# Intensive Mathematics   (#1200400)

## General Course Information and Notes

### Version Description

For each year in which a student scores at Level 1 on FCAT 2.0 Mathematics, the student must receive remediation by completing an intensive mathematics course the following year or having the remediation integrated into the student's required mathematics course. This course should be tailored to meet the needs of the individual student. Appropriate benchmarks from the following set of standards should be identified to develop an appropriate curriculum.

### General Information

Course Number: 1200400
Course Path:
Abbreviated Title: INTENS MATH
Number of Credits: Multiple Credit (more than 1 credit)
Course Type: Elective Course
Course Level: 2
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

Factoring Polynomials with Greatest Common Factor:

Learn how to factor polynomials by finding their greatest common factor in this interactive tutorial.

Type: Original Student Tutorial

Learn about different formats of quadratic equations and their graphs with experiments involving launching and shooting of balls in this interactive tutorial.

This is part 2 of a two-part series: Click HERE to open part 1.

Type: Original Student Tutorial

Quadratic Functions Part 1: Ball Games:

Join us as we watch ball games and explore how the height of a ball bounce over time is represented by quadratic functions, which provides opportunities to interpret key features of the function in this interactive tutorial.

This is part 1 of a two-part series: Click HERE to open part 2.

Type: Original Student Tutorial

Learn how to use multistep factoring to factor quadratics in this interactive tutorial.

This is part 5 in a five-part series. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

Solving Systems of Linear Equations Part 7: Word Problems:

Learn to solve word problems represented by systems of linear equations, algebraically and graphically, in this interactive tutorial.

This part 7 in a 7-part series. Click below to explore the other tutorials in the series.

Type: Original Student Tutorial

Factoring Polynomials when "a" Does Not Equal 1, Snowflake Method:

Learn to factor quadratic trinomials when the coefficient a does not equal 1 by using the Snowflake Method in this interactive tutorial.

This is part 4 in a five-part series. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

Solving Systems of Linear Equations Part 6: Writing Systems from Context:

Learn how to create systems of linear equations to represent contextual situations in this interactive tutorial.

This part 6 in a 7-part series. Click below to explore the other tutorials in the series.

Type: Original Student Tutorial

Solving Systems of Linear Equations Part 5: Connecting Algebraic Methods to Graphing:

Learn to solve systems of linear equations by connecting algebraic and graphing methods in this interactive tutorial.

This part 5 in a 7-part series. Click below to explore the other tutorials in the series.

Type: Original Student Tutorial

Factoring Quadratics When the Coefficient a Does Not Equal 1: The Box Method:

Learn how to factor quadratic polynomials when the leading coefficient (a) is not 1 by using the box method in this interactive tutorial.

This is part 3 in a five-part series. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

The Diamond Game: Factoring Quadratics when a = 1:

Learn how to factor quadratics when the coefficient a = 1 using the diamond method in this game show-themed, interactive tutorial.

This is part 1 in a five-part series. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

Solving Systems of Linear Equations Part 4: Advanced Elimination:

Learn to solve systems of linear equations using advanced elimination in this interactive tutorial.

This part 4 in a 7-part series. Click below to explore the other tutorials in the series.

Type: Original Student Tutorial

Solving Systems of Linear Equations Part 3: Basic Elimination:

Learn to solve systems of linear equations using basic elimination in this interactive tutorial.

This part 3 in a 7-part series. Click below to explore the other tutorials in the series.

Part 1: Solving Systems of Linear Equations Part 1: Using Graphs

Part 2: Solving Systems of Linear Equations Part 2: Substitution

Part 4: Solving Systems of Linear Equations Part 4: Advanced Elimination (Coming soon)
Part 5: Solving Systems of Linear Equations Part 5: Connecting Algebraic Methods to Graphing (Coming soon)
Part 6: Solving Systems of Linear Equations Part 6: Writing Systems from Context (Coming soon)
Part 7: Solving Systems of Linear Equations Part 7: Word Problems (Coming soon)

Type: Original Student Tutorial

Highs and Lows Part 2: Completing the Square:

Learn the process of completing the square of a quadratic function to find the maximum or minimum to discover how high a dolphin jumped in this interactive tutorial.

This is part 2 of a 2 part series. Click HERE to open part 1.

Type: Original Student Tutorial

Highs and Lows Part 1: Completing the Square:

Learn the process of completing the square of a quadratic function to find the maximum or minimum to discover how high a dolphin jumped in this interactive tutorial.

This is part 1 of a 2 part series. Click HERE to open Part 2.

Type: Original Student Tutorial

Identifying Parts of Linear Expressions:

Learn to identify and interpret parts of linear expressions in terms of mathematical or real-world contexts in this original tutorial.

Type: Original Student Tutorial

Movies Part 2: Whatâ€™s the Spread?:

Follow Jake along as he relates box plots with other plots and identifies possible outliers in real-world data from surveys of moviegoers' ages in part 2 in this interactive tutorial.

This is part 2 of 2-part series, click HERE to view part 1.

Type: Original Student Tutorial

Solving Systems of Linear Equations Part 2: Substitution:

Learn to solve systems of linear equations using substitution in this interactive tutorial.

This part 2 in a 7-part series. Click below to explore the other tutorials in the series.

Part 1: Solving Systems of Linear Equations Part 1: Using Graphs

Part 3: Solving Systems of Linear Equations Part 3: Basic Elimination (Coming soon)

Part 4: Solving Systems of Linear Equations Part 4: Advanced Elimination (Coming soon)

Part 5: Solving Systems of Linear Equations Part 5: Connecting Algebraic Methods to Graphing (Coming soon)

Part 6: Solving Systems of Linear Equations Part 6: Writing Systems from Context (Coming soon)

Part 7: Solving Systems of Linear Equations Part 7: Word Problems (Coming soon)

Type: Original Student Tutorial

Movies Part 1: What's the Spread?:

Follow Jake as he displays real-world data by creating box plots showing the 5 number summary and compares the spread of the data from surveys of the ages of moviegoers in part 1 of this interactive tutorial.

This is part 1 of 2-part series, click HERE to view part 2.

Type: Original Student Tutorial

Exponential Functions Part 3: Decay:

Learn about exponential decay as you calculate the value of used cars by examining equations, graphs, and tables in this interactive tutorial.

Type: Original Student Tutorial

Solving Systems of Linear Equations Part 1: Using Graphs:

Learn how to solve systems of linear equations graphically in this interactive tutorial.

Type: Original Student Tutorial

Linear Functions: Jobs:

Learn how to interpret key features of linear functions and translate between representations of linear functions through exploring jobs for teenagers in this interactive tutorial.

Type: Original Student Tutorial

Exponential Functions Part 2: Growth:

Learn about exponential growth in the context of interest earned as money is put in a savings account by examining equations, graphs, and tables in this interactive tutorial.

Type: Original Student Tutorial

Exponential Functions Part 1:

Learn about exponential functions and how they are different from linear functions by examining real world situations, their graphs and their tables in this interactive tutorial.

Type: Original Student Tutorial

Functions, Functions, Everywhere: Part 2:

Continue exploring how to determine if a relation is a function using graphs and story situations in this interactive tutorial.

This is the second tutorial in a 2-part series. Click HERE to open Part 1.

Type: Original Student Tutorial

Travel with Functions:

Learn how to evaluate and interpret function notation by following Melissa and Jose on their travels in this interactive tutorial.

Type: Original Student Tutorial

Functions, Functions Everywhere: Part 1:

What is a function? Where do we see functions in real life? Explore these questions and more using different contexts in this interactive tutorial.

This is part 1 in a two-part series on functions. Click HERE to open Part 2.

Type: Original Student Tutorial

Solving Rational Equations: Using Common Denominators:

Learn how to solve rational functions by getting common denominators in this interactive tutorial.

Type: Original Student Tutorial

Dilations...The Effect of k on a Graph:

Visualize the effect of using a value of k in both kf(x) or f(kx) when k is greater than zero in this interactive tutorial.

Type: Original Student Tutorial

A Character Reborn in The Count of Monte Cristo -- Part Three:

Explain how Edmond Dantès’ overall transformation takes the plot in a new direction as you continue to examine how the main character is reborn from a prisoner into a newly freed man in a chapter from The Count of Monte Cristo by Alexandre Dumas.

You should complete Part One and Part Two before beginning Part Three.

• Click HERE to launch Part One.
• Click HERE to launch Part Two.

Type: Original Student Tutorial

A Character Reborn in The Count of Monte Cristo -- Part Two:

As you continue to study a chapter from The Count of Monte Cristo by Alexandre Dumas, you'll examine how the main character, Edmond Dantès, is reborn from a prisoner into a newly freed man, identify his key character traits or strengths, and examine how he begins to transform as he works to secure his freedom.

This is part 2 of a three-part interactive tutorial series. Make sure to complete all three parts!

• Click HERE to launch Part One.
• Click HERE to launch Part Three.

Type: Original Student Tutorial

Solving Rational Equations: Cross Multiplying:

Learn how to solve rational linear and quadratic equations using cross multiplication in this interactive tutorial.

Type: Original Student Tutorial

A Character Reborn in The Count of Monte Cristo -- Part One:

Study a chapter from one of the most popular adventure stories of all time: The Count of Monte Cristo by Alexandre Dumas. In this interactive tutorial, you will identify key character traits or strengths of Edmond Dantès and determine how he draws on these strengths as he struggles to survive and avoid recapture.

This is part 1 of a three-part series. Make sure to complete all three parts!

Type: Original Student Tutorial

Solving Inequalities and Graphing Solutions Part 2:

Learn how to solve and graph compound inequalities and determine if solutions are viable in part 2 of this interactive tutorial series.

Type: Original Student Tutorial

Writing Equations in Two Variables:

Learn how to write equations in two variables in this interactive tutorial.

Type: Original Student Tutorial

Solving Inequalities and Graphing Solutions: Part 1:

Learn how to solve and graph one variable inequalities, including compound inequalities, in part 1 of this interactive tutorial series.

Type: Original Student Tutorial

Reflections...The Effect of k on a Graph:

Learn how reflections of a function are created and tied to the value of k in the mapping of f(x) to -1f(x) in this interactive tutorial.

Type: Original Student Tutorial

Translations...The Effect of k on the Graph:

Explore translations of functions on a graph that are caused by k in this interactive tutorial. GeoGebra and interactive practice items are used to investigate linear, quadratic, and exponential functions and their graphs, and the effect of a translation on a table of values.

Type: Original Student Tutorial

Angle UP: Player 1:

Explore the construction processes for constructing an angle bisector, copying an angle and constructing a line parallel to a given line through a point not on the line using a variety of tools in this interactive, retro video game-themed tutorial.

NOTE: This tutorial uses both the angle bisector construction and the construction to copy an angle as an extension opportunity to also construct a line parallel to a given line through a point not on the line. Students also learn to identify corresponding angles created when a transversal crosses parallel lines, and discover using Geogebra that these angles are congruent.

Type: Original Student Tutorial

The Year-Round School Debate: Identifying Faulty Reasoning â€“ Part Two:

This is Part Two of a two-part series. Learn to identify faulty reasoning in this interactive tutorial series. You'll learn what some experts say about year-round schools, what research has been conducted about their effectiveness, and how arguments can be made for and against year-round education. Then, you'll read a speech in favor of year-round schools and identify faulty reasoning within the argument, specifically the use of hasty generalizations.

