All of
Ellsworth’s current tutoring subjects are listed at the left. You
can read more about
Ellsworth’s qualifications in specific subjects below.
Algebra 1
Algebra 1 is the “gateway” course to high school math. You must master the concepts learned in this course to be successful in Geometry, Algebra 2 and beyond, since your continuing in the math curriculum assumes as much.
QUALIFICATIONS TO TUTOR THIS COURSE
• Taught this course in a high school setting in varied formats
• Tutored this subject both in school and online
• Degrees in Electrical Engineering and Computer Science
• plus… I’m pretty good at it!
WHAT *IS* ALGEBRA, ANYWAY?
Oh, *so* many answers to that question. Here are a few:
• A branch of mathematics that substitutes letters for numbers (think: Captain Crunch Super Decoder Ring!)
• A language for expressing abstract concepts (think: French, Latin, Urdu or Esperanto)
• A mathematical model of a real-life situation (think: word problems)
• A system of mathematical rules for calculating unknown numerical values which meet specified conditions (think: solving equations)
I like this last definition most, since it is closest to how we actually use it in this course, and elsewhere.
Algebra is a * -- system of rules -- * by which we get stuff done. The rules are consistent (always work the same way), no matter how simple or complex the problem; learn these rules, and how to use them productively, and they will take you far! To quote that great mathematician Buzz Lightyear, “To Infinity, And Beyond!”
MY TOP FIVE MATH “TROUBLE TOPICS”
I’ve been a high school math teacher for eight years, and in that time the same handful of topics hold students back year after year because they didn’t take the time to master them. ALL OF THEM are found in Algebra 1:
1) solving equations
2) word problems
3) systems of linear equations (2 equations, 2 unknowns)
4) equations of lines, and their graphs
5) adding and subtracting negative numbers (this one’s easily fixed – ask me how!)
I can help you become more confident in these areas, because they are so important for your mathematical success! Later courses build on them (and trust me: they lean VERY heavily on this foundation), so it’s best to master them now.
ALGEBRA 1 TOPIC LIST
Here’s a list of what most Algebra 1 courses cover:
Introduction
• variables
• order of operations
• equations
• expressions
Real Numbers and Integers
• number line
• basic four operations (add, subtract, multiply, divide )
Solving Linear Equations (this is CRUCIAL!)
• one-step and two-step equations
• multi-step equations (terms in parentheses, like terms)
• variables on both sides
• formulas and functions
Graphing Linear Equations
• what is slope?
• slope-intercept form
• solving linear equations using graphs
Writing Linear Equations
• point-slope form: y – y1 = m(x – x1)
• slope-intercept form: y = mx + b
• standard form: ax + by = c
[ ==> ??? Do these last three topics tell you that LINEAR EQUATIONS ARE IMPORTANT ??? <===]
Solving and Graphing Linear Inequalities
• (parallels section on solving linear equations, above)
Systems of Linear Equations and Inequalities
• solving by graphing
• solving by substitution
• solving by linear combination
Exponents and Exponential Functions
• multiplication and division properties of exponents
• zero and negative exponents
• scientific notation
• exponential growth/decay functions
Quadratic Equations and Functions
• simplifying radicals
• solving by finding square root
• solving by graphing
• solving by the Quadratic Formula
Polynomials and Factoring
• adding, subtracting and multiplying polynomials
• special products of polynomials
• solving polynomial equations in factored form (THIS is COOL!)
• factoring x^2 + bx + c and ax^2 + bx +c (THIS is even COOLER! I’ll show you the EASIEST WAY EVER to do this!)
• factoring using the Distributive Property (Factor By Grouping)
BUT WAIT… THERE’S MORE!
This is as far as most high school Algebra 1 courses dare go… and it’s quite a challenge for most students. But if you’re still with us, and it’s not yet June, here are some “Scenes From Upcoming Math Courses” included in the last two or three chapters of most textbooks for the more advanced student:
Rational Equations and Functions (I really pound this home in Algebra 2!)
• ratio and proportion
• direct and inverse variation
• simplifying rational expressions
• multiplying and dividing rational expressions
• adding and subtracting rational expressions
• dividing polynomials
Radicals and Connections to Geometry
• functions involving square roots
• radical expressions
• solving radical equations
• Completing The Square (this is where the Quadratic Formula comes from!)
• the Pythagorean Theorem and its converse
• the distance and midpoint formulas
• investigating similar triangles
• trigonometric ratios
You can see that Algebra 1 is a very broad course, and to be held responsible for so much varied material is a pretty tall order! As your tutor, I can help you in areas where you may need assistance.
Algebra 2
At the high school I taught at, I designed this Algebra 2 curriculum for our department, and we used it as the scope and sequence for teaching the actual course:
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REVIEW SOLVING EQUATIONS AND SYSTEMS
* 1 equation, 1 unknown and
* 2 equations, 2 unknowns
QUADRATIC EQUATIONS: We became EXPERTS at solving ax^2 + bx + c = 0. using three different methods:
1) Quadratic Formula (of course!)
