All of
Ace’s current tutoring subjects are listed at the left. You
can read more about
Ace’s qualifications in specific subjects below.
Algebra 1
Algebra is about finding the unknown or it is about putting real life problems into equations and then solving them. Unfortunately many textbooks go straight to the rules, procedures and formulas, forgetting that these are real life problems being solved.
A branch of mathematics that substitutes letters for numbers. An algebraic equation represents a scale, what is done on one side of the scale with a number is also done to the other side of the scale. The numbers are the constants. Algebra can include real numbers, complex numbers, matrices, vectors etc.
Algebra 2
Algebra 2 covers quadratic equations, graphing, complex numbers, functions, sequences and series, and exponents and logarithms. Problem solving skills are emphasized throughout, and time is devoted to advanced topics like telescoping sums and piecewise functions.
Calculus
The branch of mathematics that deals with the finding and properties of derivatives and integrals of functions. It is a particular method or system of calculation or reasoning.
Calculus is the study of change, with the basic focus being on
1. Rate of change (Differential Calculus)
2. Accumulation (Integral Calculus)
In both of these branches (Differential and Integral), the concepts learned in algebra and geometry are extended using the idea of limits. Limits allow us to study what happens when points on a graph get closer and closer together until their distance is infinitesimally small (almost zero). Once the idea of limits is applied to our Calculus problem, the techniques used in algebra and geometry can be implemented.
Chemistry
My glossary definition is short and sweet: Chemistry is the "scientific study of matter, its properties, and interactions with other matter and with energy."
An important point to remember is that chemistry is a science, which means its procedures are systematic and reproducible and its hypotheses are tested using the scientific method. Chemists, scientists who study chemistry, examine the properties and composition of matter and the interactions between substances. Chemistry is closely related to physics and to biology. Chemistry and physics both are physical sciences. In fact, some texts define chemistry and physics exactly the same way. As is true for other sciences, mathematics is an essential tool for the study of chemistry.
Differential Equations
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations play a prominent role in engineering, physics, economics, and other disciplines.
Differential equations arise in many areas of science and technology, specifically whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space and/or time (expressed as derivatives) is known or postulated. This is illustrated in classical mechanics, where the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow one (given the position, velocity, acceleration and various forces acting on the body) to express these variables dynamically as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation (called an equation of motion) may be solved explicitly.
An example of modelling a real world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air resistance. The ball's acceleration towards the ground is the acceleration due to gravity minus the deceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the ball's velocity. This means that the ball's acceleration, which is a derivative of its velocity, depends on the velocity (and the velocity depends on time). Finding the velocity as a function of time involves solving a differential equation.
Elementary Math
Elementary mathematics consists of mathematics topics frequently taught at the primary or secondary school levels. The most basic topics in elementary mathematics are arithmetic and geometry. Beginning in the last decades of the 20th century, there has been an increased emphasis on probability and statistics and on problem solving.
A mastery of elementary mathematics is necessary for many professions, including carpentry, plumbing, and automobile repair, as well as being a prerequisite for all advanced study in mathematics, science, engineering, medicine, business, architecture, and many other fields.
Geometry
Geometry is the study of the size, shape and position of 2 dimensional shapes and 3 dimensional figures. However, geometry is used daily by almost everyone. In geometry, one explores spatial sense and geometric reasoning.
It is in every part of the curriculum K-12 and through to college and university. Since most educational jurisdictions use a spiraling curriculum, the concepts are re-visited throughout the grades advancing in level of difficulty. Typically in the early years, learners identify shapes and solids, use problem solving skills, deductive reasoning, understand transformations, symmetry and use spatial reasoning. Throughout high school there is a focus on analyzing properties of two and three dimensional shapes, reasoning about geometric relationships and using the coordinate system. Studying geometry provides many foundational skills and helps to build the thinking skills of logic, deductive reasoning, analytical reasoning and problem solving to name just a few.
Linear Algebra
Linear algebra is the branch of mathematics concerning vector spaces, often finite or countably infinite dimensional, as well as linear mappings between such spaces. Such an investigation is initially motivated by a system of linear equations in several unknowns. Such equations are naturally represented using the formalism of matrices and vectors. Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations. Techniques from linear algebra are also used in analytic geometry, engineering, physics, natural sciences, computer science, and the social sciences (particularly in economics). Because linear algebra is such a well-developed theory, nonlinear mathematical models are sometimes approximated by linear ones.
