# Fundamental Explorations in Mathematics 2   (#7912115)

The course was/will be terminated at the end of School Year 2016 - 2017

## General Course Information and Notes

### General Information

Course Number: 7912115
Course Path:
Abbreviated Title: FUND EXPLORS IN MATH 2
Course Length: Year (Y)
Course Status: Terminated

## 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

Professor E. Qual Part 2: Two-Step Equations & Rational Numbers:

Practice solving and checking two-step equations with rational numbers in this interactive tutorial.

This is part 2 of the two-part series on two-step equations. Click HERE to open Part 1.

Type: Original Student Tutorial

Professor E. Qual Part 1: 2 Step Equations:

Professor E. Qual will teach you how to solve and check two-step equations in this interactive tutorial.

This is part 1 of a two-part series about solving 2-step equations. Click HERE to open Part 2.

Type: Original Student Tutorial

Balancing the Machine:

Use models to solve balance problems on a space station in this interactive, math and science tutorial.

Type: Original Student Tutorial

Driven By Functions:

Learn how to determine if a relationship is a function in this interactive tutorial that shows you inputs, outputs, equations, graphs and verbal descriptions.

Type: Original Student Tutorial

Math Soup: Creating Equivalent Expressions by Combining Like Terms :

Learn how to combine like terms to create equivalent expressions in this interactive tutorial.

Type: Original Student Tutorial

Pizza Pi: Circumference:

Explore the origins of Pi as the ratio of Circumference to diameter of a circle. In this interactive tutorial you'll work with the circumference formula to determine the circumference of a circle and work backwards to determine the diameter and radius of a circle.

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

Pizza Pi: Area:

Explore how to calculate the area of circles in terms of pi and with pi approximations in this interactive tutorial. You will also experience irregular area situations that require the use of the area of a circle formula.

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

Learn how to explain the meaning of additive inverse, identify the additive inverse of a given rational number, and justify your answer on a number line.

Type: Original Student Tutorial

Swimming in Circles:

Learn to solve problems involving the circumference and area of a circle in this pool-themed, interactive tutorial.

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

Why Does a Negative Times a Negative Equal a Positive?:

Use mathematical properties to explain why a negative factor times a negative factor equals a positive product… instead of just quoting a rule 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

## Educational Games

Rewriting Expressions: Simplifying Rational Expressions With Exponents:

In this challenge game, you will be simplifying fractional expressions with exponents. 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

Transformation Complete:

Play this interactive game and determine whether the similar shapes have gone through rotations, translations, or reflections.

Type: Educational Game

Integers Jeopardy Game:

This interactive game has 4 categories: adding integers, subtracting integers, multiplying integers, and dividing integers. Students can play individually or in teams.

Type: Educational Game

This addition game encourages some logical analysis as well as addition skills. This particular circle game uses positive and negative integers. There is only one way to combine all the given numbers so that every circle sums to zero.
(source: NLVM grade 6-8 "Circle 0")

Type: Educational Game

Estimator Four:

In this activity, students play a game of connect four, but to place a piece on the board they have to correctly estimate an addition, multiplication, or percentage problem. Students can adjust the difficulty of the problems as well as how close the estimate has to be to the actual result. This activity allows students to practice estimating addition, multiplication, and percentages of large numbers (100s). 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

Estimator Quiz:

In this activity, students are quizzed on their ability to estimate sums, products, and percentages. The student can adjust the difficulty of the problems and how close they have to be to the actual answer. This activity allows students to practice estimating addition, multiplication, or percentages of large numbers. 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

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

This virtual manipulative provides students with practice adding positive and negative integers. Students are given an addition problem and using one-to-one correspondence, the student is able to see what happens when adding negative integers. The addition problems can be computer generated or teacher generated and there is a free play mode which allows the student to practice with the chips and become familiar with the process of moving the chips around the page and creating a visual representation of an addition problem with integers.

Type: Educational Game

## Educational Software / Tools

Savings Calculator:

This manipulative is a versatile online savings calculator that calculates both simple and compounding interest. This free online calculator calculates and graphs accrued interest and total savings balance. The calculator allows for a variety of variables including interest rates, initial investment, time, compounded interest, and whether there are regular deposits made.

Type: Educational Software / Tool

Transformations Using Technology:

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

Type: Educational Software / Tool

Arithmetic Quiz:

In this activity, students solve arithmetic problems involving whole numbers, integers, addition, subtraction, multiplication, and division. This activity allows students to track their progress in learning how to perform arithmetic on whole numbers and integers. 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 Software / Tool

## Perspectives Video: Experts

Practical Use of Area and Circumference:

A math teacher describes the relationship between area and circumference and gives examples in nature.

Type: Perspectives Video: Expert

Measuring a Grid for Underwater Archeology:

Don't be a square! Learn about how even grids help archaeologists track provenience!

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

Modeling with Polygons for 3D Printers:

Understand 3D modeling from a new angle when you learn about surface geometry and 3D printing.

Type: Perspectives Video: Professional/Enthusiast

Building Scale Models to Solve an Archaeological Mystery:

An archaeologist describes how mathematics can help prove a theory about mysterious prehistoric structures called shell rings.

Type: Perspectives Video: Professional/Enthusiast

Ratios and Proportions in Mixing Ceramic Glazes:

Ceramic glaze recipes are fluid and not set in stone, but can only be formulated consistently with a good understanding of math!

Type: Perspectives Video: Professional/Enthusiast

## Presentation/Slideshow

The Pythagorean Theorem: Geometry’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

Smiles:

In this online problem-solving challenge, students apply algebraic reasoning to determine the "costs" of individual types of faces from sums of frowns, smiles, and neutral faces. This page provides three pictorial problems involving solving systems of equations along with tips for thinking through the problem, the solution, and other similar problems.

Partitioning a Hexagon:

The purpose of this task is for students to find a way to decompose a regular hexagon into congruent figures. This is meant as an instructional task that gives students some practice working with transformations.

Pennies to Heaven:

The goal of this task is to give students a context to investigate large numbers and measurements. Students need to fluently convert units with very large numbers in order to successfully complete this task. The total number of pennies minted either in a single year or for the last century is phenomenally large and difficult to grasp. One way to assess how large this number is would be to consider how far all of these pennies would reach if we were able to stack them one on top of another: this is another phenomenally large number but just how large may well come as a surprise.

Anna in D.C.:

The purpose of this task is to give students an opportunity to solve a challenging multistep percentage problem that can be approached in several different ways. Students are asked to find the cost of a meal before tax and tip when given the total cost of the meal. The task can illustrate multiple standards depending on the prior knowledge of the students and the approach used to solve the problem.

DVD Profits, Variation 1:

In this task, students are asked to determine the unit price of a product under two different circumstances. They are also asked to generalize the cost of producing x items in each case.

Glasses:

In this resource, students will determine the volumes of three different shaped drinking glasses. They will need prior knowledge with volume formulas for cylinders, cones, and spheres, as well as experience with equation solving, simplifying square roots, and applying the Pythagorean theorem.

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.

Discounted Books:

This purpose of this task is to help students see two different ways to look at percentages both as a decrease and an increase of an original amount. In addition, students have to turn a verbal description of several operations into mathematical symbols. This requires converting simple percentages to decimals as well as identifying equivalent expressions without variables.

Equivalent Expressions?:

Students are asked to determine if two expressions are equivalent and explain their reasoning.

Students are asked to write and solve an inequality to determine the number of people that can safely rent a boat.

Guess My Number:

This problem asks the students to represent a sequence of operations using an expression and then to write and solve simple equations. The problem is posed as a game and allows the students to visualize mathematical operations. It would make sense to actually play a similar game in pairs first and then ask the students to record the operations to figure out each other's numbers.

Miles to Kilometers:

In this task students are asked to write two expressions from verbal descriptions and determine if they are equivalent. The expressions involve both percent and fractions. This task is most appropriate for a classroom discussion since the statement of the problem has some ambiguity.

Shrinking:

Students are asked to determine the change in height in inches when given a constant rate of change in centimeters. The answer is rounded to the nearest half inch.

Sports Equipment Set:

The student is asked to write and solve an inequality to match the context.

