# Pre-Calculus Honors   (#1202340)

## General Course Information and Notes

### General Information

Course Number: 1202340
Course Path:
Abbreviated Title: PRE-CALCULUS HON
Course Length: Year (Y)
Course Attributes:
• Honors
Course Level: 3
Course Status: Course Approved

## Educator Certifications

One of these educator certification options is required to teach this course.

## Student Resources

Vetted resources students can use to learn the concepts and skills in this course.

## Original Student Tutorials

Where IS that Cell Tower?:

Find the location and coverage area of cell towers by completing the square to determine the center and radius of a circle given its equation in this interactive tutorial.

Type: Original Student Tutorial

Circle Up!:

Learn how to write the equation of a circle using Pythagorean Theorem given its center and radius using step-by-step instructions on this interactive tutorial.

Type: Original Student Tutorial

Hallowed Words: Evaluating a Speaker's Effectiveness:

Examine the hallowed words of Abraham Lincoln's "Gettysburg Address." In this interactive tutorial you'll identify his point of view, reasoning, and evidence in order to evaluate his effectiveness as a speaker.

Type: Original Student Tutorial

Long Division With Polynomials:

Use long division to rewrite a rational expression of the form a(x) divided by b(x) in the form q(x) plus the quantity r(x) divided by b(x), where a(x), b(x), q(x), and r(x) are polynomials.

Type: Original Student Tutorial

Around the World with Right Triangles:

Learn how to use trigonometric ratios to solve a real-world application. There are many famous monuments across the world. The measurements of these monuments were often found using trigonometric ratios. Today, there are devices that use laser beams to measure distances and heights, but trigonometric ratios are still widely used.

Type: Original Student Tutorial

Ferris Wheel Measures:

Learn about the radian measures of an angle, finding an angle measure in radians given the arc length and length of the radius, and converting between degree measures and radian measures in this interactive tutorial.

Type: Original Student Tutorial

## Educational Game

Vector Island - Boat to Island:

This is an online game where students are asked to navigate a boat using vectors to reach a goal. There are 3 different modes of the game to help students deepen their understanding of the concept. This game is to be used in conjunction with a full lesson on vectors.

Type: Educational Game

## Perspectives Video: Experts

Mathematically Exploring the Wakulla Caves:

The tide is high!  How can we statistically prove there is a relationship between the tides on the Gulf Coast and in a fresh water spring 20 miles from each other?

Type: Perspectives Video: Expert

Wing Design and Forces of Flight:

Let this video on wing design lift you up!

Type: Perspectives Video: Expert

Electromagnetism:

The director of the National High Magnetic Field Laboratory describes electromagnetic waves.

Type: Perspectives Video: Expert

Vectors and Oceanography:

Don't let the motion of the ocean cause a commotion! Learn how vectors describe ocean movement patterns.

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

Wind Vector Arithmetic for Making a Flight Plan:

Math is important to help you get where you want to go in life, especially if you plan to fly there!

Type: Perspectives Video: Professional/Enthusiast

Vectors and Flight Forces:

When you watch this video, your knowledge related to flight and physics will really take off!

Type: Perspectives Video: Professional/Enthusiast

## Presentation/Slideshow

The Pythagorean Theorem: Geometryâ€™s Most Elegant Theorem:

This lesson teaches students about the history of the Pythagorean theorem, along with proofs and applications. It is geared toward high school Geometry students that have completed a year of Algebra and addresses the following national standards of the National Council of Teachers of Mathematics and the Mid-continent Research for Education and Learning: 1) Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships; 2) Use visualization, spatial reasoning, and geometric modeling to solve problems; 3) Understand and apply basic and advanced properties of the concepts of geometry; and 4) Use the Pythagorean theorem and its converse and properties of special right triangles to solve mathematical and real-world problems. The video portion is about thirty minutes, and with breaks could be completed in 50 minutes. (You may consider completing over two classes, particularly if you want to allow more time for activities or do some of the enrichment material). These activities could be done individually, in pairs, or groups. I think 2 or 3 students is optimal. The materials required for the activities include scissors, tape, string and markers.

Type: Presentation/Slideshow

Coins in a circular pattern:

Using a chart of diameters of different denominations of coins, students are asked to figure out how many coins fit around a central coin.

Finding the area of an equilateral triangle:

This problem solving task asks students to find the area of an equilateral triangle.

Mt. Whitney to Death Valley:

This task engages students in an open-ended modeling task that uses similarity of right triangles.

Shortest Line Segment from a Point P to a Line L:

This is a foundational geometry task designed to provide a route for students to develop some fundamental geometric properties that may seem rather obvious at first glance. In this case, the fundamental property in question is that the shortest path from a point to a line meets the line at a right angle which is crucial for many further developments in the subject.

