitor to access the notes and the
assignments for chapters 1-9. The Algebra I tab also has a drop
down menu that includes a Calendar tab and a Tutorial tab. The
calendar tab will take the visitor to an embedded calendar that
informs the visitor of what is being covered on a daily basis. The
Tutorial tab informs of the tutorials that are being offered
throughout the school year.
Since technology in the classroom is essential, the Weebly website is an
excellent tool for students. Students, along with parents/guardians, can
stay informed of the lesson or lessons being taught. The website is
extremely helpful for our homebound students or for students that are
absent. It allows them to remain up to date on their assignments because
failing the course due to truancy is not an option.
Weebly for Education 2015
The Algebra I tab, as previously mentioned, has all of the notes and
assignments needed for the course. In mathematics, the Algebra I course is
the foundation for high school students. According to the dictionary, algebra
is defined as the following:
al·ge·bra
/aljəbrə/
noun
noun: algebra
the part of mathematics in which letters and other general symbols are used to
represent numbers and quantities in formulae and equations.
a system of algebra based on given axioms.
plural noun: algebras
The graduate courses of Foundations of Mathematics and Higher Algebra
give a better perspective of the material covered at the high school level. In
continuation, I will briefly discuss the objectives and the sections of each
Chapter button found in the Algebra I tab of the Weebly page. I will also
mention some of the higher mathematics that was covered in the graduate
classes. These classes gave me a better insight of more general topics that
are derived from the basic foundation I teach at the Algebra I level.
Chapter 1
The Language and Tools of Algebra
Objective: The learner will understand how Algebra can be used to express,
recognize, and use the power of symbols to represent situations. The
learner will also understand the importance of the skills required to
manipulate symbols in order to solve problems, and use the necessary
algebraic skills required to simplify algebraic expressions.
The sections in Chapter 1 are: Variables & Expressions, Order of
Operations, Open Sentences, Identity & Equality Properties, The Distributive
Property, Commutative & Associative Properties, Logical Reasoning &
Counterexamples, Number System, and Functions & Graphs.
The first chapter in Foundations of Mathematics begins with a collection of basic
mathematical concepts. Number sets is the second item in this chapter. It was
Weebly for Education 2015
discussed that there are five sets of numbers. The five sets are: natural
numbers {0, 1, 2, 3, …}, integers {…, -3, -2, -1, 0, 1, 2, 3, …}, rational
𝑥
numbers {𝑦 |𝑥 ∈ 𝑍, 𝑦 ∈ 𝑍 \ {0}}, real numbers, and complex numbers
{x + yi | x, y ∈ R}. These number sets are taught in my high school Algebra I
course at the beginning of the school year. However, complex numbers is not
mentioned in this course. High school students will first work with complex
numbers when they get to Algebra II.
The fifth item, in chapter one of Foundations of Mathematics, mentions
equivalence relation. Equivalence relation is a relation between elements of a
set that is reflexive, symmetric, and transitive. If we let r be a binary relation
on a set X, then we consider the following three conditions on r:
(r) For each element x in X, we have (x, x) ∈ r.
(s) For any two elements y and z in X with (y, z) ∈ r, we have (z, y) ∈ r.
(t) For any three elements x, y, and z in X with (x, y) ∈ r and (y, z) ∈ r, we
have (x, z) ∈ r.
If condition (r) is satisfied, then a relation is called reflexive. If condition (s) is
satisfied, the relation is called symmetric. If condition (t) is satisfied, the
relation is called transitive. As previously mentioned, a relation is called an
equivalence relation if it is reflexive, symmetric, and transitive. High school
students in Algebra I are first introduced to the basic properties of identity,
equality, distributive, commutative and associative. There are three sections
dedicated to these properties. The students have some difficulty with most of
these properties.
Weebly for Education 2015
Chapter two of Foundations of Mathematics mentions Maps. We were
introduced to domain and codomain. If r is a map from Y to Z, the set Y is
called the domain of r. The set Z is called the codomain of r.
In Algebra I, the last section (1-9) covers functions and graphs. This is the
introduction of functions. In a latter chapter, they will cover functions more in
depth.
Weebly for Education 2015
Chapter 2
Solving Linear Equations
Objective: The learner will use the necessary algebraic skills required to
solve equations in problem situations. Describe the effects of changes in
parameters of linear functions in real-world and mathematical situations.
The sections in Chapter 2 are: Writing Equations, Solving Equations by
Using Addition and Subtraction, Solve Equations by Using Multiplication and
Division, Solving Multi-Step Equations, Solving Equations with the Variable
on Each Side, Ratios and Proportions, Percent of Change, Solving for a
Specific Variable, and Weighted Averages.