Make sure to complete Part One before Part Two! Click HERE to launch Part One.

Type: Original Student Tutorial

Pennant Company Challenge: Inscribed Circles of Triangles:

Discover how easy it is for Katie to construct an inscribed circular logo on her company's triangular pennant template. If she completes the task first, she will win a \$1000 bonus! Follow along with this interactive tutorial.

Type: Original Student Tutorial

The Year-Round School Debate: Identifying Faulty Reasoning â€“ Part One:

Learn to identify faulty reasoning in this two-part interactive English Language Arts tutorial. You'll learn what some experts say about year-round schools, what research has been conducted about their effectiveness, and how arguments can be made for and against year-round education. Then, you'll read a speech in favor of year-round schools and identify faulty reasoning within the argument, specifically the use of hasty generalizations.

Make sure to complete both parts of this series! Click HERE to open Part Two.

Type: Original Student Tutorial

Meet Me Half Way:

Plan a paddle board expedition by learning how to do basic geometric constructions including copying a segment, constructing a segment bisector, constructing a segment's perpendicular bisector and constructing perpendicular segments using a variety of tools in this interactive tutorial.

Type: Original Student Tutorial

Light and Darkness in Two Artistic Mediums:

Study the poem “We Grow Accustomed to the Dark” by Emily Dickinson and view the painting The Starry Night by Vincent van Gogh to explain how each medium represents the subjects of light and darkness similarly and differently, as you complete this interactive tutorial.

Type: Original Student Tutorial

Evaluating an Argument â€“ Part Four: JFKâ€™s Inaugural Address:

Examine President John F. Kennedy's inaugural address in this interactive tutorial. You will examine Kennedy's argument, main claim, smaller claims, reasons, and evidence.

In Part Four, you'll use what you've learned throughout this series to evaluate Kennedy's overall argument.

Make sure to complete the previous parts of this series before beginning Part 4.

• Click HERE to launch Part Two.
• Click HERE to launch Part Three.

Type: Original Student Tutorial

A Square Peg in a Round Hole:

Learn how to construct an inscribed square in a circle and why certain constructions are used in this interactive tutorial.

Type: Original Student Tutorial

Evaluating an Argument â€“ Part Three: JFKâ€™s Inaugural Address:

Examine President John F. Kennedy's inaugural address in this interactive tutorial. You will examine Kennedy's argument, main claim, smaller claims, reasons, and evidence. By the end of this four-part series, you should be able to evaluate his overall argument.

In Part Three, you will read more of Kennedy's speech and identify a smaller claim in this section of his speech. You will also evaluate this smaller claim's relevancy to the main claim and evaluate Kennedy's reasons and evidence.

Make sure to complete all four parts of this series!

• Click HERE to launch Part Two.

Type: Original Student Tutorial

Where IS that Cell Tower?:

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

Type: Original Student Tutorial

Designing with Hexagons:

Learn how to construct an inscribed regular hexagon and equilateral triangle in a circle in this interactive tutorial.

Type: Original Student Tutorial

Good as New:

Learn the steps to circumscribe a circle around a triangle in this interactive tutorial about constructions. Grab a compass, straightedge, pencil and paper to follow along!

Type: Original Student Tutorial

High Tech Seesaw:

Learn how to find the point on a directed line segment that partitions it into a given ratio in this interactive tutorial.

Type: Original Student Tutorial

It's a Slippery Slope!:

Learn what slope is in mathematics and how to calculate it on a graph and with the slope formula 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 in this interactive tutorial.

Type: Original Student Tutorial

Finding the Zeros of Quadratic Functions:

Learn to find the zeros of a quadratic function and interpret their meaning in real-world contexts with this interactive tutorial.

Type: Original Student Tutorial

Introduction to Polynomials, Part 2 - Adding and Subtracting:

Learn how to add and subtract polynomials in this online tutorial. You will learn how to combine like terms and then use the distribute property to subtract polynomials.

This is part 2 of a two-part lesson. Click below to open part 1.

Type: Original Student Tutorial

Introduction to Polynomials: Part 1:

Learn how to identify monomials and polynomials and determine their degree in this interactive tutorial.

This is part 1 in a two-part series. Click here to open Part 2.

Type: Original Student Tutorial

Ready for Takeoff! -- Part Two:

This is Part Two of a two-part tutorial series. In this interactive tutorial, you'll practice identifying a speaker's purpose using a speech by aviation pioneer Amelia Earhart. You will examine her use of rhetorical appeals, including ethos, logos, pathos, and kairos. Finally, you'll evaluate the effectiveness of Earhart's use of rhetorical appeals.

Be sure to complete Part One first. Click here to launch PART ONE.

Type: Original Student Tutorial

Ready for Takeoff! -- Part One:

This is Part One of a two-part tutorial series. In this interactive tutorial, you'll practice identifying a speaker's purpose using a speech by aviation pioneer Amelia Earhart. You will examine her use of rhetorical appeals, including ethos, logos, pathos, and kairos. Finally, you'll evaluate the effectiveness of Earhart's use of rhetorical appeals.

Type: Original Student Tutorial

Analyzing Word Choices in Poe's "The Raven" -- Part Two:

Practice analyzing word choices in "The Raven" by Edgar Allan Poe, including word meanings, subtle differences between words with similar meanings, and emotions connected to specific words. In this interactive tutorial, you will also analyze the impact of specific word choices on the meaning of the poem.

This is Part Two of a two-part series. Part One should be completed before beginning Part Two. Click HERE to open Part One.

Type: Original Student Tutorial

Analyzing Word Choices in Poe's "The Raven" -- Part One:

Practice analyzing word choices in "The Raven" by Edgar Allan Poe in this interactive tutorial. In this tutorial, you will examine word meanings, examine subtle differences between words with similar meanings, and think about emotions connected to specific words. You will also analyze the impact of specific word choices on the meaning of the poem.

This tutorial is Part One of a two-part series on Poe's "The Raven." Click HERE to open Part Two.

Type: Original Student Tutorial

The Great We: Analyzing Word Choice and Tone, Part 2:

Practice using textual details and connotative meanings to determine a speaker's/narrator's tone in this two-part interactive tutorial. You'll also analyze the impact of word choices on the meaning and tone of the text in excerpts from Ayn Rand's dystopian novella Anthem.

Make sure to complete Part One before you begin Part Two. Click HERE to open Part One.

Type: Original Student Tutorial

Expository Writing: Eyes in the Sky (Part 4 of 4):

Practice writing different aspects of an expository essay about scientists using drones to research glaciers in Peru. This interactive tutorial is part four of a four-part series. In this final tutorial, you will learn about the elements of a body paragraph. You will also create a body paragraph with supporting evidence. Finally, you will learn about the elements of a conclusion and practice creating a “gift.”

This tutorial is part four of a four-part series. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

Expository Writing: Eyes in the Sky (Part 3 of 4):

Learn how to write an introduction for an expository essay in this interactive tutorial. This tutorial is the third part of a four-part series. In previous tutorials in this series, students analyzed an informational text and video about scientists using drones to explore glaciers in Peru. Students also determined the central idea and important details of the text and wrote an effective summary. In part three, you'll learn how to write an introduction for an expository essay about the scientists' research.

This tutorial is part three of a four-part series. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

The Great We: Analyzing Word Choice and Tone, Part One:

Practice using textual details and connotative meanings to help you determine a speaker's/narrator's tone in this two-part interactive tutorial. You'll also analyze the impact of word choices on the meaning and tone of a text. This tutorial series features excerpts from Ayn Rand's dystopian novella Anthem.

Part One should be completed before beginning Part Two. Click HERE to launch Part Two.

Type: Original Student Tutorial

Data and Frequencies:

Learn to define, calculate, and interpret marginal frequencies, joint frequencies, and conditional frequencies in the context of the data with this interactive tutorial.

Type: Original Student Tutorial

Finding the Maximum or Minimum of a Quadratic Function:

Learn to complete the square of a quadratic expression and identify the maximum or minimum value of the quadratic function it defines. In this interactive tutorial, you'll also interpret the meaning of the maximum and minimum of a quadratic function in a real world context.

Type: Original Student Tutorial

Graphing Linear Inequalities:

Learn to graph linear inequalities in two variables to display their solutions as you complete this interactive tutorial.

Type: Original Student Tutorial

Learn to rewrite products involving radicals and rational exponents using properties of exponents in this interactive tutorial.

Type: Original Student Tutorial

The Blueprints of Construction:

Learn to construct the perpendicular bisector of a line segment using a straightedge and compass with this interactive tutorial.

Type: Original Student Tutorial

Creating Exponential Functions:

Follow as we construct an exponential function from a graph, from a table of values, and from a description of a relationship in the real world in this interactive tutorial.

Type: Original Student Tutorial

Ninja Nancy Slices:

Learn how to determine the shape of a cross-section created by the intersection of a slicing plane with a pyramid or prism in this ninja-themed, interactive tutorial.

Type: Original Student Tutorial

Around the World with Right Triangles:

Learn how to use trigonometric ratios to find the heights of famous monuments and solve a real-world application in this interactive tutorial.

Type: Original Student Tutorial

Scout Learns Life Lessons: Analyzing How a Character Develops Themes:

Examine some of the various topics and themes present in the American classic To Kill A Mockingbird by Harper Lee. In this interactive tutorial, you'll read excerpts from the novel and examine the development of the main character, Scout. You'll analyze how her words and actions help develop the important themes of the novel. You'll wrap up the tutorial by creating your own theme statement based on the text.

Type: Original Student Tutorial

Changing Rates:

Learn how to calculate and interpret an average rate of change over a specific interval on a graph in this interactive tutorial.

Type: Original Student Tutorial

Figure it Out!:

Examine the use of hyperbole and personification in the prologue of the novel The Book Thief by Markus Zusak. In this interactive tutorial, you'll practice identifying examples of hyperbole and personification within the text. You'll also learn how these two types of figurative language help authors convey their intended meaning.

Type: Original Student Tutorial

Justifiable Steps:

Learn how to explain the steps used to solve multi-step linear equations and provide reasons to support those steps with this interactive tutorial.

Type: Original Student Tutorial

Use properties, postulates, and theorems to prove a theorem about a triangle. In this interactive tutorial, you'll also learn how to prove that a line parallel to one side of a triangle divides the other two proportionally.

Type: Original Student Tutorial

Understanding Figurative Language in Poetry:

Learn to identify figurative language within poetry, including the use of similes, metaphors, and personification. In this interactive tutorial, we'll analyze William Wordsworth’s “I Wandered Lonely as a Cloud” and William Shakespeare’s “Shall I compare thee to a summer’s day?” to discover how figurative language contributes to the meaning of each poem.

Type: Original Student Tutorial

Finding Solutions on a Graph:

Learn to determine the number of possible solutions for a linear equation with this interactive tutorial.

Type: Original Student Tutorial

Solving an Equation Using a Graph:

Follow as we learn why the x-coordinate of the point of intersection of two functions is the solution of the equation f(x) = g(x) in this interactive tutorial.

Type: Original Student Tutorial

I Scream! You Scream! We All Scream for... Volume!:

Learn to calculate the volume of a cone as you solve real-world problems in this ice cream-themed, interactive tutorial.