Prealgebra
Pre-Algebra is a common name for a course in middle school mathematics. In the United States, pre-algebra is generally taught between the fifth and eighth grades, although sometimes taught as early as the third grade for gifted students. The objective of pre-algebra is to prepare the student for the study of algebra.
Pre-algebra includes several broad subjects:
- Review of natural number arithmetic
- New types of numbers such as integers, fractions, decimals and negative numbers
- Factorization of natural numbers
- Properties of operations such as associativity and distributivity
- Simple (integer) roots and powers
- Rules of evaluation of expressions, such as operator precedence and use of parentheses
- Basics of equations, including rules for invariant manipulation of equations
- Understanding of variable manipulation
- Manipulation and arithmetic with the standard 4-quadrant Cartesian coordinate plane
Pre-algebra may include subjects from geometry, especially subjects that further understanding of algebra in applications to area and volume.
Precalculus
Precalculus, an advanced form of secondary school algebra, is a foundational mathematical discipline. It is also called Introduction to Analysis. In many schools, precalculus is actually two separate courses: Algebra and Trigonometry. Precalculus prepares students for calculus the same way as pre-algebra prepares students for Algebra I. While pre-algebra teaches students many different fundamental algebra topics, precalculus does not involve calculus, but explores topics that will be applied in calculus. It often involves covering algebraic topics that might not have been given attention in earlier algebra courses. Some precalculus courses might differ with others in terms of content. For example, an honors level course might spend more time on conic sections, vectors, and other topics needed for calculus. A college preparatory class might focus on topics used in business-related careers, such as matrices, or power functions.
In detail, precalculus deals with:
- Sets
- Real numbers
- Complex numbers
- Solving inequalities and equations
- Properties of functions
- Composite functions
- Polynomial functions
- Rational functions
- Trigonometry
- Trigonometric functions and their inverses
- Trigonometric identities
- Conic sections
- Exponential functions
- Logarithmic functions
- Sequences and series
- Binomial theorem
- Vectors
- Parametric equations
- Polar coordinates
- Matrices and determinants
- Mathematical induction
- Limits
Reading
Reading comprehension is defined as the level of understanding of a text/message. This understanding comes from the interaction between the words that are written and how they trigger knowledge outside the text/message.
Proficient reading depends on the ability to recognize words quickly and effortlessly. If word recognition is difficult, students use too much of their processing capacity to read individual words, which interferes with their ability to comprehend what is read.
During the last century comprehension lesson/s usually comprised students answering teachers' questions, writing responses to questions on their own, or both. There is not a definitive set of strategies, but common ones include summarizing what you have read, monitoring your reading to make sure it is still making sense, and analyzing the structure of the text (e.g., the use of headings in science text). Some programs teach students how to self monitor whether they are understanding and provide students with tools for fixing comprehension problems.
Instruction in comprehension strategy use often involves the gradual release of responsibility, wherein teachers initially explain and model strategies. Over time, they give students more and more responsibility for using the strategies until they can use them independently.
Spelling
Learning spelling words is important to your child’s future. Spelling is important because it aids in reading. It helps cement the connection that is shared between sounds and letters. Learning high frequency sight words also has been shown to help with both reading and writing. This is why students learn sight words during their early years. Spelling and reading also have a common factor, proficiency with language. The major goal of the English writing system is not merely to ensure accurate pronunciation of the written word – it is to convey meaning. If words that sound the same (e.g., rain, rein and reign) were spelled the same way, their meanings would be harder to differentiate. Spelling words help lay the basic foundation that your child will need throughout their education and life.
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves. The field evolved during the third century BC as a branch of geometry used extensively for astronomical studies. It is also the foundation of the practical art of surveying.
Trigonometry basics are often taught in school either as a separate course or as part of a precalculus course. The trigonometric functions are pervasive in parts of pure mathematics and applied mathematics such as Fourier analysis and the wave equation, which are in turn essential to many branches of science and technology. Spherical trigonometry studies triangles on spheres, surfaces of constant positive curvature, in elliptic geometry. It is fundamental to astronomy and navigation. Trigonometry on surfaces of negative curvature is part of Hyperbolic geometry.