This chapter belongs to linear algebra. The definition of vector space is not
introduced in Algebra I, however the basic concepts begin with this chapter.
The dictionary defines vector space as the following:
vec·tor space
noun
MATHEMATICS
a space consisting of vectors, together with the associative and commutative operation
of addition of vectors, and the associative and distributive operation of multiplication of
vectors by scalars.
Weebly for Education 2015
In Chapter 2 of Algebra I, we begin solving equations. If you look at the
calendar that is embedded in the Weebly webpage, you’ll notice that we
spend quite a bit of time teaching this concept. We drill it to our students,
because solving equations is one of the most crucial topics for their
foundation.
Example: Solve 3x−6 = 9
Start With
3x−6 = 9
Add 6 to both sides:
3x = 9+6
Divide by 3:
x = (9+6)/3
Now we have x = something,
and a short calculation reveals that x = 5
In Lecture 2 of Advanced Calculus, we are introduced to the definition of
limit. It is crucial, at this point, for students to know how to solve
equations.
Definition (Limit):
By
lim = 𝐿 it is meant that for every ∈ > 0, there is a 𝛿 > 0 such that if
𝑥→𝑎
0 < |x – a| <
𝛿 , then |f(x) – L| < ∈.
Weebly for Education 2015
Chapter 3
Functions and Patterns
Objective: The learner will understand that a function represents a
dependence of one quantity on another and can be described in a variety of
ways.
The sections in Chapter 3 are: Representing Relations, Representing
Functions, Linear Functions, Arithmetic Sequences, and Proportional and
Non-Proportional Relationships.
As previously mentioned, Chapter 1 (section 9) in Algebra I, introduces
functions. Our freshmen study functions more in depth in this chapter. Also
as previously mentioned, the fifth item in Chapter 1 of Foundations of
Mathematics, we covered binary relations. A binary relation is defined as a
subset of a Cartesian product. The following is an example of a binary
relation:
Let X := {1, 2}, and let Y := {1, 2, 3}. Then {(1, 1), (1, 3), (2, 2), (2, 3)}.
If r is a subset of a Cartesian product X x Y, we call r a binary relation
between X and Y. If r is a subset of the Cartesian product X x X, we call r a
binary relation on X. A relation is called an equivalence relation if it is
reflexive, symmetric, and transitive (as previously mentioned). There are
two more conditions on relations that we covered in this graduate class.
They are antisymmetric and connected. A relation is said to be
antisymmetric if it satisfies condition (a).
(a)
(c)
For any two elements y and z in X with (y, z) ∈ r and (z, y) ∈ r, we
have y = z.
For any two elements y and z in X, we have (y, z) ∈ r or (z, y) ∈ r.
A relation is said to be connected if it satisfies condition (c).
I also recall that in Chapter 2 of Foundations of Mathematics, we discussed
injectivity and surjectivity. We will let r be a map from Y to Z and consider
the following two conditions:
Weebly for Education 2015
(iii)
For each element z in Z, there exists at most one element y in Y
such that (y, z) ∈ r.
(iv)
For each element z in Z, there exists at least one element y in Y
such that (y, z) ∈ r.
The map is called injective if it satisfies condition (iii). It is called surjective
if it satisfies condition (iv). A map which is both injective and surjective is
called bijective.
Chapter 4
Analyzing Linear Equations
Objective: The learner will understand the meaning of slope, intercepts, and
zeros of linear functions, and interpret linear functions in real-world and
mathematical situations.
The sections in Chapter 4 are: Rate of Change and Slope, Slope and Direct
Variation, Graphing Equations in Slope-Intercept Form, Writing Equations in
Slope-Intercept Form, Writing Equations in Point-Slope Form, Scatterplots
and Lines of Fit, Parallel and Perpendicular Lines.
In Lecture 5 of Advanced Calculus, we studied the definition of the
derivative, slopes of tangent lines and secant lines, and differentiability
versus continuity. Students are first introduced to slope and rate of change
in Chapter 4 of Algebra I. The definition of derivative is the following:
The derivative of a function f(x) at a point x = a is denoted and defined as
𝑓 ′ (𝑎) =
𝑑𝑓
𝑑𝑥
(𝑎) = lim
𝑥→𝑎
𝑓(𝑥)−𝑓(𝑎)
𝑥−𝑎
.
Weebly for Education 2015
Chapter 5
Solving Systems of Linear Equations
Objective: The learner will formulate systems of linear equations, use a
variety of methods to solve them, and analyze the solutions in terms of the
situation.
The sections in Chapter 5 are: Graphing Systems of Equations, Substitution,
Elimination Using Addition and Subtraction, Elimination Using Multiplication,
and Applying Systems of Linear Equations.