Type: Original Student Tutorial

Women at Work:

Learn to analyze the same topic represented in a variety of different mediums. In this interactive tutorial, you'll analyze both visual and written representations from the 1930s and 1940s. You will learn about some of the common composition features used in visual mediums, such as photographs or paintings. You'll learn the necessary steps to analyze and compare works represented in both visual and literary mediums.

Type: Original Student Tutorial

Analyzing A Complex Character - Fahrenheit 451:

Analyze a complex character’s development in text excerpts from the novel Fahrenheit 451 by Ray Bradbury. In this interactive tutorial, you'll analyze how the main character is described and developed through his interaction with other characters.

Type: Original Student Tutorial

Cancer: Mutated Cells Gone Wild!:

Explore the relationship between mutations, the cell cycle, and uncontrolled cell growth which may result in cancer with this interactive tutorial.

Type: Original Student Tutorial

Writing Inequalities with Money, Money, Money:

Write linear inequalities for different money situations in this interactive tutorial.

Type: Original Student Tutorial

Factoring Polynomials Using Special Cases:

Learn how to factor quadratic polynomials that follow special cases, difference of squares and perfect square trinomials, in this interactive tutorial.

This is part 2 in a five-part series. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

## Educational Games

Solving Inequalities: Inequalities and Graphs of Inequalities:

In this challenge game, you will be solving inequalities and working with graphs of inequalities. Use the "Teach Me" button to review content before the challenge. During the challenge you get one free solve and two hints! After the challenge, review the problems as needed. Try again to get all challenge questions right! Question sets vary with each game, so feel free to play the game multiple times as needed! Good luck!

Type: Educational Game

Timed Algebra Quiz:

In this timed activity, students solve linear equations (one- and two-step) or quadratic equations of varying difficulty depending on the initial conditions they select. This activity allows students to practice solving equations while the activity records their score, so they can track their progress. 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: Educational Game

Algebra Four:

In this activity, two students play a simulated game of Connect Four, but in order to place a piece on the board, they must correctly solve an algebraic equation. This activity allows students to practice solving equations of varying difficulty: one-step, two-step, or quadratic equations and using the distributive property if desired. 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: Educational Game

## Educational Software / Tools

This Excel spreadsheet allows the educator to input data into a two way frequency table and have the resulting relative frequency charts calculated automatically on the second sheet. This resource will assist the educator in checking student calculations on student-generated data quickly and easily.

Steps to add data: All data is input on the first spreadsheet; all tables are calculated on the second spreadsheet

2. Input joint frequency data.
3. Click the second tab at the bottom of the window to see the automatic calculations.

Type: Educational Software / Tool

Transformations Using Technology:

This virtual manipulative can be used to demonstrate and explore the effect of translation, rotation, and/or reflection on a variety of plane figures. A series of transformations can be explored to result in a specified final image.

Type: Educational Software / Tool

## Lesson Plan

Do Credit Cards Make You Gain Weight? What is Correlation, and How to Distinguish It from Causation:

This lesson introduces the students to the concepts of correlation and causation, and the difference between the two. The main learning objective is to encourage students to think critically about various possible explanations for a correlation, and to evaluate their plausibility, rather than passively taking presented information on faith. To give students the right tools for such analysis, the lesson covers most common reasons behind a correlation, and different possible types of causation.

Type: Lesson Plan

## Perspectives Video: Experts

Type: Perspectives Video: Expert

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

MicroGravity Sensors & Statistics:

Statistical analysis played an essential role in using microgravity sensors to determine location of caves in Wakulla County.

Type: Perspectives Video: Expert

Problem Solving with Project Constraints:

<p>It's important to stay inside the lines of your project constraints to finish in time and under budget. This NASA systems engineer explains how constraints can actually promote creativity and help him solve problems!</p>

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

Base 16 Notation in Computing:

Listen in as a computing enthusiast describes how hexadecimal notation is used to express big numbers in just a little space.

Type: Perspectives Video: Professional/Enthusiast

Unit Conversions:

Type: Perspectives Video: Professional/Enthusiast

Making Candy: Illuminating Exponential Growth:

<p>No need to sugar coat it: making candy involves math and muscles. Learn how light refraction and exponential growth help make candy colors just right!</p>

Type: Perspectives Video: Professional/Enthusiast

Making Candy: Uniform Scaling:

<p>Don't be a shrinking violet. Learn how uniform scaling is important for candy production.</p>

Type: Perspectives Video: Professional/Enthusiast

Using Geometry and Computers to make Art with CNC Machining:

<p>See and see far into the future of arts and manufacturing as a technician explains computer numerically controlled (CNC) machining bit by bit.</p>

Type: Perspectives Video: Professional/Enthusiast

Correlation and Causation in a Scientific Study:

Watching this video will cause your critical thinking skills to improve. You might also have a great day, but that's just correlation.

Type: Perspectives Video: Professional/Enthusiast

Estimating Oil Seep Production by Bubble Volume:

<p>You'll need to bring your computer skills and math knowledge to estimate oil volume and rate as it seeps from the ocean floor. Dive in!</p>

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

Students explore the structure of the operation s/(vn). This question provides students with an opportunity to see expressions as constructed out of a sequence of operations: first taking the square root of n, then dividing the result of that operation into s.

Speed Trap:

The purpose of this task is to allow students to demonstrate an ability to construct boxplots and to use boxplots as the basis for comparing distributions.

Musical Preferences:

This problem solving task asks students to make deductions about the kind of music students enjoy by examining data in a two-way table.

Haircut Costs:

This problem could be used as an introductory lesson to introduce group comparisons and to engage students in a question they may find amusing and interesting.

Golf and Divorce:

This is a simple task addressing the distinction between correlation and causation. Students are given information indicating a correlation between two variables, and are asked to reason out whether or not a causation can be inferred.

The purpose of this task is to assess ability to interpret the slope and intercept of the line of fit in context.

Coffee and Crime:

This problem solving task asks students to examine the relationship between shops and crimes by using a correlation coefficient. The implications of linking correlation with causation are discussed.

Random Walk III:

The task provides a context to calculate discrete probabilities and represent them on a bar graph.

Bank Shot:

This task asks students to use similarity to solve a problem in a context that will be familiar to many, though most students are accustomed to using intuition rather than geometric reasoning to set up the shot.

Are They Similar?:

In this problem, students are given a picture of two triangles that appear to be similar, but whose similarity cannot be proven without further information. Asking students to provide a sequence of similarity transformations that maps one triangle to the other, using the definition of similarity in terms of similarity transformations.

Extensions, Bisections and Dissections in a Rectangle:

This task involves a reasonably direct application of similar triangles, coupled with a moderately challenging procedure of constructing a diagram from a verbal description.

Toilet Roll:

The purpose of this task is to engage students in geometric modeling, and in particular to deduce algebraic relationships between variables stemming from geometric constraints.

Algae Blooms:

In this example, students are asked to write a function describing the population growth of algae. It is implied that this is exponential growth.

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. (For this task, United States coins are used, but the task can be adapted for coins from other countries.)

Finding the area of an equilateral triangle:

This problem solving task asks students to find the area of an equilateral triangle. Various solutions are presented that include the Pythagoren theorem and trigonometric functions.

The Lighthouse Problem:

This problem asks students to model phenomena on the surface of the earth by examining the visibility of the lamp in a lighthouse from a boat.

Solar Eclipse:

This problem solving task encourages students to explore why solar eclipses are rare by examining the radius of the sun and the furthest distance between the moon and the earth.

A Midpoint Miracle:

This problem solving task gives students the opportunity to prove a fact about quadrilaterals: that if we join the midpoints of an arbitrary quadrilateral to form a new quadrilateral, then the new quadrilateral is a parallelogram, even if the original quadrilateral was not.

Mt. Whitney to Death Valley:

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

The Shortest Line Segment from Point P to 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.

Joining two midpoints of sides of a triangle:

Using a triangle with line through it, students are tasked to show the congruent angles, and conclude if one triangle is similar to the other.

Dilating a Line:

This task asks students to make deductions about a line after it has been dilated by a factor of 2.

Running around a track II:

The goal of this task is to model a familiar object, an Olympic track, using geometric shapes. Calculations of perimeters of these shapes explain the staggered start of runners in a 400 meter race.

Running around a track I:

In this problem, geometry is applied to a 400 meter track to find the perimeter of the track.

Paper Clip:

In this task, a typographic grid system serves as the background for a standard paper clip. A metric measurement scale is drawn across the bottom of the grid and the paper clip extends in both directions slightly beyond the grid. Students are given the approximate length of the paper clip and determine the number of like paper clips made from a given length of wire.

Ice Cream Cone:

In this task, students will provide a sketch of a paper ice cream cone wrapper, use the sketch to develop a formula for the surface area of the wrapper, and estimate the maximum number of wrappers that could be cut from a rectangular piece of paper.

How thick is a soda can? (Variation II):

This problem solving task asks students to explain which measurements are needed to estimate the thickness of a soda can. Multiple solution processes are presented.

How thick is a soda can? (Variation I):

This problem solving task challenges students to find the surface area of a soda can, calculate how many cubic centimeters of aluminum it contains, and estimate how thick it is.

How many leaves on a tree? (Version 2):

This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.

How many leaves on a tree?:

This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.

How many cells are in the human body?:

This problem solving task challenges students to apply the concepts of mass, volume, and density in the real-world context to find how many cells are in the human body.

Hexagonal pattern of beehives:

The goal of this task is to use geometry to study the structure of beehives.

Global Positioning System II:

Reflective of the modernness of the technology involved, this is a challenging geometric modeling task in which students discover from scratch the geometric principles underlying the software used by GPS systems.

Eratosthenes and the circumference of the earth:

This problem solving task gives an interesting context for implementing ideas from geometry and trigonometry.

Archimedes and the King's Crown:

This problem solving task uses the tale of Archimedes and the King of Syracuse's crown to determine the volume and mass of gold and silver.

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 angle AXB is a right angle.

Unit Squares and Triangles:

This problem solving task asks students to find the area of a triangle by using unit squares and line segments.

Triangles inscribed in a circle:

This problem solving task challenges students to use ideas about linear functions in order to determine when certain angles are right angles.

Doctor's Appointment:

The purpose of the task is to analyze a plausible real-life scenario using a geometric model. The task requires knowledge of volume formulas for cylinders and cones, some geometric reasoning involving similar triangles, and pays attention to reasonable approximations and maintaining reasonable levels of accuracy throughout.

Why Does ASA Work?:

This problem solving task ask students to show the reflection of one triangle maps to another triangle.

When Does SSA Work to Determine Triangle Congruence?:

In this problem, we considered SSA. The triangle congruence criteria, SSS, SAS, ASA, all require three pieces of information. It is interesting, however, that not all three pieces of information about sides and angles are sufficient to determine a triangle up to congruence.

Seven Circles II:

This task provides a concrete geometric setting in which to study rigid transformations of the plane.

Points equidistant from two points in the plane:

This task asks students to show how certain points on a plane are equidistant to points on a segment when placed on a perpendicular bisector.

Midpoints of the Side of a Parallelogram:

This is a reasonably direct task aimed at having students use previously-derived results to learn new facts about parallelograms, as opposed to deriving them from first principles.

Inscribing a square in a circle:

This task provides an opportunity for students to apply triangle congruence theorems in an explicit, interesting context.

Inscribing a hexagon in a circle:

This problem solving task challenges students to inscribe equilateral triangles and regular hexagons on a circle with a compass and straightedge.

