Liberal Arts Mathematics 1   (#1207300)

General Course Information and Notes

General Notes

English Language Development ELD Standards Special Notes Section:
Teachers are required to provide listening, speaking, reading and writing instruction that allows English language learners (ELL) to communicate information, ideas and concepts for academic success in the content area of Mathematics. For the given level of English language proficiency and with visual, graphic, or interactive support, students will interact with grade level words, expressions, sentences and discourse to process or produce language necessary for academic success. The ELD standard should specify a relevant content area concept or topic of study chosen by curriculum developers and teachers which maximizes an ELL's need for communication and social skills. To access an ELL supporting document which delineates performance definitions and descriptors, please click on the following link:

For additional information on the development and implementation of the ELD standards, please contact the Bureau of Student Achievement through Language Acquisition at sala@fldoe.org.

A.V.E. for Success Collection is provided by the Florida Association of School Administrators: http://www.fasa.net/4DCGI/cms/review.html?Action=CMS_Document&DocID=139. Please be aware that these resources have not been reviewed by CPALMS and there may be a charge for the use of some of them in this collection.

General Information

Course Number: 1207300
Abbreviated Title: LIB ARTS MATH 1
Number of Credits: One (1) credit
Course Length: Year (Y)
Course Attributes:
• Highly Qualified Teacher (HQT) Required
• Florida Standards 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

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

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

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.

Type: Original Student Tutorial

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

Practice identifying 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 Part One before Part Two! Click HERE to launch Part One.

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

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

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

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:

Want to learn about Amelia Earhart, one of the most famous female aviators of all time? If so, then this interactive tutorial is for YOU! This tutorial is Part Two of a two-part series. In this series, you will study a speech by Amelia Earhart. You will practice identifying the purpose of her speech and practice identifying her use of rhetorical appeals (ethos, logos, pathos, Kairos). You will also evaluate the effectiveness of Earhart's rhetorical choices based on the purpose of her speech.

Please complete Part One before beginning Part Two. Click HERE to view Part One.

Type: Original Student Tutorial

Ready for Takeoff! -- Part One:

Want to learn about Amelia Earhart, one of the most famous female aviators of all time? If so, then this interactive tutorial is for YOU! This tutorial is Part One of a two-part series. In this series, you will study a speech by Amelia Earhart. You will practice identifying the purpose of her speech and practice identifying her use of rhetorical appeals (ethos, logos, pathos, Kairos). You will also evaluate the effectiveness of Earhart's rhetorical choices based on the purpose of her speech.

Type: Original Student Tutorial

Untangling Food Webs:

Learn how living organisms can be organized into food webs and how energy is transferred through a food web from producers to consumers to decomposers. This interactive tutorial also includes interactive knowledge checks.

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

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

Comparing Mitosis and Meiosis:

Compare and contrast mitosis and meiosis in this interactive tutorial. You'll also relate them to the processes of sexual and asexual reproduction and their consequences for genetic variation.

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. This task is vital to those that work to create three dimensional objects.  Whether it is the inventor of a new toy or the architect of your next house, they must be able to convey their design on paper.  The drawings they make represent various cross sections of the finished product. Can you visualize the relationships between two-dimensional and three-dimensional objects? Imagine that Ninja Nancy will slice through this pyramid with her sword.  What two-dimensional figures will she reveal?

Type: Original Student Tutorial

Changing Rates:

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

Type: Original Student Tutorial

Justifiable Steps:

Learn how to explain the steps used to solve a simple equation 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

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:

Explain why the x-coordinate of the point of intersection of two functions is the solution of the equation f(x) = g(x).

Type: Original Student Tutorial

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

Have you ever ordered a scoop of ice cream in a cone and wondered how much ice cream actually fits inside the cone? By the end of this tutorial, you should be able to answer this question and solve other real-world problems by using the formula for the volume of a cone.