Seven Circles III:

This provides an opportunity to model a concrete situation with mathematics. Once a representative picture of the situation described in the problem is drawn (the teacher may provide guidance here as necessary), the solution of the task requires an understanding of the definition of the sine function.

Slopes and Circles:

The purpose of this task is to lead students through an algebraic approach to a well-known result from classical geometry, namely, that a point X is on the circle of diameter AB whenever ?AXB is a right angle

Setting Up Sprinklers:

This modeling task involves several different types of geometric knowledge and problem-solving: finding areas of sectors of circles, using trigonometric ratios to solve right triangles, and decomposing a complicated figure involving multiple circular arcs into parts whose areas can be found.

Neglecting the Curvature of the Earth:

This task applies geometric concepts, namely properties of tangents to circles and of right triangles, in a modeling situation. The key geometric point in this task is to recognize that the line of sight from the mountain top towards the horizon is tangent to the earth. We can then use a right triangle where one leg is tangent to a circle and the other leg is the radius of the circle to investigate this situation.

Foxes and Rabbits 3:

This problem solving task challenges students to use trigonometric functions to model the number of rabbits and foxes as a function of time.

Foxes and Rabbits 2:

This problem solving task challenges students to trigonometric functions to model the populations of rabbits and foxes over time, and then graph the functions.

As the Wheel Turns:

In this task, students use trigonometric functions to model the movement of a point around a wheel and, through space. Students also interpret features of graphs in terms of the given real-world context.

This is a simple task touching on two key points of functions. First, there is the idea that not all functions have real numbers as domain and range values. Second, the task addresses the issue of when a function admits an inverse, and the process of "restricting the domain" in order to achieve an invertible function.

U.S. Households:

The purpose of this task is to construct and use inverse functions to model a real-life context. Students choose a linear function to model the given data, and then use the inverse function to interpolate a data point.

Temperatures in Degrees Fahrenheit and Celsius:

Temperature conversions provide a rich source of linear functions which are encountered not only in science but also in our every day lives when we travel abroad. The first part of this task provides an opportunity to construct a linear function given two input-output pairs. The second part investigates the inverse of a linear function while the third part requires reasoning about quantities and/or solving a linear equation.

Trina's Triangles:

In this task students investigate and ultimately prove the validity of the method of generating Pythagorean Triples that involves the polynomial identity (x2+y2)2=(x2−y2)2+(2xy)2.

Compounding with a 100% Interest Rate:

This task provides an approximation, and definition, of e, in the context of more and more frequent compounding of interest in a bank account. The approach is computational.

A Sum of Functions:

In this example, students are given the graph of two functions and are asked to sketch the graph of the function that is their sum. The intent is that students develop a conceptual understanding of function addition.

Graphs of Compositions:

This task addresses an important issue about inverse functions. In this case the function f is the inverse of the function g but g is not the inverse of f unless the domain of f is restricted.

Crude Oil and Gas Mileage:

This task asks students to write expressions for various problems involving distance per units of volume.

Flu on Campus:

The context of this example is the spread of a flu virus on campus and the related sale of tissue boxes sold. Students interpret the composite function and determine values simply by using the tables of values.

Exponentials and Logarithms II:

In this task, students explore the inverse relationship between an exponential function and a logarithmic function. The task is to determine the relevant composite functions, their graphs, and the domain and range of each.

Combined Fuel Efficiency:

In this example, fuel efficiency of a car can be analyzed by using rational expressions and operations with rational expressions.

Compounding with a 5% Interest Rate:

This task develops reasoning behind the general formula for balances under continuously compounded interest. While this task itself specifically address the standard (F-BF), building functions from a context, an auxiliary purpose is to introduce and motivate the number e, which plays a significant role in the (F-LE) domain of tasks.

Temperature Conversions:

Unit conversion problems provide a rich source of examples both for composition of functions (when several successive conversions are required) and inverses (units can always be converted in either of two directions).

Susita's Account:

This task asks students to determine a recursive process from a context. Students who study computer programming will make regular use of recursive processes.

Summer Intern:

This task asks students to use proportions of mass and volume to create ideal brine for saltwater fish tanks. It also asks students to compare graphs.

Skeleton Tower:

This problem is a quadratic function example. The other tasks in this set illustrate MAFS.912.F.BF.1.1.a in the context of linear, exponential, and rational functions.

Rainfall:

In this task, students are asked to analyze a function and its inverse when the function is given as a table of values. In addition to finding values of the inverse function from the table, they also have to explain why the given function is invertible.

Lake Algae:

The purpose of this task is to introduce students to exponential growth. While the context presents a classic example of exponential growth, it approaches it from a non-standard point of view.