Construction of perpendicular bisector:

This problem solving task challenges students to construct a perpendicular bisector of a given segment.

Why Does SAS Work?:

This problem solving task challenges students to explain the reason why the given triangles are congruent, and to construct reflections of the points.

Reflections and Isosceles Triangles:

This activity uses rigid transformations of the plane to explore symmetries of classes of triangles.

Reflections and Equilateral Triangles II:

This task gives students a chance to see the impact of reflections on an explicit object and to see that the reflections do not always commute.

Reflections and Equilateral Triangles:

This activity is one in a series of tasks using rigid transformations of the plane to explore symmetries of classes of triangles, with this task in particular focusing on the class of equilaterial triangles

Reflected Triangles:

This task asks students to use a straightedge and compass to construct the line across which a triangle is reflected.

Centerpiece:

The purpose of this task is to use geometric and algebraic reasoning to model a real-life scenario. In particular, students are in several places (implicitly or explicitly) to reason as to when making approximations is reasonable and when to round, when to use equalities vs. inequalities, and the choice of units to work with (e.g., mm vs. cm).

Use Cavalieriâ€™s Principle to Compare Aquarium Volumes:

This task presents a context that leads students toward discovery of the formula for calculating the volume of a sphere.

Tennis Balls in a Can:

This task is inspired by the derivation of the volume formula for the sphere. If a sphere of radius 1 is enclosed in a cylinder of radius 1 and height 2, then the volume not occupied by the sphere is equal to the volume of a "double-naped cone" with vertex at the center of the sphere and bases equal to the bases of the cylinder

Two Wheels and a Belt:

This task combines two skills: making use of the relationship between a tangent segment to a circle and the radius touching that tangent segment, and computing lengths of circular arcs given the radii and central angles.

Right triangles inscribed in circles II:

Right triangles inscribed in circles I:

This task provides a good opportunity to use isosceles triangles and their properties to show an interesting and important result about triangles inscribed in a circle: the fact that these triangles are always right triangles is often referred to as Thales' theorem.

Placing a Fire Hydrant:

This problem solving task asks students to place a fire hydrant so that it is equal distance from three given points.

Why does SSS work?:

This particular problem solving task exhibits congruency between two triangles, demonstrating translation, reflection and rotation.

Building a tile pattern by reflecting octagons:

This task applies reflections to a regular octagon to construct a pattern of four octagons enclosing a quadrilateral: the focus of the task is on using the properties of reflections to deduce that the quadrilateral is actually a square.

Building a tile pattern by reflecting hexagons:

This task applies reflections to a regular hexagon to construct a pattern of six hexagons enclosing a seventh: the focus of the task is on using the properties of reflections to deduce this seven hexagon pattern.

Bisecting an angle:

This problem solving task challenges students to bisect a given angle.

Are the Triangles Congruent?:

The purpose of this task is primarily assessment-oriented, asking students to demonstrate knowledge of how to determine the congruency of triangles.

Locating Warehouse:

This problem solving task challenges students to place a warehouse (point) an equal distance from three roads (lines).

Inscribing a triangle in a circle:

This problem introduces the circumcenter of a triangle and shows how it can be used to inscribe the triangle in a circle.

Circumcenter of a triangle:

This task shows that the three perpendicular bisectors of the sides of a triangle all meet in a point, using the characterization of the perpendicular bisector of a line segment as the set of points equidistant from the two ends of the segment.

Tangent Lines and the Radius of a Circle:

This problem solving task challenges students to find the perpendicular meeting point of a segment from the center of a circle and a tangent.

Seven Circles I:

This task is intended to help model a concrete situation with geometry. Placing the seven pennies in a circular pattern is a concrete and fun experiment which leads to a genuine mathematical question: does the physical model with pennies give insight into what happens with seven circles in the plane?

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.

Angle bisection and midpoints of line segments:

This task provides a construction of the angle bisector of an angle by reducing it to the bisection of an angle to finding the midpoint of a line segment. It is worth observing the symmetry -- for both finding midpoints and bisecting angles, the goal is to cut an object into two equal parts.

Inscribing a circle in a triangle II:

This problem solving task focuses on a remarkable fact which comes out of the construction of the inscribed circle in a triangle: the angle bisectors of the three angles of triangle ABC all meet in a point.

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.

What functions do two graph points determine?:

This problem solving task challenges students to find the linear, exponential and quadratic functions based on two points.

US Population 1982-1988:

This problem solving task asks students to predict and model US population based on a chart of US population data from 1982 to 1988.

US Population 1790-1860:

This problem solving task asks students to solve five exponential and linear function problems based on a US population chart for the years 1790-1860.

Two Points Determine an Exponential Function II:

This problem solving tasks asks students to find the values of points on a graph.

Two Points Determine an Exponential Function I:

This problem solving task asks students to graph a function and find the values of points on a graph.

Taxi!:

This simple conceptual problem does not require algebraic manipulation, but requires students to articulate the reasoning behind each statement. Students are asked to verify a given linear equation from data in a table and interpret its key components in context.

Sandia Aerial Tram:

The task provides an opportunity for students to engage in detailed analysis of the rate of change of the elevation.

Rumors:

This problem is an exponential function example that uses the real-world problem of how fast rumors spread.

Rising Gas Prices - Compounding and Inflation:

The purpose of this task is to give students an opportunity to explore various aspects of exponential models (e.g., distinguishing between constant absolute growth and constant relative growth, solving equations using logarithms, applying compound interest formulas) in the context of a real world problem with ties to developing financial literacy skills. In particular, students are introduced to the idea of inflation of prices of a single commodity, and are given a very brief introduction to the notion of the Consumer Price Index for measuring inflation of a body of goods.

Newton's Law of Cooling:

The coffee cooling experiment is a popular example of an exponential model with immediate appeal. The model is realistic and provides a good context for students to practice work with exponential equations.

Linear or exponential?:

This task gives a variation of real-life contexts which could be modeled by a linear or exponential function. The key distinguishing feature between the two is whether the change by equal factors over equal intervals (exponential functions), or by a constant increase per unit interval (linear functions). The task could either be used as an assessment problem on this distinction, or used as an introduction to the differences between these very important classes of functions.

Linear Functions:

This task requires students to use the fact that on the graph of the linear function h(x) = ax + b, the y-coordinate increases by a when x increases by one. Specific values for a and b were left out intentionally to encourage students to use the above fact as opposed to computing the point of intersection, (p,q), and then computing respective function values to answer the question.

Solution Sets:

The typical system of equations or inequalities problem gives the system and asks for the graph of the solution. This task turns the problem around. It gives a solution set and asks for the system that corresponds to it. The purpose of this task is to give students a chance to go beyond the typical problem and make the connections between points in the coordinate plane and solutions to inequalities and equations. Students have to focus on what the graph is showing. When you are describing a region, why does the inequality have to go one way or another? When you pick a point that clearly lies in a region, what has to be true about its coordinates so that it satisfies the associated system of inequalities?

Quinoa Pasta 3:

This mathematical modeling task also illustrates making sense of a problem. Students are only told that there are two ingredients in the pasta and they have a picture of the box. It might even be better to just show the picture of the box, or to bring in the box and ask the students to pose the question themselves. The brand of pasta is quite commonly available at supermarkets or health food stores such as Whole Foods and even at Amazon.com. The box has the nutritional label and a reference to the website where the students can find other information about the ingredients

Quinoa Pasta 2:

This mathematical modeling task also illustrates making sense of a problem. Students are given all the relevant information on the nutritional labels, but they have to figure out how to use this information. They have to come up with the idea that they can set up two equations in two unknowns to solve the problem.

Pairs of Whole Numbers:

This task addresses solving systems of linear equations, and provides a simple example of a system with three equations and three unknown. Two (of many) methods for solving the system are presented. The first takes the given information to make three equations in three unknowns which can then be solved via algebraic manipulation to find the three numbers. The second solution is more clever, creating a single equation in three unknowns from the given information. This equation is then combined with the given information about the sums of pairs of numbers to deduce what the third number is. In reality, this solution is not simpler than the first: rather it sets up a slightly different set of equations which can be readily solved (the key being to take the sum of the three equations in the first solution). It provides a good opportunity for the instructor to show different methods for solving the same system of linear equations.

How does the solution change?:

The purpose of this task is to continue a crucial strand of algebraic reasoning begun at the middle school level. By asking students to reason about solutions without explicitly solving them, we get to the heart of understanding what an equation is and what it means for a number to be a solution to an equation. The equations are intentionally simple; the point of the task is not to test techniques in solving equations, but to encourage students to reason about them.

Do two points always determine an exponential function?:

This problem complements the problem "Do two points always determine a linear function?'' There are two constraints on a pair of points R1 and R2 if there is an exponential function f(x) = ae^bx whose graph contains R1 and R2.

Do two points always determine a linear function?:

In a geometric context, two distinct points ??1 and ??2 always determine a unique line in the Cartesian plane (this is one of the basic postulates of Euclidean geometry). Only the non-vertical lines, however, can be described by the graph of a function ??(??)=????+??. This task focuses on producing an explicit function ??(??) as long as the line is not vertical.

This problem allows the student to think geometrically about lines and then relate this geometry to linear functions. Or the student can work algebraically with equations to find the explicit equation of the line through two points (when that line is not vertical).

Comparing Exponentials:

This task gives students an opportunity to work with exponential functions in a real world context involving continuously compounded interest. They will study how the base of the exponential function impacts its growth rate and use logarithms to solve exponential equations.

Carbon 14 Dating, Variation 2:

This exploratory task requires the student to use properties of exponential functions in order to estimate how much Carbon 14 remains in a preserved plant after different amounts of time.

Carbon 14 Dating in Practice II:

This problem introduces the method used by scientists to date certain organic material. It is based not on the amount of the Carbon 14 isotope remaining in the sample but rather on the ratio of Carbon 14 to Carbon 12. This ratio decreases, hypothetically, at a constant exponential rate as soon as the organic material has ceased to absorb Carbon 14, that is, as soon as it dies.

Carbon 14 Dating in Practice I:

In the task "Carbon 14 Dating" the amount of Carbon 14 in a preserved plant is studied as time passes after the plant has died. In practice, however, scientists wish to determine when the plant died, and as this task shows, that is not possible with a simple measurement of the amount of Carbon 14 remaining in the preserved plant.

This task involves a fairly straightforward decaying exponential. Filling out the table and developing the general formula is complicated only by the need to work with a fraction that requires decisions about rounding and precision.

A Saturating Exponential:

This task provides an interesting context to ask students to estimate values in an exponential function using a graph.

Population and Food Supply:

In this task students use verbal descriptions to construct and compare linear and exponential functions and to find where the two functions intersect (F-LE.2, F-LE.3, A-REI.11).

This task is the last in a series of three tasks that use inequalities in the same context at increasing complexity in 6th grade, 7th grade and in HS algebra. Students write and solve inequalities, and represent the solutions graphically.

Braking Distance:

This task provides an exploration of a quadratic equation by descriptive, numerical, graphical, and algebraic techniques. Based on its real-world applicability, teachers could use the task as a way to introduce and motivate algebraic techniques like completing the square, en route to a derivation of the quadratic formula.