Eight Circles:

Students are asked to find the area of a shaded region using a diagram and the information provided. The purpose of this task is to strengthen student understanding of area.

Distances on the Number Line 2:

The purpose of this task is meant to reinforce students' understanding of rational numbers as points on the number line and to provide them with a visual way of understanding that the sum of a number and its additive inverse (usually called its "opposite") is zero.

Comparing Freezing Points:

In this task, students answer a question about the difference between two temperatures that are negative numbers.

Coupon Versus Discount:

In this task, students are presented with a real-world problem involving the price of an item on sale. To answer the question, students must represent the problem by defining a variable and related quantities, and then write and solve an equation.

Operations on the Number Line:

The purpose of this task is to help solidify students' understanding of signed numbers as points on a number line and to understand the geometric interpretation of adding and subtracting signed numbers. There is a subtle distinction in the Florida Standards between a fraction and a rational number. Fractions are always positive, and when thinking of the symbol ab as a fraction, it is possible to interpret it as a equal-sized pieces where b pieces make one whole.

Repeating Decimal as Approximation:

The student is asked to complete a long division which results in a repeating decimal, and then use multiplication to "check" their answer. The purpose of the task is to have students reflect on the meaning of repeating decimal representation through approximation.

Sharing Prize Money:

Students are asked to determine how to distribute prize money among three classes based on the contribution of each class.

Sand Under the Swing Set:

The 7th graders at Sunview Middle School were helping to renovate a playground for the kindergartners at a nearby elementary school. City regulations require that the sand underneath the swings be at least 15 inches deep. The sand under both swing sets was only 12 inches deep when they started. The rectangular area under the small swing set measures 9 feet by 12 feet and required 40 bags of sand to increase the depth by 3 inches. How many bags of sand will the students need to cover the rectangular area under the large swing set if it is 1.5 times as long and 1.5 times as wide as the area under the small swing set?

Art Class, Variation 1:

Students are asked to use ratios and proportional reasoning to compare paint mixtures numerically and graphically.

Chess Club:

This problem includes a percent increase in one part with a percent decrease in the remaining and asks students to find the overall percent change. The problem may be solved using proportions or by reasoning through the computations or writing a set of equations.

Comparing Years:

Students are asked to make comparisons among the Egyptian, Gregorian, and Julian methods of measuring a year.

Cooking with the Whole Cup:

Students are asked to use proportional reasoning to answer a series of questions in the context of a recipe.

Gotham City Taxis:

The purpose of this task is to give students an opportunity to solve a multi-step ratio problem that can be approached in many ways. This can be done by making a table, which helps illustrate the pattern of taxi rates for different distances traveled and with a little persistence leads to a solution which uses arithmetic. It is also possible to calculate a unit rate (dollars per mile) and use this to find the distance directly without making a table.

Finding a 10% Increase:

5,000 people visited a book fair in the first week. The number of visitors increased by 10% in the second week. How many people visited the book fair in the second week?

Friends Meeting on Bikes:

Using the information provided find out how fast Anya rode her bike.

Molly's Run:

This task asks students to solve a problem in a context involving constant speed. This task provides a transition from working with ratios involving whole numbers to ratios involving fractions. This problem can be thought of in several ways; in particular, this problem also provides an opportunity for students to work with the "How many in one group?'' interpretation of division.

Molly's Run, Assessment Variation:

Use the information provided to find out how long it will take Molly to run one mile.

Music Companies, Variation 1:

This problem requires a comparison of rates where one is given in terms of unit rates, and the other is not. See "Music Companies, Variation 2" for a task with a very similar setup but is much more involved and so illustrates MAFS.7.RP.1.3.

Music Companies, Variation 2:

This problem has multiple steps. In order to solve the problem it is necessary to compute: the value of the TunesTown shares; the total value of the BeatStreet offer of 20 million shares at \$25 per share; the difference between these two amounts; and the cost per share of each of the extra 2 million shares MusicMind offers to equal to the difference.

Robot Races:

Students should use information provided to answer the questions regarding robot races.

Quinoa Pasta 1:

This task asks students to find the amount of two ingredients in a pasta blend. The task provides all the information necessary to solve the problem by setting up two linear equations in two unknowns. This progression of tasks helps distinguish between 8th grade and high school expectations related to systems of linear equations.

Cell Phone Plans:

This task presents a real-world problem requiring the students to write linear equations to model different cell phone plans. Looking at the graphs of the lines in the context of the cell phone plans allows the students to connect the meaning of the intersection points of two lines with the simultaneous solution of two linear equations. The students are required to find the solution algebraically to complete the task.

Two Lines:

In this task, we are given the graph of two lines including the coordinates of the intersection point and the coordinates of the two vertical intercepts and are asked for the corresponding equations of the lines. It is a very straightforward task that connects graphs and equations and solutions and intersection points.

Who Has the Best Job?:

This task asks the student to graph and compare two proportional relationships and interpret the unit rate as the slope of the graph.

Coffee by the Pound:

In this example, students will answer questions about unit price of coffee, make a graph of the information, and explain the meaning of slope in the given context.

Foxes and Rabbits:

This task emphasizes the importance of the "every input has exactly one output" clause in the definition of a function, which is violated in the table of values of the two populations. Noteworthy is that since the data is a collection of input-output pairs, no verbal description of the function is given, so part of the task is processing what the "rule form" of the proposed functions would look like.

Function Rules:

This task can be played as a game where students have to guess the rule and the instructor gives more and more input output pairs. Giving only three input output pairs might not be enough to clarify the rule. Instructors might consider varying the inputs in, for example, the second table, to provide non-integer entries. A nice variation on the game is to have students who think they found the rule supply input output pairs, and the teachers confirms or denies that they are right. Verbalizing the rule requires precision of language. This task can be used to introduce the idea of a function as a rule that assigns a unique output to every input.

Calculating the Square Root of 2:

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

Sale!:

Students are asked to determine which sale option results in the largest percent decrease in cost.

Selling Computers:

The sales team at an electronics store sold 48 computers last month. The manager at the store wants to encourage the sales team to sell more computers and is going to give all the sales team members a bonus if the number of computers sold increases by 30% in the next month. How many computers must the sales team sell to receive the bonus? Explain your reasoning.

Sore Throats, Variation 1:

Students are asked to decide if two given ratios are equivalent.

Stock Swaps, Variation 2:

Students are asked to solve a problem using proportional reasoning in a real world context to determine the number of shares needed to complete a stock purchase.

Stock Swaps, Variation 3:

Students are asked to solve a multistep ratio problem in a real-world context.

Tax and Tip:

After eating at your favorite restaurant, you know that the bill before tax is \$52.60 and that the sales tax rate is 8%. You decide to leave a 20% tip for the waiter based on the pre-tax amount. How much should you leave for the waiter? How much will the total bill be, including tax and tip?

The purpose of this task is for students to calculate the percent increase and relative cost in a real-world context. Inflation, one of the big ideas in economics, is the rise in price of goods and services over time. This is considered in relation to the amount of money you have.

Track Practice:

This activity asks the student to use unit rate and proportional reasoning to determine which of two runners is the fastest.

Two-School Dance:

The purpose of this task is to see how well students students understand and reason with ratios.

Reflecting Reflections:

In this resource, students experiment with successive reflections of a triangle in a coordinate plane.

Estimating Square Roots:

By definition, the square root of a number n is the number you square to get n. The purpose of this task is to have students use the meaning of a square root to find a decimal approximation of a square root of a non-square integer. Students may need guidance in thinking about how to approach the task.

Point Reflection:

The purpose of this task is for students to apply a reflection to a single point. The standard MAFS.8.G.1.1 asks students to apply rigid motions to lines, line segments, and angles. Although this problem only applies a reflection to a single point, it has high cognitive demand if the students are prompted to supply a picture. This is because the coordinates of the point (1000,2012) are very large. If students try to plot this point and the line of reflection on the usual x-y coordinate grid, then either the graph will be too big or else the point will lie so close to the line of reflection that it is not clear whether or not it lies on this line. A good picture requires a careful choice of the appropriate region in the plane and the corresponding labels. Moreover, reflections of lines, line segments, and angles are all found by reflecting individual points.

