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# Welcome to Möbius

## Class Details

**System
Administrator**
(sre@digitaled.com)

## Units

### Welcome!

We’ve compiled a lesson for you that demonstrates the different types of content, questions, and methods that we use to deliver course materials to students. This is by no means our full content offering but it will give you an idea of what our platform looks like and the varied levels of interaction on each page.

We don’t want you to miss out on seeing what Möbius offers, but feel free to jump between sections.

### Algebra & Trigonometry

The Unit Circle and the Trigonometric Functions

In this lesson, we introduce the six trigonometric functions. The approach that we take utilizes the unit circle.

Parametric Equations

In this lesson, you will learn about how to parameterize a curve, eliminate the parameter., find a rectangular equation for a curve defined parametrically, and find parametric equations for curves defined by rectangular equations.

The Complex Plane and Polar Form

In this lesson we introduce a geometric representation for complex numbers, that will give us elegant and powerful methods for complex arithmetic.

The Ellipse

In this lesson, you will learn how to write equations of ellipses in standard form, graph ellipses centered at the origin, graph ellipses not centered at the origin, and solve applied problems involving ellipses.

Point Slope Form

Learn about parallel and perpendicular lines, point slope form for the equation of a line, and graphing linear inequalities.

Angles I

It's time to do some math. This lesson will combine a larger number of subtle, technical issues than many of the lectures that will follow. Like most of the lessons in this course, we will introduce some definitions, state and prove some results, and do quite a number of examples.

Modelling Periodic Behaviour

In this lesson, we relate real-world periodic behaviour to the properties of sinusoidal functions. We will determine equations to model the graphs of sinusoidal functions and real-world phenomena.

Introduction to Inverses

In this lesson, the concept of the inverse of a function is introduced. Inverses are constructed and analyzed for functions described using tables, mapping diagrams, or graphs. We also take a first look at inverses of functions described algebraically.

Working With \(y=mx+b\)

In this lesson, we will algebraically determine the equation of a line in the form \(y=mx+b\) given a \(y\)-intercept and a point, given a slope and a point, and given two points.

Surface Area of Pyramids and Cones

In this lesson, we will make connections between the net and the surface area of pyramids and cones. We will use different strategies to solve surface area applications.

Prime Factorization

In this lesson, students will define prime and composite numbers. They will learn to write composite numbers as a product of its prime factors.

Scatter Plots

A scatter plot is a graph consisting of points which are formed using the values of two variable quantities. Scatter plots are used to display a relationship between the two variables in question. In this lesson, we discuss the features of a scatter plot and practise creating scatter plots from paired data sets. We discuss the roles that the two variables play in a scatter plot and explore what information might be revealed when we consider the shape formed by the data points as a whole.

The Pythagorean Theorem and Trigonometry

We will touch upon a broad spectrum of geometric concepts and ideas as we work through some problems. Let's begin with the Pythagorean Theorem and trigonometry.

Comparing \(y=x^2\) and \(y=2^x\)

Students will compare the features of the graphs of \(y=x^2\) and \(y=2^x\).

### Calculus

Definition and Properties

Learn the definition of a derivative and properties of simple derivatives.

Curve Sketching and the First Derivative

In this lesson, we explore turning points of functions and introduce the first derivative test. More precisely, we will learn how to use the first derivative to find the intervals of increase and decrease of a given function.

The Tangent Plane

In this lesson, we will cover what the Tangent Plane is along with a few practice questions.

Modelling With Derivatives

In this lesson, we will consider mathematical models that involve quantities and their rates of change (expressed as functions) and their derivatives. These models are typically expressed as differential equations (i.e., equations that involve an unknown function and one or more of its derivatives).

Limits from Graphs

By the end of this lesson, students should be able to

- determine limits for a function, given the graph of the function,
- determine \(\displaystyle \lim_{x\to a^+} f(x)\) and \(\displaystyle \lim_{x\to a^-} f(x)\) in a situation where the graph of the function is given and \(f(a)\) is defined, and
- determine which descriptor, \(\infty\) or \(-\infty\), describes\(\displaystyle \lim_{x\to a^+} f(x)\) and \(\displaystyle \lim_{x\to a^-} f(x)\) in a situation where the graph of the function is given and the function has a vertical asymptote at \(x=a\).

Applications of \(y^{\prime \prime}(t)+p(t)\,y^\prime (t)+q(t)\,y(t)=g(t)\): Forced Oscillations

In the lesson, we will cover damped pendulums, damped oscillations with periodic forcing, and undamped forced oscillators.

### Financial Mathematics

Compound Interest

Learn about accumulated values and present values under compound interest.

### Linear Algebra

Norm, Dot Product, and Cross Product

In this lesson we will explore the norm, dot product, and cross product.

Orthogonality

In this lesson we will explore orthogonality and orthonormality, projections, and introduce the Gram-Schmidt Algorithm

### Precalculus

Sums and Differences of Functions

Graphs of functions, \( y=f(x) \pm g(x) \), formed by the addition or subtraction of two functions, will be investigated. Key properties of the functions will be discussed and factors influencing the behaviour of the graphs will be identified. Connections will be made to real-world applications to model and solve problems.

The Hyperbola

In this lesson, you will: locate a hyperbola’s vertices and foci, write equations of hyperbolas in standard form, graph hyperbolas centered at the origin, graph hyperbolas not centered at the origin, and solve applied problems involving hyperbolas.

Pascal's Triangle and Binomial Expansions

In this lesson, we will be introduced to a famous pattern of numbers known as Pascal’s triangle. Patterns within Pascal’s triangle will be investigated, including those that assist in expanding powers of binomials, \((a+b)^n\). The Binomial Theorem will be used to expand binomials of the form \((a+b)^n\) and to find specific terms within a binomial expansion.

### Physics

Average and Instantaneous Acceleration

By the end of this lesson, you will be able to calculate the average acceleration between two points in time, calculate the instantaneous acceleration given the functional form of velocity, explain the vector nature of instantaneous acceleration and velocity, explain the difference between average acceleration and instantaneous acceleration, and find instantaneous acceleration at a specified time on a graph of velocity versus time.

### Statistics

Probability Basics

In this lesson, we set up the rules of probability.

Using the Normal Distribution

In this lesson, we explore using the normal distribution to calculate probabilities