Cash Box:

The given solutions for this task involve the creation and solving of a system of two equations and two unknowns, with the caveat that the context of the problem implies that we are interested only in non-negative integer solutions. Indeed, in the first solution, we must also restrict our attention to the case that one of the variables is further even. This aspect of the task is illustrative of the mathematical practice of modeling with mathematics, and crucial as the system has an integer solution for both situations, that is, whether we include the dollar on the floor in the cash box or not.

Accurately weighing pennies II:

This task is a somewhat more complicated version of "Accurately weighing pennies I'' as a third equation is needed in order to solve part (a) explicitly. Instead, students have to combine the algebraic techniques with some additional problem-solving (numerical reasoning, informed guess-and-check, etc.) Part (b) is new to this task, as with only two types of pennies the weight of the collection determines how many pennies of each type are in the collection. This is no longer the case with three different weights but in this particular case, a collection of 50 is too small to show any ambiguity. This is part of the reason for part (c) of the question where the weight alone no longer determines which type of pennies are in the roll. This shows how important levels of accuracy in measurement are as the answer to part (b) could be different if we were to measure on a scale which is only accurate to the nearest tenth of a gram instead of to the nearest hundredth of a gram.

A Cubic Identity:

Solving this problem with algebra requires factoring a particular cubic equation (the difference of two cubes) as well as a quadratic equation. An alternative solution using prime numbers and arithmetic is presented.

Two Squares are Equal:

This classroom task is meant to elicit a variety of different methods of solving a quadratic equation (A-REI.4). Some are straightforward (for example, expanding the square on the right and rearranging the equation so that we can use the quadratic formula); some are simple but clever (reasoning from the fact that x and (2x - 9) have the same square); some use tools (using a graphing calculator to graph the functions f(x) = x^2 and g(x) = (2x-90)^2 and looking for values of x at which the two functions intersect). Some solution methods will work on an arbitrary quadratic equation, while others (such as the last three) may have difficulty or fail if the quadratic equation is not given in a particular form, or if the solutions are not rational numbers.

Accurately weighing pennies I:

This problem involves solving a system of algebraic equations from a context: depending how the problem is interpreted, there may be one equation or two. The main work in parts (a) and (b) is in setting up the equation(s) appropriately. Question (c) is more subtle and it requires thinking carefully about the accuracy available in a particular measurement (weight). The first two parts of this task could be used for instructional or assessment purposes while the third part should strictly be implemented for instructional purposes.

Same Solutions?:

The purpose of this task is to provide an opportunity for students to reason about equivalence of equations. The instruction to give reasons that do not depend on solving the equation is intended to focus attention on the transformation of equations as a deductive step.

Exponential growth versus linear growth I:

The purpose of this task it to have students discover how (and how quickly) an exponentially increasing quantity eventually surpasses a linearly increasing quantity. Students' intuitions will probably have them favoring Option A for much longer than is actually the case, especially if they are new to the phenomenon of exponential growth. Teachers might use this surprise as leverage to segue into a more involved task comparing linear and exponential growth.

Exponential Functions:

This task requires students to use the fact that the value of an exponential function f(x) = a Â· b^x increases by a multiplicative factor of b when x increases by one. It intentionally omits specific values for c and d in order to encourage students to use this fact instead of computing the point of intersection, (p,q), and then computing function values to answer the question.

Equal Factors over Equal Intervals:

This problem assumes that students are familiar with the notation x0 and ?x. However, the language "successive quotient" may be new.

Equal Differences over Equal Intervals 2:

This task assumes that students are familiar with the ?x and ?y notations. Students most likely developed this familiarity in their work with slope.

Equal Differences over Equal Intervals 1:

An important property of linear functions is that they grow by equal differences over equal intervals. In F.LE Equal Differences over Equal Intervals 1, students prove this for equal intervals of length one unit, and note that in this case the equal differences have the same value as the slope.

In the Billions and Linear Modeling:

This problem-solving task asks students to examine if linear modeling would be appropriate to describe and predict population growth from select years.

In the Billions and Exponential Modeling:

This problem-solving task provides students an opportunity to experiment with modeling real data by using population growth rates from the past two centuries.

Interesting Interest Rates:

This problem-solving task challenges students to write expressions and create a table to calculate how much money can be gained after investing at different banks with different interest rates.

Illegal Fish:

This problem-solving task asks students to describe exponential growth through a real-world problem involving the illegal introduction of fish into a lake.

This task asks students to write equations to predict how much money will be in a savings account at the end of each year, based on different factors like interest rates.

Identifying Functions:

This problem-solving emphasizes the expectation that students know linear functions grow by constant differences over equal intervals and exponential functions grow by constant factors over equal intervals.

Exponential growth versus polynomial growth:

This problem solving task shows that an exponential function takes larger values than a cubic polynomial function provided the input is sufficiently large. This resource also includes standards alignment commentary and annotated solutions.

Exponential growth versus linear growth II:

This task asks students to calculate exponential functions with a base larger than one.

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.

Yam in the Oven:

The purpose of this task is to give students practice interpreting statements using function notation. It can be used as a diagnostic if students seem to be having trouble with function notation, for example mistakenly interpreting f(x) as the product of f and x.

Which Function?:

The task addresses knowledge related to interpreting forms of functions derived by factoring or completing the square. It requires students to pay special attention to the information provided by the way the equation is represented as well as the sign of the leading coefficient, which is not written out explicitly, and then to connect this information to the important features of the graph.

Warming and Cooling:

This task is meant to be a straight-forward assessment task of graph reading and interpreting skills. This task helps reinforce the idea that when a variable represents time, t = 0 is chosen as an arbitrary point in time and positive times are interpreted as times that happen after that.

Using Function Notation I:

Throwing Baseballs:

This task could be used for assessment or for practice. It allows students to compare characteristics of two quadratic functions that are each represented differently, one as the graph of a quadratic function and one written out algebraically. Specifically, students are asked to determine which function has the greatest maximum and the greatest non-negative root.

The Random Walk:

This task requires interpreting a function in a non-standard context. While the domain and range of this function are both numbers, the way in which the function is determined is not via a formula but by a (pre-determined) sequence of coin flips. In addition, the task provides an opportunity to compute some probabilities in a discrete situation. The task could be used to segue the discussion from functions to probability, in particular the early standards in the S-CP domain.

The Parking Lot:

The purpose of this task is to investigate the meaning of the definition of function in a real-world context where the question of whether there is more than one output for a given input arises naturally. In more advanced courses this task could be used to investigate the question of whether a function has an inverse.

Domains:

The purpose of this task to help students think about an expression for a function as built up out of simple operations on the variable and understand the domain in terms of values for which each operation is invalid (e.g., dividing by zero or taking the square root of a negative number).

Cell Phones:

This simple task assesses whether students can interpret function notation. The four parts of the task provide a logical progression of exercises for advancing understanding of function notation and how to interpret it in terms of a given context.

Average Cost:

This task asks students to find the average, write an equation, find the domain, and create a graph of the cost of producing DVDs.

Springboard Dive:

The problem presents a context where a quadratic function arises. Careful analysis, including graphing of the function, is closely related to the context. The student will gain valuable experience applying the quadratic formula and the exercise also gives a possible implementation of completing the square.

Weed Killer:

The principal purpose of the task is to explore a real-world application problem with algebra, working with units and maintaining reasonable levels of accuracy throughout. Students are asked to determine which product will be the most economical to meet the requirements given in the problem.

The High School Gym:

This task asks students to consider functions in regard to temperatures in a high school gym.

The Customers:

The purpose of this task is to introduce or reinforce the concept of a function, especially in a context where the function is not given by an explicit algebraic representation. Further, the last part of the task emphasizes the significance of one variable being a function of another variable in an immediately relevant real-life context.

Telling a Story with Graphs:

In this task students are given graphs of quantities related to weather. The purpose of the task is to show that graphs are more than a collection of coordinate points; they can tell a story about the variables that are involved, and together they can paint a very complete picture of a situation, in this case the weather. Features in one graph, like maximum and minimum points, correspond to features in another graph. For example, on a rainy day, the solar radiation is very low, and the cumulative rainfall graph is increasing with a large slope.

Random Walk II:

These problems form a bridge between work on functions and work on probability. The task is better suited for instruction than for assessment as it provides students with a non-standard setting in which to interpret the meaning of functions. Students should carry out the process of flipping a coin and modeling this Random Walk in order to develop a sense of the process before analyzing it mathematically.

Points on a graph:

This task is designed to get at a common student confusion between the independent and dependent variables. This confusion often arises in situations like (b), where students are asked to solve an equation involving a function, and confuse that operation with evaluating the function.

Pizza Place Promotion:

This tasks asks students to use functions to predict the price of a pizza on a specific day and find which day the pizza would be cheapest according to a promotion.

Parabolas and Inverse Functions:

This problem is a simple de-contextualized version of F-IF Your Father and F-IF Parking Lot. It also provides a natural context where the absolute value function arises as, in part (b), solving for x in terms of y means taking the square root of x^2 which is |x|.This task assumes students have an understanding of the relationship between functions and equations.

Oakland Coliseum:

This deceptively simple task asks students to find the domain and range of a function from a given context. The function is linear and if simply looked at from a formulaic point of view, students might find the formula for the line and say that the domain and range are all real numbers.

Logistic Growth Model, Explicit Version:

This problem introduces a logistic growth model in the concrete settings of estimating the population of the U.S. The model gives a surprisingly accurate estimate and this should be contrasted with linear and exponential models.

Logistic Growth Model, Abstract Version:

This task is for instructional purposes only and students should already be familiar with some specific examples of logistic growth functions. The goal of this task is to have students appreciate how different constants influence the shape of a graph.

How Is the Weather?:

This task can be used as a quick assessment to see if students can make sense of a graph in the context of a real world situation. Students also have to pay attention to the scale on the vertical axis to find the correct match. The first and third graphs look very similar at first glance, but the function values are very different since the scales on the vertical axes are very different. The task could also be used to generate a group discussion on interpreting functions given by graphs.

Equations and Formulas:

In this task, students will use inverse operations to solve the equations for the unknown variable or for the designated variable if there is more than one.

Writing Constraints:

The purpose of this task is to give students practice writing a constraint equation for a given context. Instruction accompanying this task should introduce the notion of a constraint equation as an equation governing the possible values of the variables in question (i.e., "constraining" said values). In particular, it is worth differentiating the role of constraint equations from more functional equations, e.g., formulas to convert from degrees Celsius to degree Fahrenheit. The task has students interpret the context and choose variables to represent the quantities, which are governed by the constraint equation and the fact that they are non-negative (allowing us to restrict the graphs to points in the first quadrant only).

The four parts are independent and can be used as separate tasks.

Interpreting the Graph:

The purpose of this task is to help students learn to read information about a function from its graph, by asking them to show the part of the graph that exhibits a certain property of the function. The task could be used to further instruction on understanding functions or as an assessment tool, with the caveat that it requires some amount of creativity to decide how to best illustrate some of the statements.

Bernardo and Sylvia Play a Game:

This task presents a simple but mathematically interesting game whose solution is a challenging exercise in creating and reasoning with algebraic inequalities. The core of the task involves converting a verbal statement into a mathematical inequality in a context in which the inequality is not obviously presented, and then repeatedly using the inequality to deduce information about the structure of the game.

Dimes and Quarters:

Students are given a word problem that can be solved by using a pair of linear equations. This task does not actually require that the student solve the system but that they recognize the pairs of linear equations in two variables that would be used to solve the system. This is an important step in the process of solving systems.