Reflecting a Rectangle Over a Diagonal:

The task is intended for instructional purposes and assumes that students know the properties of rigid transformations described in MAFS.8.G.1.1. Note that the vertices of the rectangles in question do not fall exactly at intersections of the horizontal and vertical lines on the grid. This means that students need to approximate and this provides an extra challenge. Also providing a challenge is the fact that the grids have been drawn so that they are aligned with the diagonal of the rectangles rather than being aligned with the vertical and horizontal directions of the page. However, this choice of grid also makes it easier to reason about the reflections if they understand the descriptions of rigid motions indicated in MAFS.8.G.1.3.

Converting Decimal Representations of Rational Numbers to Fraction Representations:

MAFS.8.NS.1.1 requires students to "convert a decimal expansion which repeats eventually into a rational number." Despite this choice of wording, the numbers in this task are rational numbers regardless of choice of representation. For example, 0.333¯ and 13 are two different ways of representing the same number.

Is This a Rectangle?:

The goal of this task is to provide an opportunity for students to apply a wide range of ideas from geometry and algebra in order to show that a given quadrilateral is a rectangle. Creativity will be essential here as the only given information is the Cartesian coordinates of the quadrilateral's vertices. Using this information to show that the four angles are right angles will require some auxiliary constructions. Students will need ample time and, for some of the methods provided below, guidance. The reward of going through this task thoroughly should justify the effort because it provides students an opportunity to see multiple geometric and algebraic constructions unified to achieve a common purpose. The teacher may wish to have students first brainstorm for methods of showing that a quadrilateral is rectangle (before presenting them with the explicit coordinates of the rectangle for this problem): ideally, they can then divide into groups and get to work straightaway once presented with the coordinates of the quadrilateral for this problem.

Identifying Rational Numbers:

The task assumes that students are able to express a given repeating decimal as a fraction. Teachers looking for a task to fill in this background knowledge could consider the related task "8.NS Converting Decimal Representations of Rational Numbers to Fraction Representations."

Triangle congruence with coordinates:

In this resource, students will decide how to use transformations in the coordinate plane to translate a triangle onto a congruent triangle. Exploratory examples are included to prompt analytical thinking.

Comparing Speeds in Graphs and Equations:

This task provides the opportunity for students to reason about graphs, slopes, and rates without having a scale on the axes or an equation to represent the graphs. Students who prefer to work with specific numbers can write in scales on the axes to help them get started.

US Garbage, Version 1:

In this task, the rule of the function is more conceptual. We assign to a year (an input) the total amount of garbage produced in that year (the corresponding output). Even if we didn't know the exact amount for a year, it is clear that there will not be two different amounts of garbage produced in the same year. Thus, this makes sense as a "rule" even though there is no algorithmic way to determine the output for a given input except looking it up in the table.

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.

Kimi and Jordan:

Students are asked to create and graph linear equations to compare the savings of two individuals. The purpose of the table in (a) is to help students complete (b) by noticing regularity in the repeated reasoning required to complete the table (Standard for Mathematical Practice, MAFS.K12.MP.8.1).

Peaches and Plums:

This task asks students to reason about the relative costs per pound of two fruits without actually knowing what the costs are. Students who find this difficult may add a scale to the graph and reason about the meanings of the ordered pairs. Comparing the two approaches in a class discussion can be a profitable way to help students make sense of slope.

Running on the Football Field:

Students need to reason as to how they can use the Pythagorean Theorem to find the distances ran by Ben Watson and Champ Bailey. The focus here should not be on who ran a greater distance but on seeing how to set up right triangles to apply the Pythagorean Theorem to this problem. Students must use their measurement skills and make reasonable estimates to set up triangles and correctly apply the Theorem.

Equations of Lines:

This task asks the student to understand the relationship between slope and changes in x- and y-values of a linear function.

Find the Change:

This activity challenges students to recognize the relationship between slope and the difference in x- and y-values of a linear function. Help students solidify their understanding of linear functions and push them to be more fluent in their reasoning about slope and y-intercepts. This task has also produced a reasonable starting place for discussing point-slope form of a linear equation.

Fixing the Furnace:

Students are asked to write equations to model the repair costs of three different companies and determine the conditions for which each company would be least expensive. This task can be used to both assess student understanding of systems of linear equations or to promote discussion and student thinking that would allow for a stronger solidification of these concepts. The solution can be determined in multiple ways, including either a graphical or algebraic approach.

Extending the Definitions of Exponents, Variation 1:

This is an instructional task meant to generate a conversation around the meaning of negative integer exponents. While it may be unfamiliar to some students, it is good for them to learn the convention that negative time is simply any time before t=0.

Log Ride:

Students are asked to solve an inequality in order to answer a real-world question.

How Many Solutions?:

The student is given the equation 5x-2y=3 and asked, if possible, to write a second linear equation creating systems resulting in one, two, infinite, and no solutions.

It's Raining!!! (Compare areas of wiped windshields):

In this problem-solving task, students are challenged to determine whether the windshield wipers on a car or a truck allow the drivers to see more area clearly. To solve this problem, students must apply the Pythagorean theorem and their ability to find area of circles and parallelograms to find the answer. Be sure to click the links in the orange bar at the top of the page for more information about the challenge. From NCTM's Figure This! Math Challenges for Families.

Measuring Henry's Cabin:

This resource introduces students to the aspects a builder must think about before constructing a building. Students will study the cabin blueprint of Henry David Thoreau and then will find the surface area of the walls and how much paint would be needed. Then, students will find the volume of the cabin to determine the home heating needs. Third, students will study the blueprint and will create a 1/10 scale of it on graph paper and then will use art supplies to create a model of the cabin. Last, students will design and create models of furniture to scale for the cabin.

## Student Center Activity

Students can practice answering mathematics questions on a variety of topics. With an account, students can save their work and send it to their teacher when complete.

Type: Student Center Activity

## Tutorials

Scaling Down a Triangle by Half:

This video demonstrates the effect of a dilation on the coordinates of a triangle.

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

Pythagorean Theorem: A Carpet Example:

In this tutorial, you will practice finding the missing width of a carpet, given the length of one side and the diagonal of the carpet.

Type: Tutorial

Limits and Infinity:

We will look at more examples of limits at infinity.

Type: Tutorial

Checking Solutions to Systems of Equations Example:

This video demonstrates testing a solution (coordinate pair) for a system of equations

Type: Tutorial

Using a Graph to Analyze Solutions to Linear Systems:

This video demonstrates analyzing solutions to linear systems using a graph.

Type: Tutorial

Example of System with No Solution:

This video shows how to algebraically analyze a system that has no solutions.

Type: Tutorial

Does a Vertical Line Represent a Function?:

This video explains why a vertical line does not represent a function.

Type: Tutorial

Check if a Verbal Description Is a Function:

This video demonstrates how to check if a verbal description represents a function.

Type: Tutorial

How to Check if Points on a Graph Represent a Function:

This video shows how to check whether a given set of points can represent a function. For the set to represent a function, each domain element must have one corresponding range element at most.

Type: Tutorial

Linear Function: Spending Money:

In this tutorial, you will practice using an equation in slope-intercept form to find coordinates, then graph the coordinates to predict an answer to the problem.

Type: Tutorial

Thinking About the Sign of Expressions:

This video shows some examples that test your understanding of what happens when positive and negative numbers are multiplied and divided.

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

Solve a consecutive integer problem algebraically:

Students will learn how to solve a consecutive integer problem. Checking the solution will be left to the student.

Type: Tutorial

Age word problem:

This tuptorial shows students how to set up and solve an age word problem. The tutorial also shows how tp check your work using substitution.

Type: Tutorial

Classifying Numbers:

In this tutorial, you will practice classifying numbers as whole numbers, integers, rational numbers, and irrational numbers.

Type: Tutorial

Age word problem :

Students will learn how to set up and solve an age word problem.

Type: Tutorial

Exponent Properties Involving Products:

This video discusses exponent properties involving products.

Type: Tutorial

Exponent Properties Involving Quotients:

This video models how to use the Quotient of Powers property.

Type: Tutorial

An introduction to rational and irrational numbers:

Students will learn the difference between rational and irrational numbers.