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

Climbing Around the Hominin Family Tree:

Learn to identify basic trends in the evolutionary history of humans, including walking upright, brain size, jaw size, and tool use in "Climbing Around the Hominin Family Tree" online 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

Educational Games

Volume of a Cylinder:

Students can play a game to solve for the volume of a cylinder and work backwards using the volume to find measures of a cylinder. Students may select Teach Me to learn how to find the volume of a cylinder or measures of it prior to beginning play. Hints and feedback are provided to players.

Type: Educational Game

Construct Bisectors and Other Lines:

Play a game to construct perpendicular lines, perpendicular bisectors, midpoints and angle bisectors. Students may select Teach Me to learn about constructions prior to beginning play. Hints and feedback are provided to players.

Type: Educational Game

Population Density:

Play a game about density and population. Students may select Teach Me to learn about the density formula prior to beginning play. Hints and feedback are provided to players.

Type: Educational Game

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 / 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

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:

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!

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: Uniform Scaling:

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

Type: Perspectives Video: Professional/Enthusiast

Using Geometry and Computers to make Art with CNC Machining:

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

Type: Perspectives Video: Professional/Enthusiast

Estimating Oil Seep Production by Bubble Volume:

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!

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.

SAT Scores:

This problem solving task challenges students to answer probability questions about SAT scores, using distribution and mean to solve the problem.

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.

Should We Send Out a Certificate?:

The purpose of this task is to have students complete normal distribution calculations and to use properties of normal distributions to draw conclusions.

Do You Fit in This Car?:

This task requires students to use the normal distribution as a model for a data distribution. Students must use given means and standard deviations to approximate population percentages.

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.

Exstensions, 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.

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.

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.

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.

Seven Circles III:

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

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.

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.

Unit Squares and Triangles:

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

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.

Seven Circles II:

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

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.

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

Bisecting an angle:

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

Locating Warehouse:

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

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.

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.

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.

Taxi!:

This simple conceptual problem does not require algebraic manipulation, but requires students to articulate the reasoning behind each statement.

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?

In order to engage this task meaningfully, students must be aware of the convention that va for a positive number a refers to the positive square root of a. The purpose of the task is to show students a situation where squaring both sides of an equation can result in an equation with more solutions than the original one.

Quinoa Pasta 3:

This task is an example of a mathematical modeling problem (SMP 4) and it also illustrates SMP 1 (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 task has some aspects of a mathematical modeling problem (SMP 4) and it also illustrates SMP 1 (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 A-REI.3.6, solving systems of linear equations exactly, 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 (e.g, 6.EE.5). By asking students to reason about solutions without explicitly solving them, we get at 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 very simple; the point of the task is not to test technique in solving equations, but to encourage students to reason about them.

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.

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 mathematical practice standard MP4 (Model with mathematics), and crucial as the system has an integer solution for both situations, that is, whether or not 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.

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.

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.

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.

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.

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.

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 MAFS.K12.MP.1.1, 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.

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.

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.

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=8).

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.

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.

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 (MAFS.912.N-Q.1.1). 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.

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).

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.

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

Simplifying Square Roots Containing Variables:

This video will demonstrate how to simplify square roots involving variables.

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

Example 3: Solving Systems by Elimination:

This video is an example of solving systems 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

This video will demonstrate how to solve radical equations with additional practice problems.

Type: Tutorial

Identifying parallel and perpendicular lines:

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

Example 2: Solving Systems by Elimination:

This video shows how to solve systems by elimination.

Type: Tutorial

Function Notation:

This tutorial will help the students to understand the function notation such as f(x), which can be thought as another way of representing the y-value in a function, especially when graphing. The y-axis is even labeled as the f(x) axis, when graphing.

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 system of equations word problem using elimination.

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 and graph a system of equations. Students will see how to sketch their solution after solving the system of equations.

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:

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

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

This tutorial will help the learners to graph the equation of the quadratic function using the coordinates of the vertex of a parabola adn its x- intercepts.