Regular Tessellations of the Plane:

This task examines the ways in which the plane can be covered by regular polygons in a very strict arrangement called a regular tessellation. These tessellations are studied here using algebra, which enters the picture via the formula for the measure of the interior angles of a regular polygon (which should therefore be introduced or reviewed before beginning the task). The goal of the task is to use algebra in order to understand which tessellations of the plane with regular polygons are possible.

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.

Building a quadratic function from f(x) = x^2:

This task aims for students to understand the quadratic formula in a geometric way in terms of the graph of a quadratic function.

Building an explicit quadratic function by composition:

This task is intended for instruction and to motivate "Building a general quadratic function." This task assumes that the students are familiar with the process of completing the square.

Checking a Calculation of a Decimal Exponent:

In this example, students use properties of rational exponents and other algebraic concepts to compare and verify the relative size of two real numbers that involve decimal exponents.

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.

Dinosaur Bones:

The purpose of this task is to illustrate through an absurd example the fact that in real life quantities are reported to a certain level of accuracy, and it does not make sense to treat them as having greater accuracy.

Bus and Car:

This task operates at two levels. In part it is a simple exploration of the relationship between speed, distance, and time. Part (c) requires understanding of the idea of average speed, and gives an opportunity to address the common confusion between average speed and the average of the speeds for the two segments of the trip.

At a higher level, the task addresses MAFS.912.N-Q.1.3, since realistically neither the car nor the bus is going to travel at exactly the same speed from beginning to end of each segment; there is time traveling through traffic in cities, and even on the autobahn the speed is not constant. Thus students must make judgments about the level of accuracy with which to report the result.

Accuracy of Carbon 14 Dating I:

This task examines, from a mathematical and statistical point of view, how scientists measure the age of organic materials by measuring the ratio of Carbon 14 to Carbon 12. The focus here is on the statistical nature of such dating.

Accuracy of Carbon 14 Dating II:

This task examines, from a mathematical and statistical point of view, how scientists measure the age of organic materials by measuring the ratio of Carbon 14 to Carbon 12. The focus here is on the statistical nature of such dating.

Fuel Efficiency:

The problem requires students to not only convert miles to kilometers and gallons to liters but they also have to deal with the added complication of finding the reciprocal at some point.

How Much Is a Penny Worth?:

This task asks students to calculate the cost of materials to make a penny, utilizing rates of grams of copper.

Forms of Exponential Expressions:

There are many different ways to write exponential expressions that describe the same quantity, in this task the amount of a radioactive substance after t years. Depending on what aspect of the context we need to investigate, one expression of the quantity may be more useful than another. This task contrasts the usefulness of four equivalent expressions. Students first have to confirm that the given expressions for the radioactive substance are equivalent. Then they have to explain the significance of each expression in the context of the situation.

Runner's World:

Students are asked to use units to determine if the given statement is valid.

Harvesting the Fields:

This is a challenging task, suitable for extended work, and reaching into a deep understanding of units. Students are given a scenario and asked to determine the number of people required to complete the amount of work in the time described. The task requires students to exhibit , Make sense of problems and persevere in solving them. An algebraic solution is possible but complicated; a numerical solution is both simpler and more sophisticated, requiring skilled use of units and quantitative reasoning. Thus the task aligns with either MAFS.912.A-CED.1.1 or MAFS.912.N-Q.1.1, depending on the approach.

Calculating the Square Root of 2:

This task is intended for instructional purposes so that students can become familiar and confident with using a calculator and understanding what it can and cannot do. This task gives an opportunity to work on the notion of place value (in parts [b] and [c]) and also to understand part of an argument for why the square root of 2 is not a rational number.

Throwing a Ball:

Students manipulate a given equation to find specified information.

Paying the Rent:

Students solve problems tracking the balance of a checking account used only to pay rent. This simple conceptual task focuses on what it means for a number to be a solution to an equation, rather than on the process of solving equations.

Students extrapolate the list price of a car given a total amount paid in states with different tax rates. The emphasis in this task is not on complex solution procedures. Rather, the progression of equations, from two that involve different values of the sales tax, to one that involves the sales tax as a parameter, is designed to foster the habit of looking for regularity in solution procedures, so that students don't approach every equation as a new problem but learn to notice familiar types.

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.

Planes and Wheat:

In this resource, students refer to given information which defines 5 variables in the context of real world government expenses. They are then asked to write equations based upon specific known values for some of the variables. The emphasis is on setting up, rather than solving, the equations.

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.

In this resource, a method of deriving the quadratic formula from a theoretical standpoint is demonstrated. This task is for instructional purposes only and builds on "Building an explicit quadratic function."

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.

Profit of a Company:

This task compares the usefulness of different forms of a quadratic expression. Students have to choose which form most easily provides information about the maximum value, the zeros and the vertical intercept of a quadratic expression in the context of a real world situation. Rather than just manipulating one form into the other, students can make sense out of the structure of the expressions.

(From Algebra: Form and Function, McCallum et al., Wiley 2010)

Increasing or Decreasing? Variation 2:

The purpose of this task is to help students see manipulation of expressions as an activity undertaken for a purpose.

Variation 1 of this task presents a related more complex expression already in the correct form to answer the question.

The expression arises in physics as the reciprocal of the combined resistance of two resistors in parallel. However, the context is not explicitly considered here.

Downhill:

This task would be especially well-suited for instructional purposes. Students will benefit from a class discussion about the slope, y-intercept, x-intercept, and implications of the restricted domain for interpreting more precisely what the equation is modeling.

Ice Cream:

This task illustrates the process of rearranging the terms of an expression to reveal different aspects about the quantity it represents, precisely the language being used in standard MAFS.912.A-SSE.2.3. Students are provided with an expression giving the temperature of a container at a time t, and have to use simple inequalities (e.g., that 2t>0 for all t) to reduce the complexity of an expression to a form where bounds on the temperature of a container of ice cream are made apparent.

Sum of Even and Odd:

Students explore and manipulate expressions based on the following statement:

A function f defined for -a < x="">< a="" is="" even="" if="" f(-x)="f(x)" and="" is="" odd="" if="" f(-x)="-f(x)" when="" -a="">< x="">< a.="" in="" this="" task="" we="" assume="" f="" is="" defined="" on="" such="" an="" interval,="" which="" might="" be="" the="" full="" real="" line="" (i.e.,="" a="">

Students compare graphs of different quadratic functions, then produce equations of their own to satisfy given conditions.

This exploration can be done in class near the beginning of a unit on graphing parabolas. Students need to be familiar with intercepts, and need to know what the vertex is. It is effective after students have graphed parabolas in vertex form (y=a(xâ€“h)2+k), but have not yet explored graphing other forms.

Equivalent Expressions:

This is a standard problem phrased in a non-standard way. Rather than asking students to perform an operation, expanding, it expects them to choose the operation for themselves in response to a question about structure. Students must understand the need to transform the factored form of the quadratic expression (a product of sums) into a sum of products in order to easily see a, the coefficient of the x2 term; k, the leading coefficient of the x term; and n, the constant term.

Students are asked to interpret the effect on the value of an expression given a change in value of one of the variables.

Mixing Fertilizer:

Students examine and answer questions related to a scenario similar to a "mixture" problem involving two different mixtures of fertilizer. In this example, students determine and then compare expressions that correspond to concentrations of various mixtures. Ultimately, students generalize the problem and verify conclusions using algebraic rather than numerical expressions.

Mixing Candies:

Students are asked to interpret expressions and equations within the context of the amounts of caramels and truffles in a box of candy.

Kitchen Floor Tiles:

This problem asks students to consider algebraic expressions calculating the number of floor tiles in given patterns. The purpose of this task is to give students practice in reading, analyzing, and constructing algebraic expressions, attending to the relationship between the form of an expression and the context from which it arises. The context here is intentionally thin; the point is not to provide a practical application to kitchen floors, but to give a framework that imbues the expressions with an external meaning.

Extending the Definitions of Exponents, Variation 2:

The goal of this task is to develop an understanding of rational exponents (MAFS.912.N-RN.1.1); however, it also raises important issues about distinguishing between linear and exponential behavior (MAFS.912.F-LE.1.1c) and it requires students to create an equation to model a context (MAFS.912.A-CED.1.2).

Delivery Trucks:

This resource describes a simple scenario which can be represented by the use of variables. Students are asked to examine several variable expressions, interpret their meaning, and describe what quantities they each represent in the given context.

Traffic Jam:

This resource poses the question, "how many vehicles might be involved in a traffic jam 12 miles long?"

This task, while involving relatively simple arithmetic, promps students to practice modeling (MP4), work with units and conversion (N-Q.1), and develop a new unit (N-Q.2). Students will also consider the appropriate level of accuracy to use in their conclusions (N-Q.3).

Animal Populations:

In this task students interpret the relative size of variable expressions involving two variables in the context of a real world situation. All given expressions can be interpreted as quantities that one might study when looking at two animal populations.

Computations with Complex Numbers:

This resource involves simplifying algebraic expressions that involve complex numbers and various algebraic operations.

Operations with Rational and Irrational Numbers:

This task has students experiment with the operations of addition and multiplication, as they relate to the notions of rationality and irrationality.

Seeing Dots:

The purpose of this task is to identify the structure in the two algebraic expressions by interpreting them in terms of a geometric context. Students will have likely seen this type of process before, so the principal source of challenge in this task is to encourage a multitude and variety of approaches, both in terms of the geometric argument and in terms of the algebraic manipulation.

Selling Fuel Oil at a Loss:

The task is a modeling problem which ties in to financial decisions faced routinely by businesses, namely the balance between maintaining inventory and raising short-term capital for investment or re-investment in developing the business.

Felicia's Drive:

This task provides students the opportunity to make use of units to find the gas needed (). It also requires them to make some sensible approximations (e.g., 2.92 gallons is not a good answer to part (a)) and to recognize that Felicia's situation requires her to round up. Various answers to (a) are possible, depending on how much students think is a safe amount for Felicia to have left in the tank when she arrives at the gas station. The key point is for them to explain their choices. This task provides an opportunity for students to practice MAFS.K12.MP.2.1: Reason abstractly and quantitatively, and MAFS.K12.MP.3.1: Construct viable arguments and critique the reasoning of others.

Transforming the Graph of a Function:

This problem solving task examines, in a graphical setting, the impact of adding a scalar, multiplying by a scalar, and making a linear substitution of variables on the graph of the function f. This resource also includes standards alignment commentary and annotated solutions.

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.

Medieval Archer:

The task addresses the first part of standard MAFS.912.F-BF.2.3: "Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative)."

Growing Coffee:

This task is designed to make students think about the meaning of the quantities presented in the context and choose which ones are appropriate for the two different constraints presented. In particular, note that the purpose of the task is to have students generate the constraint equations for each part (though the problem statements avoid using this particular terminology), and not to have students solve said equations. If desired, instructors could also use this task to touch on such solutions by finding and interpreting solutions to the system of equations created in parts (a) and (b).

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.

Kimi and Jordan:

In the middle grades, students have lots of experience analyzing and comparing linear functions using graphs, table, symbolic expressions, and verbal descriptions. In this task, students may choose a representation that suits them and then reason from within that representation.

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.

Identifying Even and Odd Functions:

This task asks students to determine whether a the set of given functions is odd, even, or neither.