Weebly for Education 2015
Chapter 6
Solving Linear Inequalities
Objective: The learner will formulate inequalities based on linear functions,
use a variety of methods to solve them, and analyze the solutions in terms
of the situation.
The sections in Chapter 6 are: Solving Inequalities by Addition and
Subtraction, Solving Inequalities by Multiplication and Division, Solving
Multi-Step Inequalities, Solving Compound Inequalities, Solving Open
Sentences Involving Absolute Value, and Graphing Systems of Inequalities.
Solving inequalities is very like solving equations ... we do most of the same
things ...
... but we must also pay attention to the direction of the inequality.
Direction: Which way the arrow "points"
Some things we do will change the direction!
< would become >
> would become <
≤ would become ≥
≥ would become ≤
Weebly for Education 2015
Chapter 7
Polynomials
Objective: The learner will understand there are situations modeled by
functions that are not linear, and model the situations.
The sections in Chapter 7 are: Multiplying Monomials, Dividing Monomials,
Polynomials, Adding and Subtracting Polynomials, Multiplying a Polynomial
by a Monomial, Multiplying Polynomials, and Special Products.
In Chapter 7 of Higher Algebra, we studied Rings. Let R be a commutative
and additively written group (with neutral element 0), and assume that
there is defined a multiplication on R. Then R is called a ring if the following
two conditions hold.
(i)
(ii)
The multiplication on R is associative and has a neutral element.
For any three elements s, t, and u in R, we have
s(t + u) = st + su and (s + t)u = su + tu.
A ring R is called commutative if its multiplication is commutative, that is if,
for any two elements s and t in R, st = ts. Also, a ring R is called a ring with
1 if R possesses an element e such that er = r = re for each element r in R.
Weebly for Education 2015
In Lecture 1 of Advanced Calculus, we studied polynomials. It was
mentioned several times that even though students were first introduced to
polynomials in Algebra, by the time they are in a Calculus course, many are
still not fully comfortable with them.
In Chapter 7 of Algebra I, the freshmen students are introduced to
polynomials. They begin adding, subtracting, multiplying, and dividing of
polynomials.
Weebly for Education 2015
Chapter 8
Factoring
Objective: The learner will use algebraic skills to simplify algebraic
expressions, and solve equations and inequalities in problem situations.
The sections in Chapter 8 are: Monomials and Factoring, Factoring Using the
Distributive Property, Factoring Trinomials: x2 + bx + c, Factoring
Trinomials: ax2 + bx + c, Factoring Differences of Squares, Perfect Squares
and Factoring.
In Chapter 7 of Higher Algebra, we continued with the study of Rings.
Elements in R (commutative ring with 1) which have inverses are called
units. The ring R is called integral domain if products of elements in R \ {0}
are different from 0. It is called a field if each element of R \ {0} is a unit of
R. An ideal T of R is called prime if the following two conditions hold.
(i)
(ii)
T ≠ R.
If the product of two elements of R is in T, then one of the
factors is in T.
An element r in R is called irreducible if the following conditions hold.
(i)
(ii)
r ∈ R \ U(R) \ {0}.
If r is a product of two elements in R, then one of the two factors
is a unit of R.
Lemma 7.4 stated the following: If R is an integral domain, each prime
element of R is irreducible.
In Lecture 1 of Advanced Calculus, we worked with the factoring in
completing the square. This concept was required in order to find the radius
and the center of a circle.
Weebly for Education 2015
In Chapter 8 of Algebra I, students are introduced to factoring. Factoring is
a concept that has never been introduced before. Therefore, it takes some
practice before they become familiar with the process.
Factors
Numbers have factors:
And expressions (like x2+4x+3) also have factors:
Weebly for Education 2015
Chapter 9
Quadratic and Exponential Functions
Objective: The learner will understand there is more than one way to solve
a quadratic equation and solve them using appropriate methods.
Understand there are situations modeled by functions that are neither linear
nor quadratic, and model the situations.
The sections in Chapter 9 are: Graphing Quadratic Functions, Solving
Quadratic Equations by Graphing, Solving Quadratic Equations by
Completing the Square, Solving Quadratic Equations by Using the Quadratic
Formula, Exponential Functions, Growth and Decay.
An example of a Quadratic Equation:
Quadratic Equations make nice curves, like this one:
Weebly for Education 2015
Thank you to the committee members: Dr. Paul-Hermann Zieschang, Dr.
Taeil Yi, and Dr. James Maissen.
References:
Foundations of Mathematics by Dr. Paul-Hermann Zieschang
Higher Algebra by Dr. Paul-Hermann Zieschang
Integrating Technology by Dr. Taeil Yi
Advanced Calculus Lecture Notes by Dr. James Maissen
www.googledictionary.com
www.mathisfun.com
www.weebly.com