Type: Tutorial

Example: Evaluating expressions with 2 variables:

Evaluating Expressions with Two Variables

Type: Tutorial

How to evaluate an expression using substitution:

In this example we have a formula for converting Celsius temperature to Fahrenheit. Let's substitute the variable with a value (Celsius temp) to get the degrees in Fahrenheit. Great problem to practice with us!

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

Why aren't we using the multiplication sign?:

Great question. In algebra, we do indeed avoid using the multiplication sign. We'll explain it for you here.

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

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

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

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

Multiplying Polynomials:

This tutorial will help the learners practice multiplication of polynomials. Learners will understand that when they multiply expressions with more than two terms, they need to make sure each term in the first expression multiplies every term in the second expression.

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

In this tutorial, students will learn how to add and subtract polynomials functions using horizontal and vertical methods. In a horizontal format, like terms should be grouped together using the commutative property. In vertical format, terms should be listed by ascending degree with like terms placed below each other.

Type: Tutorial

Introduction to Polynomials:

In this tutorial students learn how to identify a polynomial, how to find the degree of a polynomial, and how to write a polynomial in standard format.

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

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

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

Slope:

"Slope" is a fundamental concept in mathematics. Slope is often defined as " the rise over the run"....but why?

Type: Video/Audio/Animation

Point-Slope Form:

Th 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

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:

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

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

Graphing Lines 1:

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

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

Which Holds More? :

This interactive manipulative will let you compare and calculate volumes of different solids.

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.  Students will see the impact of changing the height and slant of the cone and the m and b values of the plane on the shape of the graph. Students can also rotate and re-size the cone and graph to view from different angles.

Type: Virtual Manipulative

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

Type: Virtual Manipulative

Algebra Tiles (Multiplying Binomials):

This virtual manipulative is intended to allow the student to practice multiplication of binomials. The student should understand how to use algebra tiles before using this tool.

Type: Virtual Manipulative

Space Blocks:

This virtual manipulative allows students to manipulate blocks, add or remove blocks, and connect them together to form solids. They can also experiment with counting the number of exposed faces, seeing what happens to the surface area when blocks are added or removed, and "unfolding" a block to create a net .

Type: Virtual Manipulative

Histogram vs. Box Plot:

This simulation allows the student to create a box plot and a histogram for the same set of data and toggle between the two displays. Maximum, minimum, median and mean are shown for the data set. The student can change the cell width to explore how the histogram is affected.

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

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

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

Pythagorean Theorem Manipulatives:

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

Type: Virtual Manipulative

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

Algebra Balance Scales - with Negatives:

This site provides a virtual balance on which the student can represent (and then solve) simple linear equations with integer answers. Conceptually, positive weights (unit-blocks and x-boxes) push the pans of the scale downward. Negative values are represented by balloons which can be attached to the pans of the scale. The student can then manipulate the weights to solve the equation while simultaneously seeing a visual display of these effects on the equation.

Type: Virtual Manipulative

Transformations - Reflections:

The user clicks and drags a shape they have constructed to view its reflection across a line. A background grid and axes may or may not be used. The reflection may by examined analytically using coordinates. Symmetry may be displayed.

Type: Virtual Manipulative

Transformations - Rotation:

Rotate shapes and their images with or without a background grid and axes.

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

Parent Resources

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

Perspectives Video: Expert

Problem Solving with Project Constraints:

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!

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: Uniform Scaling:

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

Type: Perspectives Video: Professional/Enthusiast

Using Geometry and Computers to make Art with CNC Machining:

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

Type: Perspectives Video: Professional/Enthusiast

Estimating Oil Seep Production by Bubble Volume:

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!

Type: Perspectives Video: Professional/Enthusiast

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.

SAT Scores:

This problem solving task challenges students to answer probability questions about SAT scores, using distribution and mean to solve the problem.

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.

Should We Send Out a Certificate?:

The purpose of this task is to have students complete normal distribution calculations and to use properties of normal distributions to draw conclusions.

Do You Fit in This Car?:

This task requires students to use the normal distribution as a model for a data distribution. Students must use given means and standard deviations to approximate population percentages.

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.

Exstensions, 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.

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.

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.