Calories in a Sports Drink:

This problem involves the meaning of numbers found on labels. When the level of accuracy is not given we need to make assumptions based on how the information is reported. An unexpected surprise awaits in this case, however, as no reasonable interpretation of the level of accuracy makes sense of the information reported on the bottles in parts (b) and (c). Either a miscalculation has been made or the numbers have been rounded in a very odd way.

## Student Center Activity

Method to Multiplying Polynomials:

This video will demonstrate how to multiply polynomials.

Type: Student Center Activity

## Tutorials

Finding congruent triangles:

In this tutorial, students will use the SSS, ASA, SAS, and AAS postulates to find congruent triangles

Type: Tutorial

Dilation and scale factor:

In this tutorial, students will use a scale factor to dilate one line onto another.

Type: Tutorial

Using SSS in a proof:

This tutorial discusses the difference between a theorem and axiom. It also shows how to use SSS in a proof.

Type: Tutorial

You will learn in this video how to solve Quadratic Equations using the Quadratic Formula.

Type: Tutorial

Learning How to Complete the Square:

You will learn int his video how to solve the Quadratic Equation by Completing the Square.

Type: Tutorial

Triangle congruence postulates:

This tutorial discusses SSS, SAS, ASA and AAS postulates for congruent triangles. It also shows AAA is only good for similarity and SSA is good for neither.

Type: Tutorial

Congruent Triangles and SSS:

In this video, students will learn about congruent triangles and the "Side-Side-Side" postulate.

Type: Tutorial

Rotating polygons 180 degrees about their center:

Students will investigate symmetry by rotating polygons 180 degrees about their center.

Type: Tutorial

Line of reflection:

Students are shown, with an interactive tool, how to reflect a line segment. Students should have an understanding of slope and midpoint before viewing this video.

Type: Tutorial

Line of reflection:

This tutorial uses the midpoint of two lines to find the line of reflection.

Type: Tutorial

Points after rotation:

Students will see what happens when a figure is rotated about the origin -270 degrees. Having a foundation about right triangles is recommended before viewing this video.

Type: Tutorial

Parallel lines and transversals:

In this tutorial, students will find the measures of angles formed when a transversal cuts two parallel lines.

Type: Tutorial

Parallel lines, transversals and triangles:

This tutorial shows students the eight angles formed when two parallel lines are cut by a transversal line. There is also a review of triangles in this video.

Type: Tutorial

Parallel lines and transversal lines:

Students will see in this tutorial the eight angles formed when two parallel lines are cut by a transversal line.

Type: Tutorial

Parallel lines and transversals:

In this tutorial, students will learn the angle measures when two parallel lines are cut by a transversal line.

Type: Tutorial

Sum of Exterior Angles of an Irregular Pentagon:

In this video, students will learn how to use what they know about the sum of angles in a triangle to determine the sum of the exterior angles of an irregular pentagon.

Type: Tutorial

Example 3: Solving Systems by Elimination:

This video is an example of solving a system of linear equations by elimination where the system has infinite solutions.

Type: Tutorial

Solving Systems of Linear Equations with Elimination Example 1:

This video shows how to solve a system of equations through simple elimination.

Type: Tutorial

Inconsistent Systems of Equations:

This video explains how to identify systems of equations without a solution.

Type: Tutorial

Specifying planes in three dimensions:

In this tutorial, students are introduced to the concept that three non-collinear points are necessary to define a unique plane.

Type: Tutorial

The language of geometry:

Before learning any new concept it's important students learn and use common language and label concepts consistently. This tutorial introduces students to th point, line and plane.

Type: Tutorial

Proving vertical angles are equal:

In this tutorial, students prove that vertical angles are equal. Students should have an understanding of supplementary angles before viewing this video.

Type: Tutorial

Finding the measure of vertical angles:

Students will use algebra to find the measure of vertical angles, or angles opposite each other when two lines cross. Students should have an understanding of complementary and supplementary angles before viewing this video.

Type: Tutorial

Identifying parallel and perpendicular lines:

This tutorial is great practice for help in identifying parallel and perpendicular lines.

Type: Tutorial

Introduction to vertical angles:

In this tutorial, students will use their knowledge of supplementary, adjacent, and vertical angles to solve problems involving the intersection of two lines.

Type: Tutorial

Bhaskara's Proof of the Pythagorean Theorem:

This video demonstrates Bhaskara's proof of the Pythagorean Theorem.

Type: Tutorial

Another Pythagorean Theorem Proof:

This video visually proves the Pythagorean Theorem using triangles and parallelograms.

Type: Tutorial

Pythagorean Theorem Proof Using Similar Triangles:

This video shows a proof of the Pythagorean Theorem using similar triangles.

Type: Tutorial

This video tutorial shows students: the standard form of a polynomial, how to identify polynomials, how to determine the degree of a polynomial, how to add and subtract polynomials, and how to represent the area of a shape as an addition or subtraction of polynomials.

Type: Tutorial

Proof: Sum of Measures of Angles in a Triangle Are 180:

Lets prove that the sum of interior angles of a triangle are equal to 180 degrees.

Type: Tutorial

Triangle Angle Example 1:

Let's find the measure of an angle, using interior and exterior angle measurements.

Type: Tutorial

Example 2: Solving Systems by Elimination:

This video shows how to solve systems of equations by elimination.

Type: Tutorial

Solving Quadratic Equations by Square Roots:

In this video tutorial students will learn how to solve quadratic equations by square roots.

Type: Tutorial

This video is an introduction to the elimination method of solving a system of equations.

Type: Tutorial

Example 3: Solving Systems by Substitution:

This example demonstrates solving a system of equations algebraically and graphically.

Type: Tutorial

Substitution Method Example 2:

This video demonstrates a system of equations with no solution.

Type: Tutorial

The Substitution Method:

This video shows how to solve a system of equations using the substitution method.

Type: Tutorial

Systems of Equations Word Problems Example 1:

This video demonstrates solving a word problem by creating a system of linear equations that represents the situation and solving them using elimination.

Type: Tutorial

Using Algebra to Find Measures of Angles Formed from Transversal:

We will use algebra in order to find the measure of angles formed by a transversal.

Type: Tutorial

Figuring Out Angles Between Transversal and Parallel Lines:

We will be able to identify corresponding angles of parallel lines.

Type: Tutorial

Angles Formed by Parallel Lines and Transversals:

We will gain an understanding of how angles formed by transversals compare to each other.

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

Graphing systems of equations:

In this tutorial, students will learn how to solve and graph a system of equations.

Type: Tutorial

Solving system of equations by graphing:

This tutorial shows students how to solve a system of linear equations by graphing the two equations on the same coordinate plane and identifying the intersection point.

Type: Tutorial

Solving a system of equations by graphing:

This tutorial shows how to solve a system of equations by graphing. Students will see what a no solution system of equations looks like in a graph.

Type: Tutorial

Solving a system of equations using substitution:

This tutorial shows how to solve a system of equations using substitution.

Type: Tutorial

Graph the solution to a system of inequalities.:

This video will demonstrate how to graph the solution to a system of inequalities.

Type: Tutorial

Solving a literal equation:

Students will learn to solve a literal equation.

Type: Tutorial

Solving Percentage Problems with Linear Equations:

Many real world problems involve involve percentages. This lecture shows how algebra is used in solving problems of percent change and profit-and-loss.

Type: Tutorial

Introduction to the Coordinate Plane:

In this video, you will learn about Rene Descartes, and how he bridged the gap between algebra and geometry.

Type: Tutorial

Basic Geometry Language and Labels:

In this tutorial we will learn the basics of geometry, such as identifying a line, ray, point, and segment.

Type: Tutorial

Subtracting Polynomials with Multiple Variables:

This video explains how to subtract polynomials with multiple variables and reinforces how to distribute a negative number.

Type: Tutorial

Squaring a Binomial:

This video covers squaring a binomial with two variables. Students will be given the area of a square.

Type: Tutorial

Dependent and independent variables exercise: graphing the equation:

It's helpful to represent an equation on a graph where we plot at least 2 points to show the relationship between the dependent and independent variables. Watch and we'll show you.

Type: Tutorial

Trolls, tolls, and systems of equations:

This video tutorial discusses how to create a system of equations.

Type: Tutorial

Solving Basic Systems Using the Elimination Method:

This 8 minute video will show step-by-step directions for using the elimination method to solve a system of linear equations.

Type: Tutorial

Constructing an Equations with Two Variables - Yoga Plan:

This video provides a real-world scenario and step-by-step instructions to constructing equations using two variables. Possible follow-up videos include Plotting System of Equations - Yoga Plan, Solving System of Equations with Substitution - Yoga Plan, and Solving System of Equations with Elimination - Yoga Plan.

Type: Tutorial

Proof: Vertical Angles are Equal:

This 5 minute video gives the proof that vertical angles are equal.

Type: Tutorial

Example: Evaluating expressions with 2 variables:

Evaluating Expressions with Two Variables

Type: Tutorial

How to evaluate an expression with variables:

Learn how to evaluate an expression with variables using a technique called substitution (or "plugging in").

Type: Tutorial

Multiplying And Dividing With Inequalities:

This video discusses multiplication and division of inequalities with negative numbers to solve the inequality.

Type: Tutorial

What is a variable?:

Our focus here is understanding that a variable is just a letter or symbol (usually a lower case letter) that can represent different values in an expression. We got this. Just watch.

Type: Tutorial

Power of a Power Property:

This tutorial demonstrates how to use the power of a power property with both numerals and variables.

Type: Tutorial

Calculating Mixtures of Solutions:

This lecture shows how algebra is used to solve problems involving mixtures of solutions of different concentrations.

Type: Tutorial

Solving Inconsistent or Dependent Systems:

When solving a system of linear equations in x and y with a single solution, we get a unique pair of values for x and y. But what happens when try to solve a system with no solutions or an infinite number of solutions?

Type: Tutorial

Parallel Lines:

Parallel lines have the same slope and no points in common. However, it is not always obvious whether two equations describe parallel lines or the same line.

Type: Tutorial

Perpendicular Lines:

Perpendicular lines have slopes which are negative reciprocals of each other, but why?

Type: Tutorial

Inconsistent, Dependent, and Independent Systems:

Systems of two linear equations in two variables can have a single solution, no solutions, or an infinite number of solutions. This video gives a great description of inconsistent, dependent, and independent systems. A consistent independent system of equations will have one solution. A consistent dependent system of equations will have infinite number of solutions, and an inconsistent system of equations will have no solution. This tutorial also provides information on how to distinguish a given system of linear equations as inconsistent, independent, or dependent system by looking at the slope and intercept.

Type: Tutorial

Vertical Line Test:

A graph in Cartesian coordinates may represent a function or may only represent a binary relation. The "vertical line test" is a visual way to determine whether or not a graph represents a function.

Type: Tutorial

Solving Systems of Equations by Elimination:

Systems of two equations in x and y can be solved by adding the equations to create a new equation with one variable eliminated. This new equation can then be solved to find the value of the remaining variable. That value is then substituted into either equation to find the value of other variable.

Type: Tutorial

Solving Systems of Equations by Substitution:

A system of two equations in x and y can be solved by rearranging one equation to represent x in terms of y, and "substituting" this expression for x in the other equation. This creates an equation with only y which can then be solved to find y's value. This value can then be substituted into either equation to find the value of x.