Type: Tutorial

Negative exponents:

This tutorial shows students the rule for negative exponents. Students will see, using variables, the pattern for negative exponents.

Type: Tutorial

Converting repeating decimals to fractions :

Students will learn how to convert difficult repeating decimals to fractions.

Type: Tutorial

Converting repeating decimals to fractions :

This tutorial shows students how to convert basic repeating decimals to fractions.

Type: Tutorial

Negative exponents:

In this tutorial, students will learn about negative exponents. An emphasis is placed on multiplying by the reciprocal of a number.

Type: Tutorial

Converting a fraction to a repeating decimal:

Students will learn how to convert a fraction into a repeating decimal. Students should know how to use long division before starting this tutorial.

Type: Tutorial

Finding the square root of a decimal:

Students will learn how to find the square root of a decimal number.

Type: Tutorial

Finding cube roots:

Learn how to find the cube root of -512 using prime factorization.

Type: Tutorial

Introduction to cube roots:

Students will learn the meaning of cube roots and how to find them. Students will also learn how to find the cube root of a negative number.

Type: Tutorial

Introduction to square roots:

Students will earn about the square root symbol (the principal root) and what it means to find a square root. Students will also learn how to solve simple square root equations.

Type: Tutorial

Impact of a radius change on the area of a circle:

This tutorial shows how the area and circumference relate to each other. Students will investigate how changing the radius of a circle affects the area and circumference.

Type: Tutorial

Circles: Radius, Circumference, Diameter and Pi:

A circle is at the foundation of geometry. In this tutorial, students are shown the parts of a circle and how the radiius, diameter, circumference and Pi relate to each other. Students will also learn how to find the area and circumference of a circle.

Type: Tutorial

Circumference of a circle:

This tutorial shows how to find the circumference, the distance around a circle, given the area. Students will build upon their knowledge of the parts of circle.

Type: Tutorial

Area of a circle:

In this example, students solve for the area of a circle when given the diameter. The diameter is the length of a line that runs across the circle and through the center.

Type: Tutorial

Factor a Linear Expression by Taking a Common Factor:

This video demonstrates how to factor a linear expression by taking a common factor.

Type: Tutorial

Basic Linear Equation Word Problem:

This video shows how to construct and solve a basic linear equation to solve a word problem.

Type: Tutorial

Proportion Word Problem:

This video demonstrates how to write and solve an equation for a proportional relationship.

Type: Tutorial

Adding and Subtracting Numbers in Different Formats:

In this example, you will work with three numbers in different formats: a percent, a decimal, and a mixed number.

Type: Tutorial

Changing a Fraction to Decimal Form:

In this video, you will practice changing a fraction into decimal form.

Type: Tutorial

Multiplying and Dividing Even and Odd Numbers of Negatives:

You will learn how multiplication and division problems give us a positive or negative answer depending on whether there are an even or odd number of negative integers used in the problem.

Type: Tutorial

Interpreting Graphs of Proportional Relationships (Examples):

This video shows how to read and understand graphs of proportional relationships.

Type: Tutorial

Combining Like Terms Introduction:

This video teaches about combining like terms in linear equations.

Type: Tutorial

Find the Volume of a Ring:

Find the volume of an object, given dimensions of a cube filled with water, and the incremental volume after the object is dropped into the cube

Type: Tutorial

Solving a Problem Involving the Volume of a Rectangular Prism:

A problem involving packing a larger rectangular prism with smaller ones is solved in two different ways.

Type: Tutorial

Find the Volume of a Triangular Prism and Cube:

We will practice finding the volume of a triangular prism, and a cube by appying the formula for volume.

Type: Tutorial

Simplifying Expressions with Rational Numbers:

In this tutorial, you will simplify expressions involving positive and negative fractions.

Type: Tutorial

Making Sense of Complex Fractions:

In this tutorial, you will see how to simplify complex fractions.

Type: Tutorial

Solving a proportion with an unknown variable :

Here's a great video where we explain the reasoning behind solving proportions. We'll put some algebra to work to get our answers, too. This video shows three different methods for solving proportions.

Type: Tutorial

Setting up proportions to solve word problems:

This video shows some examples of writing two ratios and setting them equal to each other to solve proportion word problems.

Type: Tutorial

Determining Rates with Fractions:

This video demonstrates finding a unit rate from a rate containing fractions.

Type: Tutorial

Rate Problem With Fractions:

One common application of rate is determining speed. Watch as we solve a rate problem finding speed in meters per second using distance (in meters) and time (in seconds).

Type: Tutorial

Rational Number Word Problem with Decimals:

This video demonstrates adding and subtracting decimals in the context of an overdrawn checking account.

Type: Tutorial

Multiplying and dividing inequalities :

Students will solve the inequality and graph the solution.

Type: Tutorial

Negative Signs in Numerators and Denominators:

In this tutorial, you will evaluate fractions involving negative numbers and variables to determine if expressions are equivalent.

Type: Tutorial

Dividing Negative Fractions:

In this tutorial, you will see how to divide fractions involving negative integers.

Type: Tutorial

Multiplying Negative and Positive Fractions:

In this tutorial you will practice multiplying and dividing fractions involving negative numbers.

Type: Tutorial

Multiplying Positive and Negative Numbers:

In this tutorial, you will learn rules for multiplying positive and negative integers.

Type: Tutorial

Dividing Positive and Negative Numbers:

In this tutorial you will learn how to divide with negative integers.

Type: Tutorial

Why a Negative Times a Negative Makes a Positive:

In this tutorial you will use the repeated addition model of multiplication to help you understand why multiplying negative numbers results in a positive answer.

Type: Tutorial

Why a Negative Times a Negative is a Positive:

In this tutorial, you will use the distributive property to understand why the product of two negative numbers is positive.

Type: Tutorial

Substitution with negative numbers:

Practice substituting positive and negative values for variables.

Type: Tutorial

Finding the absolute value as distance between numbers:

In this video, we will find the absolute value as distance between rational numbers.

Type: Tutorial

Even More Negative Number Practice:

This video uses the number line to find unknown values in subtraction statements with negative numbers.

Type: Tutorial

Adding Negative Numbers on Number Line Examples:

This video asks you to select the model that matches the given expression.

Type: Tutorial

Negative Number Word Problem:

Use a number line to solve a word problem that includes a negative number.

Type: Tutorial

Finding Initial Temperature from Temperature Changes:

In this video, we figure out the temperature in Fairbanks, Alaska by adding and subtracting integers.

Type: Tutorial

Linear Equations:

This tutorial will help you to explore slopes of lines and see how slope is represented on the x-y axes.

Type: Tutorial

Percent Word Problem Example 1:

We're putting a little algebra to work to find the full price when you know the discount price in this percent word problem.

Type: Tutorial

Converting Decimals to Percents:

This video demonstrates how to write a decimal as a percent.

Type: Tutorial

This video demonstrates how to add and subtract negative fractions with unlike denominators.

Type: Tutorial

This video demonstrates use of a number line and absolute value to add negative numbers.

Type: Tutorial

This video demonstrates use of a number line to add numbers with positive and negative signs.

Type: Tutorial

Subtracting a Negative = Adding a Positive:

Find out why subtracting a negative number is the same as adding the absolute value of that number.

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

Negative Number Practice:

This video demonstrates adding and subtracting integers using a number line.

Type: Tutorial

The Meaning of Percent over 100:

This video demonstrtates a visual model of a percent greater than 100.

Type: Tutorial

Exponents and Powers:

This tutorial reviews the concept of exponents and powers and includes how to evaluate powers with negative signs.

Type: Tutorial

Power of a Power Property:

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

Type: Tutorial

Absolute Value:

This tutorial will help you understand the concept of absolute value. Take the quiz after the lesson to practice!

Type: Tutorial

Solving One-Step Equations Using Multiplication and Division:

This tutorial will help you to solve one-step equations using multiplication and division. For practice, take the quiz after the lesson!

Type: Tutorial

Multiplying Integers:

This tutorial demonstrates the number line method of multiplying integers. You will encounter four different combinations when multiplying integers: (1) Positive times positive, (2) Positive times negative, (3) Negative times negative, (4) Negative times positive. The lesson is available in video format, and there is a quiz for practice.

