Math

Mathematical Models

Connecting Graphs and Equations of Quadratic Relations
Connecting Graphs and Equations fo Exponenetial Realtions
SOlving Problems Incolving Exponenetial Relations

Personal Finance

Solving Problems Involving Compound INtersets
Comparing Financial Services
Owning and Operating a Vehicle

Geometry and Triganometry

Representing 2D Shapes and 3D Ficulres
Applying the Sine and Cosine Laws in Acute Triangles

Data Management

Working with 1-Variable Data
Applying Probability

Number Sense and Algebra

Operating with Exponents

Review the exponential notations and practice evaluating natural-number exponents with rational-number bases. Use the multiplication rule to derive the exponent rules: $$x^m \times x^n = x^{m+n}$$ $$\frac{x^m}{x^n} = x^{m-n}$$ $$(x^m)^n = x^{m\times n}$$ $$x^{-n} = {1\over{x^n}}$$ $$x^0 = 1$$

The purpose of this project is to produce a report on the sunspot cycle and how you developed a mathematical model of it. The report should be 2-3 pages long and include

• a graph of the data,
• your mathematical model and an explanation of the amplitude, period, phase shift and offset,
• a story of how you developed and tested it.
• A discussion on how well your model matches the data and what you could do to improve the model.
• A discussion of why or why not your model matches the current sunspot cycle.
  

Specific Expectations

2.5 determine, through investigation using technology, the roles of the parameters a, k, d, and c in functions of the form y =af (k(x – d)) + c, where f(x) =sinx or f(x) =cosx with angles expressed in degrees, and describe these roles in terms of transformations on the graphs of f(x) =sinx and f(x) =cosx

Materials

art supplies, computer access

Important Terminology

Background Knowledge

Hook

Lesson Sequence

Specific Expectations

2.4 sketch the graphs of f(x) =sinx and f(x) =cosx for angle measures expressed in degrees, and determine and describe their key properties
2.6 determine the amplitude, period, phase shift, domain, and range of sinusoidal functions whose equations are given in the form f(x) = asin(k(x – d)) + c or f(x) = acos(k(x – d)) + c
2.7 sketch graphs of y = af (k(x – d)) + c by applying one or more transformations to the graphs of f(x) =sinx and f(x) =cosx, and state the domain and range of the transformed functions

Specific Expectations

2.5 determine, through investigation using technology, the roles of the parameters a, k, d, and c in functions of the form y =af (k(x – d)) + c, where f(x) =sinx or f(x) =cosx with angles expressed in degrees, and describe these roles in terms of transformations on the graphs of f(x) =sinx and f(x) =cosx
2.4 sketch the graphs of f(x) =sinx and f(x) =cosx for angle measures expressed in degrees, and determine and describe their key properties

Materials

Internet, graph paper or a printer, radian to degrees cheat-sheet

Important Terminology

Specific Expectations

2.1 describe key properties of periodic functions arising from real-world applications, given a numeric or graphical representation
2.2 predict, by extrapolating, the future behavior of a relationship modeled using a numeric or graphical representation of a periodic function

Materials

Computer Lab

Important Terminology

amplitude, period, sunspot, solar cycle

Background Knowledge

None beyond previous lessons in unit.

Specific Expectations

2.3 make connections between the sine ratio and the sine function and between the cosine ratio and the cosine function by graphing the relationship between angles from 0º to 360º and the corresponding sine ratios or cosine ratios, defining this relationship as the function f(x) =sinx or f(x) =cosx, and explaining why the relationship is a function
2.4 sketch the graphs of f(x) =sinx and f(x) =cosx for angle measures expressed in degrees, and determine and describe their key properties

Materials

Spaghetti, yarn, paper, glue.

Rationale

The universe if full of phenomenon that are cyclical or follow a repeating pattern. Many of these can be modeled using sinusoidal functions.

Expectations

2.1 describe key properties of periodic functions arising from real-world applications given a numeric or graphical representation
2.2 predict, by extrapolating, the future behavior of a relationship modeled using a numeric or graphical representation of a periodic function

Heretofore, you have been dealing with scalar quantities. These are things like numbers, mass, length, that just have a size (magnitude). We will now formally introduce the vector. A vector is a mathematical object that has both a magnitude and a direction. Vectors are used to describe things like position, forces, weight, and hence play a big part in physics. In order to describe the direction of a vector, we need a frame of reference.