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

Special Products of Binomials:

The video tutorial discusses about two typical polynomial multiplications. First, squaring a binomial and second, product of a sum and difference.

Type: Tutorial

An Antihero of One's Own :

In this very engaging animated video from TEDed, you will learn about antiheroes. Antiheroes can be hard to distinguish from typical heroes. However, through this video you will be able to identify what an antihero is and understand how these complex characters with often unclear motivations play such important roles in great literature.

Type: Tutorial

The Art of the Metaphor:

In this animated video from TEDed, you will learn about the power of metaphors in your reading and in your writing. The video explores questions like: "How do metaphors help us better understand the world?", as well as "What makes a good metaphor?"

Type: Tutorial

Multiplying Bionomials:

Binomials are the polynomials with two terms. This tutorial will help the students learn about the multiplication of binomials. In multiplication, we need to make sure that each term in the first set of parenthesis multiplies each term in the second set.

Type: Tutorial

Linear Equations in One Variable:

This lesson introduces students to linear equations in one variable, shows how to solve them using addition, subtraction, multiplication, and division properties of equalities, and allows students to determine if a value is a solution, if there are infinitely many solutions, or no solution at all. The site contains an explanation of equations and linear equations, how to solve equations in general, and a strategy for solving linear equations. The lesson also explains contradiction (an equation with no solution) and identity (an equation with infinite solutions). There are five practice problems at the end for students to test their knowledge with links to answers and explanations of how those answers were found. Additional resources are also referenced.

Type: Tutorial

Linear Inequalities:

Upon completing this lesson, the student should be able to use the addition, subtraction, multiplication, and division properties of equality to solve linear inequalities, write the answer to an inequality using interval notation and draw a graph to give a visual answer to an inequality problem.

The lesson begins with explanations of inequality signs and interval notation and then moves on to demonstrate addition/subtraction and multiplication/division properties of equality. The site demonstrates a strategy for solving linear inequalities and presents three problems for students to practice what they have learned.

There is also a link to a previous tutorial which covers solving linear equations of one variable for students who need the review.

Type: Tutorial

## Video/Audio/Animations

What is a Function?:

This video will demonstrate how to determine what is and is not a function.

Type: Video/Audio/Animation

Solving Quadratic Equations using Square Roots:

This video will demonstrate how to solve a quadratic equation using square roots.

Type: Video/Audio/Animation

Relations and Functions:

This video demonstrates how to determine if a relation is a function and how to identify the domain.

Type: Video/Audio/Animation

Real-Valued Functions of a Real Variable:

Although the domain and codomain of functions can consist of any type of objects, the most common functions encountered in Algebra are real-valued functions of a real variable, whose domain and codomain are the set of real numbers, R.

Type: Video/Audio/Animation

Roots and Unit Fraction Exponents:

Exponents are not only integers. They can also be fractions. Using the rules of exponents, we can see why a number raised to the power " one over n" is equivalent to the nth root of that number.

Type: Video/Audio/Animation

Rational Exponents:

Exponents are not only integers and unit fractions. An exponent can be any rational number expressed as the quotient of two integers.

Type: Video/Audio/Animation

Radical expressions can often be simplified by moving factors which are perfect roots out from under the radical sign.

Type: Video/Audio/Animation

Solving Mixture Problems with Linear Equations:

Mixture problems can involve mixtures of things other than liquids. This video shows how Algebra can be used to solve problems involving mixtures of different types of items.

Type: Video/Audio/Animation

Using Systems of Equations Versus One Equation:

When should a system of equations with multiple variables be used to solve an Algebra problem, instead of using a single equation with a single variable?

Type: Video/Audio/Animation

Systems of Linear Equations in Two Variables:

The points of intersection of two graphs represent common solutions to both equations. Finding these intersection points is an important tool in analyzing physical and mathematical systems.

Type: Video/Audio/Animation

Why the Elimination Method Works:

This chapter presents a new look at the logic behind adding equations- the essential technique used when solving systems of equations by elimination.

Type: Video/Audio/Animation

Domain and Range of Binary Relations:

Two sets which are often of primary interest when studying binary relations are the domain and range of the relation.

Type: Video/Audio/Animation

Point-Slope Form:

The point-slope form of the equation for a line can describe any non-vertical line in the Cartesian plane, given the slope and the coordinates of a single point which lies on the line.

Type: Video/Audio/Animation

Two Point Form:

The two point form of the equation for a line can describe any non-vertical line in the Cartesian plane, given the coordinates of two points which lie on the line.

Type: Video/Audio/Animation

Linear Equations in the Real World:

Linear equations can be used to solve many types of real-word problems. In this episode, the water depth of a pool is shown to be a linear function of time and an equation is developed to model its behavior. Unfortunately, ace Algebra student A. V. Geekman ends up in hot water anyway.

Type: Video/Audio/Animation

Solving Literal Equations:

Literal equations are formulas for calculating the value of one unknown quantity from one or more known quantities. Variables in the formula are replaced by the actual or 'literal' values corresponding to a specific instance of the relationship.

Type: Video/Audio/Animation

Parallel Lines 2:

This video shows how to determine which lines are parallel from a set of three different equations.

Type: Video/Audio/Animation

Parallel Lines:

This video illustrates how to determine if the graphs of a given set of equations are parallel.

Type: Video/Audio/Animation

Perpendicular Lines 2:

This video describes how to determine the equation of a line that is perpendicular to another line. All that is given initially the equation of a line and an ordered pair from the other line.

Type: Video/Audio/Animation

Example of Solving for a Variable - Khan Academy:

This video takes a look at rearranging a formula to highlight a quantity of interest.

Type: Video/Audio/Animation

Basic Linear Function:

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

Type: Video/Audio/Animation

This video gives a more in-depth look at graphing quadratic functions than previously offered in Quadratic Functions 1.

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

Graphing Lines 1:

Khan Academy video tutorial on graphing linear equations: "Algebra: Graphing Lines 1"

Type: Video/Audio/Animation

Fitting a Line to Data:

Khan Academy tutorial video that demonstrates with real-world data the use of Excel spreadsheet to fit a line to data and make predictions using that line.

Type: Video/Audio/Animation

Annotated Proof of the Pythagorean Theorem :

This resource gives an animated and then annotated proof of the Pythagorean Theorem.

Type: Video/Audio/Animation

Averages:

This Khan Academy video tutorial introduces averages and algebra problems involving averages.

Type: Video/Audio/Animation

## Virtual Manipulatives

Circumscribe a Circle About a Triangle:

In this GeoGebraTube interactive worksheet, you can watch the step by step process of circumscribing a circle about a triangle. Â Using paper and pencil along with this resource will reinforce the concept.

Type: Virtual Manipulative

Inscribe a Regular Hexagon in a Circle:

This geogebratube interactive worksheet shows the step by step process for inscribing a regular hexagon in a circle. There are other geogebratube interactive worksheets for the square and the equilateral triangle.

Type: Virtual Manipulative

3-D Conic Section Explorer:

Using this resource, students can manipulate the measurements of a 3-D hourglass figure (double-napped cone) and its intersecting plane to see how the graph of a conic section changes. Â Students will see the impact of changing the height and slant of the cone and the mÂ and bÂ values of the plane on the shape of the graph.Â Students can also rotate and re-size the cone and graph to view from different angles.Â
Â

Type: Virtual Manipulative

Combining Transformations:

In this manipulative activity, you can first get an idea of what each of the rigid transformations look like, and then get to experiment with combinations of transformations in order to map a pre-image to its image.

Type: Virtual Manipulative

This resource will assess students' understanding of addition and subtraction of polynomials.

Type: Virtual Manipulative

Solving Quadratics By Taking The Square Root:

This resource can be used to assess students' understanding of solving quadratic equation by taking the square root. A great resource to view prior to this is "Solving quadratic equations by square root' by Khan Academy.

Type: Virtual Manipulative

Slope Slider:

In this activity, students adjust slider bars which adjust the coefficients and constants of a linear function and examine how their changes affect the graph. The equation of the line can be in slope-intercept form or standard form. This activity allows students to explore linear equations, slopes, and y-intercepts and their visual representation on a graph. 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

Cross Section Flyer - Shodor:

With this online Java applet, students use slider bars to move a cross section of a cone, cylinder, prism, or pyramid. This activity allows students to explore conic sections and the 3-dimensional shapes from which they are derived. 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

Linear Function Machine:

In this activity, students plug values into the independent variable to see what the output is for that function. Then based on that information, they have to determine the coefficient (slope) and constant(y-intercept) for the linear function. This activity allows students to explore linear functions and what input values are useful in determining the linear 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

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

Box Plot:

In this activity, students use preset data or enter in their own data to be represented in a box plot. This activity allows students to explore single as well as side-by-side box plots of different data. 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

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 Matching:

This is a graphing tool/activity for students to deepen their understanding of polynomial functions and their corresponding graphs. This tool is to be used in conjunction with a full lesson on graphing polynomial functions; it can be used either before an in depth lesson to prompt students to make inferences and connections between the coefficients in polynomial functions and their corresponding graphs, or as a practice tool after a lesson in graphing the polynomial functions.

Type: Virtual Manipulative

Normal Distribution Interactive Activity:

With this online tool, students adjust the standard deviation and sample size of a normal distribution to see how it will affect a histogram of that distribution. This activity allows students to explore the effect of changing the sample size in an experiment and the effect of changing the standard deviation of a normal distribution. Tabs at the top of the page provide access to 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

This is an online graphing utility that can be used to create box plots, bubble graphs, scatterplots, histograms, and stem-and-leaf plots.

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

Curve Fitting:

With a mouse, students will drag data points (with their error bars) and watch the best-fit polynomial curve form instantly. Students can choose the type of fit: linear, quadratic, cubic, or quartic. Best fit or adjustable fit can be displayed.

Type: Virtual Manipulative

Equation Grapher:

This interactive simulation investigates graphing linear and quadratic equations. Users are given the ability to define and change the coefficients and constants in order to observe resulting changes in the graph(s).

Type: Virtual Manipulative

Line of Best Fit:

This manipulative allows the user to enter multiple coordinates on a grid, estimate a line of best fit, and then determine the equation for a line of best fit.

Type: Virtual Manipulative

A Plethora of Polyhedra:

This program allows users to explore spatial geometry in a dynamic and interactive way. The tool allows users to rotate, zoom out, zoom in, and translate a plethora of polyhedra. The program is able to compute topological and geometrical duals of each polyhedron. Geometrical operations include unfolding, plane sections, truncation, and stellation.

Type: Virtual Manipulative

Histogram Tool:

This virtual manipulative histogram tool can aid in analyzing the distribution of a dataset. It has 6 preset datasets and a function to add your own data for analysis.

Type: Virtual Manipulative

Multi Bar Graph:

This activity allows the user to graph data sets in multiple bar graphs. The color, thickness, and scale of the graph are adjustable which may produce graphs that are misleading. Users may input their own data, or use or alter pre-made data sets. 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

Histogram:

In this activity, students can create and view a histogram using existing data sets or original data entered. Students can adjust the interval size using a slider bar, and they can also adjust the other scales on the graph. This activity allows students to explore histograms as a way to represent data as well as the concepts of mean, standard deviation, and scale. 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

## Worksheet

Inscribing a circle in a triangle I:

This problem solving task shows how to inscribe a circle in a triangle using angle bisectors.

Type: Worksheet

## Parent Resources

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