Type: Tutorial

Direct and Inverse Variation:

This video provides assistance with understanding direct and inverse variation.

Type: Tutorial

Solving Two-Step Equations:

This short video uses both an equation and a visual model to explain why the same steps must be used on both sides of the equation when solving for the value of a variable.

Type: Tutorial

Raising Exponential Expressions to Powers:

If a term raised to a power is enclosed in parentheses and then raised to another power, this expression can be simplified using the rules of multiplying exponents.

Type: Tutorial

Raising Products and Quotients to Powers:

Any expression consisting of multiplied and divide terms can be enclosed in parentheses and raised to a power. This can then be simplified using the rules for multiplying exponents.

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

Pre-Algebra - Fractions and Rational Numbers:

The first fractions used by ancient civilizations were "unit fractions." Later, numerators other than one were added, creating "vulgar fractions" which became our modern fractions. Together, fractions and integers form the "rational numbers."

Type: Tutorial

Pre-Algebra - Multiplying Negative Numbers:

When number systems were expanded to include negative numbers, rules had to be formulated so that multiplication would be consistent regardless of the sign of the operands.

Type: Tutorial

Pre-Algebra - Commutative & Associative Properties of Addition:

A look behind the fundamental properties of the most basic arithmetic operation, addition

Type: Tutorial

Subtracting Integers:

This tutorial will help the learner to understand the concept of subtracting the positive and negative integers with the help of a number line. Learners can also take a quiz after the concept is internalized.

Type: Tutorial

Students will be able to see examples of addition of integers while watching a short video, and practice adding integers using an online quiz.

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

Using the Proportion Method to Solve Percent Problems:

This site explicitly outlines the steps for using the proportion method to solve three different kinds of percent problems. It also includes sample problems for practice determining the part, the whole or the percent.

Type: Tutorial

Solving Equations With the Variable on Both Sides.:

This video models solving equations in one variable with variables on both sides of the equal sign.

Type: Tutorial

Solving Equations with One Variable :

This Khan Academy presentation models solving two-step equations with one variable.

Type: Tutorial

Converting Speed Units:

In this lesson, students will be viewing a Khan Academy video that will show how to convert ratios using speed units.

Type: Tutorial

Multiplying Fractions:

The video describes how to multiply fractions and state the answer in lowest terms.

Type: Tutorial

## Video/Audio/Animations

Solving Motion Problems with Linear Equations:

Based upon the definition of speed, linear equations can be created which allow us to solve problems involving constant speeds, time, and distance.

Type: Video/Audio/Animation

Solving Problems with Linear Equations:

How do we create linear equations to solve real-world problems? The video explains the process.

Type: Video/Audio/Animation

Irrational Numbers:

Although the Greeks initially thought all numeric quantities could be represented by the ratio of two integers, i.e. rational numbers, we now know that not all numbers are rational. How do we know this?

Type: Video/Audio/Animation

Converting Fractions to Decimal Numbers:

Any fraction can be converted into an equivalent decimal number with a sequence of digits after the decimal point, which either repeats or terminates. The reason can be understood by close examination of the number line.

Type: Video/Audio/Animation

Exponents of One, Zero, and Negative:

Integer exponents greater than one represent the number of copies of the base which are multiplied together. hat if the exponent is one, zero, or negative? Using the rules of adding and subtracting exponents, we can see what the meaning must be.

Type: Video/Audio/Animation

Simplifying Multiplied Exponential Expressions:

Exponential expressions with multiplied terms can be simplified using the rules for adding exponents.

Type: Video/Audio/Animation

Simplifying Divided Exponential Expressions:

Exponential expressions with divided terms can be simplified using the rules for subtracting exponents.

Type: Video/Audio/Animation

Simplifying Mixed Exponential Expressions:

Exponential expressions with multiplied and divided terms can be simplified using the rules of adding and subtracting exponents.

Type: Video/Audio/Animation

Averages:

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

Type: Video/Audio/Animation

## Virtual Manipulatives

Algebra Balance Scales-Negatives:

This virtual manipulative allows the learners to solve simple linear equations through the use of a balance beam. Unit blocks and x-boxes are placed on the pans of a balance beam to balance it.

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

Percentages:

This virtual manipulative allows the student to enter any two of the three quantities involved in percentage computation: the whole, a part and the percent. This manipulative can also be used for the discussions of relations among fractions, decimals, ratios and percentages.

Type: Virtual Manipulative

Converting Units Through Dimensional Analysis:

Using this virtual manipulative, students apply dimensional analysis (AKA factor-label method or unit-factor method) to solve unit conversion problems. There is also the opportunity to create your own unit conversion problems.

Type: Virtual Manipulative

Circle Tool:

This applet allows students to investigate the relationships between the area and circumference of a circle and its radius and diameter. There are three sections to the site: Intro, Investigation, and Problems.

• In the Intro section, students can manipulate the size of a circle and see how the radius, diameter, and circumference are affected. Students can also play movie clip to visually see how these measurements are related.
• The Investigation section allows students to collect data points by dragging the circle radius to various lengths, and record in a table the data for radius, diameter, circumference and area. Clicking on the x/y button allows students to examine the relationship between any two measures. Clicking on the graph button will take students to a graph of the data. They can plot any of the four measures on the x-axis against any of the four measures on the y-axis.
• The Problems section contains questions for students to solve and record their answers in the correct unit.

(NCTM's Illuminations)

Type: Virtual Manipulative

Linear Function Machine:

In this activity, students plug values into the independent variable to see what the output is for that function. Then based on that information, they have to determine the coefficient (slope) and constant(y-intercept) for the linear function. This activity allows students to explore linear functions and what input values are useful in determining the linear function rule. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet.

Type: Virtual Manipulative

Mixtures:

In this online activity, students apply their understanding of proportional relationships by adding circles, either colored or not, to two different piles then combine the piles to produce a required percentage of colored circles. Students can play in four modes: exploration, unknown part, unknown whole, or unknown percent. This activity also includes supplemental materials in tabs above the applet, 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

Color Chips - Subtraction:

This virtual manipulative guides the student in the use of color counters to model subtraction of integers.

Type: Virtual Manipulative

Function Machine:

This animation helps students to understand the function concept through the machine metaphor. The domain elements are dragged into the machine, which then goes through some process and outputs the range element corresponding to the input. The user is asked to complete the outputs for the remaining inputs.

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

Number Cruncher:

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

Type: Virtual Manipulative

Pythagorean Theorem Manipulatives:

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

Type: Virtual Manipulative

Curve Fitting:

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

Type: Virtual Manipulative

Transformations - Translation:

The user can demonstrate or explore translation of shapes created with pattern blocks, using or not using a coordinate axes and lattice points background, by changing the translation vector.
(source: NLVM grade 6-8 "Transformations - Translation")

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 - Dilation:

Students use a slider to explore dilation and scale factor. Students can create and dilate their own figures. (source: NLVM grade 6-8 "Transformations - Dilation")

Type: Virtual Manipulative

Volt Meter (positive and negative numbers):

The user drags batteries to create a circuit. The voltage of the batteries that are placed will be displayed on the voltmeter, and an equation will be displayed in a list on the right, giving an example of how positive and negative numbers work together.

Type: Virtual Manipulative

Transformations - Rotation:

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

Type: Virtual Manipulative

Rotation of a Point:

This virtual manipulative is an interactive visual presentation of the rotation of a point around the origin of the coordinate system. The original point can be dragged to different positions and the angle of rotation can be changed with a 90° increment.

Type: Virtual Manipulative

## Parent Resources

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

## Educational Games

This addition game encourages some logical analysis as well as addition skills. This particular circle game uses positive and negative integers. There is only one way to combine all the given numbers so that every circle sums to zero.
(source: NLVM grade 6-8 "Circle 0")

Type: Educational Game

This virtual manipulative provides students with practice adding positive and negative integers. Students are given an addition problem and using one-to-one correspondence, the student is able to see what happens when adding negative integers. The addition problems can be computer generated or teacher generated and there is a free play mode which allows the student to practice with the chips and become familiar with the process of moving the chips around the page and creating a visual representation of an addition problem with integers.