2) Factoring (the NEATEST and SIMPLEST trick I EVER saw makes this EASY and FOOLPROOF!)
3) Completing the Square (for those who need/want the extra rigor)
FACTORING HIGHER ORDER EXPRESSIONS: After mastering quadratics, we expanded our factoring repetoire to higher-order functions (x^3, x^4 and beyond) by building up our "bag of tricks" for reducing these big-boys to products of first- and second-order expressions. Some of the tools we developed:
* Synthetic Division
* Factor by Grouping
* Greatest Common Factor (GCF)
* Difference of Squares
* Sum/Difference of Cubes
And all of this was to prepare you for...
RATIONAL EXPRESSIONS AND EQUATIONS: Around the end of the first semester we finally arrive at the mountain top -- being able to:
* find least-common denominator (LCD) and greatest common factor (GCF) of two rational expressions
* simplify rational expressions (using our factoring methods developed above!)
* solve rational equations (including identifying extraneous solutions)
RADICAL EQUATIONS AND EXPRESSIONS: The same "simplify expression/solve equation" duality we used for quadratics and rational expressions is harnessed to tame equations containing radicals.
* Eliminating Radicals (where possible)
* Rationalizing the Denominator
* Simplifying Radical Expressions
* Solving Radical Equations
COMPLEX NUMBERS: Remember the "lie" you were told in second grade about how you can't evaluate 3 - 7, since 7 is bigger than 3? Well, one day you suddenly *could*, and "negative numbers" were born!
Similarly, the statement "you can't take the square root of negative numbers" is also going into the dust bin, because we are again expanding our numbering system, this time to include "i", the square root of -1.
* Using Complex Conjugates
* Getting "i" out of the Denominator (or, as I pithily put it, "Keeping It Real"... nyuk nyuk...)
* Simplifying Complex Expressions
* Solving Complex Equations
* Solving Quadratic Equations with Complex Roots (you know... the ones you "couldn't" solve before, because the quantity under the radical was negative)
LOGARITHMS AND EXPONENTIALS: Finally, after losing a month to state testing, we got to end the year in a subject which was long neglected at my school: logarithms and exponentials. Besides learning each of these topics inside and out, I also emphasize their dual nature, namely how every logarithmic equation can be written in an exponential form (and vice versa), and how to exploit the fact that logs and exponentials are inverse functions of each other.
Specifics:
* What is a logarithm -- REALLY?
* Simplifying log equations
* Log identities
* Exponential equations (solving)
* Applications (e.g. population)
ALL THE OTHER STUFF: Sadly, everything we'd like to see in every Algebra 2 course won't fit... which is why I can also tutor on many of the things we were forced to leave out:
* Conic Sections
* Sequences and Series
* Using a Calculator to Characterize Functions
If you don't see what you need listed, please ask. If I can, I'll give you some ideas or point you in the right direction.
Geometry
Geometry can be a difficult class for many students. Lodged in most high school curricula between Algebra 1 and Algebra 2, geometry is different from those classes because it calls upon more than just the pure mathematical aptitude the others require.
But make no mistake: you DO need solid algebra skills to succeed in geometry, but in my experience you also need to be creative, logical and be able to “think” logically, and visualize in 2 and 3 dimensions. I’ve taught this class twice, mostly to 10th graders, and it’s been a challenge for many of them to make that leap from just using formulas.
Precalculus
When you get to “Pre-Cal”, you’re getting SERIOUS about math! Most students bail after Algebra 2, because that’s the last math class they are required to take, so if you’re still around for Pre-Cal, it’s because you WANT to be here, and my hat is certainly off to you!
QUALIFICATIONS TO TUTOR THIS COURSE:
• Taught this course in a high school setting
• Tutored this subject both in school and online
• Degrees in Electrical Engineering and Computer Science
• plus… I’m pretty good at it!
USEFUL PREREQUISITES:
Before you jump in to pre-calculus, I’d recommend brushing up on some prerequisites you’ll need from earlier courses:
• Solving Equations Algebraically (Algebra 1 and 2) -- you NEED to MASTER this!
• Graphical Representation of Data (Algebra 1)
• Graphs of Equations (Algebra 1, Geometry)
• Lines in the Plane (Algebra 1, Geometry)
• Solving Equations Algebraically and Graphically (Algebra 1 and 2)
• Solving Inequalities Algebraically and Graphically (Algebra 1 and 2)
THE COURSE ITSELF:
Here’s a subject list for a typical modern pre-calculus class.
The first topics (1-5) are a review of material covered in Algebra 2 (or before), but also given a more advanced in-depth treatment.
Starting with 6) you’re largely into new concepts.
This is roughly how the class is typically broken up between two semesters in high school.