Invertible or Not?:

This task illustrates several components of standard MAFS.912.F-BF.2.4.c: Find Inverse Functions. Here, instead of presenting two functions and asking the students to decide which on is invertible, students are asked to complete a table of input-output pairs for the functions in such a way that one of the functions is invertible and the other one is not.

The Canoe Trip, Variation 2:

The primary purpose of this task is to lead students to a numerical and graphical understanding of the behavior of a rational function near a vertical asymptote, in terms of the expression defining the function.

The Canoe Trip, Variation 1:

The purpose of this task is to give students practice constructing functions that represent a quantity of interest in a context, and then interpreting features of the function in the light of the context. It can be used as either an assessment or a teaching task.

## Tutorials

The basics of trigonometry:

This tutorial shows the basics of trigonometry, sine, cosine, and tangent. Students will determine oppopsite sides, adjacent sides and the hypotenuse of right triangles.

Type: Tutorial

Redefining a Function by Finding a Limit to Make the Function Continuous:

In this video students are introduced to an algebraic technique for rewriting a rational function in order to find a limit of the function.

Type: Tutorial

Determine the Point a Function is Continuous:

This video will help you determine at which point a function is continuous.

Type: Tutorial

Introduction to the unit circle:

This tutorial gives an introduction to the unit circle. It also extends the students knowledge of SOH CAH TOA so that they can define trigonometric functions for a broader class of angles.

Type: Tutorial

Binomial Theorem:

This video tutorial gives an introduction to the binomial theorem and explains how to use this theorem to expand binomial expressions.

Type: Tutorial

Pascal's Triangle for Binomial Expansion:

This tutorial shows students how to use Pascal's triangle for binomial expansion.

Type: Tutorial

Limit and Function Defined at Point of Discontinuity:

In this video we will determine if a limit exists at a point of discontinuity.

Type: Tutorial

Limits to Define Continuity:

We will use limits to define continuity.

Type: Tutorial

Limit by Factoring Cubic Expression:

Le'ts find a limit by factoring a cubic expression.

Type: Tutorial

Limit Properties:

Let's look at some examples of limits using various properties.

Type: Tutorial

Limits and Infinity:

We will look at more examples of limits at infinity.

Type: Tutorial

Using Trigonometry to solve for missing information:

This tutorial will show students how to use trigonometry to solve for missing information in right triangles. This video shows worked examples using trigonometric ratios to solve for missing information and evaluate other trigonometric ratios.

Type: Tutorial

Basic Trigonometry:

This tutorial gives an introduction to trigonometry. This resource discusses the three basic trigonometry functions, sine, cosine, and tangent.

Type: Tutorial

Example of a limit:

This video will demonstrate how to find a limit through sums, differences, products, and quotients.

Type: Tutorial

Introduction to limits:

This video will provide an introduction to the intuition behind limits through graphing.

Type: Tutorial

How to determine a limit at a point of discontinuity:

Tihis video demonstrates how to determine a limit at a point of discontinuity.

Type: Tutorial

How to find One-sided limits from graphs:

You will be able to find one-sided limits from graphs.

Type: Tutorial

Determining which limit statements are true:

This video demonstrates how to determine which limit statements are true.

Type: Tutorial

Two-sided limit from a graph:

Two-sided limit from graph

Type: Tutorial

Projectile at an angle:

This video discusses how to figure out the horizontal displacement for a projectile launched at an angle.

Type: Tutorial

Refraction of Light:

This resource explores the electromagnetic spectrum and waves by allowing the learner to observe the refraction of light as it passes from one medium to another, study the relation between refraction of light and the refractive index of the medium, select from a list of materials with different refractive indicecs, and change the light beam from white to monochromatic and observe the difference.

Type: Tutorial

Human Eye Accommodation:

• Observe how the eye's muscles change the shape of the lens in accordance with the distance to the object being viewed
• Indicate the parts of the eye that are responsible for vision
• View how images are formed in the eye

Type: Tutorial

Concave Spherical Mirrors:

• Learn how a concave spherical mirror generates an image
• Observe how the size and position of the image changes with the object distance from the mirror
• Learn the difference between a real image and a virtual image
• Learn some applications of concave mirrors

Type: Tutorial

Convex Spherical Mirrors:

• Learn how a convex mirror forms the image of an object
• Understand why convex mirrors form small virtual images
• Observe the change in size and position of the image with the change in object's distance from the mirror
• Learn some practical applications of convex mirrors

Type: Tutorial

Color Temperature in a Virtual Radiator:

• Observe the change of color of a black body radiator upon changes in temperature
• Understand that at 0 Kelvin or Absolute Zero there is no molecular motion

Type: Tutorial

Solar Cell Operation:

This resource explains how a solar cell converts light energy into electrical energy. The user will also learn about the different components of the solar cell and observe the relationship between photon intensity and the amount of electrical energy produced.