Type: Educational Game

## Educational Software / Tool

Savings Calculator:

This manipulative is a versatile online savings calculator that calculates both simple and compounding interest. This free online calculator calculates and graphs accrued interest and total savings balance. The calculator allows for a variety of variables including interest rates, initial investment, time, compounded interest, and whether there are regular deposits made.

Type: Educational Software / Tool

## Perspectives Video: Experts

Practical Use of Area and Circumference:

A math teacher describes the relationship between area and circumference and gives examples in nature.

Type: Perspectives Video: Expert

Measuring a Grid for Underwater Archeology:

Don't be a square! Learn about how even grids help archaeologists track provenience!

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

Modeling with Polygons for 3D Printers:

Understand 3D modeling from a new angle when you learn about surface geometry and 3D printing.

Type: Perspectives Video: Professional/Enthusiast

Building Scale Models to Solve an Archaeological Mystery:

An archaeologist describes how mathematics can help prove a theory about mysterious prehistoric structures called shell rings.

Type: Perspectives Video: Professional/Enthusiast

Ratios and Proportions in Mixing Ceramic Glazes:

Ceramic glaze recipes are fluid and not set in stone, but can only be formulated consistently with a good understanding of math!

Type: Perspectives Video: Professional/Enthusiast

Smiles:

In this online problem-solving challenge, students apply algebraic reasoning to determine the "costs" of individual types of faces from sums of frowns, smiles, and neutral faces. This page provides three pictorial problems involving solving systems of equations along with tips for thinking through the problem, the solution, and other similar problems.

Partitioning a Hexagon:

The purpose of this task is for students to find a way to decompose a regular hexagon into congruent figures. This is meant as an instructional task that gives students some practice working with transformations.

Pennies to Heaven:

The goal of this task is to give students a context to investigate large numbers and measurements. Students need to fluently convert units with very large numbers in order to successfully complete this task. The total number of pennies minted either in a single year or for the last century is phenomenally large and difficult to grasp. One way to assess how large this number is would be to consider how far all of these pennies would reach if we were able to stack them one on top of another: this is another phenomenally large number but just how large may well come as a surprise.

Anna in D.C.:

The purpose of this task is to give students an opportunity to solve a challenging multistep percentage problem that can be approached in several different ways. Students are asked to find the cost of a meal before tax and tip when given the total cost of the meal. The task can illustrate multiple standards depending on the prior knowledge of the students and the approach used to solve the problem.

DVD Profits, Variation 1:

In this task, students are asked to determine the unit price of a product under two different circumstances. They are also asked to generalize the cost of producing x items in each case.

Glasses:

In this resource, students will determine the volumes of three different shaped drinking glasses. They will need prior knowledge with volume formulas for cylinders, cones, and spheres, as well as experience with equation solving, simplifying square roots, and applying the Pythagorean theorem.

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.

Discounted Books:

This purpose of this task is to help students see two different ways to look at percentages both as a decrease and an increase of an original amount. In addition, students have to turn a verbal description of several operations into mathematical symbols. This requires converting simple percentages to decimals as well as identifying equivalent expressions without variables.

Equivalent Expressions?:

Students are asked to determine if two expressions are equivalent and explain their reasoning.

Students are asked to write and solve an inequality to determine the number of people that can safely rent a boat.

Guess My Number:

This problem asks the students to represent a sequence of operations using an expression and then to write and solve simple equations. The problem is posed as a game and allows the students to visualize mathematical operations. It would make sense to actually play a similar game in pairs first and then ask the students to record the operations to figure out each other's numbers.

Miles to Kilometers:

In this task students are asked to write two expressions from verbal descriptions and determine if they are equivalent. The expressions involve both percent and fractions. This task is most appropriate for a classroom discussion since the statement of the problem has some ambiguity.

Shrinking:

Students are asked to determine the change in height in inches when given a constant rate of change in centimeters. The answer is rounded to the nearest half inch.

Sports Equipment Set:

The student is asked to write and solve an inequality to match the context.

Eight Circles:

Students are asked to find the area of a shaded region using a diagram and the information provided. The purpose of this task is to strengthen student understanding of area.

Distances on the Number Line 2:

The purpose of this task is meant to reinforce students' understanding of rational numbers as points on the number line and to provide them with a visual way of understanding that the sum of a number and its additive inverse (usually called its "opposite") is zero.

Comparing Freezing Points:

In this task, students answer a question about the difference between two temperatures that are negative numbers.

Coupon Versus Discount:

In this task, students are presented with a real-world problem involving the price of an item on sale. To answer the question, students must represent the problem by defining a variable and related quantities, and then write and solve an equation.

Operations on the Number Line:

The purpose of this task is to help solidify students' understanding of signed numbers as points on a number line and to understand the geometric interpretation of adding and subtracting signed numbers. There is a subtle distinction in the Florida Standards between a fraction and a rational number. Fractions are always positive, and when thinking of the symbol ab as a fraction, it is possible to interpret it as a equal-sized pieces where b pieces make one whole.

Repeating Decimal as Approximation:

The student is asked to complete a long division which results in a repeating decimal, and then use multiplication to "check" their answer. The purpose of the task is to have students reflect on the meaning of repeating decimal representation through approximation.

Sharing Prize Money:

Students are asked to determine how to distribute prize money among three classes based on the contribution of each class.

Sand Under the Swing Set:

The 7th graders at Sunview Middle School were helping to renovate a playground for the kindergartners at a nearby elementary school. City regulations require that the sand underneath the swings be at least 15 inches deep. The sand under both swing sets was only 12 inches deep when they started. The rectangular area under the small swing set measures 9 feet by 12 feet and required 40 bags of sand to increase the depth by 3 inches. How many bags of sand will the students need to cover the rectangular area under the large swing set if it is 1.5 times as long and 1.5 times as wide as the area under the small swing set?

Art Class, Assessment Variation:

Art Class, Variation 1:

Students are asked to use ratios and proportional reasoning to compare paint mixtures numerically and graphically.

Art Class, Variation 2:

Giving the amount of paint in "parts" instead of a specific standardized unit like cups might be confusing to students who do not understand what this means. Because this is standard language in ratio problems, students need to be exposed to it, but teachers might need to explain the meaning if their students are encountering it for the first time.

Use the information provided to answer the questions regarding Carlos and his bananas

This is a task where it would be appropriate for students to use technology such as a graphing calculator or GeoGebra, making it a good candidate for students to engage in Standard for Mathematical Practice 5 Use appropriate tools strategically. A variant of this problem is appropriate for 8th grade; see Coffee by the Pound.

Tom wants to buy some protein bars and magazines for a trip. He has decided to buy three times as many protein bars as magazines. Each protein bar costs \$0.70 and each magazine costs \$2.50. The sales tax rate on both types of items is 6½%. How many of each item can he buy if he has \$20.00 to spend?

Chess Club:

This problem includes a percent increase in one part with a percent decrease in the remaining and asks students to find the overall percent change. The problem may be solved using proportions or by reasoning through the computations or writing a set of equations.

Comparing Years:

Students are asked to make comparisons among the Egyptian, Gregorian, and Julian methods of measuring a year.

Cooking with the Whole Cup:

Students are asked to use proportional reasoning to answer a series of questions in the context of a recipe.

Gotham City Taxis:

The purpose of this task is to give students an opportunity to solve a multi-step ratio problem that can be approached in many ways. This can be done by making a table, which helps illustrate the pattern of taxi rates for different distances traveled and with a little persistence leads to a solution which uses arithmetic. It is also possible to calculate a unit rate (dollars per mile) and use this to find the distance directly without making a table.

Finding a 10% Increase:

5,000 people visited a book fair in the first week. The number of visitors increased by 10% in the second week. How many people visited the book fair in the second week?

Friends Meeting on Bikes:

Using the information provided find out how fast Anya rode her bike.

Molly's Run:

This task asks students to solve a problem in a context involving constant speed. This task provides a transition from working with ratios involving whole numbers to ratios involving fractions. This problem can be thought of in several ways; in particular, this problem also provides an opportunity for students to work with the "How many in one group?'' interpretation of division.

Molly's Run, Assessment Variation:

Use the information provided to find out how long it will take Molly to run one mile.