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.

Seven Circles III:

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

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.

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.

Unit Squares and Triangles:

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

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.

Seven Circles II:

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

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.

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

Bisecting an angle:

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

Locating Warehouse:

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

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.

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.

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.

Taxi!:

This simple conceptual problem does not require algebraic manipulation, but requires students to articulate the reasoning behind each statement.

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?

In order to engage this task meaningfully, students must be aware of the convention that va for a positive number a refers to the positive square root of a. The purpose of the task is to show students a situation where squaring both sides of an equation can result in an equation with more solutions than the original one.

Quinoa Pasta 3:

This task is an example of a mathematical modeling problem (SMP 4) and it also illustrates SMP 1 (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 task has some aspects of a mathematical modeling problem (SMP 4) and it also illustrates SMP 1 (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 A-REI.3.6, solving systems of linear equations exactly, 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 (e.g, 6.EE.5). By asking students to reason about solutions without explicitly solving them, we get at 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 very simple; the point of the task is not to test technique in solving equations, but to encourage students to reason about them.

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.

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 mathematical practice standard MP4 (Model with mathematics), and crucial as the system has an integer solution for both situations, that is, whether or not 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.

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.

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.

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.

Sum of Angles in a Polygon:

This problem provides students with an opportunity to discover algebraic structure in a geometric context. More specifically, the student will need to divide up the given polygons into triangles and then use the fact that the sum of the angles in each triangle is 180°.

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.

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.

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 MAFS.K12.MP.1.1, 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.

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.

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.

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=8).

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.

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.

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 (MAFS.912.N-Q.1.1). 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.

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).

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.

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.

Tutorials

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

Multiplying And Dividing With Inequalities:

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

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

Multiplying Polynomials:

This tutorial will help the learners practice multiplication of polynomials. Learners will understand that when they multiply expressions with more than two terms, they need to make sure each term in the first expression multiplies every term in the second expression.

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

In this tutorial, students will learn how to add and subtract polynomials functions using horizontal and vertical methods. In a horizontal format, like terms should be grouped together using the commutative property. In vertical format, terms should be listed by ascending degree with like terms placed below each other.

Type: Tutorial

Introduction to Polynomials:

In this tutorial students learn how to identify a polynomial, how to find the degree of a polynomial, and how to write a polynomial in standard format.

Type: Tutorial

Multiplying Polynomials:

This resource is a step-by-step tutorial on how to multiply polynomials.

Type: Tutorial

Video/Audio/Animations

Graphing Lines 1:

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

Type: Video/Audio/Animation

Averages:

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

Type: Video/Audio/Animation

Virtual Manipulatives

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

Type: Virtual Manipulative

Algebra Tiles (Multiplying Binomials):

This virtual manipulative is intended to allow the student to practice multiplication of binomials. The student should understand how to use algebra tiles before using this tool.

Type: Virtual Manipulative

Space Blocks:

This virtual manipulative allows students to manipulate blocks, add or remove blocks, and connect them together to form solids. They can also experiment with counting the number of exposed faces, seeing what happens to the surface area when blocks are added or removed, and "unfolding" a block to create a net .

Type: Virtual Manipulative

Histogram vs. Box Plot:

This simulation allows the student to create a box plot and a histogram for the same set of data and toggle between the two displays. Maximum, minimum, median and mean are shown for the data set. The student can change the cell width to explore how the histogram is affected.

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

Pythagorean Theorem Manipulatives:

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

Type: Virtual Manipulative

Algebra Balance Scales - with Negatives:

This site provides a virtual balance on which the student can represent (and then solve) simple linear equations with integer answers. Conceptually, positive weights (unit-blocks and x-boxes) push the pans of the scale downward. Negative values are represented by balloons which can be attached to the pans of the scale. The student can then manipulate the weights to solve the equation while simultaneously seeing a visual display of these effects on the equation.

Type: Virtual Manipulative

Transformations - Rotation:

Rotate shapes and their images with or without a background grid and axes.

Type: Virtual Manipulative