Review/Advanced Treatment:
1) Functions and Their Graphs
• Function Basics
• Graphs of Functions
• Shifting, Reflecting, and Stretching Graphs
• Combinations of Functions
• Inverse Functions
2) Polynomial and Rational Functions
• Quadratic Functions
• Polynomial Functions of Higher Degree
• Real Zeros of Polynomial Functions
• Complex Numbers
• The Fundamental Theorem of Algebra (polynomials with complex coefficients)
• Rational Functions and Asymptotes
• Graphs of Rational Functions
3) Exponential and Logarithmic Functions
• Exponential and Logarithmic Functions (and their graphs!)
• Properties of Logarithms
• Solving Exponential and Logarithmic Equations
3) Trigonometric Functions
• Radian and Degree Measurement
• Trigonometric Functions: The Unit Circle
• Right Triangle Trigonometry
• Trigonometric Functions of Any Angle
• Graphs of Sine and Cosine Functions
• Inverse Trigonometric Functions
4) Analytic Trigonometry
• Using Fundamental Identities
• Verifying Trigonometric Identities (VERY algebra-intensive!)
• Solving Trigonometric Equations (gotta be creative here!)
• Sum and Difference Formulas
• Multiple-Angle and Product-Sum Formulas
5) Systems of Equations and Inequalities
• Solving Systems of Equations
• Systems of Linear Equations in Two Variables
• Multivariable Linear Systems
• Systems of Inequalities
• Linear Programming (constraint optimization, using a system of linear inequalities)
New Stuff:
6) Advanced Topics in Trigonometry
• Law of Sines (and Cosines)
• Vectors and Dot Products
• Trigonometric Form of a Complex Number (believe it or not, they ARE related!)
7) Matrices and Determinants
• Matrices and Systems of Equations
• Operations with Matrices
• Inverse of a Square Matrix
• Determinant of a Square Matrix
• Applications of Matrices and Determinants (e.g. Cramer’s Rule for solving linear systems)
8) Sequences and Series
• Arithmetic Sequences and Partial Sums
• Geometric Sequences and Series
• Mathematical Induction
9) Probability (This is a whole course in itself! But we’re just introducing it here)
• Introductory Concepts (counting, experiments, outcomes, sample spaces, …)
• Counting Principles
• The Binomial Theorem
10) Advanced Topics In Pre-Calculus
• Conic Sections (parabolas, ellipses and hyperbolas)
• Systems of Quadratic Equations
• Parametric Equations
• Polar Coordinates (and their graphs)
11) Three-Dimensional Analytic Geometry
• The Three-Dimensional Coordinate System
• Vectors in Space
• Cross Product of Two Vectors
• Lines and Planes in Space
12) Limits and an Introduction to Calculus
• What ARE Limits, anyway?
• Techniques for Evaluating Limits
• The Tangent Line Problem
• Limits at Infinity
• Limits of Sequences
• The Area Problem (you’re actually doing calculus here, but you’re not supposed to know that yet! Shhhh…)
Trigonometry
Trigonometry (sine, cosine, tangent, etc.) is usually first encountered in Geometry, and then amplified in either Algebra 2 or Pre-Calculus. It is typically used in the context of the angles of triangles, and how they are related to the lengths of the triangle’s sides.
Wherever you encounter it, I can help you understand this material from the ground up.
QUALIFICATIONS TO TUTOR THIS COURSE:
• Taught this course in a high school setting
• Tutored this subject both in school and online
• Degrees in Electrical Engineering and Computer Science
• plus… I’m pretty good at it!
I can start out, if needed, by illustrating how sine, cosine and tangent are the ratios of certain pairs of triangle sides (opposite, adjacent and hypotenuse). From there we can use the following road-map to develop our understanding. We can jump in at an appropriate point if you are an advanced student, or need specific help:
BASIC TRIGONOMETRY (Geometry)
• Using Trigonometric Functions: Find angles, given side lengths
• Solving Right Triangles: You can find all the missing angles and sides when you are given two sides, or one side and one angle
• Inverse Trigonometric Functions (arcsin, arccos, arctan): Given the side lengths in a right triangle, find the angles!
• Special Right Triangles: The triangles 45-45-90 and 30-60-90 have special properties which make finding missing side lengths a snap, even if you are given only one side!
• Sum and Difference Formulas (including Double Angle): Find, for example, cos(2x) or tan(a – b)
INTERMEDIATE TRIGONOMETRY (Geometry, Algebra 2)
• Using Fundamental Identities: your “survival kit” of basic trig identities
• Radian and Degree Measurement: Angles may be measured in either degrees or radians. Learn to do both.
• The Unit Circle: A circle of radius 1, centered at the origin, can tell us so much about trigonometry…
• Graphs of Sine and Cosine Functions: Everything you wanted to know about amplitude, phase shift, period, offset… and how to get from graphs to equations, and vice versa
ADVANCED TRIGONOMETRY (Algebra 2, Pre-Calculus)
• Law of Sines (And Cosines): Now we’re not restricted to right triangles! Use these rules to relate angles and side lengths in a more general way.
• Verifying Trigonometric Identities: VERY algebra-intensive! Only “Algebra Ninjas” need apply… but I’ll help you out…
• Solving Trigonometric Equations
As usual, if you don’t see what you need… ask!