Music Companies, Variation 1:

This problem requires a comparison of rates where one is given in terms of unit rates, and the other is not. See "Music Companies, Variation 2" for a task with a very similar setup but is much more involved and so illustrates MAFS.7.RP.1.3.

Music Companies, Variation 2:

This problem has multiple steps. In order to solve the problem it is necessary to compute: the value of the TunesTown shares; the total value of the BeatStreet offer of 20 million shares at \$25 per share; the difference between these two amounts; and the cost per share of each of the extra 2 million shares MusicMind offers to equal to the difference.

Robot Races:

Students should use information provided to answer the questions regarding robot races.

Ants versus humans:

This task requires students to work with very large and small values expressed both in scientific notation and in decimal notation (standard form). In addition, students need to convert units of mass. The solution below converts the mass of humans into grams; however, we could just as easily converted the mass of ants into kilograms. Students are unable to go directly to a calculator without taking into account all of the considerations mentioned above. Even after converting units and decimals to scientific notation, students should be encouraged to use the structure of scientific notation to regroup the products by extending the properties of operations and then use the properties of exponents to more fluently perform the calculations involved rather than rely heavily on a calculator.

Quinoa Pasta 1:

This task asks students to find the amount of two ingredients in a pasta blend. The task provides all the information necessary to solve the problem by setting up two linear equations in two unknowns. This progression of tasks helps distinguish between 8th grade and high school expectations related to systems of linear equations.

Sore Throats, Variation 2:

The purpose of this task is to show how the ideas in the RP and EE domains in 6th and 7th grade mature in 8th grade. Parts (a)-(c) could easily be asked of 7th grade students. Part (a) asks students to do what is described in 7.RP.2.a, Part (b) asks students to do what is described in 7.RP.2.c, and Part (c) is the 7th grade extension of the work students do in MAFS.6.EE.3.9.
On the other hand, part (d) is 8th grade work. It is true that in 7th grade, "Students graph proportional relationships and understand the unit rate informally as a measure of the steepness of the related line, called the slope". However, in 8th grade students are ready to treat slopes more formally: 8.EE.5 says students should "graph proportional relationships, interpreting the unit rate as the slope of the graph" which is what they are asked to do in part (d).

Cell Phone Plans:

This task presents a real-world problem requiring the students to write linear equations to model different cell phone plans. Looking at the graphs of the lines in the context of the cell phone plans allows the students to connect the meaning of the intersection points of two lines with the simultaneous solution of two linear equations. The students are required to find the solution algebraically to complete the task.

Two Lines:

In this task, we are given the graph of two lines including the coordinates of the intersection point and the coordinates of the two vertical intercepts and are asked for the corresponding equations of the lines. It is a very straightforward task that connects graphs and equations and solutions and intersection points.

Who Has the Best Job?:

This task asks the student to graph and compare two proportional relationships and interpret the unit rate as the slope of the graph.

Coffee by the Pound:

In this example, students will answer questions about unit price of coffee, make a graph of the information, and explain the meaning of slope in the given context.

Foxes and Rabbits:

This task emphasizes the importance of the "every input has exactly one output" clause in the definition of a function, which is violated in the table of values of the two populations. Noteworthy is that since the data is a collection of input-output pairs, no verbal description of the function is given, so part of the task is processing what the "rule form" of the proposed functions would look like.

Function Rules:

This task can be played as a game where students have to guess the rule and the instructor gives more and more input output pairs. Giving only three input output pairs might not be enough to clarify the rule. Instructors might consider varying the inputs in, for example, the second table, to provide non-integer entries. A nice variation on the game is to have students who think they found the rule supply input output pairs, and the teachers confirms or denies that they are right. Verbalizing the rule requires precision of language. This task can be used to introduce the idea of a function as a rule that assigns a unique output to every input.

Calculating the Square Root of 2:

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

A Rectangle in the Coordinate Plane:

This task provides an opportunity to apply the Pythagorean theorem to multiple triangles in order to determine the length of the hypotenuse; the converse of the Pythagorean theorem is also required in order to conclude that certain angles are right angles.

Bird and Dog Race:

The purpose of this task is for students to use the Pythagorean Theorem as a problem-solving tool to calculate the distance between two points on a grid. In this case the grid is also a map, and the street names can be viewed as defining a coordinate system (although the coordinate system is not needed to solve the problem).

Sale!:

Students are asked to determine which sale option results in the largest percent decrease in cost.

Selling Computers:

The sales team at an electronics store sold 48 computers last month. The manager at the store wants to encourage the sales team to sell more computers and is going to give all the sales team members a bonus if the number of computers sold increases by 30% in the next month. How many computers must the sales team sell to receive the bonus? Explain your reasoning.

Sore Throats, Variation 1:

Students are asked to decide if two given ratios are equivalent.

Stock Swaps, Variation 2:

Students are asked to solve a problem using proportional reasoning in a real world context to determine the number of shares needed to complete a stock purchase.

Stock Swaps, Variation 3:

Students are asked to solve a multistep ratio problem in a real-world context.

Tax and Tip:

After eating at your favorite restaurant, you know that the bill before tax is \$52.60 and that the sales tax rate is 8%. You decide to leave a 20% tip for the waiter based on the pre-tax amount. How much should you leave for the waiter? How much will the total bill be, including tax and tip?

The purpose of this task is for students to calculate the percent increase and relative cost in a real-world context. Inflation, one of the big ideas in economics, is the rise in price of goods and services over time. This is considered in relation to the amount of money you have.

Track Practice:

This activity asks the student to use unit rate and proportional reasoning to determine which of two runners is the fastest.

Two-School Dance:

The purpose of this task is to see how well students students understand and reason with ratios.

Reflecting Reflections:

In this resource, students experiment with successive reflections of a triangle in a coordinate plane.

Estimating Square Roots:

By definition, the square root of a number n is the number you square to get n. The purpose of this task is to have students use the meaning of a square root to find a decimal approximation of a square root of a non-square integer. Students may need guidance in thinking about how to approach the task.

Point Reflection:

The purpose of this task is for students to apply a reflection to a single point. The standard MAFS.8.G.1.1 asks students to apply rigid motions to lines, line segments, and angles. Although this problem only applies a reflection to a single point, it has high cognitive demand if the students are prompted to supply a picture. This is because the coordinates of the point (1000,2012) are very large. If students try to plot this point and the line of reflection on the usual x-y coordinate grid, then either the graph will be too big or else the point will lie so close to the line of reflection that it is not clear whether or not it lies on this line. A good picture requires a careful choice of the appropriate region in the plane and the corresponding labels. Moreover, reflections of lines, line segments, and angles are all found by reflecting individual points.

Reflecting a Rectangle Over a Diagonal:

The task is intended for instructional purposes and assumes that students know the properties of rigid transformations described in MAFS.8.G.1.1. Note that the vertices of the rectangles in question do not fall exactly at intersections of the horizontal and vertical lines on the grid. This means that students need to approximate and this provides an extra challenge. Also providing a challenge is the fact that the grids have been drawn so that they are aligned with the diagonal of the rectangles rather than being aligned with the vertical and horizontal directions of the page. However, this choice of grid also makes it easier to reason about the reflections if they understand the descriptions of rigid motions indicated in MAFS.8.G.1.3.

Converting Decimal Representations of Rational Numbers to Fraction Representations:

MAFS.8.NS.1.1 requires students to "convert a decimal expansion which repeats eventually into a rational number." Despite this choice of wording, the numbers in this task are rational numbers regardless of choice of representation. For example, 0.333¯ and 13 are two different ways of representing the same number.

Is This a Rectangle?:

The goal of this task is to provide an opportunity for students to apply a wide range of ideas from geometry and algebra in order to show that a given quadrilateral is a rectangle. Creativity will be essential here as the only given information is the Cartesian coordinates of the quadrilateral's vertices. Using this information to show that the four angles are right angles will require some auxiliary constructions. Students will need ample time and, for some of the methods provided below, guidance. The reward of going through this task thoroughly should justify the effort because it provides students an opportunity to see multiple geometric and algebraic constructions unified to achieve a common purpose. The teacher may wish to have students first brainstorm for methods of showing that a quadrilateral is rectangle (before presenting them with the explicit coordinates of the rectangle for this problem): ideally, they can then divide into groups and get to work straightaway once presented with the coordinates of the quadrilateral for this problem.