Type: Tutorial

Electromagnetic Wave Propagation:

• Observe that light is composed of oscillating electric and magnetic waves
• Explore the propagation of an electromagnetic wave through its electric and magnetic field vectors
• Observe the difference in propagation of light of different wavelengths

Type: Tutorial

Basic Electromagnetic Wave Properties:

• Explore the relationship between wavelength, frequency, amplitude and energy of an electromagnetic wave
• Compare the characteristics of waves of different wavelengths

Type: Tutorial

Geometrical Construction of Ray Diagrams:

• Learn to trace the path of propagating light waves using geometrical optics
• Observe the effect of changing parameters such as focal length, object dimensions and position on image properties
• Learn the equations used in determining the size and locations of images formed by thin lenses

Type: Tutorial

## Video/Audio/Animations

Will an Ice Cube Melt Faster in Freshwater or Saltwater?:

With an often unexpected outcome from a simple experiment, students can discover the factors that cause and influence thermohaline circulation in our oceans. In two 45-minute class periods, students complete activities where they observe the melting of ice cubes in saltwater and freshwater, using basic materials: clear plastic cups, ice cubes, water, salt, food coloring, and thermometers. There are no prerequisites for this lesson but it is helpful if students are familiar with the concepts of density and buoyancy as well as the salinity of seawater. It is also helpful if students understand that dissolving salt in water will lower the freezing point of water. There are additional follow up investigations that help students appreciate and understand the importance of the ocean's influence on Earth's climate.

Type: Video/Audio/Animation

Basic Linear Function:

This video demonstrates writing a function that represents a real-life scenario.

Type: Video/Audio/Animation

MIT BLOSSOMS - Fabulous Fractals and Difference Equations :

This learning video introduces students to the world of Fractal Geometry through the use of difference equations. As a prerequisite to this lesson, students would need two years of high school algebra (comfort with single variable equations) and motivation to learn basic complex arithmetic. Ms. Zager has included a complete introductory tutorial on complex arithmetic with homework assignments downloadable here. Also downloadable are some supplemental challenge problems. Time required to complete the core lesson is approximately one hour, and materials needed include a blackboard/whiteboard as well as space for students to work in small groups. During the in-class portions of this interactive lesson, students will brainstorm on the outcome of the chaos game and practice calculating trajectories of difference equations.

Type: Video/Audio/Animation

## Virtual Manipulatives

Cool Tool For Ellipses:

Use this interactive GeoGebraTube tool to see how the foci and other graph characteristics are related to the equation of the ellipse.  Make sure you use the sliders to change the characteristics of your ellipse and pay attention to how the graph relates to its equation each time.

Type: Virtual Manipulative

Right Triangle Solver:

This virtual manipulative will help the students in understanding that the relationships found in right triangles can be used to solve many applied problems in science and engineering. The right triangle solver manipulative displays a triangle with some its sides and angles given. The student is then asked to determine values of the remaining sides and angles by choosing a workable strategy.

Type: Virtual Manipulative

Triangle Solver:

The triangle solver manipulative displays a triangle with some of its sides and angles given. The students are then asked to determine values of the remaining sides and angles. Students are motivated to choose a workable strategy such as using the Pythagorean theorem, the sine, cosine, tangent relationships, the law of sines, or the law of cosines. They are directed through the key steps of the chosen strategy to find the unknown sides and the angles.

Type: Virtual Manipulative

Graphing Lines:

Allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the line and the slope can be observed.

Type: Virtual Manipulative

Data Flyer:

Using this virtual manipulative, students are able to graph a function and a set of ordered pairs on the same coordinate plane. The constants, coefficients, and exponents can be adjusted using slider bars, so the student can explore the affect on the graph as the function parameters are changed. Students can also examine the deviation of the data from the function. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

Function Flyer:

In this online tool, students input a function to create a graph where the constants, coefficients, and exponents can be adjusted by slider bars. This tool allows students to explore graphs of functions and how adjusting the numbers in the function affect the graph. Using tabs at the top of the page you can also access supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

Number Cruncher:

In this activity, students enter inputs into a function machine. Then, by examining the outputs, they must determine what function the machine is performing. This activity allows students to explore functions and what inputs are most useful for determining the function rule. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

Pythagorean Theorem Manipulatives:

This web address, from the National Library of Virtual Manipulatives, will help teachers and students validate the Pythagorean Theorem both geometrically and algebraically. It can be used interactively with the Smartboard and the Promethean Board to create a better understanding of the topic.

Type: Virtual Manipulative

Fractal Tool:

Students investigate shapes that grow and change using an iterative process. Fractals are characterized by self-similarity, smaller sections that resemble the larger figure. From NCTM's Illuminations.

Type: Virtual Manipulative

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this course.