Identifying Rational Numbers:

The task assumes that students are able to express a given repeating decimal as a fraction. Teachers looking for a task to fill in this background knowledge could consider the related task "8.NS Converting Decimal Representations of Rational Numbers to Fraction Representations."

Calculating and Rounding Numbers:

In this task, students explore some important mathematical implications of using a calculator. Specifically, they experiment with the approximation of common irrational numbers such as pi (π) and the square root of 2 (√2) and discover how to properly use the calculator for best accuracy. Other related activities involve converting fractions to decimal form and a concrete example where rounding and then multiplying does not yield the same answer as multiplying and then rounding.

Triangle congruence with coordinates:

In this resource, students will decide how to use transformations in the coordinate plane to translate a triangle onto a congruent triangle. Exploratory examples are included to prompt analytical thinking.

Comparing Speeds in Graphs and Equations:

This task provides the opportunity for students to reason about graphs, slopes, and rates without having a scale on the axes or an equation to represent the graphs. Students who prefer to work with specific numbers can write in scales on the axes to help them get started.

Area of a Trapezoid:

The purpose of this task is for students to use the Pythagorean Theorem to find the unknown side-lengths of a trapezoid in order to determine the area. This problem will require creativity and persistence as students must decompose the given trapezoid into other polygons in order to find its area.

US Garbage, Version 1:

In this task, the rule of the function is more conceptual. We assign to a year (an input) the total amount of garbage produced in that year (the corresponding output). Even if we didn't know the exact amount for a year, it is clear that there will not be two different amounts of garbage produced in the same year. Thus, this makes sense as a "rule" even though there is no algorithmic way to determine the output for a given input except looking it up in the table.

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.

Applying the Pythagorean Theorem in a Mathematical Context:

Three right triangles surround a shaded triangle; together they form a rectangle measuring 12 units by 14 units. The figure used shows some of the dimensions but is not drawn to scale. Understand and apply the Pythagorean Theorem.

Areas of Geometric Shapes with the Same Perimeter:

This problem is part of a very rich tradition of problems looking to maximize the area enclosed by a shape with fixed perimeter. Only three shapes are considered here because the problem is difficult for more irregular shapes. For example, of all triangles, the one with fixed perimeter P and largest area is the equilateral triangle whose side lengths are all P3 but this is difficult to show because it is not easy to find the area of triangle in terms of the three side lengths (though Heron's formula accomplishes this). Nor is it simple to compare the area of two triangles with equal perimeter without knowing their individual areas. For quadrilaterals, a similar problem arises: showing that of all rectangles with perimeter P the one with the largest area is the square whose side lengths are P4 is a good problem which students should think about. But comparing a square to an irregularly shaped quadrilateral of equal perimeter will be difficult.

Kimi and Jordan:

Students are asked to create and graph linear equations to compare the savings of two individuals. The purpose of the table in (a) is to help students complete (b) by noticing regularity in the repeated reasoning required to complete the table (Standard for Mathematical Practice, MAFS.K12.MP.8.1).

Peaches and Plums:

This task asks students to reason about the relative costs per pound of two fruits without actually knowing what the costs are. Students who find this difficult may add a scale to the graph and reason about the meanings of the ordered pairs. Comparing the two approaches in a class discussion can be a profitable way to help students make sense of slope.

Running on the Football Field:

Students need to reason as to how they can use the Pythagorean Theorem to find the distances ran by Ben Watson and Champ Bailey. The focus here should not be on who ran a greater distance but on seeing how to set up right triangles to apply the Pythagorean Theorem to this problem. Students must use their measurement skills and make reasonable estimates to set up triangles and correctly apply the Theorem.

Equations of Lines:

This task asks the student to understand the relationship between slope and changes in x- and y-values of a linear function.

Find the Change:

This activity challenges students to recognize the relationship between slope and the difference in x- and y-values of a linear function. Help students solidify their understanding of linear functions and push them to be more fluent in their reasoning about slope and y-intercepts. This task has also produced a reasonable starting place for discussing point-slope form of a linear equation.

Fixing the Furnace:

Students are asked to write equations to model the repair costs of three different companies and determine the conditions for which each company would be least expensive. This task can be used to both assess student understanding of systems of linear equations or to promote discussion and student thinking that would allow for a stronger solidification of these concepts. The solution can be determined in multiple ways, including either a graphical or algebraic approach.

Extending the Definitions of Exponents, Variation 1:

This is an instructional task meant to generate a conversation around the meaning of negative integer exponents. While it may be unfamiliar to some students, it is good for them to learn the convention that negative time is simply any time before t=0.

Log Ride:

Students are asked to solve an inequality in order to answer a real-world question.

How Many Solutions?:

The student is given the equation 5x-2y=3 and asked, if possible, to write a second linear equation creating systems resulting in one, two, infinite, and no solutions.

## Teaching Idea

A Penny Saved is a Penny at 4.7% Earned:

There are lots of ways to receive income, and lots of ways to spend it. In this EconomicsMinute teaching idea, students will develop two budgets, or plans, to help them decide how to allocate their income.

Type: Teaching Idea

## Tutorials

Exponents and Powers:

This tutorial reviews the concept of exponents and powers and includes how to evaluate powers with negative signs.

Type: Tutorial

Power of a Power Property:

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

Type: Tutorial

Absolute Value:

This tutorial will help you understand the concept of absolute value. Take the quiz after the lesson to practice!

Type: Tutorial

Subtracting Integers:

This tutorial will help the learner to understand the concept of subtracting the positive and negative integers with the help of a number line. Learners can also take a quiz after the concept is internalized.

Type: Tutorial

Students will be able to see examples of addition of integers while watching a short video, and practice adding integers using an online quiz.

Type: Tutorial

Solving Equations With the Variable on Both Sides.:

This video models solving equations in one variable with variables on both sides of the equal sign.

Type: Tutorial

Multiplying Fractions:

The video describes how to multiply fractions and state the answer in lowest terms.

Type: Tutorial

## Video/Audio/Animations

Story of Pi:

This video dynamically shows how Pi works, and how it is used.

Type: Video/Audio/Animation

Averages:

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

Type: Video/Audio/Animation

## Virtual Manipulatives

Algebra Balance Scales-Negatives:

This virtual manipulative allows the learners to solve simple linear equations through the use of a balance beam. Unit blocks and x-boxes are placed on the pans of a balance beam to balance it.

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

Percentages:

This virtual manipulative allows the student to enter any two of the three quantities involved in percentage computation: the whole, a part and the percent. This manipulative can also be used for the discussions of relations among fractions, decimals, ratios and percentages.

Type: Virtual Manipulative

Converting Units Through Dimensional Analysis:

Using this virtual manipulative, students apply dimensional analysis (AKA factor-label method or unit-factor method) to solve unit conversion problems. There is also the opportunity to create your own unit conversion problems.

Type: Virtual Manipulative

The Circle:

This interactive lesson introduces students to the circle, its attributes, and the formulas for finding its circumference and its area. Students then perform a few calculations to practice finding the area and circumference of circles, given the diameter.

Type: Virtual Manipulative

Color Chips - Subtraction:

This virtual manipulative guides the student in the use of color counters to model subtraction of integers.

Type: Virtual Manipulative

Function Machine:

This animation helps students to understand the function concept through the machine metaphor. The domain elements are dragged into the machine, which then goes through some process and outputs the range element corresponding to the input. The user is asked to complete the outputs for the remaining inputs.

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

Transformations - Translation:

The user can demonstrate or explore translation of shapes created with pattern blocks, using or not using a coordinate axes and lattice points background, by changing the translation vector.
(source: NLVM grade 6-8 "Transformations - Translation")

Type: Virtual Manipulative

Transformations - Dilation:

Students use a slider to explore dilation and scale factor. Students can create and dilate their own figures. (source: NLVM grade 6-8 "Transformations - Dilation")

Type: Virtual Manipulative

Transformations - Rotation:

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

Type: Virtual Manipulative